ABSTRACT DEVELOPMENT AND APPLICATION OF A GAMMA-BASED INVENTORY MANAGEMENT THEORY by E. Martin Basic The purpose of this study is to determine if the pat- tern of demand for the products marketed by the metal ser- vice center industry may be approximated by the gamma proba- bility function, and if so, if inventory management procedures may be developed based on this gammaabased inventory theory. One major hypothesis was posed, and two sub-hypotheses: Major Hypothesis.--The pattern of demand for the pro— ducts marketed by the metal service center industry may be approximated by the gamma probability function. Sub-Hypothesis I.--A gamma-based theoretical framework may serve as a basis for developing internal operating pro- cedures for the firm, including tables and charts tailored to the individual firm's needs. Sub-Hypothesis II.--A gamma-based theoretical framework may serve as a basis for developing computer programs which will enable a computer to provide optimum decisions based on actual or projected demand information furnished to it. The metal service center industry was selected for this study because of E. MARTIN BASIC l. the present lack of scientific decision making in inventory management in this industry. 2. the relative importance of inventory management in the metal service center industry. 3. the desire of the business communtiy for assistance in the area of inventory management. 4. characteristics of the industry which make it part— icularly conducive to research by scientific methods: a. standard undifferentiated products. b. large number of items offered by the firm. c. lack of obsolescence, permitting the use of statistical methods in research, using rela- tively large amounts of data. d. the ability to determine stockout costs without estimating dollar values for customer loyalty and goodwill, due to the widespread practice of purchasing out-of—stock items from compet- itors at a higher than mill price. The gamma probability function was selected for the sim— ulation because it has the characteristic of never going below zero on the x—axis, thus meeting an important requirement of the demand pattern which the normal distribution, for example, cannot. The gamma probability function left tail terminates at the origin, or on the y—axis, and then rises relatively sharply to a maximum, then decreases toward the x—axis with a gentler slope, thus creating a skewed distribution curve. There is a separate gamma distribution for every combination of mean and variance, as with the normal distribution. For extremely small values of mean, the gamma approximates the exponential distribution. For extremely large values of mean, the gamma approximates the normal distribution. Sales data consisting of pieces per month for 59 months were obtained for twelve typical metal service center items. E. MARTIN BASIC For each item the average and standard deviation was calcu- lated. These figures were then substituted into gamma equa— tions to obtain a gamma distribution. This theoretical distri- bution was plotted in curve form, and the actual data super— imposed on the same chart to test for goodness of fit. The standard chi—square test was applied. The major hypothesis was emphatically validated. Procedures and computer programs were developed and prOposed new systems offered. Sample data was run through the computer to simulate the procedures and use of the tables prepared by computer programs to test their feasibility. As far as was possible, calculations were eleminated or minimized. Instructions are included in the dissertation for modifying the computer programs to obtain tables with different ranges of parameters, should needs change. A chart was developed for reading the order point direct— ly in one operation. A computer program was developed which enables the com— puter to calculate the reorder point, thus transferring the decision making from a company official to the computer. Four procedures were proposed for using this particular program. The presenting of tables as aids in decisionmaking, the establishment of procedures utilizing tables and charts, and the running of trials on the decision making program, all served to validate our two sub-hypotheses. DEVELOPMENT AND APPLICATION OF A GAMMA-BASED INVENTORY MANAGEMENT THEORY By E. Martin Basic A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Business Department of Management 1965 ‘1. \- \) ‘1 3:; ACKNOWLEDGMENTS A deep debt is owed to my dissertation committee chair— man, Professor Claude Mc millan, Jr., for his encouragement and guidance, and the many hours spent on behalf of this study. A note of thanks to the other members of my dissertation committee, Professor Richard Gonzalez and Professor Frank Mossman, for their cooperation, encouragement and help, and to Dr. J. H. Stapleton, Associate Professor of Statistics, for his time spent in checking my curve fitting work. Further thanks are extended to the executives of the several metal service centers who generously furnished valu— able data from confidential records. Further thanks to the members of the Steel Service Center Institute, and particularly the officials of this organiza- tion, for many hours of interviews and plant tours generously given. Finally, a note of appreciation and gratitude to my wife, Janine, for her encouragement and patience during the research and writing of this publication. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS. . . . . . . . . . . . . . ii LIST OF ILLUSTRATIONS. . . . . . . . . . . . iv LIST OF APPENDICES . . . . . . . . . . . . vi Chapter I. INTRODUCTION . . . . . . . . . . . . I II. SURVEY OF LITERATURE . . . . . . . . . 3 III. DEVELOPMENT OF THE GAMMA DISTRIBUTION. . . . 16 IV. TEST OF THE MAJOR HYPOTHESIS. . . . . . . 41 V. DEVELOPMENT OF FIRST SUB—HYPOTHESIS . . . . 90 VI. DEVELOPMENT OF THE SECOND SUB-HYPOTHESIS. . . 131 VII. SUMMARY AND CONCLUSIONS . . . . . . . . 139 BIBLIOGRAPHY. . . . . . . . . . . . . . . 147 APPENDICES . . . . , . . . . . . . . . . 152 iii LIST OF ILLUSTRATIONS Illustration Subroutine CAMDEN Line Print. Program Program Program Program DATA Line Print DATA Computer Printout Table CAMDEN Line Print. CAMDEN Computer Printout Table Subroutine GAMTAB Line Print. Program Program Program Program GAMCUM Line Print. GAMCUM Computer Printout Table CURVES Line Print. CURVES Computer Printout Table Cumulative Probability Charts Density Program Program Program Program Program Program Program Program Program Program Pattern Probability Charts. GAMCON Line Print. GAMCON Computer Printout Table RANDL Line Print RANDL Computer Printout Table. ORDER Line Print ORDER Computer Printout Table. MCHART Line Print. MCHART Computer Printout Table LAMBDA Line Print. LAMDA Computer Printout Table. iv Page 23 24 25 27 36 37 38 A3 A7 66 78 101 10A 11A 115 117 1.19 122 123 124 126 Illustration Reorder Program Program Program Program Program Program Point Chart. REORDR Line Print. REORDR Computer Printout NORMX Line Print NORMX Computer Printout Table. NORMI Line Print NORMI Computer Printout Table. LIST OF APPENDICES Appendix A. CORRELATING GAMMA CUMULATIVE PROBABILITY CHART WITH GAMMA CUMULATIVE PROBABILITY TABLE. B. DEVELOPMENT OF NORMAL PROBABILITY INTEGRAL C. PROOF OF MATHEMATICAL RELATIONSHIP vi 161 CHAPTER I INTRODUCTION Inventory management is one of the most important areas in the field of physical distribution. The typical manufact- uring company in this country spends about half its sales dollar on materials and services. This is roughly twice as much as it spends on total payrolls. For example, in 1959 the 100 largest manufacturing companies spent an average of 52.3% of their sales dollar on purchased materials and services. The average manufacturing corporation has about 24% of its total assets invested in inventories.1 Inventory management is thus a major element in management decision—making. A survey of the literature in the field reveals: a. universal agreement on the importance of the activity in the business enterprise. b. an awareness that the body of knowledge is inadequate for satisfactory decision making. This awareness is present in both the business and academic communities. c. the desire of the business community for assist- ance in. improving the level of decision making in this area. lDean Ammer, Materials Management (Homewood, Ill.: Richard D. Irwin, Inc., 1962), pp. 3-6. l d. efforts by some individuals to build a theoreti- cal statistical base on which systematic pro- cedures could be built. e. efforts by some individuals and organizations to take advantage of the capabilities of a modern digital computer to aid in decision making. It is clear that the proliferation of directions in which research is being conducted in this field, the continued pub— lication of literature on the subject, the lack of a common hypothesis behind which everyone can stand, and the wide diversity of inventory control methods in the business com- munity indicate the need for much further study, and the likeli— hood that future contributions will exceed the measure of the present body of knowledge. This area, then, is an obvious choice within which to select a topic for research and public— ation of a doctoral dissertation. Within the broad areas of application of inventory management, it was decided that attention should be concentrat- ed on a business sector within which the inventory management activity is a more vital activity than it is in other types of business operations. There are certain industries in which in— ventory management is the dominant element in the business operation. One of these, the metals service center industry, was an obvious choice for a research project. First, it is a large and important industry in the U. S., with sales exceeding 3 billion dollars per year, supplying 16% of the steel used in the industrial steel market. There are metal service centers in every state in the union, with 800 companies operating 1300 establishments. Second, inventory management is the dominant internal activity in steel service centers. 85% of all customer orders are received by telephone. It is accepted industry practice that orders are shipped on either the same day they are received, or else within 2A hours. The customer is a sophisticated purchasing agent, ordering an undifferentiated product which meets certain standard specifications. There is little or no price competition among competitors. Avail- ability of ordered items in inventory is thus the key factor in the company's operations. In most small and medium size companies in this industry, the specific detailed inventory management decisions are made by the company's top operating executive. A 1963 survey revealed this to be true even in l A 1960 survey of steel ser— some fairly sizeable companies. vice centers conducted by the U. S. Steel Corporation dis- closed that the overall average investment in inventories was 6A% of the firm's current assets, and A8.A% of the total assets. For firms not owning their own buildings and property, inventories representing 65% of the total assets were common. Third, the majority of firms in the industry do not utilize inventory management procedures which would be rated as scientific even with respect to the present inadequate lJohn Demaree, Inventory Management (unpublished Doctoral Dissertation, Michigan State University, East Lansing, Michigan 1964). body of knowledge. The 1963 questionnaire referred to pre— viously indicated that 61% of steel service centers make their inventory management decisions mainly on the impressions formed by noting the past inventory records of the particular item. Fourth, the industry desires assistance in this area. In the same 1963 questionnaire were many comments on this matter, of which a brief sampling follows: Our present problem is to set up proper guides for reordering points and reorder quantities. We know this is the area where cash can be released for other use. We must learn more about the best methods for accomplishing this. Present problem--lack of knowledge on which to base inventory management decisions. In our company we need a formula for determining ordering point and quantity. Maintaining prOper inventory levels to get max— imum return on dollars invested. Finally, characteristics of steel service center inven— tory management make it particularly conducive to research by scientific methods: 1. The characteristic of standard, undifferentiated products. 2. The characteristic of the large number of items offered by the firm. 3. The characteristic of lack of obsolescence of products, permitting the use of statistical methods in research, using relatively large amounts of data. A. The characteristic of a basic product with a relatively stable market acceptance. 5. The characteristic of a relatively large number of orders per month. 6. The virtual non-existence of the major problem facing mathematical deveIOpment of Optimum quantita— tive determination of "stockout cost," needed to determine optimum inventory levels, weighing costs of too much inventory against costs of too little inventory. As a rule this figure must include such intangible factors as relative customer loyalty, personal relationships, goodwill, etc. No satis— factory method has yet been devised to represent these factors quantitatively in a mathematical equation. This problem is rare in the metal ser— ice center industry, since the firm invariably informs the customer that the items are available. Determination of stockout costs can therefore be determined quantitatively by measuring such factors as the costs involved in purchasing the out—of— stock items from a competitor at a higher than mill price, and in picking up the items at the competitor's plant. The key element in inventory management is the estab- lishment of inventory levels, taking into consideration the expected sales. Since of course the sales volume per month varies, an important parameter is the average, or mean, sales rate. If sales volume is so uniform that variations from the mean are negligible, then this mean figure and the lead time, i.e., the time lapse between deciding to order and receipt of the order, would be the only parameters necessary to estab- lish proper inventory levels. The complexity of inventory management ensues from the non-uniform sales volume, making it difficult to decide on the amount of stock to keep on hand. For example, monthly sales of 1" round 0—1018 bar 12' long might vary from 30 pieces to 200 pieces, with records of the other months showing figures distributed in some fashion between these two extremes. It is of great importance to have a good knowledge of the manner in which these individual sales records are distributed between these two extremes, because this knowledge would then enable us to assign a quantitative probability to any particular sales rate for the following month or the following lead time. Upon deciding on a particular inventory level at which we will reorder a replenishment of our stock, we will then know the probability of running out of stock before the replenishment stock arrives. We would not wish to adopt a reorder point such that this probability is zero, because this would necessitate keeping on hand sufficient stock to handle the largest possible sales volume during leadtime. Since this sales volume is a rare occurence, we are nearly always burdened with the heavy cost of carrying an unnec— eSsarily large inventory. 0n the other hand, neither would we want this probability to be 1, because to always run out of stock during lead time would result in heavy stockout costs. The subject is introduced here to show the importance of knowing the probabilities of sales distribution during lead— time, based on some combination of analysis of historical re- cords and sales forecasting techniques. It is this probability distribution on which the newest inventory management theories are based. But what is this distribution likely to be? Because the concept of basing inventory management theories on the probability distribution during leadtime is so new, this is an area which has not as yet received sufficient attention. Among the few who are probing this new area, perhaps the best known is Robert Gocdell Brown of Arthur D. Little, Inc. While he is making our standing contributions in this area, unfortunately his work is based heavily on normal (i.e., bell—curve) distribution theory. The position taken in this dissertation is that the consideration of some distribution other than normal might be fruitful, and, based on another distribution theory, a system of procedures could be promulgated to enable the metal service center industry to conduct its inventory manage— ment operations in a more scientific manner. The consideration of a distribution other than the normal appeared worthy of study. For example, an analysis of the distribution of 1% round C1018 bars at a local steel ser— vice center showed mean sales of 39.2 pieces per month, with a standard deviation (a measure of the average amount that actual sales for the month.differed from 39.2) of 38.1 pieces If these sales records were assumed to be normally dis— tributed (i.e., a bell-curve of 38.1 standard deviation drawn symmetrically on both sides of the 39.2 mean point) then approximately 15% of the bell—curve would extend to the left of zero sales, i.e., 15% of sales were for a negative quantity. Since this is patently impossible, it is not.possible that the distribution follows the normal curve. The study of some distribution pattern which never goes below zero was indicated. The most general case of this type of distribution is the one which has been assigned the label "gamma distribution." The gamma distribution probability density functioncfi‘the form. f(x) = Ar e‘AX xr’”l Ir—II! and was originally derived to determine the probability of x units of length between one occurrence and the rth suc- ceeding occurrence, assuming a mean of A occurrences per unit of length. In the most general case, this density function starts at the origin, rises to a maximum, and then tapers down to approach the abscissa. With the left end tied to the origin, the curve is thus considerably skewed, with a relatively sharp rise to the maximum and a more gradual taper to the right, unlike the perfectly symmetrical normal curve. It is not a requirement of this distribution that the left end of the curve must go through the (0,0) origin. Depending on the selection of parameters, the curve could originate at some positive point on the Y axis (the probability measure) and even taper down gradually from that point toward the abscissa. It appeared persuasive that actual sales during lead- time follows some pattern of gamma distribution. The fact that this pattern of gamma distribution is probably differ- ent for every item in stock, and changes with time, was no hardicap to our pursuit of procedural methods, since for— tunately any type of gamma distribution i.e., any values of A and r, can be completely specified by its mean and stand— ard deviation, as in the case of the normal distribution. The specifying or adjusting of these two easily determined parameters is thus merely a matter of detail. The following procedure was adcpted: l. A study of the present literature in this area 2. The solicitation of assistance from steel service centers in the form of actual data. 3. The testing of actual data against the hypothesis that demand during leadtime can be approximated by a gamma distribution, using standard acceptable statistical methods. A. When the hypothesis proved to be true, research was conducted with the use of the CDC36OO computer to test the validity of two subhypotheses: a. First sub-hypothesis. A gamma-based theoretical framework may serve as a basis for developing internal operating procedures for the firm, including tables and charts tailored to the firm's needs. b. Second sub—hypothesis. A gamma-based theoretical framework may serve as a basis for developing computer programs which will enable a computer to provide op~ timum decisions based on actual and projected demand information furnished by the firm. The results of the researCh conducted in the course of testing these two sub-hypotheses make up the major part of this dissertation. Some of the research had to be conducted prior to the testing of the major hypothesis, in order to develop computer programs which could solve the gamma density probability function and the gamma cumulative probability function equa— tions. A computer program, code name CURVES, was then IC developed to provide tables for the purpose‘of drawing the various gamma curves plotted on the same chart with the actual data. This enabled us to proceed with the test of the major-hypothesis. The validation of the major hypothesis enabled us to proceed with the research required to test the two sub-hy- potheses. This necessary procedural sequence accounts for the particular sequence of chapters adopted. Chapter III develops the equation solving research. Chapter IV, with the aid of computer program CURVES, covers the test of the major hypothesis. Chapter V develops the first sub-hypothesis. Chapter VI develops the second sub-hypothesis. CHAPTER II SURVEY OF LITERATURE The first research step which was taken in this pro— ject was a survey of the literature. There are many hundreds of articles and books on the subject of inventory management, and many dozens of publications covering probabilistic models, but comparatively few which combine the two. All of the books and articles listed in the bibliography at the end of this dissertation serve a useful purpose with— in their own area, or, in some cases, specialty. They have been chosen for listing principally because they are well known, and the student of inventory control is expected to have knowledge of them and their contents. They properly belong in this dissertation because of the necessary back- ground they provide for further research. With regard to out primary purpose, the search for specific aid in conducting our project, only a handfull of outside sources was helpful. Following is a list of the publications in the area specifically covered by this dissertation which were of spec— ial interest. 1. Brown, R, Smoothing, Forecasting, and Prediction (Englewood Cliffs, New Jersey: Prentice—Hall, Inc., 1963) 2. Holt, C., Modigliani, F., Muth, J., and Simon, H. Planning Production,_lnventory, and Work Force (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1963) 3 11 3. McMillan, C., and Gonzalez, R., Systems Analysis (Homewood, Illinois: Richard D. Irwin, Inc., 1965) The first book on the above list will not be commented on here, since it is cited several times in the body of the dissertation. The second book on the list contains at least one point of difference with the first, since in their'model described on pages 251-253, they assume that sales forecast errors have a gamma distribution, whereas Brown makes a specific point of noting that he uses the normal distribution for this factor. Of special interest to us here is that this second book includes an abbreviated version of the P. R. Winters expan- sion of the gamma cumulative probability function. It will be noted in Chapter III that our development of this function is based on the Winters expansion. Of additional interest to us is the analysis of the demand patterns of 25 products manufactured by the Westing- house Electric Co., covering a period of 18 months. They found that the normal distribution was unsatisfactory for simulation of the demand pattern. They found that the log- normal distribution best fitted the demand pattern for cook- ing utensils, electric motor parts, and other products. There would appear to be some logic in applying the log— normal distribution to demand patterns, since it is certainly possible that demand may be due to the product, rather than the sum, of some of the factors which influence demand. Generally, they found that lognormal and gamma distri- butions fitted the demand patterns best, with the Poisson distribution somewhat less satisfactory; which is to be expected, considering the relative inflexibility of the Poisson distribution (mean and variance always equal). The third book on the above list contains the clearest exposition of the behavior of the gamma function. Other than the above three, few publications make a study in depth of inventory control with probabilistic demand. Arrow, K., Karlin, S., and Scarf, R., Studies in the Mathematical Theory of Inventory and Production, (California: Stanford Press, 1958), contains a probabilistic inventory model on pages 322-32A, which assumes a Poisson distribution. Buchan, R., and Koenigsberg, C., Scientific Inventory Management, (Englewood Cliffs, New Jersey: Prentice—Hall, Inc., 1963) contains a model on page 345 to which they apply, in turn, the uniform (i.e., rectangular), the normal, and the exponential distributions. Our own research indicates that the gamma could have been substituted for both the eXponential and the normal, since the gamma function approximates the exponential functions when the mean-square to variance ratio is less than 1, and approximates the normal function when the mean-squared to variance ratio is greater than 10. Neither the exponential nor 11‘. distribution functions are of much value for our purpose be- tween these limits. (Note: this is a conclusion which is amplified in the final chapter). Morse, F., Queues,fiInventories, and Maintenance (New Jersey: John Wiley & Sons, 1962), includes a mzdel on page A51, which uses the Poisson distribution. Morse has some in— teresting criteria for inventory decision making, and his mathematical development of these are well done. The book is highly recommended. No dissertation could be located which covered our area. A well known monograph on the application of computers to inventory control is the IMPACT Manual, Inventory Manage- ment Program And Control Techniques. It is distributed as an International Business Machines Corporation manual, but seems to bear the imprint of Robert Goodel; Brown‘s work. This manual uses the mean absolute deviation rather than the standard deviation. For the normal distribution, MAD is approximately equal to sigma divided by 1.25. The manual con- tains the curious statement that the substitution of MAD for sigma is for the purpose of simplifying calculations of large amounts of data, since its equation contains no squared terms, in contrast to the sigma equation. But a study of its equation: Zl(x - Z) n MAD = with its necessity for a subtraction operation for every :5 piece of data, makes it obvious that with the particular form of the sigma equation explained in another chapter of this dissertation, and with a late mcdel office calculator which compiles both X X and 2X2 with one entry of data, MAD could be much more rapidly obtained by solving for sigma and then dividing by 1.25. IMPACT is based on the normal distribution. In summary, all of the above cited works were of some value, but as sources of expansion of our framework of knowledge, not as a means of specific assistance. 16 CHAPTER III DEVELOPMENT OF THE GAMMA DISTRIBUTION A. General Description of the Gamma Function The gamma function is sometimes called the factorial function, since by definition: r (n) = (n-l)! Example: T (5) = (5-1)! :11! = A x 3 x 2 x l = 2A The behavior of this function for values of n from —A to +4 is shown in Figure 6.1 on page 255 of the National Bureau of Standards Handbook of Mathematical Functions (Washington, D. C.: U. S. Government Printing Office, 1965), hereinafter referred to as the NBS Handbook. However, we have only a minimal interest in the gamma factorial function itself in this dissertation. Our principal concern is with the probability density distribution of the gamma function and its accompanying cumulative distribution. The probability density function of the gamma distribution is r —xl r—l e x (r—l)! A f(x) Equation (1) The above equation shows the probability of x units of length (or time, etc.) between one occurrence, of an event under study, and the rth succeeding occurrence, given a mean of A (lambda) occurrences per unit of length (or time, etc.). Thus by definition neither r nor A can be less than zero, with the result that all possible con— figurations of this function are contained within the first quadrant, a very important characteristic for our purposes. The mean, u, (mu) and standard deviation, 0, (sigma) of the probability distribution are related to r and A as follows: 2 _u 1" C7 u A: 67' Thus a gamma distribution, like a normal distribution, is completely determined if its mean and its standard deviation are known. Since the cumulative function is F(x) = éf/kf(x) dx then given the previously stated probability density function, it follows that the cumulative gamma distribution function must be r-l r x _ 1 -x>\ F(x) - _(F:ITT— “g/ e x dx Equation (2) Plotting the density distribution in curve form is a relatively straightforward (although very time-consuming- matter, by inserting the proper parameters in Equation (1) and solving algebraically, for each value of x. 18 Plotting the cumulative gamma distribution in curve form by solving Equation (2) for each value of x would be so vastly timeeconsuming as to be impractical. Since it is necessary to do this in the course of deveIOping a com— puter program for solving this equation (see Appendix C) the following series expansion of Equation (2) is offered, based on a similar equation (No. 567.9) on page 127 of H. Dwight's Table of Integrals (New York: Macmillan Book Co., 19u7): I" X A —xA r-l F(x) = e x dx = Ar o~-xA (xr l (r—l) xr"2 + (r—1)(r-2) x1"-3 (r—l)! “ —A " —A2 I -A5 ....... + H)“2 Jig—34" + <-1>I”‘l iiéllfl Equation (3) _)\ ' _ This equation is tractable, and is used in the proof contained in Appendix C. We will now turn our attention to the task of developing computer programs for handling Equations (1) and (2). B. Development of a Computer Program of the Gamma Density Probability Function Distribution It was shown in Part A that the equation for the above is r —xA r-l f(X) = A Tr-l)? 1. Equation (1) 19 which can be rewritten as r e-xA xr-l . 1 f(x) = A W for FORTRAN computer programming, let DEN = f(x) ALAM = A RR = r S = x EXPF = e = 2.71828 ..... PFACT = r21 1 PFACT is segmented as follows: Q = r21 ! when all factorial values of r are greater than 1 RFACT = r21 when the factorial value of r is less than 1 P = rel VAR = 02 = variance, which is the square of the standard deviation AVE = u = mu, the mean In FORTRAN language, Equation (1) is then written DEN = ALAM**RR * EXPF(—ALAM*S) * S**P * PFACT which is straightforward algebra except for the last term, PFACT. The difficulty with PFACT ensues from the fact that r is unlikely to be an integer, making it ordinarily necessary to take the factorial of a decimal number. Thus if r is A.8 then P r-l = 3.8 and PFACT n *‘S I - FJH H l 1 91 a} m a: f.— on a: 'cO 1 ’ 3.8 ~ 2.8 . 1.8 - .8: RFACT Equation (A) ll £7 The problem now becomes one of handling factorial RFACT, when r has been successively reduced factorially to .8. The choice of solutions is between use of the method of Chebyshev polynomials or the Davis series expansion. The latter is the method adopted here. From H. T. DaVis, Tables of Mathematical Functions (Bloomington, Indiana: Principia Press, 1935): “Hi—1 = c + C r + 03r + ....... C -r ..... Equation (5) The values for C are given for 26 terms in Davis's tables. They can also be found on page 256 of the NBS Handbook. The series is carried out to 15 terms here, with values of C as follows: C( 1) = 1. C( 2) = .57721566 C( 3) = - 65587807 C( A) = —.ou2oo263 C( 5) = .16653861 C( 6) = -.ou219773 C( 7) = -.00962197 C( 8) = .0072189A c( 9) = —.oc116517 C(10) = —.ooo2152u C(11) = .00012805 C(12) = —.oooo2013 C(13) = -.00000125 C(1u) = .00000113 C(15) = —.oooooo2o Consider the following series of FORTRAN statements: 21 P = RR-l. By definition. IF(P) 2,1,2 Determine if r is an integer, in PFACT = 1. which case PFACT = 1 and the decimal GO TO 9 factorial procedure is bypassed. R = P Each value of r after being reduced by l retains the variable name R. In other words, R is a working number within the factorial procedure. Statement 2 starts R out as r-l. Q = l. Establishes the left part of the Q RFACT equation as I l at the moment. IF(R—l.) 6,4,5 This is the top of a loop at the bottom of which we successively reduce R by 1. Returning to the top of the loop here we test the new size of R. RFACT = 1. If R is now exactly 1 then RFACT is GO TO 8 now 1 and we bypass the factorial procedure. Q = Q/R An algebraic device to build up the Q part of the Q ° RFACT equation. Example: if r = 4.8 then statement 5 establishes Q at the moment to be I J. 378 On the next cycle of the loop this statement will include the next value of R, and will become 1 318 - 2.8 and so on. DO 7 I = 1,15 We escape from the loop when R becomes a decimal, and now proceed with the Davis series expansion to solve the factorial decimal number. N = 1—1 RFACT“= RFACT + C(I) 'R**N Equation (5) in FORTRAN language. PFACT 2 Q - RFACT Equation (4) in FORTRAN language. Knowing PFACT, we can now proceed to solve Equation (1) 22 9 DEN = ALAM**R * EXPF (-ALAM*S)**P * PFACT Equation (1) in FORTRAN language. RETURN Return to the main program to start the whole procedure over again for the next point along the x axis of our distribution. This solution of Equation (1) is included as a sub- routine in complete programs DATA and GAMDEN. These programs produce tables from which the gamma density prob- ability distribution may be plotted, DATTA using data obtained from a metal service center in Michigan, and GAMDEN reproducing the gamma curves on page 713 of R. Schlaiffer's Probability and Statistics for Business Decisions (New York: McGraw—Hill Book Company, 1959. The purpose of producing tables which reproduce Schlaiffer's curves is to furnish evidence of the validity of the claim made herein that the GAMDEN and DATA programs will furnish accurate gamma density probability tables. Unlike the cumulative probability computer programs developed in the following section of this chapter, tables produced by the density probability distribution computer program cannot be compared with existing tables to demonstrate the validity of the programs. It appears, after a thorough search, that there are no gamma density probability distribution tables available Outside of this dissertation. One possible reason may be Schlaiffer's statement on page 229 of the above citation: 23 § 451776 BASIC 1 "PI""‘SUéfioofINE GAMDEN (AVE. VAR. So DEN) DIMENSION C(15) C(l"lo C(2)'5o7721566 E-l C(3)8-6o5587807 E-l C(4)'-4o2002635 E-Z Cl5)31o6653861 E-l C(6)=-4o2197735 E-Z C(7)'-9o6219715 E-3 C(BD8702189432 E-3 C(9)'-l.1651676 E-3 C(10)3-201524167 E-4 c1113-1.zaosoza E-4 C(12)8-200134858 E-S C(13)I-lo2504935 E-6 C(i4)8101330272 E-6 C(15)8-2o0563384 E-7 RFACT'O. ALAN CAVE/VAR RR 8 ALAM * AVE p3RR- 1. IF (P) 20 lo 2 l PFACT 8 to GO TO 9 2 R i P 0 8 lo 3-IF(R - lo) 60 4o 5 “RFACT: ‘0 GO TO 8 5 O 8 O/R R3R-io GO TO 3 6 DO 7 I = 10 15 N 8 I - l 7 RFACT a RFACT + C(l) * RiiN 8 PFACT I RFACT * G 9 DEN - ALAM**RR * EXPFt-ALAM*S) * S**P * PFACT ' RETURN END END 24 O 45I776 BASIC I PROGRAM DATA DIMENSION AIIZ). VIIZIO SOIIOOIO DIIOOOIZI PRINT 96 96 FORMAT IIMII PRINT 69 69 FORMAT'IIHO. I3H PROGRAM DATA) I 1 PRINT TI TIOFORMAT (IMO. 6IH THEORETICAL DENSITY DISTRIBUTIONS FOR I2 TYPICAL ISTOCK ITEMS) AI I) P 95.27 AI 2) P 39.19 AI 3) P 3708I AI P, P IO063 AI 5) P I808“ AI 6) P ZIQDI AI 7) P QOQI9 AI 8, P I38064 AI 9) P 42.05 AIIO, P ZBoIT AIIII P 30093 AIIZI P I609° VI I) P IBQQQ VI 2, P I552. VI 3) P 49Io VI ‘7 I 853 VI 5’ P 37.. VI 6’ P I950 VI 7) P 2369. VI DI P 49as. VI 9) P 7010 ”VTTOT'P' P593 'MHW VIII) P 3090 VIIZI P 1620 00 I00 J I IOIZ AVE P AIJI VAR P VIJ) '“DO”TOOH"1 P IOIOO 55 P I . SOIII P 55 P 20 S P SOIII CALL GAMDENIAVEQ VAR. So DEN! IOO DIIOJI P DEN “’*'*“”PRTNW“72” ‘ ‘ " " ' TZOFORMAT IIHOO 99M A B C D E IF 6 U/ I J K L. I I PRINT IOIo ISOIIIQ IDIIOJIO J 8 IOIZIO I P 1080) IOI FORMAT IF6QOO IZFDQAI PRINT 97 25 PIHGPA' 0474 YNPOPE'I'AI "ENSITV "I‘I‘IRUIInN‘ '09 12 IVVICIL STOPI IYE‘S 0 u C O E f 0 H I J K L 0. .0000 .0203 .0010 .0001 .0404 .0041 .0411 .0000 .0010 .0100 .0413 .0270 4. .0000 .0290 .0042 .0075 .0431 .0100 .0200 .0000 .0040 .0100 .0049 .0300 4. .0000 .0215 .0040 .0001 .0303 .0040 .0240 .0000 .0000 .0274 .0000 .0473 0, .0001 .0200 .0040 .0515 .0347 .0310 .0200 .0001 .0000 .0245 .0120 .0420 10. .0001 .0107 .0117 .0411 .0300 .0143 .0102 .0001 .011: .0205 .0100 .0412 12. .0002 .0109 .0142 .0307 .0274 .0303 .0103 .0002 .0131 .0297 .0202 .0300 14. .0003 ."100 .0100 .0292 .9245 .0300 .0140 .0002 .0147 .0204 .0220 .0391 10. .0005 .0172 .0101 .0230 .0220 .0337 .0130 .0003 .0100 .0240 .0207 .0309 10. .0007 .0104 .0105 .0103 .0100 .0310 .0120 .0000 .0170 .0239 .025. .0200 20. .0010 .0107 .0204 .0109 .0170 .0200 .0117 .0000 .0177 .0220 .0209 .0240 22. .0013 .0140 .0210 .2120 .0100 .0270 .0100 .0007 .0101 .0217 .0200 .021! 24. .0010 .0142 .0213 .0100 .0104 .0344 .0102 .0000 .0103 .0200 .0200 .0100 20. .0020 .01\5 .0212 .0000 .0130 .0710 .0000 .0010 .0104 .0101 .0202 .0101 20. .0094 .0199 .0210 .0004 .0117 .0104 .0000 .0010 .0102 .0170 .0240 .0130 30. .0029 .0122 .0205 .0001 .0105 .0171 .000! .0014 .0100 .0100 .0227 .0110 30. .0033 .0110 .0109 .0040 .0090 .0100 .0000 .0010 .0170 .0194 .0213 .0101 34. .0030 .0111 .0102 .0032 .0000 .0130 .0070 .0010 .0171 .0142 .0100 .0000 30. .0043 .0100 .0100 .0020 .0077 .0113 .0072 .0020 .0100 .0131 .0103 .0073 30. .0040 .0100 .0174 .0070 .0070 .0000 .0000 .0022 .0100 .0121 .0100 .0001 40. .0093 .0000 .0105 .0010 .0003 .0004 .000! .0024 .0103 .0111 .0193 .0092 42. .0050 .0000 .0105 .0013 .0097 .0072 .0002 .0027 .0140 .0102 .0137 .0044 44. .0002 .0000 .0145 .0010 .0051 .0102 .0000 .0020 .0130 .0003 .0125 .0037 40. .0007 .0001 .0135 .0000 .0040 .0003 .0000 .0031 .0132 .0005 .0112 .0031 40. .0071 .0077 .0120 .0000 .0002 .0045 .0004 .0033 .0129 .0070 .0101 .0020 50. .0070 .0073 .0117 .0005 .0030 .0010 .0091 .0030 .0110 .0071 .0000 .0022 52. .0079 .0070 .0100 .0004 .r034 .0012 .0040 .0037 .0111 .0004 .0000 .0010 54. .0003 .0040 .0000 .0003 .0031 .0077 .0047 .0034 .0104 .0000 .0471 .0010 00. .0000 .0003 .0001 .0002 .0020 .0023 .0049 .0041 .0000 .0053 .0002 .0013 50. .0000‘ .0000 .0004 .0002 .0025 .0019 .0043 .0043 .0001 .0040 .005! .0011 00. .0092 .0007 .0077 .0001 .0022 .0010 .0041 .0040 .0005 .0044 .0040 .0009 0?. .0090 .0004 .0070 .0001 .0020 .0014 .0040 .0047 .0000 .0040 .0042 .0007 04. .0097 .0001 .0004 .0001 .0010 .0011 .0030 .0040 .0074 .0030 .0037 .0000 40. .0090 .0040 .0000 .0001 .0017 .0010 .0010 .0000 .0000 .0032 .0032 .0005 00. .0100 .0040 .0003 .0001 .0015 .0000 .0015 .0051 .0044 .0090 .0470 .0004 70. .0101 .0044 .0040 .0000 .0013 .0007 .0034 .0093 .0000 .0020 .0025 .0003 72. .0101 .0041 .0043 .0000 .0012 .0000 .0032 .0054 .0000 .0024 .0021 .0003 74. .0102 .0039 .0039 .0000 .0011 .0005 .0001 .0000 .0001 .0022 .001. .0002 70. .0102 .0037 .0035 .0000 .0010 .0000 .0030 .0057 .0047 .0010 .0010 .0002 70. .0102 .0035 .0002 .0000 .0009 .0003 .0020 .0050 .0043 .0010 .0010 .0002 00. .0101 .0034 .0090 .0000 .0000 .0003 .0027 .0000 .0040 .0010 .0012 .0001 02. .0101 .0032 . .0020 .0000 .0007 .0002 .0020 .0000 .0037 .0014 .0010 .0001 04. .0100 .0030 .0023 .0000 .0007 .0002 .0025 .0000 .0034 .0013 .0009 .0001 00. .0099 .0020 .0021 .0000 .0000 .0002 .0024 .0001 .0031 .0012 .0000 .0001 00. .0007 .0097 .0010 .0000 .0005 .0001 .0024 .000? .0090 .0010 .0007 .0001 90. .0090 .0020 .0017 .0000 .0000 .0001 .0023 .0002 .0094 .0000 .0044 .0001 0?. .0004 .0094 .0015 .0000 .0004 .0001 , .0072 .0003 .0094 .0000 .0400 .0400 7‘. .0093 .0023 .0013 .0000 .0004 .0001 .0071 .0003 .0022 .0000 .0000 .0000 90- .0021 .0022 .0012 .0000 .0000 .0001 .0020 .0004 .0020 .0007 .0000 .0000 00. .0009 .0021 .0010 .0000 .0003 .0100 .0020 .0004 .0010 .0000 .0003 .0000 100. .0007 .0070 .0009 .0000 .0003 .0000 .0019 .0004 .0017 .0005 .0003 .0000 102. .0004 .0019 .0000 .0000 .0003 .0000 .0010 .0004 .0010 .0009 .0002 .0000 I... 400'? 4’03. 4""‘7 o"0”° 4"002 .0000 000‘. 4”“6. 400“ OPOOP on... a”... 100. .0000 .0017 .0007 .0000 .0002 .0000 .0017 .0004 .0013 .0004 .0002 .0000 100. .0077 .0010 .0000 .0000 .0002 .0000 .0010 .0004 .0012 .0004 .0001 .0000 710. .0075 .5015 .0005 .0000 .0002 .0000 .0010 .0004 .0011 .0003 .0001 .0000 '12. .0073 .0014 .0005 .0000 .0002 .0000 .0010 .0003 .0010 .0003 .0001 .0000 114. .0070 .0014 .0004 .0000 .0001 .0000 .0010 .0041 .0000 .0003 .0441 .0444 110. .0000 .9013 .0004 .2000 .0001 .0000 .0010 .0003 .0000 .0002 .0001 .0000 ‘10. .0005 .0012 .0003 .0000 .0001 .0000 .0014 .0002 .0007 .0002 .0001 .0000 120. .0003 .0012 .0003 .0000 .0001 .0000 .0013 .0002 .0007 .0002 .0001 .0000 12?. .0001 .0011 .3003 .0000 .0001 .0000 .0013 .0001 .0000 .0002 .0000 .0000 ‘3‘4 400‘! 4""‘1 42022 .0000 .6001 4"”"0 4n°It 0006‘ 40.06 00.03 .fl... 4.... I250 40096 400‘“ 420”? 40020 onooI 40000 40°12 4006“ .00“! 0.001 0'... 4.... 170. .0090 .0009 .0002 .0000 .0001 .0000 .0011 .0090 .0000 .0001 .0000 .0040 100. .0091 .0009 .0002 .0000 .0001 .0000 .0011 .0090 .0004 .0001 .0000 .0000 132. .0040 .0000 .0001 .0000 .0001 .0000 .0011 .0000 .0004 .0001 .0000 .0000 134. .0047 .0000 .0001 .0000 .0001 .0000 .0010 .0097 .0001 .0041 .0000 .0000 130. .0040 .0000 .0001 .0000 .0000 .0000 .0010 .0007 .0003 .0001 .0000 .0000 ‘3'. 40", 40007 4‘0‘1 400”“ 4‘000 4"050 400‘“ 000’. 00003 002.3 0.... 0.... 100. .0041 .0007 ‘ .0001 .0000 .0000 .0000 .0000 .0000 .0003 .0001 .0000 .0400 107. .0039 .0007 .0001 .0000 .0000 .0000 .0000 .0004 .0000 .0001 .0000 .0000 104. .0037 .0000 .0001 .0000 .0000 .0000 .0000 .0093 .0002 .0000 .0000 .0000 140. .0030 .0004 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 100. .0030 .0000 .0000 .0000 .0000 .0000 .0000 .0092 .0000 .0000 .0000 .0000 100. .0032 .0000 .0000 .0000 .0000 .0000 .0000 .0001 .0000 .0000 .0000 .0000 '97. .0030 .0005 .0000 .0000 .0000 .0000 .0000 .0090 .0001 .0000 .0000 .0000 754. .0029 .0005 .0020 .0020 .0000 .0000 .0007 .0040 .0001 .0000 .0000 .0000 ‘50- .009: -0004 .0000 .0000 .0000 .0000 .0007 .0040 .0001 .0404 .0444 .0444 100. .0020 .0004 .0000 .0000 .0000 .0000 .0007 .0047 .0001 .0000 .0000 .0000 140. .007! .0004 .0000 .0000 .0000 .0000 .0007 .0044 .0001 .0000 .0000 .0000 ZI‘FHI(0IV 1‘04 .luVFVQ I0“ “xv-ml(l‘ {\~ It- - 26 .ou oa .m><.zmn¢o 4440 ...am . m N. 0 mm . ...om . . mm mo; 4 _ boasbp .5.> . 14> .n.< . u>< n.. . a oo. on .n .m.> .c .4.> .n pn~> .m .~.> .. ...> .n .n.< .c .¢.< .n .n.< .~- 1_w~n- .. .... .44.:Iznuouz.mzo.m.umo mmmz.m3m mom mu.»m.»<»m oz< >0.J.m».4.m0.mzmo <:: ..n.¢ zo.mzux.o . - --, w- .. -::l+.‘L‘£;0, , ,,,,, ,. .-;: 1:3-1 : :1 - . ,2mnxnwlunnnpnntll:a:l « . u.m i=0 Equation (6) where n and i have the usual algebraic function of specifying the algorithm for the terms, and p = ud — l. The reader iS 02 reminded_that the symbolix, (capital Pi) is the convention “Nfiikiting the multiplication of the terms, in contrast to the additive symbol 2 , (capital Sigma)“ 31 Equation (6) has the advantage of being more tractable than Equation (3). Its disadvantage, its lack of perfect accuracy outside certain limits, does not deter us, since the limits far exceed any values that will ever be encountered in data concerning the metal service center industry. The limits are: l. p cannot be less than —l. Does not apply to our real—life situation. 2. p cannot exceed Al. It is improbable that we will ever encounter a value of p higher than ten in the metal service center industry. In the highly unlikely event that we should ever encounter a value greater than Al, we would proceed to our solution by assuming a normal, rather than a gamma, distribution. Several procedures have been published for handling cases of normal distri— bution, which serve a useful function for the rare exceptional cases when the normal distribution applies. 3. x cannot be less than zero. Does not apply to our real—life situation. 4. X cannot exceed 110. Since in our development we specify x in terms of Inultiples of lambdas, which is invariably an Ethremely small number in the metal service center lindustry, it is highly unlikely we would ever encounter 'this upper limit in this industry. 32 For FORTRAN programming, let R = r PROB = F(x) 2 P = %2 - 1 PFACT = '_TF%T7T Segmented into Q and RFACT as in part B. ALAM = A (lambda) QTY = the demand quantity whose probability is being determined. EXPF = e = 2.71828. VNR—Cg . - . . 5- - variance, the square of the standard deViation. AVE = u = the average, or mean S 2 x Xn+1 Z ._ nzo n (p+i+l) Y = i x B “b-b(m) E = .0000001 = an arbitrarily selected number which determines the number of terms to which we carry our series expansion. The smaller the value of E, the greater the number of terms and therefore the greater the accuracy, because our program will continue to solve the expansion until it eventually reaches a term whose value is less than E. Consider the following series of FORTRAN statements: P = R—l. By definition. .ALAM = AVE/VAR Fundamental gamma relationship. See part A of this chapter. lOO twm U! 33 X = QTY ° ALAN Appendix C contains the mathematical proof of this relationship. IF (p+l) l, l, 2 Testing for limitation number I. PRINT 100, P, X Print P and X if any of the four limitations has been exceeded. FORMAT (lHO, 9H ERROR P = , Fl5 3, AH x = , F15.9) ERROR = 773A. A device for having a PRINT GO TO 18 statement in the main body of the program print all the other variables if a limitation has been exceeded, and stop the running of the program. IF (P-Al) 3, l, 1 Testing for limitation number 2. IF(X) l, l, A Testing for limitation number 3. IF(X—llO) 5, l, 1 Testing for limitation number A. IF(P) 7, 6, 7 If P is an integer we bypass the factorial procedure. PFACT = 1. GO TO 1A RR = P Statements 5 through 13 are the Q = l. factorial procedure explained in part B. IF (RR—i.) 11, 9, 10 IKFACT = 1, (30 TO l3 10 11 13 IA 15 16 17 18 3A Q = Q/RR RR = RR-l. GO TO 8 DO 12 I = 1, 15 N = N—l RFACT = RFACT + c(I) a RRaaN PFACT = Q * RFACT S = O. Returning these three variables to Y = 0. their original values for the new B = 10 point along the x axis now E = ,OOOOOOl being calculated. B = B % (X/(P+Y+l.fihese series of statements build up S = S + B S, term by term, until cut off by IF(B-E)17, 17 16 the extremely small value of the 3 latest term calculated. Y = Y + 1. GO TO 15 PROB = PFACT * S * EXPF(-X) * X**P This is Equation (2) in Fortran language per the Equation (6) expansion. CONTINUE Return to the main program to start IRETURN the entire procedure over again for the next point along the x axis. UThis solution of Equation (2) is included as a sub- r . I O O 0 . OutlIlEB, code name GAMTAB, With minor variations, 1n 35 computer programs GAMCUM, GAMCON, CURVES, LAMBDA, MCHART, MPLOT, and REORDR. These programs are devices for utilizing the gamma cumulative probability distribution in different ways. With the exception of program GAMCUM, all of the above programs perform important services in our proposed new metal service center procedures and will be analyzed in detail in other chapters of this dissertation. The only purpose in developing program GAMCUM was to determine the validity of our solution to Equation (2). Unlike the case twith.gamma probability density tables, tables of the gamma outnulative probability distribution have been published, ali:hough none in a form usable for our purpose. Program GAAKZUM was developed to determine if our solution of ZEquzation (2) can reproduce existing tables. The most ccnng>rehensive tables are in the NBS Handbook, labelled Table 26-"7'. These tables show the cumulative probabilities of vari;ous combinations of m and v. {Mkenlis our QTY, and v is ggiven as equal to 2 times c, where c is our r. Since tun? (development is in terms of r rather than v, and, in fact;, must remain in terms of r for the purpose of deVeloping the metal service center procedures in later ChaFVteers, it was more meaningful for our purpose to CONS131uuct tables showing cumulative probabilities of various Cmmmfiiriations of m and r, rather than m and v. Since V = 22I“, this does not complicate the comparison. 36 I 451776 BASIC 1 12 13 14 _ ‘5. 16 17 18 DIMENS1ON C(15) C(11‘1o C(21'507721566 E‘1 C(31'-605587807 E-1 C(41'-‘02002635 5-2 C(513106653861 5-1 C(613-Q02197735 E-Z C(71"906219715 E-3 C(81'702189432 5-3 C191.-1o1651676 E-3 C(101'-201520167 5-4 C(1113102805028 E-4 C(1213-200134858 E's C(1313-102504935 E-6 C(1Q1'101330272 5-6 C(151'-200563384 5-7 RFACTSOO R i R - 10 E ' 00000001 ALAN . 1o X'OTYR‘LA" : lFCP+1o110102 PR1NT 1000POX FORMAT (1H0. 9H ERROR p.0F150304H ERROR I 7734. GO TO 18 IF(P‘Q10130101 1FCX11010Q IF(X‘1100150101 1F19170697 PF‘CT'1. GO To 1‘ RR 3 P 0'10 IF(RR-lo) 110 90 10 RFAC7310 GO TO 13 0 8 OIRR RR 8 RR - 10 GO TO 8 DO 12 1.1015 Nat-l RFACT I RFACT + C(l) i RRiiN PFACT'O'RFACT 3'00 YIOO 9:10 B‘B’CX/‘P+10+Y11 355¥B ‘ ' 17(0-2117917016 V3V*1o GO TO 13 PROBOFFACT§S§EXPFC-X)§X§§P CONTINUE 'RETURN" 1 END END SUBROUTINE GAHTAB(R9 QTY. PROS. ERROR) X80F15091 37 02w .mo0m0~ oaoom. Fk0 .OO oflavmb0 u A~VX< m oO~\hG u >PO n u u #0 0".“ u a O“ 00 ad u .5.U< oon\ha n ma 1 u ha 0m .~ u 3 Cu 00 on u 00 . «om Oh no Scam a to mu MDJ<> mom « Oh «- £03m OM~Q<> Z Ik~3 mMJfl I 3% a) .IJI. . .1... t an... rccocc. 39 For example, for m = .20 and v = 6, Table 26,7 gives a cumulative probability of 0.99885. The table produced by program GAMCUM shows, for m = .20 and r = 3, exactly the same figure. In fact, all corresponding values of the two tables are identical, proving that program GAMCUM is an exact reproduction of Table 26,7, thus proving the validity of our solution. It should be noted that the lowest value of v provided by Table 26.7 is l (i.e., r = .5) whereas program GAMCUM shows values of r down to .1, Thus the table produced by program GAMCUM is superior to Table 26.7, since values of r as low as .2 are not unusual in the study of gamma probabilities. Secondly, Table 26.7 varies v in increments of l (i.e., varies r in increments of .5) whereas the GAMCUM table gives r in steps of .l. Thirdly, the producing of probability values in program GAMCUM to 5 decimal places was selected arbitrarily, to exactly coincide with the values in Table 26.7. Program GAMCUM is offered as a tool to those who in the future wish to construct their own tables, or desire the cumulative probability up to any point. Extending the range of program GAMCUM, for either higher or lower values of the parameters than those shown, requires a modification of the statement RR = RT/lO, U0 Producing tables showing still finer increments requires modifying the above statement plus the statement DO 10 J = 1,50. Producing tables to more decimal places requires a modification of statement 101 FORMAT (F901, lOF9.5). For those who are used to working with normal distri« butions, it should be pointed out that, unlike the normal equation, gamma cumulative equations represent the area of the left tail of the density distribution, rather than the usual right tail common to the normal distribution. In the case of gamma, it is more logical and usual to determine the area to the left of x, since the function originates at the Y axis and continues to plus infinity. Of course, for anyone interested in the right tail, the usual 1 - P relationship can be used. CHAPTER IV TEST OF THE MAJOR HYPOTHESIS The Major Hypothesis.--The demand pattern in the metal service center industry may be approximated by the gamma function. A. Methodology Step Number 1. The assistant general manager of a metal service center located in Michigan selected twelve inventory items which in his opinion were important to nearly all metal service centers. These were relatively high volume basic items which a metal service center is expected to have on hand to satisfy customer needs, and which have a profit mar— gin such that dealing in these items contributes to the firm's profitability. He selected items which in his opinion show- ed no significant seasonal or trend influences on demand. The selected items were: a. 1” round steel bar, 01018, 12' long b. 1%" round steel bar, 01018, 12' long c. 1%“ round steel bar, 01018, 12' long d. 1 3/4" round steel bar, 01018, 12' long e. %" square steel rod, 01018, 12' long f. 1" square steel rod, 01018, 12' long g. k” x 3/4" rectangular steel flat, 01018, 12' long h. %" x 1" rectangular steel flat, 01018, 12' long i. k" x u" rectangular steel flat, 01018, 12' long j. a" x 1" rectangular steel flat, 01018, 12' long k. g" X 3" rectangular steel flat, 01018, 12' long 1. a" x 6" rectangular steel flat, 01018, 12' long' Step Number 2. Company records were inspected to de- termine the demand for each item, in units of pieces per U1 M2 month. This data was noted for 59 months, from January, 1960 through November, 196M. Step Number 3. The mean and standard devoatopm were deter— mined for 59 pieces data for each item. It should be recalled that these two values completely determine a gamma distribution, since the two parameters, r and l have a very specific relation to the mean and standard deviation. Step Number u. 0omputer program CURVES, developed for this purpose, produced a table for each item which showed the gamma comulative probability distribution for a gamma dis— tribution with the same mean, standard deviation, and range of demand as was noted for the actual data. A cumulative probability curve was plotted for each item. Step Number 5. The cumulative distribution of the actual 59 pieces of data was tabulated. The points of this distribution were plotted on the same chart with the gamma curve. Step Number 6. The goodness of fit of the points repren senting the actual distribution to the gamma cumulative probability curve was tested by the standard chi-square test. Step Number 7. As a second visual—aid for comparing actual with theoretical data, another set of charts was prepared, showing a density type of distribution, i.e., the demand throughout the range of monthly values, from zero demand to the maximum historical demand, both gamma 43 0~W0(Q3 In. I C4) OuU01QD 0000- I U)( .COOuUUO AWOJDOIO—U0ofl)¢o!30!€0 JJ‘U _ I >h0 NICO.“ I - on 00 u~u0h0I¢<>oU>(.IDUICG JJho ouoncmoo. I _ ON 00 m~m0¢t3 . INnQ— I at) m—w04n3 O—IDH I N>¢ ma .D~NO(QDImOZDon onIOZDOQ 102— awha130 uzo 02¢ U20 10¢ cot—ob(!¢0k Non h2~¢n 00310-00 hZ-Gl amomno>FOICC>IU>(~IDUI(0 JJ‘U . I >ho o_.~_~.o_ I _ . on <.mo< coo. uwo .<.mob.»2<30 z»_J_oh~hz<30 to .01.. hz_23 Ion Io:_.h(:¢ou ~o~ hz.¢n ..I»3:._z<.4o_ooz.hJox nmN motn zo_»30_ahm_o <11 4<.co»ua3u xu 4 14 S300 Y=OI 3310 I5 B=B*(X/IP+II+Y)) 5=S+B IF(B‘E117017016 l6 'Y=Y+II GO TO 15 . '17 PROB=PFACT*S*EXPF(-X)*X**P 18 CONTINUE RETURN END END 47 cmvco. can ndmuo. cow mnooo. cow mnsro. cow naaco. cud cooro. cod vcaoo. cud comma. any sauce. and nears. cu“ mommo. cur cameo. ecu omoum. co wmaav. co moocn. cu oomum. co onowa. cm nonco. cc ownmo. ;n osvca. cu vmaco. cu «'m04nDImczzon nn.az:c¢ 102. uzc >»~»z<:c zII» mmua me e» 4.30m mqum >»uo_a»~»2I:o w chA <>24axoo 4mm»w 44mmm>~za Expax._zI.ao_ach».oz Ina no.1 2c.»:c_c»m~c .xxIc mw>¢3c t<¢ac¢t 1+8 mmooo. fiasco. cocoa. noooo. «coco. «mnoo. Iuaoo. ncoco. cacao. omega. nacho. «noon. aaomo. «saga. «nmwo. «cuca. combo. osvno. mecca. «wows. anono. nmomm. snow». oomam. a: 23 S: :3. ...: E: a: c2. c3 3... c: a: :2 a: a: 2.. .3 i .3 .3 S 3 .3 3 mawe<13Imczaoa chozacm ICZ~ awhmIDO wzo 024 wzo 49 ONE AND ONE HALF INCH ROUN0I72 POUNDSIUPAGEIC 2 .0007? 4 .00481 6 .01399 8 .02892 10 .04960 12 .07561 14 .10629 16 .14088 16' .17854 20 .21849 22 .2599! 24 .30226 26 .34479 26 .38704 30 .42856 32 .46900 34 .50806 36 .54555 36 .53128 40 .61515 42 .64711 44 .67712 46 .70518 46 .73132 50 .75559 9? .77805 54 .79878 56 .81784 56 .83535 60 .8‘136 62 .86602 64 .87937 66‘ .89152 66 .90256 70 .91257 72 .92164 74 .99983 76 .93723 76 .94390 80 .94990 82 .95530 84 .96015 66 .96450 86 .96840 90 .97189 92 .97501 94 .97780 50 ONE AND THPEF QUARTEQS INCHES ROUNDIUPAGF10098LBS 2 .1159: 4 .25434 6 .3324? a - .49410 10 .53860 1? .66726 14 .73199 15 .78484 18 . .8?773 20 . .86238 22 .89027 24 , .91265 26 , .93056 26 ‘ .94487 so .95628 32 .96536 34 .97258 36 .97831 33 .98286 40 ‘ .98646 42 .98931 44 .99157 46 .99335 45 .99476 50 .99587 51 ONE QUARTER INCH SQUARFI7ISS POUNDS.HPAGE 1E 10 20. 30 40 50 on 70 an _9o 100 INCH SQUARFI4O.8 anMDQDUPAGF 1F I4?120 .65681 .79535 .87760 .92666 .95600 .97358 .98412 .99045 .90426 .onezo .0355: .on23o .14066 .2n631 .27673 .34731 .41622 .49187 .5432: .59970 .65102 .69718 .73833 .77474 .80675 .83472 .85904 .83010 .89825 .91335 .9?720 .93860 .94831 .95655 .96354 .96944 .97443 .97863 .98216 .98512 .98760 ONF QUARTEP nY THRFE QHAQTCRS FLAT.UDAGE16'10.21p 10 .35363 2n .49802 30 .59749 4" [.67193 5n .77934 60 .77593 7" .81316 8" .84365 90 .86856 ion .89926 {in .9n649 12" .92088 isn .93294 140 .94308 150 .95162 ion .95883 17n .96493 tan . .97010 19" .97448 20a .97820 21a .99137 22o .9R407 23h . .9A637 24a .99632 250 .99000 ’6" .99143 ?7" .99265 23" .99369 ?90 .99459 son .9¢535 31“ .99601 320 .99657 330 .99705 ONE QUARTER RY ONE INC“ FLKTI17 pOUNDSIUF’AGEiH in an an 4n 5n 6n 7n an 90 ion 110 120 130 14a 150 ion- 170 13" 190 200 210 220 230 240 250 260 270 28a 290 300 310 320 330 340 350 360 370 380 .00028 .00335 .01303 .03216 .06192 .10202 .1510? .20710 .26785 .33115 .39501 .45776 .51808 .57500 I6?787 .67630 .7?014 .75940 .79422 .82485 .85159 .87477 .89474 .91186 .92643 .93880 .94924 .95802 .96537 .97151 .97661 .98085 .90435 I98724 .99961 .99156 .99315 .99446 ONE QUARTER RY FOUR INfiH FLATI48 POUNDSIUPAGElI 1O .20 30 4O 50 60 7n an 90 10" 110 120 i:n m 15" 160 ONE ‘ 5 10 15 Zn 25 30 39 40 45 5O 55 60 65 70 75 an 85 90 95 100 SA .05307 .20366 .36551 .55313 .68853 .76966 .86138 .91039 .94297 .96418 .97774 .98630 .99163 .99492 .99694 .9981? HALF BY ONF INCH FIAYI14 POUNDSIUPAGfilJ .05855 .16992 .20440 .4145: .52255 .61568 .69379 .75802 .8100? .8517? .88487 .91094 .9313: .94727 .95960 .96913 .97646 .9n209 .98639 .98968 55 ONE HALF BY THREE INCH FLATI109 POUNOSIUPAGElK 2 .000:7 4 .00643 : .0194: a .0409: 10 .07079 12 .10794 14 .1511: 1: .19::0 1: .2495: 20 .30204 22 .35502 24 .40752 2: .45870 2: .5n794 :0 .5547: :2 .59::: 34 .63998 3: ' .67607 3: .71311 4: .74515 42 .77427 44 .000:1 4: .8243: 4: .84560 50 , .::4:0 52 .88151 54 .:9:52 5: .909ao 5: .92152 :0 .931:2 :2 .940:7 :4 .94::0 :: .9557: 66 ' .96177 70 .9:70: 72 .97160 74 .97557 7: .97901 78 .9:19: 80 - .96455 one HALF av 51x INCH rtAr.204 Pouwns.021051L ? 4 6 6 1O 12 14 16 16 20 2? 24 26 28 36 32 34 36 36 40 42 44 46 46 50 52 54 56 .03403 .10151 .18266 .24840 .35260 .43231 .5058! .57242 .63190 .68446 .73051 .77059 .80526 .83510 .86069 .88254 .90114 .91694 .93031 .94162 .95115 .9591? .96592 .97157 .97631 .98028 .98360 .9863? probable and actual demand, plotted on the same chart, the gamma distribution shown as a smooth curve and the actual data as a histogram. B. Testing For Goodness of Fit This section of the chapter is an expansion of Step Number 6. It is assumed that the reader is familiar with the standard chi~ quare test for goodness of fit. For the in— cidental reader who may not be, reference may be made to any standard statistics textbook, a few of which are noted in the bibliography. This is a traditional, orthodox measure of discrepancy of frequency distributions based on the theory of least-squares. The chi—square tests are shown on the following pages, with the step—by-step procedure tabulated in six columns. IFOllowing is an explanation of the columns: Column Column Column Column 1: Range. Range of demand. Example: first range for item A shows a range from 0 to 45 pieces per month. Observed. Number of actual occurrences within that range, i.e., number of instances of demand per month which fell within this range. Gamma area. Cumulative area of gamma probability curve up to this point, i.e., the maximum point in this range. These values were read directly from program CURVES. Theoretical total. The gamma cumulative expected number of 58 occurrences up to this point. This figure obtained by multiplying the value in column 3 by 59. Column 5: Expected. The expected number of occurrences within each range, according to gamma probability. Figure is obtained by subtracting from the value in column 4 the previous value in column M (i.e., on the line above). Column 6: (O _ E)2 E The discrepancy between Observed and Ex— pected in terms of X2 Once the value of chi—square has been obtained, it is checked against a X2 table to determine the level of con— fidence, for the apprOpriate number of degrees of freedom. For most of the tests, the area under the curve was segmented into five sub—areas and thus five sets of data obtained. Since there are two independent variables in the gamma func— tion, r and A, we lose two degrees of freedom, so in most of the following tests we have 5 — 2 = 3 degrees of freedom. For the items coded B, E, and G, four instead of five sets of hdensed tables in a textbook. 59 A. 1" Round Steel Bar, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (O — E)2 E 0 — A5 7 .09326 5.50 5.50 .009 A6 — 9O 22 .51058 30.10 20.60 .275 91 — 135 21 .83331 A9.20 19.10 .185 136 — 180 6 .95735 56.50 7.30 .231 181 — 225 3 1 00000 59.00 2.50 .100 59 59.00 x2 = 1 200 NBS Handbook shows confidence level of 75.3%. That is, a chi-square value of 1.200 or larger will be obtained 75.3% of the time even when the hypothesis is true. We may therefore reasonably conclude that the major hhypothesis is true. Note that it is standard practice to accept the major hypothesis at a 1% confidence level, or 5% if it is wished to adOpt a rigorous standard. A confidence level of 75.3% 118 astonishing, and is the strongest possible proof that our major hypothesis is true. 60 B. 1%" Round Steel Bar, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 — E)2 0 — 60 50 .785 A6.30 AO.30 .295 61 — 120 7 .956 56.00 10.10 .950 121 — 180 1 .991 58.50 2.10 .570 181 - 245 1 1.000 59.00 .50 .500 59 59 00 A2 = 2.31 NBS Handbook shows confidence level of 32%. C. 1%" Round Steel Bar, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 - E)2 E 0 - 20 14 .21809 12.9 12.9 .090 21 — MO 23 .61515 36.3 23.: .007 Al - 60 11 .85138 50.2 13.9 .605 61 — 80 8 .94990 56.0 5.8 .83A 81 - 100 _3 1.00000 59.0 3.0 0 59 59 0 X2 = 1.5A0 NBS Handbook shows confidence level of 67%. 61 D. 1 3/A" Round Steel Bar, 01018,,12' Long GAMMA THEORETICAL RANGE OBSERVED .AREA TOTAL EXPECTED (O - E)2 E 0 - 10 32 .58860 3A.7 34.7 .210 ll - 20 20 .86238 50.8 16.1 .9A5 21 - 3O 5 .95628 56.4 5.6 .OOA 31 - MO 1 .986A6 58.2 1.8 .355 Al - 50 _1 1.00000 59.0 .8 .050 59 59 0 X‘ = 29 NBS Handbook shows confidence level of 6A%. E. 5" Square Steel Rod, 01018,,l2' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 — E)2 ' E O — 25 no .73510 02.6 02.6 .159 26 — 50 15 .92666 53.7 11.1 1.370 51 - 75 2 .97952 56.8 3.1 -389 76 — 108 _1_ 1.00000 58.0 1.2 .033 58 58.0 x2 = 1.951 NBS Handbook shows confidence level of 37%. Note: total of 58 months because of deletion of 1 month extremely high figure due to an unusual circumstance. 62 F. 1” Square Steel Rod, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 — E)2 E 0 — 12 17 .27673 16.3 16.3 .0300 13 — 2A 22 .65102 38.4 22.1 .0005 25 - 36 10 .859OA 50.7 12.3 .0310 37 — 48 7 .9A831 55.9 5.2 .6210 A9 — 6A _3 1.00000 59.0 3.1 .0032 59 59 O X2 = 1.0857 NBS Handbook shows confidence level of 79%. G. l/A" x 3/4" Rectangular Steel Flat, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (O —E)2 ‘“‘E“‘ 0 — 50 A: .72984 A2.3 42.3 .77 51 — 100 5 .88926 51.5 9.2 1.91 101 — 150 3 .95162 55.2 3.7 .13 151 - 331 2 1.00000 58.0 2.8 .22 58 58 0 2 X = 3.03 ‘ .NBS Handbook shows confidence level of 22% lNote: Total of 58 months because of deletion of 1 month extremely high figure due to an unusual circumstance. H. k" x l" Rectangular Steel Flat, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (O — E)2 g, E 0 — 80 9 .20710 12.2 12.2 .804 81 — 160 33 .67630 39.8 27.6 1.030 161 - 240 12 .91186 53.7 13.9 .260 241 - 320 4 .98085 57.8 4.1 .002 321 - 384 _1 1.00000 59.0 1.2 .033 59 59 O X2 = 2 165 NBS Handbook shows confidence level of 55%. I. 5" x 4" Rectangular Steel Flat, 01018, 12' Long GAMMA THEORETICAL . RANGE OBSERVED AREA TOTAL EXPECTED (0 - E)2 E 0 - 3O 24 .38551 22.4 22.4 .110 31 — 6O 25 .78966 47.7 25.3 .003 61 - 9O 6 .94297 54.6 6.9 .117 91 — 120 2 .98630 57.1 2.5 .100 121 - 165 1 1.00000 58.0 .9 .001 5 58.0 X2 = .33T NBS Handbook shows confidence level of 88%. Note: Total of 58 months because of deletion of 1 month because of 1 month extremely high figure due to an unusual circumstance. 64 J. %" X l" Rectangular Steel Flat, ClOl8, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (O — E)2 E 0 - 20 25 .41453 24.4 24.4 .001 21 — 40 17 .75802 42.7 18.3 .092 41 - 60 13 .91094 53.7 11.0 .363 61 — 80 2 .96913 57.2 3.5 .640 81 — 100 _g 1.00000 59.0 1.8 O22 59 59 0 X2 = 1.118 NBS Handbook shows confidence level of'76%. K. g" X 3" Rectangular Steel Flat, C1018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 — E>2 __ E 0 — 16 9 .19880 11.7 11.7 .62 17 — 32 28 .59883 35.3 23.6 .81 33 — 48 11 .84560 49.8 14.5 .84 49 - 64 7 .94880 56.0 6.2 .10 65 — 88 _3 1.00000 59.0 3.0 .33 59 59 O X2 = 2.70 \ NBS Handbook shows confidence level of 44%. 65 L. B” X 6" Rectangular Steel Flat, 01018, 12' Long GAMMA THEORETICAL RANGE OBSERVED AREA TOTAL EXPECTED (0 - E)2 E 0 — 12 25 .43231 25.5 25.5 .009 13 — 24 21 .77059 45.5 20.0 .050 25 — 36 7 .91694 54.0 8.5 .264 37 — 48 4 .97157 57.3 3-3 -148 49 - 60 _g 1.00000 59.0 1 7 .053 59 59.0 X2 = .524 NBS Handbook shows confidence level of 91%. 66 PROGRAM CURVES >~ .- 900 A K o s o .90 - 8 . 5: |$.80 - 3’; 3 .70 L % 3.6m- \I § CUMULATIVE 85°" — GAMMA I3 PROBABILITY .3540- ACTUAL 0: . DEMAND ; PATTERN {4.30- It COMPARING ACTUAL DE- E ' MAND PATTERN WITH In .20 - CUMULATIVE GAMMA E PROBABILITY. k 3 I" ROUND STEEL BAR, g-IOP ClOl8, Iz' LONG. a I l l l l l l I I l 020 40 60 80 |00 I20 I40 l60l80 200220 QUANTITY (IN PIECES) CUMULAT/VE PERCENT SALES EQUAL T0 0}? LESS THAN QUANTITY .90 .80 .70 .60 .50 .40 .30 .20 .I0 OPROGRAM CURVES 67 COMPARING ACTUAL DEMAND PAT- TERN WITH CUMULATIVE GAMMA PROBABILITY. I'“ ROUND STEEL BAR, CIOIB I2' LONG. ICUMULATIVE GAMMA PROBABILITY OACTUAL DEMAND PATTERN J I I I1 I I I I I II 0 20 40 60 80 l00 I20 I40 |60 l80 200220 240 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL T0 OI? LESS THAN QUANTITY .30 .20 .I0 68 PROGRAM CURVES 00 _ c O .90 - .80 - .70 — CUMULATIVE GAMMA 50 - PROBABILITY ACTUAL .50 _ . DEMAND PATTERN .40 _ COMPARING ACTUAL DE- MAND PATTERN WITH CUMULATIVE GAMMA PROBABILITY. II"ROUND STEEL BAR, ClOl8, I2' LONG. lllllllll 0I02030405060708090 QUANTITY (IN Pl ECESI I I00 CUMULATIVE PERCENT SALES EQUAL TO OR LESS THAN QUANTITY 69 PROGRAM CURVES D I.00 .90 - .80 - .70 - .60 '- ’ COMPARING ACTUAL DEMAND .50.. PATTERN WITH CUMULATIVE GAMMA PROBABILITY. .40 _ I%“ ROUND STEEL BAR, ClOl8, I2‘ LONG. .30 - CUMULATIVE ' GAMMA PROBABILITY .20 - ACTUAL 0 DEMAND .IO - PATTERN I I I I J 0 IO 20 30 4O 50 QUANTITY (IN PIECES) 7O PROGRAM CURVES XLOO H t E . . . k 2 § .90- Q E .80- E o; ‘G .70- \I g Q .60 "" K COMPARING ACTUAL DEMAND L. PATTERN WITH CUMULATIVE ‘I _ GAMMA PROBABILITY. S .50 B A A SQUARE STEEL BAR, I44 _ ClOl8, I2' LONG. q .40 55 k CUMULATIVE g .30- GAMMA a PROBABILITY o: (“f .20- ACTUAL m . DEMAND Q PATTERN k 3 .IO S S k) I I l l l l l l l l l 0 IO 20 30 4O 50 60 70 80 90 IOO ”0 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TO OR LESS THAN QUANTITY. I.OO .90 .80 .70 .60 .5 O .40 .3 O .20 .IO 71 PROGRAM CURVES COMPARING ACTUAL DE- MAND PATTERN WITH CUMULATIVE GAMMA PROBABILITY. I" SQUARE STEEL BAR, ClOl8, I2'L0NG. CUMULATIVE GAMMA PROBABILITY ACTUAL 0 DEMAND PATTERN I I I I I I - IO 20 30 4O 5O 60 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TQ QR LESS THAN QUANTITY. |.00 .90 .80 .70 .60 .50 .40 .30 - .20 72 PROGRAM CURVES ._ COMPARING ACTUAL DEMAND PATTERN .WITH CUMULATIVE GAMMA PROBABILITY. X M- " STEEL FLAT, CIOIB, I2' LONG. «Mon CUMULATIVE GAMMA PROBABILITY o ACTUAL DEMAND PATTERN IIIIIIIIIIIILIII 0 50 I00 I50 200 250 300 350 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TQ QR LESS THAN QUANTITY. I00 .90 .80 .70 .60 .50 .40 .30 .20 .IO 73 PROGRAM CURVES I- H COMPARING ACTUAL DEMAND PATTERN WITH .. CUMULATIVE GAMMA PROBABILITY. _. {'x I" STEEL FLAT, ClOl8, I2' LONG. _ CUMULATIVE GAMMA PROBABILITY ACTUAL 0 DEMAND _. PATTERN I I I I l I I O 50 I00 I50 200 250 300 350 400 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TO OR LESS THAN QUANTITY l.OO in o .80 .70 .60* .50 .40 .30 .20 O 74 PROGRAM CU RV ES COMPARING ACTUAL DEMAND PATTERN WITH CUMULATIVE GAMMA PROBABILITY. Fx 4" STEEL FLAT, ClOl8, I2' LONG. _CUMULATIVE GAMMA PROBABILITY 0 ACTUAL DEMAND PATTERN fl I I I I I I I I I l6 32 48 64 8O 96 "2 l28 I44 I60 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL T0 QR LESS THAN QUANTITY. l.00 .90 .80 .70 .60 .50 .40 .30 75 PROGRAM CURVES F J ' _ COMPARING ACTUAL DEMAND PATTERN WITH CUMULATIVE GAMMA PROBABILITY. 'E"XI" STEEL FLAT, ClOl8, I2' LONG. — CUMULATIVE GAMMA PROBABILITY 0 ACTUAL DEMAND PATTERN III‘IIIIII 0 IO 20 30 40 5O 60 7O 80 90 I00 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TO OR LESS THAN QUANTITY |.OO .90 .80 .70 .60 .50 .40 .30 .20 .I0 76 PROGRAM CURVES K COMPARING ACTUAL DEMAND — PATTERN WITH CUMULATIVE GAMMA PROBABILITY. I{'xs" STEEL FLAT, ClOl8, I2' LONG. _ — CUMULATIVE GAMMA PROBABILITY. 0 ACTUAL DEMAND PATTERN. g I I I I I I I I I 0 IO 20 3O 40 50 60 70 80 90 I00 QUANTITY (IN PIECES) CUMULATIVE PERCENT SALES EQUAL TO OR LESS THAN QUANTITY. LOO .90 .80 .7 O .60 .50 .40 .30 .20 .IO 0 77 PROGRAM CURVES I. COMPARING ACTUAL DEMAND PATTERN WITH CUMULATIVE GAMMA PROBABILITY. {'x G” STEEL FLAT, ClOl8, I2' LONG. — CUMULATIVE GAMMA PROBABILITY 0 ACTUAL DEMAND PATTERN I I I I I I I I I I 6 l2 IS 24 30 36 42 48 54 60 QUANTITY (IN PIECES) 78 x k \ qu‘mbtl v€<§ 36 IR \\: Etlwkk v.04 \C\m.\<.wQ S<3§NQ .VSK b? 35% vQ§Qb Ammowi z: >..._._.Z._._l__mL._._.Z._._I__m._._._.z<30 00. 00 00 0» 00 on 00 0m 0m 0. 0 1 _ L L _ _ U _ _ 4 L 4 ....I. L. _ 1.1. L . . lo - 4 /. .L . //. L .09 /. x . / . I o/* 0‘ x. .L . lo \ I / . I . ./ ..L .0. A D 0‘ 024.200 .2304 I / LL . C_.__m§mx§§9u Amwowi z: >._._._.ZF_4_m._._._.Z._._I_.Z._._._.Z23: L.......z4....o 000 own 090. OmN OVN OON GO. ON. Om 0.... O 024550 44:...04 I >I.._I._m4mOmn. 45.5.40 .I.l ‘ 'N. 85 ’- ‘. ~.‘. 00. O_. m.. ON. 0N. Om. mm. .1 N27953:! ... 2.. 00402 020.2 .0. 0.0.0 0.4.... ..005 ... x ..2... xkxx0$LI..._...Z4DO 00. 0.... 00. 00. 00 02 00 on 0. 0 ]III.LII.I 2 _ 4 _ 3 . I — #DIIID — \¢o. a. II, 0‘ 024.200 ..40204 I ././ .L. 00. 22....040000 42.240 ..i /./ 0.. w I 0.. .L I 00. . ..L l. \ I 0N. loll-\C I 00. I. Nm. . Iwm. 2200400 020.. .0. 0.0.0 .24.... 20020 ..4 x ..w O0. 1 NJQUJO’ 87 .xCmeQQQtOL 0.30006 2:44: EQMLKKYQ 5&thst Q2034NQ <03ka .3504340490 .mwowE z: EFT—.2430 00. 00 00 02 00 00 04 on 00 0. 0 I IL L L L L L L L 4 lira, .I. I. I. Iv I. I ’0 0’. ’0 / 9/ i 024200 ..40204 I ..r L 22.204000“. 42240 I... _I/ .L x. L x . I . L / o . .L .2 .L o/( 4‘ I 2. . .L / I» —. ./. ..s 2 00400 020.. .0. 0.0.0 .24.... ..0020 ... x ...m /.I.\ I no. 0.. ON. ON. .I NJQUSd 88 kaWL..._...Z4DO 04 04 00 40 0. . . d a . I. I 024200 ..40204 I _ Lu 22.204000“. 4.2.240 Ii /./ L. ./. - \ /.I..1.I \.\. v— 2.00400 .0 x m. .OZOI. .N. .90.0 ....4.._0. ...wm...m no. 0.. 0.. ON. mN. On. on. O0. .1 N JOE/27d XK\<\Q04QQQQ 03.3403 03:»: Etmkk‘l xkmexIwQ QL<0SP._._._.2430 89 00 40 02. 04 00 0.... 40 0. 0. 0 0 ..I _ L L A _ H _ . 0 1 .-.-.I.I—I.I — I... . I. /. I l. /. 024200 ..40204 II ..II . 222040000 42240 I... IS I ./ .— ... 00400 020.. .0. 0.0.0 .24.... 00020 ..0 x ..0. 9 .I N39U3d CHAPTER V DEVELOPMENT OF FIRST SUB—HYPOTHESIS f]; 0 Restatement of First Hypothesis A gamma—based theoretical framework may serve as a basis for developing internal operating procedures with— in the firm, includirg tables and Charts tailored to the firm's needs. E. Basis For Determining Reorder Point' With regard to the inventory itself, there are two parameters involved in the decision making: reorder quan— tit and reorder point. As the quantity in stock decreases, due to demand, a point is reached at which we must make the decision to replenish our stock. This level of inventory is called the reorder point, R. At the reorder point, the decision is also made as to the quantity to be ordered, called the order quantity, Q. There is a very large body of literature devoted to the study of the econom'cal EOQ, the economic (I) order quantity, usually referred to a order quantity. Virtually every book and article listed in the bibliography of this dissertation devotes close attention to this subject. In View of this, it would serve no pur— pose to cover the subject here. We will therefore assume that the EOQ has been determined and that we may accept the 90 91 figure Q, as given. The remainder of this dissertation will be devoted to the determination of the second para- meter, the reorder point, R, All of the early writing in inventory management assumed that Q and R were mutually independent, and many present simplified models make this assumption, often with very satisfactory results. Nearly all present literature, however, assumes some type of mutual dependence, either equal or the dependence of R on Q. The point of View adopted in this dissertation is that Q and R are not mu— tually independent. Since the form of inter-dependence is not crucial to our hypotheses, we will adopt the convention that R is dependent on Q. It is not crucial because objec- tions by those who believe the dependence is mutual can be easily met by suggesting that after determining R by means of the procedures developed in this dissertation, they use this new value of R to calculate a new value of Q (since several modern sophisticated EOQ equations include R as a variable) and continue to perform the iteration until, with reference to total annual cost equations, they are satisfied that they have determined the best possible combination of Q and R. This fairly simple-minded iteration can be easily programmed for a computer, so there is little point in de— voting undue attention to mutual independence theory here. It was pointed out in Chapter I that a balance must be struck between too large an inventory, resulting in high 92 holding costs, against too low an inventory, resulting in high stockout costs. Unit holding costs, referred to as UHC, expressed as percent of the dollar value of one unit, is another item of inventory management which has been studied in great detail, so will not be covered here. For illustrative purpose only, a fairly typical UHC breakdown is included here (from G. Carson, Production Handbook (New York: Ronald Press, 1960). In Section 4—58 of this citation: Interest on investment 3.0% Shrinkage (waste, scrap, losses, theft, etc.) 5.0% Storage (rent, heat, light, janitor service, etc.) 2.0% Taxes 1.5% Insurance .5% Depreciation of Capital Assets 2.0% Material Handling 0 Record Keeping 4.0% 18.0% For example, an item purchased at a cost of $1 has cost $1.18 after being held for one year. The figure used in the metal service center industry appears to range from 15% to 25%. Stockout costs in the metal service center industry may be calculated directly, Since the major factors are tIe purchase of the out—of—stock items from a competitor at a higher—than—mill price, and the cost of picking the material up at the competitor's plant. The reader is referred to the bibliography included in this dissertation for more com- prehensive coverage of the subjects of unit holding cost and stockout cost. 93 It was pointed out in Chapter I that to never have a stockout would require an inventory large enough to always be prepared to handle the maximum demand. Since this maximum demand is a rare occurence, it is clearly too costly to be always prepared for it. This leads us to the conclusion that the optimum inventory policy requires a condition of stockout a certain percentage of the time. Assuming that the metal service center will always act rationally, inventory will not decrease to zero with- out the management immediately reordering. This, then, gives us the condition that stockout always occurs during a lead— time. Rather than the determination of the optimum percent— age of time the metal service center Should be out of stock, it is more meaningful for decision making to determine in what percentage of leadtimes a stockout condition should occur. The determination of this percentage has been devel— Oped in several recent publications. The most straightfor— ward derivation appears to be the development on pages 375- 377 in Robert Goodell Brown's Smoothing, Forecasting, and Prediction (Englewood, New Jersey: Prentice—Hall, 1963). The derivation is unfortunately marred by an error in the ”expected annual cost of shortages” equation on page 376 which (corrected) should be Dl : 1 i (k — t) p(t) dt 94 Since Brown concludes with the correct final solution on page 377, the error may be typographical. It is mentioned here to assist those who may wish to Study the dervation of the "Optimum probability of a stockout juring leadtime" equation. His derivation is straightforward in that he takes the derivative of the total annual cost equation and sets it to zero (i.e., set the tangesnt to the total annual cost curve at its minimum point) to determine the probability at the minimum point of the total cost curve. This turns out to be (page 377) 'D 1 - F(k) =f.. p(t) dt : l l + TSCl/er) Equation (7) where s = demand per year Cl: stockout cost per unit short r= unit holding cost v= cost per unit The clearest exposition of an "Optimum probability of stockout during leadtime" equation appears to be one used in various publications prepared for the Steel Service Center Institute by Dr. Claude McMillan Jr., Professor of Management, University of Colorado: Optimum Probability = UHC (CPU) Q STKCOS(DPY) + UHC(CPU)Q Equation (8) where UHC = unit holding cost, expressed as a year cent (in decimals) of the dollar value of one unit 95 CPU = cost per unit (delivered cost of the item in inventory) in dollars Q STKCOS reorder quantity stockout cost per unit short, in dollars DPY = expected demand per year An illustrative example: if UHC = .18 CPU = $11.20 Q = 100 pieces STKCOS = $5 DPY = 289 then .18 x 11.20 x 100 5 x 289 + .18 x 11.20 x 100 II Optimum Probability = .12 In other words, for the above parameters, the optimum inventory policy is to have a stockout occur in 12 out of every 100 leadtimes. If for this particular item we average 4 replenishments of stock per year, 100 replenishments would cover a period of 25 years. Twelve stockouts in 25 years is approximately one stockout condition per two years, for this item. Obviously the way to obtain this optimum probability is by selection of the exact reorder point which will cause this proportion of stockout conditions to occur. The following section will be devoted to developing procedures for determining this reorder point. 96 (7 Development of Procedures for Determining Reorder Point Our development follows orthodox statistical pro- cedures for solving this type of problem. Those who are familiar with the normal distribution will find that the argument has a familiar ring, the only novel feature being that attention is drawn to gamma, rather than normal, tables. Development of General Gamma-based Tables to Determine Reorder Point for Any Parameters of the Demand Pattern In part B we addressed ourselves to the problem of deter— mining the reorder point for an item such that the probability of a stockout during leadtime would be .12. Since this is ob— viously a problem in cumulative probabilities, we know we can find the solution in the tables produced by computer program GAMCUM. It will be recalled that GAMCUM was developed to reproduce an existing gamma cumulative prObability table. Eefore solving the specific problem before us, let us first familiarize ourselves with the unique parameters of gamma cumulative probability tables. If we know the mean and the standard deviation of the item under study, we can determine the gamma parameters r and lambda, since 97 Program GAMCUM lists cumulative probabilities for various values of r and x, where x is the quantity along the x—axis whose probability is under study (see Equation (2)). The parameter r is listed vertically, ranging from .1 to 5.0 in this particular table. The value of x is plotted horizontally. To present tables which are universally applicable, x is stated in terms of lambdas and labelled m, where m = x°x. The pur- pose of this convention is to provide universal tables, since for a certain value of r there are obviously an infinite number of combinations of x and Awhich will have the same cumulative probability. For this set, the product of x and A;is a certain fixed value, called m. Let us take a gamma distribution such as that u = 15 and 02 = 750 for the particular leadtime required for replenishment. 2 2 Then r = u --——LL:L- = .30 and 02 _ ISO _ u = 5 A — '7;T‘— "rfl§3—— = .02 What is the probability of the occurence of a value of 45 or greater during this leadtime? m = A 'x .02 -45 = .90 On the table produced by progam GAMCUM for r = .30 and m = .90 read .09775 as the probability. The probability is .09775 that a value of 45 or more will occur. Note that 98 we have defined u as the average figure for total demand during leadtime, i.e., on the average, demand during lead- time totals 15. Thus the statement "the probability is .09775 that a value of 45 or more will occur" means that the probability is .09775 that total demand during leadtime will be “5 or more. If we fixed our reorder point at “5, the probability is .09775 that this figure will be exceeded. i.e., that we will have a stockout. If we fixed our reorder point at 45, the probability is that we will have a stockout 9.775% of the leadtime. Now let us suppose that the item we are studying also happens to be the very item which in part B we found required a reorder point such that the probability of stockout was .12. We have noted that in the tables produced by program GAMCUM probabilities are found in the table, which is con- structed for certain combinations of r and m. Since we know that r = .30 we go horizontally along the r = .30 line to find a probability of .12. Unfortunately we do not find a pro- bability of .12 listed. The nearest we can come to .12 is either .13314 for m = .70 or .11376 for m = .80. A rough interpolation would suggest that .12 would correspond to an m of .75. Using this value of m, then since m = A-x, then x = m = = 37.5 units A If we set our reorder point at 37.5 pieces, the pro- .75 O2 bability is .12 that demand during leadtime Will be 37.5 or greater. If we set our reorder point at 37.5 pieces, the probility is that a stockout will occur 12 out or every 100 leadtimes. 99 We have thus solved our problem and determined the proper reorder point, but with mixed emotions. While it is gratifying to obtain a correct solution to a problem, it is clear that the use of program GAMCUM tables is a very cumber— some operation. Since the values of r and optimum pro— bability are known, and we are attempting to determine m so that we can divide it by lambda to obtain the reorder point, then clearly the table to use is one which gives values of m for various combinations of r and optimum pro— bability. A study of Equation (2) will make it obvious that con— structing this table is no simple matter. I" . A X _ A _ . As a first step it will be necessary to rearrange the equation algebraically to solve for x, which is clearly im— possible, since x appears three times within the integral, as an exponent, as a variable raised to a power, and as a limit. The only solution appears to be to accept Equation (2) as it stands and use the subroutine GAMTAB developed in part 0 of Chapter III as a base on which to design an iterative type of computer program, which will assign, in order, the desired values of probability to the left side of the equation and then by iteration search for the value of m, called QTY in lOO our program, which will cause the right side of the equation to equal the left side. This is program GAMCON. The line print of this computer program, as with programs GAMCUM, MCHART, and REORDR, appears in this dissertation without its subroutime GAMTAB, which in actual practice would follow immediately after the main body of each of the above programs. This was done because the subroutine GAMTAB is common to several programs, and repetition would have been needless. Program GAMCON begins by assigning a value of .2 to r, and then, successively, values of .01 to .50 to pro— bability, at each value of probability searching for the m (or QTY) which will give that probability. For each pro- bability, a value of l is first attempted for QTY, without knowing whether this initial figure is too high or too low. Should this prove to be too low, QTY is increased to 2 and so on. Once the correct value of QTY has been exceeded, 1 is subtracted from the tentative built up value of QTY and the search continues in increments of .1. This procedure is repeated for increments of .01, .001, .0001, .00001, and .000001. When the correct value of QTY is exceeded on the last search, the search ends and QTY printed out in the body of the table. The process is repeated for the next value of probability, which is .02, and so on. After com- pleting the first column (r = .2) through a probability of .50, the next column is claculated, this time with r = .u, 101 5 45I776 BASIc‘ ' I RROORAM GAMCON DIMENSION sIZOFNIso.IOI. AaIso.IOI. AREIIOI. OPTPRB(50) DIMENSION AEEEISO.20I. RROBABISOI PRINT 77 77 FORMAT IIHII RRINT 69 69 FORMAT (IHO. 15H RROORAN GAMCON) PRINT 70 7OOFORMAT IIHO.IO9H THIS TABLE snows THE VALUE OF M FOR EACH OPTIMUM IPROBABILITY OF srocxour OURINO LEAD TIME. FROM .OIO TO .500) PRINT 7) 71 FORMAT (44H (THAT )5. FROM 1 TO 50 PERCENT PROBABILITY)) PRINT 72 72 FORMAT (1H0. 31H FOR VALUES OF R FROM .5 TO 201) PRINT 75 75 FORMAT (1H0. 84H THE SECOND COLUMN FOR EACH VALUE OF R SHOWS THE F IACTOR WHICH WHEN DIVIDED BY LAMBDA) PRINT 76 76DFORMATC57H G)VES THE EXPECTED AMOUNT OF STOCKOUT PER EACH LEADTIM 1E) AREC )) = 05 ARE( 2) 3 06 ARE( 3) = 07 ARE! 4) 3 08 ARE‘ 5) = 09 ARE‘ 6) = 10) ARE! 7) 3 )03 ARE( 8) 3 )05 ARE‘ 9) = 108 ARECIO) 3 20) ERROR 8 O. D\ 100 J 8 lo )0 R 8 ARE‘J) QTY 3 00 DO )OO L 8 lo 50 PRBLTY 3 L PROBAB(L) 8 PRBLTY § .010 l s 49 + L ) QTY 8 QTY + )0 CALL GANTAB (R0 QTY. PROS. ERROR) STEP 3 )0 IF (ERROR - Do) 20. )OO). 20 100) N 8 PROD * 10000. [F (N - 1*100) lo 990 2 2 QTY 3 QTY - lo 3 QTY 8 QTY + 0) CALL GANTAB (R0 OTYO PROBO ERROR) STEP 3 20 1002 1005 10 )1 1006 12 13 1007 99 100 200 102 (F (ERROR - O.) 20. N 3 PROB * 10000. (F (N - (3)00) 3. QTY 3 QTY.- .1 QTY 3 QTY + .0) CALL GAMTAB (R. STEP 3 3. IF (ERROR - O.) 20. N 3 PROB * 10000. [F (N - 1*100) 5. QTY 3 QTY - .0) QTY 3 QTY + .00) CALL GAMTAB (R. STEP 3 4. (F (ERROR - O.) 20. N 3 PROB * 10000. (F (N - 1*100) 7. QTY 3 QTY - .00) QTY 3 QTY + .0001 CALL GAMTAB (R. QTY. STEP 3 5. (F (ERROR - O.) 20. N 3 PROB * (0000. IF (N - (*IOD) 9. QTY 3 QTY - .OOOl QTY 3 QTY + .0000) CALL GAMTAB (R. QTY. STEP 3 6. )F (ERROR - O.) 20. N 3 PROB * 10000. )F (N - )*100) l). 99. QTY 3 QTY - .OOOO) QTY = QTY + .OOOOO) CALL GAMTAB (R. QTY. STEP 3 7. IF (ERROR - 0.) 20. N 3 PROB * )OOOO. TE (N - I*)OO) l3. SIZOFM(L.J) 3 QTY CONTINUE DO ZOO J 3 l. 10 DO ZOO K 3 50. 99 I 3 (DO ‘ K L 3 K - ‘9 AB((.J) 3 SIZOFM(L.J) DO 300 J 3 1.10 DO SOD I 3 l. 50 K 3 (J52) - ) ABEE((.K) 3 AB(I.J) 1002. 20 99. 4 QTY. PROB. ERROR) 1003. 20 99. 6 QTY. PROB. ERROR) 1004. 20 99. 8 PROB. ERROR) 1005. 20 99. IO PROB. ERROR) 1006. 20 )2 PROB. ERROR) 1007. 20 99. 99 103 I 102m - .~I_. h»O .a .ambm .oc »z_aa ON on be no: .n.mu .m.hu .n.mu N .n.hu .n.mm .nohu .m.mm .mohm .n.mm onohu .m.mu .m.hu .n.mu— .n.bm .m.mm .n.hm .n.mm .n.>m .n.mm .n.>m_.x_.n.mu. p_a aux: xu_xx acsucs .xs nxoxn a so mag.) xo>~4~cccocu tat—pgo 39.0 to; I so m34.> as» .103. .40.» an!» :oox.. ...-ac. 105 so on. The complexity of the program is shown by the fact taht the total time to construct the chart, plus the initial compiling of the program, consumes a total of “5 seconds, a relatively long program for the CDC3600 computer. Following is an example of the use of program GAMCON: Let us assume an item which has a demand pattern during leadtime such that u = 38 and 02 = 722. Then r = u,E = (3?) = 2.0 and O 722 A = u 38 = .0527 02 722 Let us assume, further, that the Optimum probability of stock during leadtime is .11, as determined from Equation (8). From GAMCON, for r = 2.0 and optimum probability of .11, read m = 3.770. x = m A 3.770 .0527 = 72 = optimum reorder point. Note that the only parameters required to determine the optimum reorder point, other than those used in Equation (8.), are the mean and the standard deviation. All other values are based on these two. To start the new systemon a firm, these two parameters, can be obtained by using historical data to solve for the mean and variance (the square of the standard deviation). 0f the several forms of the variance equation, the following version is the most conducive to rapid calculation with an Office calculator. 106 2 ()Xi)2 O2 _ jig—i - N —' N - 1 Equation (9) A modern office calculator will compile 5x2 and Ex with one entry of the data. It is only necessary to square the com— piled value of fix to obtain (fx)2, divide this value by N, and subtract the result from 2X. The final step is to divide this figure by N—l. If company records are accumulated monthly, these figures would give the mean and variance for one month. For cases where the leadtime equals one month, these figures may be used directly in claculating r and lambda. For leadtimes for more or less than one month, u changes line— arly and lambda remains the same. For example, if leadtime is two months, and u has been obtained from historical monthly records, multiply u by 2 to obtain u for the leadtime period. Lambda does not change. values of u, variance, and optimum reorder point should be recalculated several times per year. Once the new system has been installed, it may develop that raw historical data is not satisfactory for forecasting future demand patterns. The two parameters, mean and variance, may then be forecast by more modern methods. The reader is referred, for example, to the previously cited R. G. Brown, Smoothing, Forecasting, and Prediction, which handles this subject at a very high level of sophistication. 107 Since it is immaterial to the new proposed procedure what forcasting techniques are used, this subject will not be dealt with here. Any number of different tables may be constructed by minor modifications in the GAMCON program. Additional tables with different values of r require only the substituting of other values for the ARE statements. Tables showing finer increments of r also need other ARE statments in place of those in the original. Tables carried out to more decimal places require modification of the 102 FORMAT statement. Tables showing finer increments of optimum probability require modification of the statement D0 100 L = l, 50, This statement may also be modified to extend the tables beyond 50%, although it is unlikely that for high volume items it would be the best policy to have a shortage more than half of the leadtimes. D. Determination of the Expected Amount of Stockout During Leadtime. The procedure developed in part C is complete in it- self. However, it may be a matter of interest, and an aid in policy making, to know not onlytflmaproportion of lead- times which will have stockouts, but, in addition, on an overall long range basis, what stockout quantities will occur (i.e., how many units will we be out of stock with whis reorder policy?). The most meaningful manner of presenting this figure, so that it may be compared with shortage quantities at other optimum probabilities of 108 stockout, is to find the expected value of the demand during leadtime in excess of the reorder point, in the traditional statistical sense of expected value, i.e., ERDLLT R). This figure represents the expected amount of shortage per each leadtime, even though most leadtimes will have no shortages. For example if E(DDLT>R) = .17, then we should expect that for a large number of le dtimes the total of all quantities short, when divided by this large number of leadtimes will average out to .17 units per leadtime. We would expect a total of 17 unites short over a period of 100 leadtimes, although shortages will occur in only perhaps 11 or 12% of the leadtimes (as determined by the reorder point). A moment's reflection will show that once we have found our reorder point on our cumulative probability curve and run a vertical line up to the curve, then the area of the right tail must be the quantity of expected shortage, 1.8., the E(DDLI)R). This is because if we had plotted this vertical line on our density distribution curve, the area of this curve's right tail would represent the average amount in excess of this particular reorder point. But the point to the right of this reorder point also has its right tail, and so on. The sum of these density areas is the average quantity in excess of our fixed reorder point. But the sum of these density areas is precisely what a cumulative area represents. 109 If we were to segment the cumulative curve by running a vertical line from each integer reorder point up to the curve, then the area of one segment would be approximately l(i.e., the horizontal distance between reorder point (1) and reorder point (2))times ( Pl+P2), that is,the average of the probabilities on the Y axis associated with these reorder points, in other words the average height of the segment. If we proceeded along the x axis for every segment to the right of the main reorder point (the optimum for the case) we will have determined the area of the curve to the right of the reorder point, R, and have thus determined E(DDLT>R). A computer program based on this principle was developed and incorporated as part of program GAMCON, the only differ- ence being that instead of segmenting by integer reorder points, it was more convenient, (but more difficult for explaining the theory) to segment by those reorder point values which gave integer probabilities. In either case, the area of the first segment to the right of the main reorder point, R, is P + P = - 1 Area1 (R2 R1) ( 2 ) But since R = ___EL__. (from section C of this chapter) we A may rewrite the above equation as follows: P2 * Pl Area, = (£2 — m} > <——.,—> 110 Area2 = (_Ei —._E§_) (_E3_g_E§) and so on. A A The sum of all the areas equals E(DDLT)R), which can be written, in general form, as , ,P + P E(DDLT)R) =._l_ (1n_3__n:l (mn_l — mn)) + ....etc. A _—. _ Equation (10) The computer program which will do this is incorporated into program GAMCON from statement DO MOO k = 2, 20, 2 through #00 CONTINUE. The purpose of statement 402 CUMTRM = .15 * ABEE (I, K-l) % OPTPRB (l) is to include the area from one per cent to zero per cent. Since the GAMCON table does not show a value for m for zero per cent probability, it was determined empirically to be approximately 15% larger than the value of m for one per cent. The appropriate figures are included in the GAMCON table as the second column under each value of r, the first column under each r, of course, being m. For those who are curious as to how one programs a computer to interlace two matrices, the method used here will be explained. The first matrix, a 10 column x 50 row matrix, is es- tablished in statement 200 AB ( K,J) = SIZOFM(L,J). This matrix contains the values of m. The 10 column x 50 row matrix is then expanded to a 20 column x 50 row matrix by the following series of statements: DO 300 J = 1.10 DO 300 I = 1,50 K = (J r 2) -l ABEE(I,K) = AB(I,J) The values of m have been transferred from the AB(I,J) matrix to the ABEE(I,K) matrix. Note from the K = (J * 2) — 1 statement that K consists of odd numbers only, so the m matrix is in the odd numbered colums and the even numbered columns are vacant, for the moment, in the computer memory. (The stockout figures are established in the same AEEE(1,K) matrix but in even numbered columns only, by means of the statements DO MOO K = 2, 20, 2 and ABEE(1,K) = EMCUM The writing of the 102 FORMAT statement proved cumbersome, as can be seen, because of the desire to group the two columns under the same value of r closer together than to the columns for the adjacent values of r, which required the specification of field width for each of the columns. With regard to program GAMCON, the number in the second column is the value inside the brackets in Equation (10). In other words, we must divide this number by lambda to obtain E(DDLT)R). The purpose of this convention is to give the table general applicability, for any gamma distribution. We will now proceed to prove the validity of the figures in these second columns. To reduce needless 112 algebraic calculating time, let us take a gamma distribution with the parameters r = l and A = 1, and we desire the integral from 1% to 50% probability. Then F(x) 2 -A r -x* xi“1 dx (r—l)§ e A Equation (2) 1 (mso — 1 ol 1 1 d (1-1): Jmi e X X X mso = )( e"X dx II (D l >4 L__ B S W U"! o From the GAMCON table, under r = 1, read m .693 for 50% and 4.612 for 1%. Then ... L e .693 = -4.612 e .501 = .010 .491 = area between 1% and for these \n 0 b3 parameters. The figures in the second column are .494 for 50% and .003 for 1% .491 = area between 1% and 50% for these parameters. -he figures check, proving the validity of the second column. The second example will check for values with small increments. For the area between 32% and 35%: -1 050 35% read m = 1.050 e . 2 = 359 32% read m = 1.140 9-13149 = 323 033 The second column figures give .344 for 35% and L314 for 32% .030 Check. 113 E. The Development of Special Tables Program RANDL It has been pointed out that the meanewwlvariance, along with the optimum probability of stockout, are the only parameters needed to determine the reorder point. Knowing the mean and variance, the next step is to calculate the gamma parameters, r and lambda, where r is the square of the mean divided by the variance, and lambda is the mean divided by the variance. Since r and lambda ensue directly from the mean and variance, it is a simple matter to develop a computer pro- gram which will solve these two equations for any given combinations of mean and variance. This has been done in program RANDL. In the version appearing in this dissertation, values of mean have been selected ranging from 20 to 320, and values of variance frdm 20 to 300. For any combination of given values of mean and variance within this range, the computer prints out two values, r and lambda. Program RANDL, as with the other programs included in this dissertation, is offerred as a time—saving tool for any— one working with any aspect of gamma distributions. As many different tables as desired may be constructed with this program. Different values of mean are obtained by modifying the statements 114 O ASIIJO BASIC I "m- H. PROGRAMTRANDL DIMENSION VARNCEIITIO THEMNCIT). HLAMDAII6). RIIb). MEANIIT) ‘ DIMENSION JVARNCIITI PRINT 6? 67 FORMAT IIHII PRINT 69 69 FORMAT ‘IHOO IAH PROGRAM RANDLI PRINT TO TOOFORMAT (1H0. 65H SHOWING R AND LAMBDA FOR GIVEN COMBINATIONS OF ME IAN AND VARIANCE) PRINT TI TIOFORMAT IIHOO 54H VALUES FOR MEAN LISTED HORIZONTALLY AT TOP OF COL ’IUMNSI V PRINT 72 TZOFORMAT (IHO. 58H VALUES FOR VARIANCE LISTED VERTICALLY IN LEFT HAN ID COLUMN) PRINT 73 73OFORMAT IIHOo 84H FOR EACH SET OF DOUBLE FIGURES THE UPPER FIGURE I IS R A30 TFE LO‘ER FIGURE IS LAMBDAI ’ ‘ " ' “ I‘DING 3 O. DO 3 K 3 20 I7 VARNCEIII 3 200 JVARNCII) 3 20 VARNCEIK) 3 VARNCEIK-I) + 20. JVARNCIKI 3 VARNCEIKI DO 8 J 3 20 I7 THEMNCII 3 20. MEANII) 3 20 THEMNIJI 3 THEMNCJ-II + 20. MEANIJI 3 THEMNIJI _HLAMDAIJ-II 3 THEMNIJ-I) I VARNCEIK-I) RIJ-I) 3 HLANDA(J-I):3 THEMN(J-I) CONTINUE IF(HEDING - O.) 20 60 2 PRINT To (MEANINIQ N I I0 I6) FORMAT (IMO. 4X9 16I8I WING 3 7733. PRINT A. JVARNCCK‘IIO (RILIE L I I. I6) FORMAT (IHOO I60 IOFOoZI PRINT 50 (HLAMDAIMI. M 3 Io I6) 5 FORMAT (TX. IOFOoZI 3 CONTINUE ’ PRINT‘OO 68 FORMAT IIHII END END ON ~10 U 115 ...a ...-an u-.u nu.ucn cu.“ as...» no.« on...» anon u..o~v ovou Ivonne ...u ...Nun us.“ ...-on 00.0 ...-v. ouon av.unn so.~ na.nno cu.» o..d~o- 0a.. ao.oo~« 50.00su 90.300« 09.0« ...ouuu on» :0 o~.u.~ on.“ ...-on up.” noouun ou.u nuoovn nu.“ oo.nsn onou ...... onou co.oov boon ...-on coon on.~on «u.~ o-.«co on.“ 0..-ts co.» co..:. rs.» oo.nnuw ao.ocnu on.~ no.9ruu co.nw a..oonv can :0 ...aou no. nu.uo~ coca ooooou ...u vnoucn bu.u $0.03» an.“ 00.0an ovou oooaon on.u anon». anon no.00v no.“ nu.nro co.~ co.vcn co.» cc.oco so.v 50.09nu co.h cc.o44¢w> au»v_4 muz<_c«> «on um3444¢>zc~.rcz cu»v_4 :«u: «on om34<> wuz..a<> c2. 74accuu¢u> auuo_4 caczca so. nuauc0 uzxzuou .0 0o» uc >aucuao~u0oa noun-4 c 0c. unauc0 cooxcu as. c .0 zcuyczucxou zu>uu m2» can uzuo0 can-on: xztuuno m2» nw>uo mun.» aux» 4w»m4 wo_>0un m>c0c mxu can uzuuuug nu guano 20199.0 120 n1 uniquely associated with each value of r. Since this value cxf m must be divided by lambda to obtain the reorder point, ijt should be a simple matter to develop a computer program vvtiich will make this simple calculation for all the values (DI? lambda which are liable to be encountered, thus eliminating ‘tlie need to make this calculation. This is done with prrogram ORDER. Program ORDER tabulates for a given optimum purobability of stockout during leadtime, the optimum reorder puaint for various combinations of r and lambda. One table is; required for each optimum probability. Since there are 5C) optimum probabilities listed on program GAMCON tables, 5C) ORDER tables would completely eliminate the need for Larwagram GAMCON. In actual practice, the firm will probably ffirld that the optimum probability is confined to a small rwarlge, so that far less than SO ORDER tables are required ‘tc: eliminate GAMCON. Note that program ORDER is tabulated for combinations CDf‘ r and lambda. But it will be recalled that program YRAIIDL gave us values of r and lambda from the basic mean aruj variance characteristic. The step-by—step procedure is ttman.as follows: 1. Determine the mean and variance characteristics Of“this particular inventory item, either by means of Equéition (9) or by some form of forecasting technique. 2. Look up r and lambda in a RANDL table. 3. For these values of r and lambda, look up the Optinnlm reorder point in a program ORDER table. 121 Not only does this procedure eliminate calculations c>f r and lambda, it also eliminates m from the procedure. UThe only items requiring calculating or forecasting are the rnean and the variance, which are parameters easy to under- sstand and visualize. Eliminated are calculations involving I“, lambda, and m, which are somewhat abstract in concept 23nd therefore difficult to visualize. It would not take long for a firm to determine which Isanges of tables it requires, and modify the various pyrograms to obtain the exact tables desired. F“. Development of the Reorder Point Chart. Let us turn our attention once more to program GAMCON. [A study of the values of m which are tabulated therein sldows a pattern of the variation of this parameter. For exxample, if we were to note the position of the value m = 2 ill each column, we would see that as we moved from left to Ifiight, the position of m = 2 drops gradually. If we were ‘tCD make a mark in each column at the position m = 2, CCJnnecting these points witha line would produce a line of cc1nstant m = 2. This is precisely what has been done to obtain Ccnistant-m lines on the Reorder Point Chart. With regard tC) the constant-m lines, the Reorder Point Chart is a grYiphic presentation of a GAMCON table. The actual lines W91“? obtained by development of program MCRART, which is 122 02m quu. F(IUOL #0 F0 hZuflQ unoouou .u0MLv F(ZKOL ~0— uomou u 5 ..Ouou u x 0u¥036PZURUQ6 oufivmak0 oflumk0 V n u¥vzm Ouou u x Ou.OO uhaw4 wc~>¢wm wxupacwm 02¢ gut—h aqw... .mcz:c<> .z.wx Lo mm:a<>.2m>—e to... «scam: 14:601- 136 SERVLV = the service level, in per cent, 1. e., the probability of stockout during leadtime, determined from Equation (8). The procedure is as follows: 1. The values of the four variables are keypunched, as for example: THEMN = 18.84 VARNCE = 37.M THELDT = 2. SERVLV = 12. These four cards are placed on top of the program deck (or as near to the top as possible, depending on the computer procedure). 2. The computer calculates the exact value of r, which is equal to the value of r for one month times the leadtime: = THELDT .. THEI‘«HT**2 / VARN’CE will The GAMCON type of search procedure takes place to find the m which corresponds to the particular value of r and SERVLV. A. Lambda is calculated: HLAMDA = THEMN / VARNCE 5. The calculated m is divided by lambda to obtain the reorder point. REORDR = SIZOFM / HLAMDA 137 6. The computer prints the four given parameters plus the optimum reorder point. It should be noted that program REORDR provides the exact answer, in contrast to the GAMCON tables, which may require some interpolation between the r columns. There are various ways in which program REORDR can serve a useful function for a metal service center: 1. For a firm large enough to have its own computer, program REORDR is a convenient means of changing over rapidly to use of gamma-based inventory control policies, since the four variables: mean, variance, lead time, and optimum probability are easily determined or forecast. 2. For a firm too small to own a computer, the program can be furnished to the nearest data processing center for supplying the firm with the proper reorder point for any item. 3. As a statement of reorder point policy, which can be transferred to every inventory card in the small firm, as follows: Let us assume, for an example, that the firm has 2,000 different items in inventory. The variance, optimum probability, average leadtime, and monthly mean can be keypunched for each item. One statement is added near the top of the program and one near the bottom: DO 96 J = 1, 2000 138 96 CONTINUE to form a DO loop. With slight modification of the PRINT and FORMAT statements the program will determine the optimum reorder point in a few seconds for every item in stock. The procedure would be repeated several times per year, to keep the reorder point determination current. CHAPTER VII SUMMARY AND CONCLUSIONS A. Validity of the Hypothesis It will be recalled that we originally embarked on this program because logic impelled us toward the belief that; l. demand in the metal service center industry could be approximated by some probability function, and 2. the function should be one which never went below zero. The most general case of this type, and the one whose distribution appeared most logical to apply to metal service center demand, was the gamma function. We then postulated the hypothesis that demand in the metal service center industry could be approximated by the gamma function. Twelve items were selected (by a metal service center) as ”typical" in their behavior. We therefore, have every reason to believe that these l2 items represent the characteristics of metal service center inventory items in general. In other words, if our major hypothesis proved valid for these 12 items, we felt we would be justified in asserting that the hypothesis describes a characteristic of metal service centers in general. 139 1A0 In view of the impressive validation of the hypothesis for these 12 items, as presented in Chapter IV, we now make this assertion, and aver that evidence is strong that demand in the metal service center industry can be approximated by the gamma function. E. Applicability of the Gamma Distribution The procedures proposed in this dissertation, with their accompanying computer programs, have general applica— bility over all ranges of demand patterns. Once we make this sweeping statement, we are immediately confronted with the fact that in enveloping the entire range of demand 'behavior we are not entering a vacuum but, rather, an area thich in recent years has been filled by the application of riormal distribution theory to the demand pattern. Does our claim of universal range applicability carry .ixiherently with it the implication that the previously eesstablished normal distribution applications have been in earsror and now must be discarded? A study of the characteristics of the gamma density (ii.stribution will furnish the answer to this question. It was stated in Chapter I that the gamma density Ciimstribution rises relatively sharply from the origin, rweenches a peak, and then tapers more gradually down to the I”iE§trt, or, in other words, that it is skewed, in contrast PC) Tihe normal (bell-curve) which is perfectly symmetrical. 141 However, as we increase the mean, u, i.e., as we go to the right a greater distance from the Y axis, with modest values of the variance the skewness decreases. Finally, for extremely large values of u and very small values of variance, a point is reached such that by eye it is extremely difficult to determine whether the distribution is gamma or normal. In these cases it is immaterial whether one assumes a gamma or normal distribution, since the distri- butions themselves are nearly identical. Specifically, then, for a high ratio of mean to variance, the normal distribution may be applied to obtain accurate results. Note that the ratio of mean to variance lias, when the mean is squared, a specific parameter, Vlhich is r. Since r is the mean squared divided by the \zariance, then when we speak of the ratio of the mean, or nuean squared, to the variance, we are talking about the Eyize of the parameter r. Research conducted in the course CDf‘ this dissertation project disclosed that normal distri- IDthion theory works well for values of r in excess of ten. An analysis of the 12 items selected by the metal fseeravice center as having typical behavior disclosed a range C31? 1? values from .31 to 5.7, with the range from 1 to 2 mOst common. This would indicate that normal distribution tklewory as applied to demand must be confined to relatively latpége firms with stable, uniform, sales rates. 142 In summary, then, to determine whether or not to use one of the presently established normal distribution appli- it is necessary to determine the value of r. If cations, For very large it exceeds ten, the normal may be used. values of r, its solutions are as accurate as the gamma's. Evaluation of Contribution C. The only contribution of this dissertation which can be evaluated is the demonstration of the validity of the major hypothesis, i.e., the work presented in Chapter IV. It is hoped that the validation of the major hypothesis has added a small increment to the yet meager knowledge of the 'behavior of demand in the metal service center industry. Since our attention has been concentrated in this iaidustry, and our range restricted to parameters character- i.stic of this industry, we make no comment with regard to 'tlie general applicability of the gamma function or its To the contrary, zsp>ecific applicability to other industries. ciLle to the unique characteristics of the metal service czeeriter industry, as enumerated in Chapter I, we aver that eawridence of the applicability of the gamma function to crther industries could not have been used to hypothesize its applicability to this industry. With regard to the two sub-hypotheses, we feel confident th81: we have proved their validity by presenting proposed pIVDCNEdures and computer programs which take advantage of the 1A3 gamma characteristics of the industry's demand pattern. Since the procedures produce a desired conclusion, we feel that we have proved that inventory management procedures can be based on the unique gamma function characteristics. However, no evaluation of these procedures can be made, since, of course, they have not yet come to the attention of those to whom they are directed. Until such a time, the systems and procedures developed here would be better described as offerings rather than contributions, and therefore cannot be evaluated. In addition, we are certainly aware that lack of total knowledge always makes uncertain the question of duplication of the efforts of others, so 'that in the following list of matters offered in this ciissertation, we have no certainty that they have not appeared ealsewhere, but, to the contrary, feel that some, such as the Iqtarmal table development, most certainly must have. A. Development of Subroutine CAMDEN, the computer programming of the gamma density probability function. E. Computer programs DATA and CAMDEN 1. Example of the use of Subroutine CAMDEN to produce tables of the gamma probability density distribution. 2. Instructions for modifying these programs to: a. produce tables of different gamma parameters, lAA b. change scale, both in range and increments, c. increase the number of decimal places of listed values. Development of Subroutine GAMTAB, the computer programming of the gamma cumulative probability function. Development of computer program CURVES Example of computer program which will construct tables for producing gamma cumulative probability curves. Development of comuter program GAMCUM 1. Example of the use of Subroutine GAMTAB to produce gamma cumulative probability tables. 2. Instructions for modifying the program to: a. Construct tables with different values of parameters. b. extrapolate beyond existing tables. 0. produce finer increments in the tables. d. increase the number of decimal places in the listed values. Demonstration of determination of optimum reorder point using existing previously published gamma cumulative probability tables. Development of program GAMCON. 1. Development of computer program which produces gamma tables from which m can be read directly. 1A5 2. Instructions for modifying the program to: a. Construct tables for different values of the parameters. b. Construct tables with finer increments, c. Increase the number of decimal places in listed values. 3. Development of computer program for obtaining Expected Value of Demand During Leadtime in excess of {km Reorder Point. a. Demonstration of use of E(DDLT> R) tables b. Demonstration of interlacing two matrices. Development of computer program RANDL. Example of computer program which will construct table tabulating r and lambda for combinations of mean and variance. Development of computer program ORDER. Example of a computer program which will construct a table from which reorder point can be read directly. Reorder Point Chart. 1. Development of computer program MCHART. 2. Development of computer program LAMBDA. 3. Explanation of how to find reorder point directly on chart. 146 Development of computer program REORDR. 1. Instructions for handling program deck. 2. Various proposed procedures incorporating computer program REORDR. Proof of Gamma relationship x = m -A Development of Normal Probability Integral Tables. 1. Dwight's —x to x (computer program NORMX) 2. Parzen's —«>to x (computer program NORMX) BIBLIOGRAPHY 1147 BIBLIOGRAPHY Books Ammer, D. Materials Management. Homewood, Ill.: Irwin, 1962. Anyou, G. Managing an Integrated Purchasing Process. New York: Holt, Rinehart, Winston, 1963. Arken, N., and Carlton, E. Statistical Methods. New York: Barnes and Noble, 1963. Arrow, K., Karlin, S., and Scarf, H. Studies in the Mathematical Theory of Inventorykand Production. Stanford, California: Stanford Press, 1958. Bowman, E., and Fetter, R. Analysis for Production Management. Homewood, Illinois: Irwin, 196T. Broom, H. Production Management. Homewood, Illinois: Irwin, 1962. Brown, R. Statistical Forecasting for Inventory Control. New York: McGraw—Hill Book Company, 1959. Buchan, R., and Koenigsberg, G. Scientific Inventory Management. Englewood Cliffs, New Jersey: Prentice— Hall, 1963. Buffa, E. Modern Production Management. New York: Wiley, 1965. Carson, G. Production Handbook. New York: Ronald Press, 1958. Demming, W. Statistical Adjustment of Data. New York: Wiley, 1948. DiRoccaferrera, G. Operations Research Models. Cincinnati, Ohio: South-Western Publishing Co., 1964. Dixon, W., and Massey, F. Statistical Analysis, New York: McGraw—Hill, 1957. Dwight, H. Tables of Integrals and Other Mathematical Data. New York: Macmillan Co., 1947. 148 149 Eilon, S. Production Planning and Control. New York: Macmillan Co., 1962. Fetter, R., and Dalleck, W. Decision Models for Inventory Management. Homewood, Illinois: Irwin, 1961. Fisher, F., and Swindle, G. Computer Programming Systems. New York: Holt, Rinehart, Winston, 1964. Farrente, V. Industrial Dynamics. New York: John Wiley and Sons, 1961. Freund, V. Mathematical Statistics. Englewood Cliffs, New Jersey: Prentice—Hall, 1964. Goldberg, S. Probability. Englewood Cliffs, New Jersey: Prentice—Hall, 1962. Guest, P. Numerical Methods of Curve Fitting. Cambridge, Massachusetts: Cambridge Press, 1961. Hanssmann, F. Operations Research in Production and Inventory Control. New York: Wiley, 1962. Hildebrand, A. Numerical Analysis. New York: McGraw- Hill Book Co., 1956. Holt, C., Modigliani, F., Muth, J., and Simon, H. Planning ProductionyyInventoryy_and Work Force. Englewood," Cliffs, New Hersey: Prentice-Hall, 1963. Horowitz, S. Quantitative Business Analysis. New York: McGraw-Hill, 1965. Howell, V., and Teichnoew, D. Mathematical Analysis for Business Decisions. Homewood, Illinois: Irwin, 1963. Johnson, R., Kart, F., and Rosenzwerg, V. Theory_and Management of Systems. New York: McGraw—Hill, 1963. Koepke, C. Plant Production Control. New York: Inter- national Textbook Co., 1963} MacNiece, E. Production Forecasting, Planning, and Control. New York: Wiley, 1961. Malcolm, D., and Rowe, A. Management Control Systems. New York: Wiley, 1960. 150 Mayer, Ré Production Management. New York: McGraw-Hill, 19 2. McMillan, C., and Gonzalez, K. System Analysis. Homewood, Illinois: Irwin, 1965. Moore, F. Manufacturing Management. Homewood, Illinois: Irwin, 1961. Morse, P. Queues, Inventories, and Maintenance. New York: Wiley, 1962. Optner, S. Systems Analysis for Business Management. Englewood Cliffs, New Jersey: Prentice-Hall, 1960. Parzen Eé Modern Probability Theory. New York: Wiley, 19 4. Putman, A., Barlow, E., and Stiban, G. Unified Operation Management. New York: McGraw—Hill, 1965. Rago, L. Production Analysis and Control. New York: International Textbook Company, 1963. Raiffa, H., and Schlaiffer, R. Applied Statistical Decision Theory. Cambridge, Massachusetts: Harvard Press, 1961. Reinfeld, N. Production Control. Englewood, New Jersey: Prentice-Hall, 1961. Scheele, E., Westerman, W., and Wimmert, R. Principles and Design of Production Control Systems. Englewood Cliffs, New Jersey: Prentice-Hall, 1960. Schlaiffer, R. Probability and Statistics for Business Decisions. New York: McGraw-Hill, 1959. Schlaiffer,R. Statistics for Business Decisions. New York: McGraw-Hill, 1961. Schuchman, A. Scientific Decision-Making in Business. New York: Holt, Rinehart, and Winston, 1963. Siegel, S. Non-Parametric Statistics for the Behavioral Sciences. New York:'”McGraw-Hi11, 1956. Smykay, E. Bowersox, D., and Mossman, F. Physical Distri— bution Management. New York: Macmillan, 1961. Starr, M, and Miller, D. Inventory Control: Theory and Practice. Englewood Cliffs, New Jersey: Prentice— Hall, 1962. 151 Timms, H. The Production Function in Business. Homewood, Illinois: Irwin, 1962. Underwood, B., Duncan, C., Taylor, V., and Cotton, J. Statistics. New York: Appleton-Century-Craft, 1959. Welch, W. Tested Scientific Inventory Control. Homewood, Illinois, Irwin, 1962. Articles Brownell, C. ”The Computer Center Distribution,” hm Industrial Distribution, Vol. 59, No. 2 (February, 1964), pp. 44-4D. - a Disney, R. NA Review of Inventory Control Theory," The E: Engineering Economist, Vol. 6, No. 9 (Summer, 1961). Exclusive Survey of Production and Inventory Control, Vol. 119, No. 4 (April, 1961), pp. 80-87. Morgan, V. "Questions for Solving the Inventory Problem,” HBR, Vol. 42, No. 4 (July—August, 1963), pp. 95-110. Newberry, T. C. "A Classification of Inventory Control Theory,” Journal of Industrial Inventory, Vol. XI, No. 5 (September—October, 1960), pp. 391-399. Simon, H. "On the Application of Servo Mechanism Theory in the Study of Production Control,” Geometrica, Vol W, No. 2 (April, 1952), pp. 247-268} Hadley, G., and White, T. Analysis of Inventopy Systems. Englewood Cliffs, Prentice—Hall, 1963. Holt, C., and Simon, H. Optimum Decision Rules for Production and Inventory Control. Proceedings of the conference on operations research in Production and Inventory Control. Cleveland, Ohio: Case Institute of Technology, January, 1954, pp. 73-89. APPENDICES 152 APPENDIX A CORRELATING GAMMA CUMULATIVE PROBABILITY CHART WITH GAMMA CUMULATIVE PROBABILITY TABLE A chart of the gamma cumulative probability function can be found on page 713 of R. Schlaiffer's Probabiligy and Statistics for Business Decisions (N. Y.: McGraw- Hill, 1959). Although a chart is not as accurate as a table, it has the important advantage of showing graphically how the probability changes with changes in the parameters. Following is the method of correlating the chart readings with the gamma cumulative probability table in the NBS Handbook: Schlaiffer m = Handbook m Handbook v = 2 x Schlaiffer r To convert, subtract the Handbook reading from 1 to get the chart reading. Example 1: For Schlaiffer r = 1, and m = 1, read .63 on chart. Page 978 in Handbook: V = 2 x r = 2 x 1 = 2 For v = 2 and m = 1, read 0.36788 in table. 1.00000 .036788 = .63 153 Example 2: For Schlaiffer r = Handbook: For v 1.000 V = 2 x 10 = 20 and m = .005 = .995 10, 2O 20, 154 and m read = 20, read .995 on chart .OO5OO in table. APPENDIX B DEVELOPMENT OF NORMAL PROBABILITY INTEGRAL In the course of our research it became necessary to obtain values of the normal distribution of an accuracy not readily available. It was therefore decided to develop a computer program which would construct such tables. x 1 t2 e __ The normal probability integral = <73'?’ -x 2 dt Page 129 in Dwight's Tables of Integrals, previously cited, shows the following expansion of this equation: NPI = x(2>1/2 I? + x2 + Xi“L + X6 + . . E! W ‘ 2 3 2 11-3 ° ' 2 21 5 2 '31'7 Which appears to be the algorithm: n 2n PCUM = X( 2 >1/2 1 + ‘1 ‘X . 1 Equation (10) 1’1 1’1. T 2 (2n+1) This equation was then developed into computer program NORMX, which constructs standard normal distribution tables fFOUI—X to +x. The advantage of this program is its ability 155 156 45I776 BASIC I 96 69 7O 73 71 PROGRAM NORMX DIMENSION PCUM(IO) PRINT 96 FORMAT (IHII PRINT 69 FORMAT (1H0. 14H PROGRAM NORMXI PRINT 70 FORMAT (1H0. 43H DEVELOPMENT OF NORMAL PROBABILITY INTEGRAL) PRINT 73 ' FORMAT (1H0. 23H FROM MINUS X TO PLUS X) PRINT 71 FORMAT (IHO. 35H USING ALGORITHM OF SERIES EQUATION) PRINT 72 TZOFORNAT (IHO.I03H .00 .0I .02 .03 33 32 31 40 20 97 I.09 .05 .06 .07 .08 .09. / I DO 2 J = I. 41 XX 8 J - I EX 8 XX * .I DO 20 K = I. IO ECKS 8 K - I EKS = ECKS § .01 X = EX + EKS SERCUN 3 0. DO 40 N = I. 99 R 8 N RTFACT 8 I. IF(R-I.) 3I. 31. 32 RTFACT 8 RTFACT / R R 8 R - I. GO TO 33 T 8 N SERTRM 8 (-I.**N) * X**(2.*T) * RTFACT / (2.**T * (2.*T + I.)) SERCUM 8 SERCUM + SERTRM PCUM(K) 8 X * (2./3.1416)**.5 * (I. + SERCUM) CONTINUE PRINT 4. EX. (PCUMIK). K.= I. I0) FORMAT (IX. F401. IOFIO.5I CONTINUE PRINT 97 FORMAT (IHI) END END 1537 0..... 0..... ...... ...... ...... .0.... .0.... on.... ...... one... 0..... «as... .u.... n..... .0.... 0..... «as... mu»... 0..... ..us.. can... at»... ...“... roou.. ...... one... an»... henna. romeo. ...... ...N.. nth... uncut. «cash. «do... «0.... .00.». och-n. Odom.- .0.... and... noooo. noooo. ...... ...... ...... ...... ...... s~.oo. ...... moo... ...... «as... «so... 0..... ...... «no... oo.o.. ..no.. on»... ...... ...... gnu... ...... ...... ...... ...... nauco. «.Nno. o..o.. com... .0.... ...... .«uuo. ...... ..nom. coo... spoon. nocou. «noun. .ou... on»... 0..... no.... ...... ...... ...... ...... ...... ...... ...... ave... 0..... was... .0.... .n.... «on... no.... ...... «aw... ...... ...... mnao.. and... «mono. .mnuo. can... can... ...... «n.~o. «or... use... muons. ...... osouo. one... .u.... «nun.. ...on. «coon. .onau. o..nu. wanna. he. ...... ...... ...... ...... ...... noooo. ...... «no... ...... ~.o.o. ...... woo... 0..... us.... can... ”no... «we... ...... mac... one... one... ope... «as».. ...N.. can... .«u... «some. sacuo. nnu... ..nns. nocan. ...... «node. ..«n.. ...... ~n.«.. ...... nuuow. .«oam. «as... .o.... 0:. ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... vac... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 00.... ...... ...... ...... ...... ...... ...... .thn. ...... ...... ...... ...... ...... ...... one... woo... nsooo. coo... ...... ...... o..... «9:... ...... «so... ...... non... ...... «.coo. annoo. Nucoo. ...... ...... moo... «on... n~.no. ...... ...... ...... macro. ...... wares. «no... 0..... ...... ...... ...... nos... .0.... ooc.n. .«oou. ...... .n.... wounc. cc. ...... u..... ...... «0.... Nu.... 0..... ...... ”do... 0..... mac... 0..... .00... can... hon... ova... ...... ...... .«o... nae... ”no... vow... .novo. nswno. snow.. 0..... 0.»... can... ovoac. onuos. «nu... ..o... no»... 0.»... «ovum. and... .onov. ocwnn. cocoa. «.dod. nvnow. n.nmc. no. ...... ...... n..... c:.... ...... ...... poo... ...... ...... ...... ...... ...... our... ..n... ...... can... c..o.. .0.... can... ...... «co... ...... ...n.. ...«.. ...... ....o. .n.... can... was... ...n.. ...... ...... ...... ...... ...... ...... anon». moron. ...... «or... ...... «c. ...... ...... ...... ...... ...... ...... .n.... ...... nod... mac... ans... «ac... .90... .w»... 1.5... n..... .a.... ...... ocws.. ...... «mm... hon... csomo. rnwno. cow... ...oc. cc»... cacao. «an... conns. canoe. ousnt. .cNom. cnwum. ...... ..oon. oucan. .vnvu. mac... oosoc. ...oo. #9. ...... coo... moo... ...... oo.o.. nnooo. «no... no.... ...... coo... on»... .u.... .o.... can... one... can... can... was... .«uso. su.... ...... so“... .uouo. soouo. coo... «no... ...no. cvcoo. 0..... son... .ouoc. coonc. owe... socum. ...... «...». .ooun. woman. mac... ...... cacao. ... zc...:aw ma.amm Lo sxpuooo4< oz.o: x maan c» x max-t ton. qucwrz. ...Juucocmn 4431c: LC .zuxnoqw>mc xxxoz th_z_L2~ m:z.1 Lean >L.._m.:czc Ocazcz LC Lzuxoodw>wc _Laoz ICQQCQ. APPENDIX C PROOF OF MATHEMATICAL RELATIONSHIP The development of Equation (2), the gamma cumulative probability function, in part C of Chapter III includes the equation X = QTY * ALAM, where X = the distance along the x—axis, in terms we are specifying in the above equation. QTY = the demand quantity whose probability is being determined. ALAM = lambda,A Since in certain cases it does not destroy the generality of the relationship to specify lambda = 1, then X and QTY are identical in value, in those cases. However, to be absolutely certain of the relationship and to have confi— dence in the computer program development of Equation (2), it is necessary to determine that the relationship X = QTY *ALAM is true. This may be done by substituting specific values of r and lambda in Equation (2) to find the probability figure which can then be located in the gamma cumulative probability table 26.7 in the NBS Handbook to find X (called m in the Handbook). If the value of X turns out to be the same as the product of QTY and ALAM, we have proved the relationship. 161 162 Example: For r = 3 and A = 2, Page 127 of Dwight's Tables of Integrals, cited in the bibliography, has an equation, numbered 567.9, which can be adapted as an expansion of Equation (2). A simplified version of this equation, for the special case of r = 3, appears on the same page, and is numbered 567.2. Then 2 ‘)[;2 eax dx = eax X _ 2X + 2 Equation a a2 a3 (557-2) By referring to Equation (2) in Chapter III, part C, it will be noted that a = - = -2 For limits from 0 to 2, i.e., for gamma cumulative proba- bility from 0 to QTY = 2, we have "-11 A >< V II —2x x2 2X . 2‘ u e - + -2 u —8 = .7620 To transfer this probability to the right tail, to corre- spond with the NBS Handbook, subtract from 1.0000, which gives us .2380. On page 980 of the NBS Handbook for r = 3 (read into table at v = 6, since v = 2r) read across to locate .23810. Note the value of X (labelled m in Handbook) 163 at the top of the column which contains .23810. This is found to be 4.0. Using the x from the Handbook and the QTYand h from our example, we substitute all values in the equation X 2 QTY * ALAM and see if the left side of the equation equals the right side. X = QTY * ALAM A = 2 x 2 Check It will be recalled from Chapter III, section C, that another check was the reproducing of Table 26.7 in the NBS Handbook by means of computer program GAMCUM, which was based on our development, and noting that the computer program reproduced the table exactly.