OPTIMAL UTELEZATEON 05F LIMITED FLOOR SPACE. IN A LAYING ENTERPRlSE Thesis {or ”19 Degraa of M. 5. MICHIGAN STATE UNWERSITY Michel A. Gervais 1957 Tum-r9 LIBRA R Y Michigan State University —h— *‘F. ... ll OPTIMAL UTILIZATION OF LIMITED FLOOR SPACE INA LAYING ENTERPRISE By Michel A. Gervais A THESIS Submitted to the College of Agriculture of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of 'i "MASTER OF SCIENCE Department of Agricultural Economics 1957 Approved bM/Wém r l— "if; c A A ' thee m fiery; 81‘ hive bee; TV 1&5 ~. , meg ACKNOWLEDGMENTS The author wishes to express his heartfelt thanks to Dr. Dean E. McKee who suggested the problem and provided help and encouragement at every step. Without his invaluable guidance this thesis would never Thave been completed. Thanks are also due to Dr. Richard G. Wheeler and Dr. Henry E. iLarzelere for providing the basic data. The author also wishes to express his most sincere thanks to Miss Alfreda.Abell who read the thesis and corrected the author's "English". Special thanks are due both to Mrs. Margaret waldmeir of the Tabulating Department who carried out the computations and Mrs. Joann Prendergast who typed the final draft. They spent many extra hours in bringing this thesis to its completion. The author would like to take this opportunity of thanking Dr. Lawrence B0ger and the Michigan State University Scholarship Committee as well as the Institut National de la Recherche.Agronomigue for making possible his year of study in the United States. Wé-Pé-‘kkHfi-ififié-‘k ii u.“ \vl s. "“ ,_...'i .u“ ABSTRACT An.optimum plan is established for the management of a leghorn iflock under a limitation of available floor space equal to 10,000 square jfeet in each of the 12 months of the year. Twelve possible dates of liatch (at the beginning of each of the 12 months) were considered. For each date of hatch, 18 different lengths of laying period (7 month, 8 month . . . to 2b month-lay) were possible. Thus the optimum plan was to be selected among 18 x 12 = 216 different activities. The problem 'being.essentially one of allocating limited resources between a finite number of alternatives, the linear programming technique appeared to be the tool best fitted to attack it. As a result of the computation, four different flocks are the components of the optimum plan. The dates of hatch are fixed between July and November, and the laying periods vary between 19 and 22 months. In three flocks out of four, a heavy culling occurs one or two months before the flock is sold to free space for the brooding of the replace- ment flock. .As the replacement birds grow'and require more floor space, more of the old hens are culled and sold. This process continues until the twofiyear-old hens have been completely replaced by the pullets. This management system does not use up the available floor space in 5 of the 12 months of the year. It brings a return.of'$3119 which is three times bigger than the one yielded by the usual April hatching and 12-month lay period, under similar conditions of price and egg production. It appears as an elaboration on the lS-lé-month lay system '1 recommended by Marble and Jeffrey fbr the modern strain of leghorn. 1D. R. Marble and F. P. Jeffrey, Commercial Poultry Production, New York, The Ronald Press Company, 195?, p. 202. iii ‘F‘ *A .I 1-. ~4- The egg production cycle determined under such a plan follows the IxriCe movements in the case of medium and small eggs but differs notably jirom it in the case of large eggs, as a high volume of eggs obtained lzy'the shift in the date of hatch from mid-winter to late summer more ‘bhan compensates the contrary action of prices. As it is, this study deals with a very restricted case of the lay- ing flock management problem. The consideration of only one kind of limiting factor (floor space) strongly limits its scope. The diffi- culties encountered in gathering the data are also limitational. A number of extensions are possible by adjunction of new limitations (labor, capital) or comparison with alternative enterprises present on the farm, and competing with the poultry enterprise. iv ‘m .. an.“ 7"... "'n.‘ lr‘r- .g . L “‘— TABLE OF CONTENTS INTRODUCTIONOIOOOOIOOO0.00.0.0...OOOOOOOOOOOOOOOOOO. 0000000 0...... 1 METHOD OFANALYSIS-cococoo-00009000000000.0090.coo oooooooo 900000.. 7 ChOj-ce 0f the MethOdooooooool00.0.0000...000000.00.0000...0000. 7 Description of Linear Programming.............................. 12 Assumptions ofLinearProgramming....... ....... 17 Limitations of Linear Programming.............................. 20 METHODOLOGY ...... ....... 2s Restatement of the Problem in Linear Programming Form. . ..... . . . 25 Choice of the Numerical Data 30 RESULTS...... 36 Description of the Optimum Plan. ...... 36 Discussion of the Results.............. .......... 149 CONCI—JUSIONSOOOOOOOOOOOOOIDO 000000 OOOOOOOOOOIOOOOCOOOOOO0.0.0.00... 63 APPmDHAOOOOOIIOOOOOOOOOIIOOIOOIOOOOOOOOOOOIOIOOOOOOOOOOOOOOOOO. 65 APPmDH BOOOOIOIOOOOOOOOOOOQIOOOOOOOOOOIOOOOOIOOO oooooooo oo ooooo o 82 v‘ a... ‘r v}. '1 r. ’11 LIST OF TABLES TABLE Pa ge l. 2. Optimal Organization of the Laying Flock 37 Number of Birds According to the Age of the Flock for Each of the Seven Selected Activities................................ [:0 Number-of Birds per Month ..... 142 Use of Floor Space Per Month 15 Pattern of Monthly Egg Production From the Optimum Plan ...... Sh APPENDIX TABLES 1. List of Activities..........................................6S-66 2. Rates of Decrease in the Number of Birds..................... 67 3. Number of Birds in the Flock According to the Month of Life. 68 )4. Input Output Coefficients for x1 To x18............ ..... 69 S . Total Egg Production Per Bird Per Month in Dozens. . . . . . . . . . . . 70 6. Egg Production According to the Month of Hatch... . ..... 71-73 7 . Price of Grade A White Eggs Detroit Marketa-Receiver Buying Prices, Cents Per Dozen...................................... 7h Estimates for the Market Price of Small Eggs. . . . . . . . . . . . . . . . . 7S Margins Used} to Obtain the Prices of Eggs on the Farm. .. . . . . . 76 Monthly Price of Grade A White Eggs at the Farm, 1953-1956. . . 7.7 Monthly Prices of Hens at the Farm, 1953-1956, for Leghorns.. 78 .Prices Used in the cj's Computations....... ..... 79 Values Of the Gj‘scooooogooooooooooooooooooooooooouooooooooOBO-al vi u-v u- LIST OF FIGURES FIGURE Page :L . Iiypothetical representation of the factor-factor relation- sship........................................................ 2 , Number of birds of each category per month.................. [£3 Allocation of the floor space among the four flocks. . . . . . . . . 146 Use of floor space for brooders and layers per month. . . . . . . . )47 Ddcnflkfly'price and monthly production of small, medium and iliarge eggs under the optimum organization...... ...... ....... SS vii INTRODUCTION In Commercia_l_ Poultry Production, Marble and Jeffrey speak of the economics of poultry farming in the following terms: One approach to the subject is to assume that a specified amount of floor space is available for housing the birds. The question then arises as to what use can be made of it. The next assumption is that this floor space is adequate for a one- man or a one-family enterprise. Making this assumption renders it unnecessary to enter into the calculation of labor costs.1 A similar approach is used here. Labor and capital being avail- able in sufficient quantities, the specific objective of the study is to determine the length of time that the hens will be kept in the laying flock and the timing of their replacement which will result in the maximum net income per square foot of available floor space. Marble and Jeffrey explain that "for many years the management program followed the conventional pattern of spring hatching, housing 0f the pullets the next fall, selling off the whole or a large part of 2 the la-ying flock the following summer . . . ." They then note that When poultry men began shifting to an earlier date of hatch, they tended no t to sell off the old hens in early summer. The result has been the differentiation of several management systems, the most important of Which are: The utilization of three to four hatches a year, a seven- months lay system, a fifteen-sixteen-months lay system, and a two-years \ New 113. R. Marble and F. P. Jeffrey, Commercial Poultry Production, ~ York, The Ronald Press Company, 1935, p. 331. 22I‘hid., p. 202. ‘- lay system. They try to determine which system is the best under various sets of circumstances. The utilization of three to four hatches a year is a simple matter if the farmer can use three or four separate buildings or floors. He "starts the chicks, rears them to maturity and carries them as a laying flock in the same building. He merely rotates, disposing of the laying flock by culling and shifting to pens in the other buildings to bring them up to capacity."1 The seven-months lay system is recommended for heavy breeds, and for poultrymen having only one building. This system would not be advisable for leghorns because of their higher depreciation. Their rearing cost is almost as much as that of heavy breeds, but the value of the birds at the end of seven months of lay is much smaller. This makes it much more difficult for leghorns to pay for this depreciation on only seven months of production. The development of modern strains of leghorn which lay through 15 or 16 months or more without melting, led to the fifteen-sixteen months lay system. It is less recommended with heavy or cross breeds because many strains of heavies or crosses tend to slow'up in their intensity of lay'after six to eight months. The practice of carrying layers for two years is considered hazard- ous. When the cost of growing out a pullet is high and the market price of fowl is low, there is a better chance of realizing a greater return 11bid., p. 203. from old hens than from pullets, but this practice can pay one year and not the next. Marble and Jeffrey conclude: All of the above systems have some merits and are in common use by poultrymen. Each operator must study his own conditions and adopt the system which will make him the most money. This may require shifting from one to the other as conditions change or it may even involve the use of a combination of two different systems on a particular farm.1 While essentially correct, these recommendations give little concrete basis upon which the producer can base a decision. More useful conclusions might result from a more precise analysis of the question. The present study will deal with white leghorn chickens since it is the breed most commonly used in egg production. The floor-space and feed requirements per bird are known. The death rate, , the culling rate of the flock and the monthly production pattern, as well as the monthly prices of eggs, poultry, and feed are also given. What problems does an egg farmer face? He has a certain amount of floor space and can choose among a certain number of alternatives. The farmer can purchase day-old sexed chicks from which the r ePLB-cement flock will be produced in any of the 12 months of the year. For the sake of simplicity, it is assumed that the date of hatch will be the first day of the month (12 possible dates of hatch in a year). When Placed in the laying flock, the birdscan be kept in lay seven months"? eight months, nine, . . . up to twenty-four months ,3 that is, _-.¥ 1 . .Ib‘iclu p. 206. 2Fliminnm indicated by Marble and Jeffrey, _p_. cit., p. 202. ‘3!th indicated by Marble and Jeffrey, _p. cit., p. 202. 18 different lengths of laying period may be considered for each hatch- ing date. This gives the farmer 12 x 18 = 216 different alternatives among which to choose. The total profit derived from any one of these alternatives will be different from any other, because the total production differs from one enterprise to any other, and also because seasonal variations, both in the price of the products (eggs, poultry), and in the price of the inputs (mainly feed), change the revenue and cost per bird from one month to the next. -As for the floor space requirements, the problem is the follow- ing: As the birds move along their growth curve from pullets to layers, they require more and more floor space. On the other hand, as the months pass, mortality and culling frees floor space. Lastly, for enterprises having a productive life of more than 12 months there occurs a. doubling up of the floor space requirement when the annual use of floor space is taken into consideration. As a result of the differences in dates of hatch and of lengths of laying period, the floor space required by any enterprise in a particu- lar month will vary. It is the existence of these two distinct series 0f Variations (profit and floor space) which creates the managerial Pmblem of choosing a production plan which will bring the maxinnm prof it With a given floor space. To solve this problem, the farmer could act as follows: First a Comparison of the total profit per hen from each of the possible enter- prises C=<3~uld be made and the one enterprise resulting in the greatest ne , t remrn selected. The floor space requirements of the enterprise I‘I . 1.0.. ‘r‘ la a“ 4.; is" t" u I- . I... ~ an--_ :ST‘I ~._.\_ (I! with the highest profit would then be compared with the available floor space . The selected enterprise will not need all the available floor space in every month since the monthly requirement varies because of the growth of the birds and because of the mortality in the flock. To further increase total profit, the farmer may try to more fully utilize the available floor space. For this purpose he might raise a second flock of a size, date of hatch, and length of the laying period different from those of the first flock. This second flock will use some of the remaining floor space. It will also make necessary a reduction in the size of the first flock if both are to be included in the plan, due to competition among the enterprises for the available floor space in the limiting months. Thus, the introduction of a second flock raises the question of the opportunity cost of each available square foot. The sacrifice of revemie resulting from reducing the size of the first flock might be more than compensated by the added revenue fr‘Om the second one. These considerations can lead to a reallocation 0f floor space between the two flocks. The two flocks may still not use all the available floor space. A third flock using previously meted space may further increase the total revemle. Its presence in the production plan leads to a new real-]-°<38.‘l:.ion of floor space. Adjustment of flock sizes would continue until no further opportunity for increasing net income exists by reallocating the available floor space among the possible alternatives. . Thus the problem is: Which date of replacement and which length of the lay period or which combination of flocks of different sizes-- each one having different dates of replacement and/or different laying period s--will bring the maximum net revenue per available square foot of floor space? As previously discussed, the length of the laying period is here assumed to be greater than, or equal to seven months but less than or equal to twenty-four months. The month is the time unit. The date of replacement will be at the beginning of any of the 12 months of the year . The first question which has to be answered refers to the choice of the method utilized to solve such a problem. Among the techniques of analysis currently used in farm management research, which one best conforms to the theoretical framework of this problem? A description of the selected technique with a review of its advantages and limitations must follow. Next comes an appraisal of the methodological problems encountered in this particular case and a justification of the solu- tions which will be proposed. The empirical results will then be presented and their interpretation offered and discussed before conclud- ing With an attempt to judge the usefulness and limitations of the analysis which has been carried out. METHOD OF ANALYSIS Choice of the Method At the present time, three methods of analysis are used in farm maJaggement: Budgeting, production-function analysis, and linear pro gramming. These three techniques are merely tools with which the economist can tackle the problems he faces. The same static theory underlies them. As King indicates in the'Journal of Farm Economics of December 1953,:L in the three cases, the problem is one of estimating the surface relating inputs to output which is usually represented by isoproduct lines in the factor-factor graphs of marginal analysis and selecting the desired points from among the available alternatives. Budgeting limits itself to the study of a small rmmber of points on the production surface. It compares them and determines which among them gives the highest return. If one used budgeting, one would study, for itistance, the points A, B, and C on the graph (Figure 1). One among these three points would be selected as the most profitable combination 0f f3-3 tors X1 and X2. In the problem under study, the economist would Choose from among the 216 possible enterprises, four or five combinations Wh10h, according to his estimates, must be among the most profitable. He W7C>11:|.d budget them and arrive at an answer. The greater the experi- ence and intuition of the economist, the more reliable his conclusions. 1R , A. King, "Some Applications of Activity Analysis in Agri- cultural Economics ," Journal of Farm Economics, Vol. XXXV (1953) p. 823. ENrEEP / ‘ ,zacflafl X: # ENTEBPQISE II ENTEBPQ/55 II. c \ ISOPeooocT {’3 \ - \fiomooucr P2 IsoPRoDucr I? _+ o rAcroe x2 Figure l. Hypothetical representation of the factor-factor relationship. But under the most favorable circumstances, there is no way of rigor— ously determining the high profit point on theproduction surface, for it is impossible to establish 216 different budgets and combine them in an unlimited number of proportions, two by two, three by- three, etc., in order to compare returns of alternative organizations. The ideal solution would be to find the equation of the total sur- . x the . xn), y being the product, and x1 . n’ face y = f (X1 ' complete list of inputs used in producing y. This is the aim of pro- duction function analysis or, in its most usual form, of the use of the b1 b2 Cobb —Douglas function. For example, a function of the type y = a x1 x2 would be computed for the problem described in Figure 1. It would then b1 b2 be easy to fix y = P1, forinstance, and to have a function P1 = a x1 x2 representing the iSoproduct line P1 which, together with a price line, would allow the economist to find the optimum combination of x1 and x2 at level P1. The same thing can be done at levels P2, P3, etc. The high profit point for the production of P1 is thus rigorously determined. - 1 Bush , as Dorfman points out, the production function "is a tool for emiibiting and comparing different but related processes. What it fails to present adequately, is the consequence of using several processes in parallel" and it is just this action of several processes used together which is of interest here. In order to study this action, the PI‘Oblem of combining production functions must be solved. “L the FIR . Dorfman, A lication of Linear Progx; amming to the Theory of 533111 University 0 Ca ornia Press, Berke ey and Los AngEles, T9317?":5, 10 In the present study the following steps would have to be taken: First, 216 different subproduction functions must be computed. Each corresponds to one of the possible enterprises, i.e., y1 is the product of a_ laying flock hatched on January 1 and kept laying 7 months, y2 the product of a laying flock hatched on January 1 and kept laying 8 months, etc . In the present study, the inputs in the production function are related only to floor space utilization. The problems arising from the utilization of inputs other than (the one concerning floor space are assumed to be solved. The floor space inputs are assumed to be fixed for the farm as a whole . There are 12 of these inputs (one for the floor space used in each month of the year). ' So the 216 subproduction ftmctions are y1 = f (x1 . . . x12) x Yale = f (X1 - ° ' x12) The Optimum amounts of the x2,L (i = l . . . 12) used in producing each of the y). (j = l . . . 216) are defined by the following system of equations: - 216 production functions - l profit equation of the type: weiYsr ._y.j- .lCiMPP j-l j i=1 jsl - 215 x 12 equations of the form: 3% = MPP P -MPP P = O iY xy yj 3513’ y 3's 1 3' 3+1 3+1 along with 12 equations of the form: xiy:L + . . . + Xiyzle = Ci “ ‘5 (i=1...12) * (j=l...2l6) ‘ This system has the same number of unknowns as equations; it can be solved, but it would be a difficult task. The problem has been simplified by reducing it to the point where only inputs fixed for the farm as a whole enter the picture. Conse- quently, attention is focused only on the problem of opportunity costs. It renders quite uneconomical the use of a method elaborated to study the problems of production in their most general form. Vihat is needed here is a tool especially designed for the considera- tion of opportunity costs, a tool whose purpose is to determine how Scarce or limitational resources should be allocated, amng competing alternatives, to give the greatest profit. Even if this tool were more restrictive in its assumptions than production function analysis, it is might handle (the same problem, without oversimplifying reality, much mre easily than the latter. thear'progmmn: is , in fact, designed for this very, purpose. It allows the selection of the best combination of specified activities requiring constant input ratios. It can be described as a choice between combinations of enterprises I, II, and III, as shown in Figure h; but it 12 smuld be added that in applying this technique, the production surface is replaced by an inscribed volume such that its intersection with a horizontal plane replaces the isoproduct contours by the broken lines drawn on Figure )4. This assumption of linearity is discussed below. But, it is already quite reasonable to think that, in spite of restrictiv? assumptions, linear programming seems particularly well designed for V problems such as the present one. As Plaxico has ascertained, linear pro gralmning . . . provides the tool needed to better consider total farm, house, and intra-industry inter-industry allocations problems. Previously such questions could be approached by budgets and by other methods but linear pr0gramming is a more powerful tool because it is easier to apply, nd it provides the optimum solution from anrmg the possibi ities considered.1 Linear programming, then, potentially appears as the correct frame-J work in dealing with the question now under discussion. A closer look at what linear programming is, what its advantages and its limitations are , will permit a redefinition of the problem. Description of Linear Programming A’s Charnes2 expresses it, "In short, linear programming is concerned / With the problem of planning a complex of interdependent activities in the best possible fashion." The problem that the firm has to solve being one of selecting optimum capacity levels, this is accomplished by Wzing 0r minimizing a linear criterion (profits or costs) ___‘ ment. lPlaxico, "Discussion: Linear Programming as a Tool in Farm Manage- Analysis," Journal of Farm Economics, Vol. Down (1955) p. 1260. 2A . Charnes, w. w. Cooper, and A. Henderson, An Introduction to 1W, New York, John Wiley and Sons, 19§3, p. l. 13 subject to such restraints as resource supplies, product requirements or definitional restrictions as may be appropriate. This presentation of linear programming would be better understood after rapidly going over Dorfnan's set of definitions: Linear programming is a mathematical study of a certain kind of optimization . . . and applies whenever the facts of an economic situation fulfill, to a sufficiently good approximation, the mathematical postulates of the method. As a mathematical matter, linear programming has been defined to be the study of the maximization or minimization . . . of a mathematical function subject to linear inequalities.1 The basic concepts which need to be defined are the following: A resourcg is a class of homogeneous elements bought by the firm; a product is a class of homogeneous elements produced by the firm; a pgoductive process or activity is a "physical event or series of events in which men participate purposefully in order to transform some 2 . resources into some product." It is specified that "two productive events are instances of the same process if they consume the same resources in the same proportions and produce the same outputs in the 3 same Proportions." Thus, by definition, a process permits only one type of variation, that of over-all scale. In these conditions, the productive problem is one of choosing Which processes to use and the level at which to use each of them. We have to find the intensity vector, X, to which corresponds the largest attainable measure of desirability f(x_) subject to the limitations \ 3R . Dorfm, in site, pl 12. 2 Qt, p“ 1h. 3 I‘-*1<3,. X > O A K _<_ S I > o expresses the fact that any process cannot be used at a negative level. A x 5 S expresses the fact that, given the technical ratios1 determining the transformation of resources into product, which are unique, certain and constant for a given process, the intensity vector x,L of ansr process “must be such that the firm remains within the limit established by the total quantity of resources (8) over which it exercises its control. Or, in short, as Hildreth2 explains it: Assume a firm using m fixed ‘ ' resources used in n different processes (m O - their omission from the production plan creates a loss of revenue as their Opportunity (3°31? per unit (zj) is less than the profit per unit they create (cj) - these Processes are successively brought into the plan. This procedure is stopped when all c‘j -- Zj < O. No further increase 0f revezme is possible. At that time, the production plan is composed 0f m Processes which can be either disposal or active, and the production Pmblem ‘ or the firm is solved. 17 Assumptions of Linear Programming Having set forth the basic principles of linear programming, it is now necessary to state the set of assumptions which are implied in this type of analysis. In accordance with the usual economic theory, linear programming assumes rationality of the economic agents and ascertains as a guiding objective of economic decision, the maximization of some measurable function of the variables under the control of the decision unit. ‘ It also establishes special postulates of its own: 1 . The production opportunities of a farm are defined by, and limited .to, the resources available and the products which can be produced. Both the amount of one or more resources and the mimber of enterprises are limited. 2 - Any process is used at any positive level consistent with the supply of resources available. Returns to scale are assumed to be constant . . 3 . It is possible to use several production processes--in which case the consumption of each resource is equal to the sum of the consul"P‘bions‘of this resource in each process, and the total output is equal to the sum of the outputs of each process. These assumptions are emphasized by Dorfman under four headings: a) Linearity . . . each process is characterized by certain ratios of the quantities of the inputs to each other and to the quantities of each of the outputs. These ratios are defined to be constant and independent of the extent to which the process is fused. (Constant returns to scale.) ‘ b) Divisibility . . . any process can be used to any- positive are supposed to be single-valued and certain. 18 extent so long as sufficient resources are available. Indivisi- bilities and 'lumpiness' in production are ignored. 0) Additivity . . . if several processes are used simul- taneously, the quantities of output and inputs equal the sum of the quantities used in each process individually. d) Finiteness . . . the number of processes available is finite.1 It is important to stress the fact that the input-output coefficients Each of these assumptions introduces some limitation in the SCOpe of linear programming. At first glance the concept of linearity seems the most objectionable since one unconsciously over-emphasizes the fact that it apparently contradicts the law of diminishing returns; but as KOOpmans says: Linearity of linear programming relates only to: a) the assumptions of proportionality of inputs and outputs in each elementary productive activity, and b) the assumption that the result of simultaneously carrying out two or more activities is the sum of the results of the separate activities. These assumptions imply constant returns to scale in all parts of technology but they do not imply linearity of production functions, only its homogeneity of degree one.2 Indeed this postulate states that if we. start with a point on the production function and multiply each of the inputs and outputs by the same positive quantity, the new point will also be on the production 3 . ftulctipn - Koopmans says, "The production function so obtained, fully exPresses the phenomenon of decreasing returns to proportional increase in the liltljp'uts of some but not all primary factors of production." ‘ lR- Dorfman, pp. cit., p. 81. Alloc 2T3 filling KOOpmans, editor, Activity Analysis of Production and w, Cowles Commission Monograph No. T3,fiW'iley, 1951, P- 5- 3K0O‘pmans, pp. cit., p. 6. 19 "The divisibility assumptions," Dorfman1 states, "is not seriously restrictive except where the product of an enterprise or economy con- sists of a few indivisible items as, for example, in shipbuilding." It is not so clear that the divisibility assumption cannot be bother- some . Two kinds of difficulties may arise from its use. The first is the one mentioned by Dorfman. The consumption of inputs by some I processes my encounter certain indivisibilities (use of machinery, etc . ,) . On the other hand, trouble can arise frbm the fact that the operational units are, in. practice, more indivisible than the pro- ductive unit. This could easily happen in poultry management. But even if some caution is needed, the trouble is not great-enough to prevent the use of the method .. As can be seen from the last quotation of Koopman,2 the additivity assumption is part of the linearity one. Dorfman:3 states that "if two or more identical processes are carried out simultaneously, the physical “511115 will be additive. The additivity assumption extends this postu- late to the case where the processes are not identical." It also implies that the processes are independent of each other so that the inclusion or one in the program will not affect the input-output ratios of any Other Process already in the program. Consequently, processes must be caL1‘3‘1‘ULLZLy defined if the additivity assumption is to hold. lDorflmn, pp. 91.3., p. 83. 2m p. 18. 3D0rfman, _p_. git” p. 83. 20 The assumption of finite alternatives might be considered as greatly limiting but, for all practical purposes, the firm's "range of choice does not lie along a continuous scale but involves a selection among discrete alternatives ,"1 and in the short run at least only a finite number of productive processes are available. Moreover, the use of a large though finite number of discrete processes approximates the continuity of the theoretical production function accurately enough for the requirements of most problems. Limitations of Linear Programming The basic mathematical limitations of linear programming have already been noted: Linearity, divisibility, additivity, and finiteness. . These limitations compel the economist to be very careful in applying linear programming only to cases where their postulates hold good. For example, theproblem of studying the response to rates of fertilization of a given crop in a certain type of soil or the levels of feeding on dairy Production does not clearly imply a selection among discrete alternatives. Production function analysis is better adapted to this kind of problem than is linear programing- Amother group of limitations might be called practical ones. It would inelude: a) The expression of objectives and constraints in measurable terms. This limitation is of greater or less importance depending on the type of operation studied. It is eSpecially important in dealing with non-monetary objectives. \ 1Dorfman, pp. 923., p. 11. 21 b) The difficulties encountered in the determination of suitable numerical values for the input-output coefficients. Because of the rigorous way linear programming defines the optimum solution, these coefficients have to be determined with as much accuracy as possible. Since linear programming is a more powerful tool than budgeting, it includes a greater number of points on the production surface. It therefore requires more data. A shortage of data is one of the great— est obstacles to be overcome in applying linear programming to agri- cultural economics . c) The importance of the computational labor required to execute, numerically, large-scale linear programming problems. This leads to the difficulty of establishing a global plan for a farm through linear Prograrmrning. The present problem deals only with 12 inputs in one enterprise of a farm and it already groups 216 different processes. It is not exaggerated to think that every enterprise on a farm needs some 50 inputs (labor, mechinery, fertilizer, etc.). 'A farm which would add grain production and another cash crop to its poultry enterprise would not be exceptional. Because of the law of diminishing returns, at least three different levels of output would have to be considered f°r each two-variable subproduction function. For each level of produc tion, at least three different processes expressing potential mbsti‘tnltabilities would be used. The complete farm would then comprise C 5%) x 9 x 3,. i.e., more than 33,000 different processes. Even if the disposal activities are supposed, as in the present case, to be only equal to 50, the matrix so constituted is practically impossible to 2 22 mm: with. Therefore, it becomes necessary to reduce the number of inputs as much as possible by aggregating them. This solution is reasonable, but it is an additional source of error. A last group of limitations stems from the theoretical framework used in linear programming. The problems arising from the construction of the input-output coefficients when economies of scale exist belong to this category. The solution of any farm management problem obtained through the use of linear programing is only meaninng in a given I. physical or technological set-up. The increase in the size of enter- - prises, alone, may lead to the necessity of determininga new set of innit-output coefficients. How can one relate this change in enter- prises to the size of the production model analyzed through linear PrOgraIm'ning? A correlated problem is the one of selecting the length of run for which the analysis applies. Linear programming is based on the utiliza- tion 0f existing limitations on certain factors. These limitations are bound to be removed when considering long-run planning. How long will. the conclusion of a given linear programming problem remain valid? H°V Will it be replaced when previously fixed factors become variable as the problem is considered for a longer time period? Another type of obstacle may arise from the impossibility of limit- ing the results to meaningful dimensions. Let us suppose for example that 111 the present problem of laying flock nanagement, the final solu- tion Contains a few enterprises at a level of intensity corresponding to noBks of under 50 or even 100 birds. It is impossible for a farmer 23 to follow completely a management plan containing such activities. But in the potential solution it is necessary to keep the possibility for each enterprise to remain at the zero level, thus, it is impossible to define a lower limit of, say, 1% birds at the intensity of enter- prise actually carried on. It is impossible to introduce in the computation the limitation that the intensity level of any enterprise will be zero 21; greater than 100, and by doing so, to eliminate the danger of arriving at a production plan which cannot, in practice, be applied on a farm. V A correlated obstacle comes from the fact that if linear programming rigorously defines the best plan of production among as many alternatives as the computer can handle, it leaves the production economist with no idea about what would be the second best plan. The technique used does not lead to the high profit point by following the optimum path on the Production surface. As a result, if the final solution is not satis- factory for technological or other reasons, no second solution can be found from the preceding computation. The problem must be redefined along new lines and. it must be recomputed. But linear programming also has important advantages. It permits the silm-lltaneous consideration of a mmber of alternatives and their int"art'elation together with the stocks of available resources in arriv- ing at ”the optimum plan. It gives the marginal value of the limiting factors Without extra computations. It by-Ipasses the problem of assum- ing eatPile it values on resources not priced in the market by taking int” <3°Ilszideration the opportunity cost of these resources, as used in 2h the firm. It frees the researcher from the great bulk of computational operations which are reduced to a clerical job. Finally, one is certain that the resulting allocation is necessarily the optimum position relative to the maximization of net cash returns for the alternatives considered and the limitations specified. In spite of its limitations, the advantages of linear programming permit a simple formulation of the laying flock management problem, for it has been shown that the underlying ‘theory involved in the present study does, in fact, deal with. the interrelations among different alternatives and the limiting factors. r_4 Tr METHODOLOGY Restatement of the Problem in Linear Programming Form The quantities which must be defined are: The different activities, , xj, and the related returns, cj, the different limitative factors, bi’ the ianit-output coefficients, aij’ expressing the quantity of the limiting factor, bi’ necessary to produce one unit of xj. As the problem has been defined in this case the choice of a ‘ management plan for a layer-hen enterprise is made fromamong 216 dif- ferent alternatives.1 Tl'Ius, there are 216 different processes or activities numbered from x1 to 3:216. The complete table of activities is given in Appendix A. (Table 1, pages 65-66.) A few examples suffice: x1 corresponds to a flock for which the replacement chicks are Wrens-Bed on the first of Jamary and with a laying period equal to seven months. The pullets, bought as sexed chicks 'on January 1, begin to lay during the month of June and are sold during the following December (seven-month lay). x2 corresponds to a flock with the chicks purchased on the same I date as x1 (January 1) but with a laying period of eight months. ' The Old hens are sold during the following January. ~ 1% p. h. 26 The activities from x1 to x18 all refer to alternatives with chicks purchased on January 1. x1.3 corresponds to a laying flock with replace- ments hatched on January 1 and with a laying period extending over 214' months . Old hens are sold at the end of May of the second year after the chicks were bought. x19 corresponds to a laying flock where replacement chicks are purchased on February 1 and with a laying period of seven months. The activities from x19 to x:36 inclusive have the same date of purchhse of replacement chicks but the length of the laying period is increased by one month each time the index of the activity increases by one. One way of simplifying reality in so defining the activities may be open to criticism. Considering the purchase of sexed chicks on the first of the month, the sale of old hens on the last day of the last month of lay, and including all the eggs laid during this last month in the PI‘Oduction, does not allow the farmer sufficient time to prepare his henhouse, etc. But this is a somewhat literal interpretation. In fact , the chicks might be bought during the first weeks of one month and the old hens sold during the last weeks of the preceding one with- out materially affecting the basic assumptions of this study (prices, pmduction, etc.). The unit level of intensity for each process is defined as a flock °f a thousand birds at the beginning of the laying. period (beginning of their 8ELI-nth month of life). This unit of a thousand birds is managed in each case as indicated in the table of‘activities. The questions of mortality and culling are discussed below. Thus, t he net profit cj related to the activity xJ. will be the total return 27 from the production of a flock of 1,000 birds at the beginning of the laying period, hatched and kept laying in accordance with the descrip- tion of activities which is given in Appendix A (Table 1, pages 65-66). As an example, c is the total return secured from a flock of hens, 1 purchased as one-day-old sexed chickens on the first of January, in a quantity sufficient to constitute a one-thousand-bird laying flock on the first of June, when the birds begin to. lay. Allowance is made for a certain death and culling rate. The hens are sold at the end of the following December. Generally all Cj's will be constituted as follows: c3.j =21 large eggs produced during a given month by a flock numbering 1,000 at the beginning of the laying period, whose number / decreases at the culling and mortality rate described below, multiplied by the price of large eggs during this particular month. +2 of medium eggs produced in the same conditions during a given month of the laying period, times price of medium eggs during this month. _ *2: of small eggs produced in the same conditions during a given month of the laying period, times the price of small eggs during this month. + weight of one hen at the time of the selling of the old flock, times the number of remaining birds multiplied by the price of hens during the month of sale. \ the 1A1]. the sums extend over the total number of months constituting par'ticzular laying period . ' 28 + weight of one bird at culling time, times the number of culled birds times the price of hens during the month of culling. —- price of chicks, times the number of chicks bought as one-day- old sexed pullets. - .price of starting-chick. mash, times total consumption of starting- chick mash. - price of layer mash, times total consumption of layer mash. The justification for the way in which the cj's are constituted and the mmerical values chosen will be found in the second part of this chapter. The only limiting factor considered in this study is the floor space available on the farm for brooding. and egg production. The total available floor space has been fixed at 10,000 square feet.1 The single global limiting factor, namely the floor space, has been divided into twelve independent ones, i.e., a limitation on floor space for each one Of the twelve months of the year. Fro“! one month to the next, the growth of the birds, on the one hand, the mortality and culling, on. the other, bring about differences / in the floor space required by any laying flock. _ Through the sub- ' division into monthly components of the limiting factor, these dif- ferences influence the choice of the final production plan which results In the Optimal utilization of the available floor space. ‘ 1M5 the fl ble and Jeffrey, 92- 9311., p. 332, give 10,368 square feet as Agri 001‘ space needed for a one-family egg farm. R. G. Wheeler of the for cultural Economics Department considers 10,000 square feet typical 8‘ c’ol'I'lmercial poultry farm in southern Michigan. 29 The different limiting factors are: b1 the limitation on floor space available in January = 10 ,000 sq. ft . b2 the limitation on floor space available in February = 10,000 sq. ft. b_.3 the limitation on floor space available in March = 10,000 sq. ft. etc - up to b12 the limitation onfloor space available in December = 10,000 square feet. I The existence of these twelve limitational factors leads, as explained on pages 12 through 16 (Description of Linear Programming) to the adjunctionv of 12 disposal activities number x217 to x225. (See the description of activities in Appendix A, pages 65 and 66, Table 1.) For example, x217 is the non-use of the available floor space during the month of January. x218 the ‘nonduse of the available floor space during the month of February, etc. At the unit level these - diaposal activities correspond to the non-use of one square foot of floor Space in the corresponding month. 1 ThIIS, the matrix A of coefficients a - will have 228 columns and 1.1 12 rows . The quantity of floor space needed during the month related to the 1 index, by~ a layer flock managed in accordance with the rules th mrized under the 3' activity, is here symbolized by a ij ' Before determining the numerical values which have to be given to this Series of symbols , it will be useful to quickly determine if the problem Stated respects the linear programming postulates, previously outlined . ¥ 13111333., p. 11;. 30 The attribute of finiteness is fulfilled as there are only 216 processes. There is also no question about additivity. It is clear that in comparing several different ways of producing eggs with respect to their demand for floor space, the total quantities of input (floor space) and of outputs (eggs produced or hens sold) will be equal to the sum of the quantities used and produced in each process individually. The assumption of fixity of the coefficients and their independence from the intensity with which the processes are carried out are also sufficiently valid to permit the use of linear programming.- The indi- vidual floor space requirement remains the same whether 10 or 10,000 birds mks up the flock, once the rearing technique has been chosen. Moreover, the floor space requirements are specified exactly. More laxity is introduced in the computation of profit both because of the uncertainty of prices and because of the more or less conventional definitions of egg production and feed consumption. The divisibility of the process does not seem to create any mortalit problem. It can, however, produce unmanageable solutions and it would seem, off hand, that the only serious trouble which might arise would stem from the divisibility assumption. This question was discussed on page 19. Choice of the Numerical Data It has been said that bi = 10,000- square feet with i = 1,. 2, . . . 12. The xj , s are the unknowns. Twelve of the xj's corresponding to disposal and/0r ac tive processes will be selected to go into the optimum plan. 31 The numerical values for the aij and the c. remain to be found. 1. Determination of the aij's' Any of these aij coefficients are computed on a monthly basis. It is necessary to know a) what are the floor space requirements for birds between zero and 30 months of age, and b) how many birds will make up the flock in any of the 30 months. 3.) Floor space requirements according to age: The growth curve of the chicks has been approximated1 as indicated here: I From the date of hatch to 3 months old = 1 square foot per bird From 3 to 5 months old = 2 square feet per bird From 5 months old and above == 3 square feet per bird. b) Number of birds in any of the 30 months: The basis for this computation is the fixed number of birds at 5 months old (beginning to lay) = 1,000. The number of birds in each of the other months is deduced from this number by taking into account. the death arid culling rates. To simplify the computations, the effects of the death and culling rate have been separated and are. considered as acting on different phases of the life of the flock as explained in Appendix B, page 82; The number of birds constituting the flocks from the month of hatch t° the 30th month of life (2hth month of lay) is obtained as a result of these successive decreases. The numbers are given in Appendix A (Table 3: P. 68). \ Stat 1Dr~ Wheeler of the Agricultural Economics Department, Michigan “utiEnUn-iversity, has been consulted on the advisability of this approxi- 32 For each month, the multiplication of the floor space requirements per bird according to the age, by the number of birds, will give the monthly floor space requirements for the initial flocks. The final floor space requirement for any month will be computed by taking into account the possible presence of two or more flocks at different moments of evolution. To illustrate, several particular a representing all ij's the possible cases are computed in Appendix B, page 83.. As a result of these computations, it is possible to establish a table of coefficients for processes from x1 to xla inclusive, (shown in Appendiac A, Table 11,, p. 69) . The processes from x19 to x36 differ from those from x1 to 3:18 in that the date of hatch is February 1 instead of January 1 but with this exception, each has the same characteristic as its homologeous counterpart in the 3:1 to x18 sequence. As a result, the following relation exists between the a. .: 1.] aim = 814,de and 812,3 = :11,de J = 1, 2, . . .18 Or, to put it in another way, the table of coefficients for the processes x19, x20 . . . x36 repeats the table of coefficients for the processes x1, x2 . . . x18 after a circular permutation of the rows, each taking the place of the one just below. The same reasoning applies to x37 to x54. 1 a0 = = 1+1’31 ai+l*1,j'+18 j' 19, 20, o . o 36 812:3“ = aisj'+18 o r a circular ,permutation of the rows on the table for the processes 33 x19, x20 . x36, each row taking the place of the one just below, gives the table of coefficients for processes 1:3,, x38 . . . x54 . and so on up to x216. For each group of 18 processes, the table of coefficients is deduced from the immediately preceding one by a circular permutation of the rows. An objection concerning this way of defining the aij's might be made. The above definition implies that the adjustment of pen size is nude every month for each flock as mortality reduces the number of birds in the flock. ‘This seems improbable. This assumption was made in order to give as clear a picture as possible of the influence of monthly variations in eggs and poultry prices on the management of a laying flock. Since the problem considered is one of a specialized commercial egg farm, the burden‘created by the monthly re-evaluation of floor space allocation seems on the other hand justifiable. 2 - Determination of the cj's. Four types of questions must be answered before one can compute any of the cj's. a) The question of the quantities of products which can be Sold; b) the question of the prices of these products; c) the question of the quantities of feed consumed; and d) the question of the price of these inputs. a) Questions of the quantities of products which can be sold. These ProduCts' are the eggs, the culled chickens, and the old hens. 1) Pro- duction of eggs. One needs the monthly egg production per bird on a 2 “month laying period and, for flocks hatched, in any month of the year. 314 The source for these data and the way the computation was handled. may be found in Appendix B, page 86. 2) The culled birds and the old hens. It is necessary to determine the weight of the birds when they are sold. The weights of the birds have been defined as follows: old hens - five pounds culled birds - four pounds1 b) The question of the price of the output. Because of the exist- ence of a two-year cycle in the price of eggs and poultry, the «prices. used will be monthly averages for the last four years (from 1953 to 1956 inclusive). The sources for the prices and the way their determination was handled may be found in Appendix B, page 86 ff. c) -Questions of consumption. The problem is to fix the quantities of feed which will be used in computing the cost of production.’ The basis for the rough estimate needed here is: 25 pounds of chicken feed per bird for the five months preceding the laying period. To siMplify the problem these 25 pounds per bird are assumed to be eaten “113' by the 1,111 birds which constitute the flock on the fifth month 0f its life. This is considered a reasonable approximation since the f1°°k is already reduced to 1,181 during the first month of its life. The 70 birds which die progressively during the remaining four months can be fed on the 25 x 111 = 27,775 pounds of feed which constitute the total es"l‘aZ'Lmate for chicken feed without any trouble. The feed consumed by the 1ayers was estimated to be eight pounds a month for each layer. N 1This weight has been chosen after consulting Dr.. Wheeler and Dr. Larzelere of the Agricultural Economics Department, Michigan State University, . 35 (3) Questions of costs. The prices involved here are the prices of sexed chicks, the price of growing mash and the price of laying mash. The source utilized and the way the computations were handled may be found in Appendix B, page 89 ff. These four series of preliminary equations having been solved, one can now compute the cj‘s according to the model given on page 27. Two examples of these computations are in Appendix B, page 90 ff- The table of the cj's is also ‘in Appendix A (Table 13, pages 80 and 81). All the elements of the simplex tableau are now available. The successive reductions remain to be done, and pursued up to the time V 1 when all 03. - zj become negative. 1A description of the simplex method of computation for linear ngra-mming may be found in A. Charnes, W. W. COOper and A. Henderson, 22? fie, p. 8 ff. I V 36 RESULTS Description of the Optimum Plan The combination of the 216 possible alternatives previously described which will make the most profitable use of the. available 10,000 square feet of poultry house floor space is given in Table 1. The optimal plan specifies that the replacement chicks will be purchased in the late summer and fall months and will be brooded in ‘four separate groups over a nine-month period. The number of pullets entering the laying flock at the end of the five-month brooding period is given in the last oolumn of Table 1. The length of time that the birds are kept in the laying flock before being replaced varies from 19 to 22 months. In the. case of activity X121, enough chicks-are purchased from a 'July 1 hatch so that 29 pullets will be available to enter the laying flock at the end of the brooding period and are kept in the laying flock for 19 mnths . Activity x122, differs from x121 only in that the birds are kept in the laying flock 20 months instead of 19. The birds involved in theSQ two activities constitute the first group'of chicks brooded. Other ac tivities having the same date of hatch can be regarded in a Similar manner. The plan presented in Table 1 results in floor space being used at less than full capacity, during the months of May, June, September, November and December. The amount of idle floor space in each of these men ths is given in the last, column of the second part of the table. 3'? The return over feed and chick cost resulting from this organization of the layer enterprise amounts to $3,119. Table 1. Optimal Organization of the Laying Flock — V_T Activities in Date of Hatch Length of the Activity Level _§olution Laying Period - - - - - - ----- Active Activities -------- Number of Hens} '3 X121 July 1 19 months 29 X122 July 1 20 months LL89 X141 August 1 21 months 23h X14:2 August 1 22 months 732 X177 October 1 21 months 2h3 X194 November 1 20 months 20h X196 November 1 22 months 39 Disposaljictivities Nonduse of floor space in: Square feet X221 May 200 X222 ' . June ' 393 x225 September 235 X 2 27 November 222 X 2 as December 886 Return over feed and chick cost $3,119 .34. War of hens at the beginning of the laying period. This is the exact solution obtained from the programming computations, 38 A better understanding of how this organization of the alternatives functions, can be obtained by examining Table 2 where the number of birds in the flock in each month, at the various stages of their develop- ment, are shown. The underlined numbers refer to the number of birds in the replacement flock prior to the time they begin to lay. The remaining numbers refer to the number of laying hens. Taking the first flock, which comprises activities X121 and X122, in the month of July, there are 6hh chicks in the first month of rearing, (36 + 608). At the same time there are 150, (25 + h25) hens in' their first year of lay which were purchased from a July hatch the year before and 33h hens in their second year of lay which were hatched in July two years earlier. A similar interpretation applies to each of the other flocks in each of the other months. Following activity X121 through its complete cycle, 36 chicks are obtained from a July hatch and are brooded through November. The decline in numbers taking place over this period is due to mortality, 5% in the first month of brooding and 2% in each month thereafter. At the end of November, 10% of the pullets are culled and the remaining 29 enter the laying flock. Over the following 19 months the number of birds declines at the rate of 2% per month (mortality). At the end of June of the second year after hatching, this group of old hens is sold. (Table 2 clearly shows how the groups of chickens from the four different dates of hatch‘enter the optimum plan. The farmer is follow- ing four different timetables in the operation of his laying enterprise. Th 18 leads to a much more complex managerial problem than that of a 39 single—hatch system. In any one month the farmer has no fewer than eight ciigfferent age groups of chickens in his operations although several. (if these will differ in age by only a month or two. On the other Inarni, never does he have more than ten different age groups. Because of the differences in ages, it is necessary to provide separate pens for each of the groups. However, with large numbers of chickens this is a. practice which would be recommended anyway. In order to clarify the cycle of activities which stems from the adoptiorl (if the optimum plan, the following over-all timetable has been drawn lip--it is nothing more than a summary of Table 2. January Flock II: 966 young hens 5-months-old begin to lay. Block III: 275 chicks h-months-old must be moved from 1 to 2 sq1are feet per bird. Flock IV: 25 old hens 26-months-old are sold. Fm Flock IV: 276 chicks h-months-old must be moved from 1 to 2 square feet per bird. M Flock III: 2h3 young hens 5-months-old begin to lay. A r11 Flock IV: 21:3 young hens 5-months-old begin to lay. 191 All flocks lay. 3.1122 Flock I: Culling of 20 birds 23-months-old. lull Flock I: Selling of the 331; old hens 2h-months-old, but chicks hatched. All-ELSA Flock II: 1200 chicks hatched. W Flock II: Culling of 156 birds 25-months-old. O . 43% Flock I: 588 chicks h-months-old moved from 1 to 2 square feet per bird. Flock II: Selling the h79 remaining old hens 26emonths-old. 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No comm how xoofim one No owe. mg on mnflohoooe. mohfim Ho hon; «N wands I'll“!!! I l' hl November Flock II: 1095 chicks h-months-old moved from 1 to 2 square feet per bird. Flock IV: 302 chicks hatched, culling of 139 birds 2h-months- old. December Flock I: 518 young hens 5-months-old begin to lay. Table 3 indicates the monthly variations in the composition of the different flocks for the entire year as well as the total number of chicks and layers present on the farm in every month. These data have been graphed on Figure 2. The first important feature which appears on Figure 2 is the fact that the number of birds in the laying flock is maintained at a high and nearly constant level.‘ The layer flock has more than 2,000 birds in 11 months and during the 12th one (November), it still numbers 1,9148 birds. From Jamary to J111Y inclusive, 3,000 layers or more are on the farm. If one remembers that the available floor space is limited to 10,000 square feet and that the brooders also are reared on the farm, the presence of such a large. War of layers shows the efficiency of the optimum plan. Figure 2 also shows that Flock II constitutes the most important Part of the flock. It is interesting to note that the activity with the highest c3. (x142 - 0142 = $1,662.90) is included in Flock II. A3 for the replacements, one sees in Figure 2 that none are reared in the months of April, May, and June, but that they are present in each or the nine other months. Thus, the replacement of the laying flock Occurs over a period of months rather than with an abrupt shift from old. hens to pullets. Egg production is maintained at a more n early constant level and the sharp break which would occur while waiting h2 8mm ._ osmw mean bows mess bass msms scam sown smmm Nmsm moon ooasp_sss anew osms oosm ,oeom. swam oNHm sown; boom sown, «mam ommm mmomll nachos osos ommm soon oses . News sso oem osm omm ososso Hosea omm was son mam oos mos has mms. mms mos mes . mom Hosea . nwm. own. .sow .Won oos .mms, eds“, mas was was mms mam cabana pom mom oem oem How assess pH soosa mos smo _oao son sen oos mos oss mas mos sos mes asses mom oonlw own, son ssh, oos (mos cos, mas yams. saa oms steaks How boo mom oem mam usosse HHH seeds msos sows sosm soom seem emms omms moms mobs. mmoa mobs muss Hosea new! sax. sows moss .omss emms .amms moms macs mmoa mobs mmms nomads Naos moos sass osss ooms assess HH sodas moo oao msos NmoH mmos muss moss one mmm mom was mom Hosea mmm mmm sms «ms, ass .nsme mam mmm mmm awm mam momir mhmgwfi . we so so as as cases . H xoofim Q . z O m 4 H. H. . S d. 2 .m H. seed: ace steam so sooesz an oases 14.3 .3205 son showcase comm mo moafip mo $0,552 . m madman a .. , success to» H too: I _ «I III.“ .llmlj H.209: E IN V H s93“. ‘H n Tull w a a a a to: I i N . . - IIJIJ \ a a. m V x a b .V m _ m. “View 0 \< o m. V 6 xx t ¥ k 5 s x x \ \ s x a s s N Till N w lull: « , n . N . .lnllnvilwl .N u N m a m N4 \. .IIIII V a {will IN .llle V IIIIIJIIIIJ . V m. m1 m. m. .HWIIH ”I . ‘ , V _ V . . [Fifi *Vlv WNWX‘V Ilan‘ ' Mt for pallets to come into full production is at least partially avoided. If the replacements were brooded in a large single batch, approximately the same amount of floor space would be required for brooding but it ‘ would be concentrated in a shorterwperiod of time‘. Since only a limited amount (51' floor space is available, it could only be obtained by reduc- ing the size of the laying flock. in these months and thus cutting egg produc tion more severely than would the type of organization outlined here. Finally, it should be noted that, essentially, a two-year-laying system is‘ used, and, consequently, only about one-half of the laying flock is replaced in any one year. The composition of the different flocks making up the optimum plan has been studied. It is time now to describe what allocation of the available floor space results from this plan. Table )4 shows the amount 0f floor Space used by the brooders and the‘layers' of each flock and the totai brooding and laying space required in each month of the year. The Smll'differences between the total floor space actually. needed and the 10,000 square feet which were used as a basis of the computation are due to the rounding of the number of birds present on the farm in each month of the year. The floor space requirements have been graphed in Figures 3 and )4. Figure 3 essentially indicates the global use of floor 3118.33,, It clearly. shows what has been said about the use of floor 8138.ce at the beginning of this description. Though the most ‘ efficient use of floor space has been the aim of the present study, the Optimum Solution keeps a certain amount of floor space-out of usein fi ve months of the year. These quantities going to "waste" are not 5 Id. FAI‘ ma.aama ma.mmmm mama ma.moooa moooa aoooa mo.aoooa am.aamm . Noooa. hmma moooa ommm annoy ma. 0mm ma. mmm ma. mmm me. man am. mma .pmmafnoz aaam Naam mmmm comm moooa aoooa moom mama Noooa . mmmm Noooa. omam ten: ameoa mman sawm, Jada maow ”mam oeam monm. mmam ammoa .nmaa oamm mmam ensue nemea mama mama mama oaaa .mama nan _ cam Nmoa amm meson weaeoonn aueom a l1hmm4141 omaa awaa maaa ooaa laama mama mama mmaa maaa maaa mom. eon: aaeoa ntlmmh waaa mmaa .maaa ooNa aaaa mama mama mmaa mam .liamw awn meson nemea gm Non 04m mmm amm mmaaon wfiooohn _ >a xeoam malamm omaa cana amaa mmaa. OONa aaaa mama mama mama mmaa aaaa tens ampoa hon aoaa wmaa ‘mwamp, Nmaa ooNa tamma mama mama mmma Nam} .mamwl meson gamma amm 0mm men . cam 0mm ensue weaeoonn . aaa xeoam 3a: awn; meme 3mm 8mm am: :3 ans mean mama 30W Noam e8: 33 mama aoma .Nmmal aoaa, mnaa. amm: .aan: -ana: mom: mama atom! meam ensue hasa 34am omaa .aaaa oaaa ooNa ensue meaeooen aa aeoaa waam mmMN111 maaa nmma mama nmma awam coma Inmmma. aoom anew maam ..emns anema Inwaam mama mama. omma mama mmaalm amam mama mmmm .aonm anew , maaa. than: nmaua omda 2.: 000 mac .30 canon mfiooopof . a xeoam i I! '11 a .z o I 1 1 . a . m RH H. E... Villa,» d H. H. S < E i»! npnoz hum modmm .aoofim no em: savanna. 1:6 .mxooam pdoa on» mcoem woman pooaa map mo coapwooaa< «.3203. Q h .m 8&2 t H“ v33». I .33.; Queuk 03$ >6? D .3. 23¢ BU H 1004M E H . XUOQK g x\ k \\\ IR. \e. a «$5 + i-‘# 4- 4- i- 4L 4L .+ §.-+ 4-‘t \\ \\t ++ ++ ++ ++ ++ ++ ++ ++ 4+ ++ ++ +... +$_++. 35,06.e one 000 ¥::::j‘.‘* + ‘l 1' +- +* +4 ++ 4+ ++ ++ ++ t... $+ ...+ ++ ++ ++ ++ eta... ++ ++ ++ ++ one 000 92 ._ A *++++4,$+ +**++4++ .ficoa mom .98sz now mhmpoopn .aoa woman .80: mo mm: .4 magmam $8». cl an w in H. NU‘R W W?\Q0me 'm: n: w H53 E H 28: m N1 n. N m h2g3... a \H Moo: 5 IN N N | \1 N N . v \ a \ a a \ 7 e. m o z o w a. b b x. a. ¥ u a \ a \ l j \ \ \ \ \ \ \ N I rIN N r m.1 N m. N N V1 N Ma 1.? IIII, N N r h. m. V N N N N h j V 1.1.] m. to n. m. V _l.|l N' v V m: IllL1|ll m: ll'llll. ox: WO‘QW %\<\\:\JV b8' negligible: in December, 886 square feet are not used. This is com- parable to the area occupied in the same month by Flock III or by Flock IV. The presence of this unused floor space in the optimum plan will be discussed later. Figure h deals with the allocation of floor space between brooders 37“ and layers of the different flocks. In November, 3,928 square feet of ; brooding space are required. In April no brooding space is necessary and the layers utilize the whole available surface (actually 10,002 square feet as a result of rounding the number of birds). Thus, it is necessary to be able to change some laying quarters into brooding. quarters or vice versa from one season to the other. This will certainly create numerous practical problems. To end this presentation of the results, some light can be thrown on the relative importance of the limitational factors. Five factors limit the production. The linear programming computations give the MVP 01' these limitational factors.1 Thus, the profits derived from the use Of the last unit of each limiting factor can be compared. The factor having the highest MVP will appear as the one constituting the most important limitation to the enterprise under the present conditions. The MVP's of the limitational factors are: Flecr space in January x217 MVP 28.17 a per 1000 square feet February X218 MVP 53.87 -\ ennprelThe estimate of the MVP arrived at by the programming technique limitsses the addition to net income of a one-unit increase in the Placeing factor taking into account the adjustment which would take add-1 t in all activities in the solution in taking advantage of the 118.1 quantity of the limiting factor. 119 March X219 MVP = 234.69 3 per 1000 square feet April X220 MVP = 56.21; July x223 MVP = h7.88 August X224 MVP = 85.00 October X226 MVP = 15.85 August is the month where the last unit of available floor spaCe brings the highest return. Consequently, the introduction of some means of increasing the available floor space in that period would materially increase income .' It could be a temporary installation which would not change the Quantity of floor space available outside the summer months. Discussion of the Results In discussing the results, three different points must be dealt with. First, why is the optimum plan organized in the way which has been described? Second, how does the plan arrived at through linear ngra-mning compare with the systems usually recommended by poultry production specialists? The optimum plan will be compared with Marble and Jeffrey's recommendations.l Third, what are the limitations of the present Study? A‘ What determines the choice of the optimum plan? When looking at Table 1, page 37, a mimber of questions might be raised ‘ Why are several different enterprises combined in the optimum plan Whereas the poultrymen have customarily recommended only a single hat ' ch management system? Why do the selected enterprises fix dates of \ . 1 Marble and Jeffrey, op. cit., p. 202, fetal. ~"1a '. 50 hatch in Autumn and laying periods of 20 to 22 months? How can the use of floor space at less than full capacity in five months out of twelve be justified? The combining of different systems of production instead of having a one-hatch-a-year system was envisaged in the introduction. How the Tm» multiple-hatch pattern affects the profit is easy to measure. Let it be noted that the activity which brings the highest revenue at the unit level intensity appears in the final plan. This activity is X142 in which the date of hatch is fixed on August 1 and the layers are kept for 22 months. If the farmer chose to carry only one activity,.it is logical to assume that he would select X142. He would then carry it at the maximum level compatible with a floor Space limitation of 10,000 square feet each month. If one comIl’lltes the ratios of the available floor space to the monthly- require— ments for X142, and compares them to each other, one sees that the available floor space in August would be the limiting factor. It would 311°" x142 to be carried at an intensity level equal to: 10 000 W93— : 1.69693 The table of the cj's (Appendix A, Table 13, pp. 80-81) shows that at the unit level of intensity the net revenue from X142 is $1,662'50' The Prof it. from such an enterprise would then be: 1662.50 x 1.69693 = 2821.15 dollars. The profit from the optimum plan is 383,119-33 01‘ 3”98-18 more than from the on-hatch system, i.e., approximately 10% 0f th e tot-8.1 profit. This is not negligible. 51 mt would be the total profit if instead of choosing the single enterprise with the highest net returns, the farmer was following the old pattern of hatching a single flock in April and keeping the layers 12 months (activity X60)? According to the present floor space require- ments the floor space in July would be limitational. It would be possible to have an enterprise consisting of 2,119 layers at the beginning of the laying period (highest level of intensity attainable with X60: 2.1191). With the assumed pattern of production and prices, the net revenue over feed and chick cost would be: 3.1413381 x 2.1191 = $875.99. The usual management plan would bring less than one-third of the revenue derived from X142 and only one-fourth of the profit from the Optmn plan. This simple computation clearly shows how much it pays to have a combination of enterprises instead of a single enter- prise. It also shows that it pays to change the date of hatch, and extend the length of the laying period. That a long laying period bring-‘3 more profit is not surprising. A longer laying period means a larger VOlume of product. Then, the cost of rearing layers will be more eaBilly compensated by a long laying period than by a short one. As the leghorn hens cannot be sold for a good price even if they have been la-Ying only for‘a few months, it is more profitable to keep them in produc tion as long as possible. In addition to the cost considera- tions, 1She necessity of making roam for the replacement flock on a limited amount of floor space, plays its part in determining the length of . the laying period. However, the plan advises laying periods of more \ _-_ 1 52 than 20 mnths. The modern strain of leghorn can easily lay 15 to 16 months tiri'thout molting and Marble and Jeffrey indicate1 that "many (birds) will still be in heavy production at the end of 15 to 16 months." The new element here is the recommendation of laying periods five months longer than the recognized possibilities of the current strains. This does not seem so unrealistic if one notes that because of the date of hatch chosen, the birds attain their 16th month of lay between April and July, i.e., during the period of the year the most favorable for egg production. The influence of the weather added to the natural POSSib ilities recognized by Marble and Jeffrey:3 permit the extension of the laying period till the following fall, to attain a 20— or 22-month lay. ' ' - ‘ A counterpart of the possibility of extending the laying period is the necessity of culling the flock at the end of its life. This culling which also makes room for the replacement flocks is indicated in the Optimum plan by the presence of several enterprises having the same date of hatch but different lengths of laying periods; for example, x121 and X122 are hatched on Julyl but X121 has a l9-months laying period aJild X122 a 20-months laying period. In practice the farmer has a flock of layers of the same age and at the end of 19 months of lay he culls 20 birds out of 31,0 4. 20 a 360, i.e., approximately 1/18 of K 1 Marble and Jeffrey, pp. cit., p. 201;. 3399 quotation above. 53 The culling rates for the old layers vary from one flock to the next. From the second flock (X141 and X142), 156 birds out of 156 + 1489 a 6J45, i.e., approximately l/h are sold one month before the others. The third flock (X177) is not culled but the fourth one (X194 and X196) looses 139 birds, i.e., 33% of more than 8 out of 10 layers, two months before being sold. These cullings take place between July and December, at. a time when the production patterns utilized in this study (see Appendix A, Table 5, p. 70) show a net decrease in the number of eggs laid, regard- less of the date of hatch. It is also the period chosen to rear the chicks of the replacement flock - A comparison of the egg production pattern resulting from the Optimlm plan with the seasonal variations in egg prices may help to understand why summer and fall dates of hatch and long laying periods have been selected. Table 5 shows the 1110an quantities 0f large, medium, and small eggs produced under the optimum plan. The curves of production of large, medium, and small eggs have been Plotted on Figure 5, against the price curves1 for the same c0"““0dities. The production curve for small eggs follows the corres- pending Price curve relatively closely. The production curve for medium eggs is still better correlated with its price curve. Under the 0mm“ Plan, the returns from small and medium eggs are the largest whi Ch can be obtained. On the other hand, the production curve for \ 1 Table 3‘3 data needed to draw these price curves were taken from ’ Page 79, in Appendix A. Sh o~.amoa ce.mmaa o~.mmem ow.wamm oa.aoem oo.maaa oe.mmea oa.omms oo.maom 0;.mom: oo.mmms om.smam. Hausa eoneam - - - omumw No.ana mo.omm, ea.eae .om.aaollam.oao 3&2w os.aae . _ fleece . u u . sm.mm loo.-m mm.mHH ::.mo~ oo.amm : n . >H sodas - a - . mm.mm oe.mm mm.ee. mm.maa oa.oaa Na.mmm . . HHH sodas. - - - - - - mo.oma om.ooa mm.oam mfl.mam.. :a.moo No.0mm HH asses om.oam . - - n u - . ms.oe ae.mm aH.:mH we.mem H aooaa .Hassm Henmam om.oma em.maa, om.mam, Hm.m~HH awnmoma ow.maaa wN.ooHH n~.aamly am.oanw om.mmma om.aao Hosea Ho.ea mw.mmy mo.moa os.owa mm.mwa o~.mwa as.mma_ mm.nma on.mm . . mH.wH >H sodas 8.0m woém 2.6a 8.2a fitmma oméma 3.03 3.29 mm.mma 3.02 n u HHH noose . ms.mm mN.HoN oa.mma ma.amm on.amm Ho.eaa oo.amm ma.mma em.oma om.ome mo.~om HH anode om.ama - ao.os oH.noH oo.eow ,oo.aem on.mem om.mam oo.mm~ mm.emm oe.omm om.om: H sodas ”assess mmwomma NM.HoeH {wa.mwn~ sm.mmam mm.oonm mo.omomy Hm.owmm wm.~mam, am.mamn, -.N:mm Ha.omm~ i:n.mmaa Hopes mmnnom ms.mom mo.maa owwomn sermon. om.mmn no.2mm mH.HmN ::.saw oo.mma oo.mma Noawoa_ sH roses oo.HoH ms.aam em.mnm oa.aem as.mmm mm.mem am.amm oo.amm wH.HmN 4;.eam oo.aafl oo.maa HHH sodas oa.mao mm.a:e mm.mmfla oo.wmma mH.mmmH oe.sema H:.Hmoa om.mmoa No.4mea ma.moma om.aHHH om.wam HM «some . o o . o o o o . . . . ow Sm 8 Na 3 E as am 8 N: s E S. as e E s we age mos swam H use: “amass . own . >02 . poo . pmmm . was. inflow mag. E . has. . Ha: . nth . ~35 Ansonoo 53 no.3 game can Scam scapegonm mmm Eugene: Mo unoppdm am canon. SS 82] . Soar 33.. Goo no _ 933$? xJ\/ .:0HpmNfiommao Essflpmo one soon: mmmo mmpma pom ssfipos «Hamsm mo coapooUOQQ zanpnofi poo define hdnpooz .m ohsmflm ace“ 3% a 3% Asia. co ~83 mosh J.\ ¥ 9‘ _ . m3 / fife 3 u _ _ its; to rezone Ix Jw \ I u .ion \/ nun J. I. \K a}... moon 32%;. no abnszoolvm i“ 10V \ LN‘ \ x 1!. \ \ i3 AK . nh‘ \ anW How‘s \T 1.x “1:333 a e . . . . LNuxfiw . fin. 56 large eggs reaches a maximum between April and July at a time of low large egg prices. The period of high prices (August to October) corres- ponds to a. slowing down in the decrease in large egg production but does not induce a peak in production. It seems that a better plan would be realized by moving the peak of production for large eggs, from May and June, to September and October. This would have a counter- effect on the relationship between the production and price of medium and small eggs. But, this movement would only be possible by changing the date of hatch from the beginning of the fall to December or January. In so doing, the layers would reach their 15th month of lay in the fall. Accordirlg to the production data on which this study relies (see Appendix A, Table 5, p. 70) the decrease in production at this time of the Y ear ‘ is general. It would then no longer be profitable to keep the birds 20 or 22 months. The decrease in large egg production which wolld accompany this change would balance the gain obtained from more fen(”able large egg prices. Thus , the optimum plan creates a situation highly favorable to the development of the quantity of large eggs produced. Because of the possibilities of development of this production, the counter-effect on large egg. prices is offset. At the same time the fact that the medium and small egg production cannot vary in quantity is taken into account. The profit from their production is not maximized by acting on their Volume but by adapting their pattern of production to their price Variations as closely as possible. Given the present conditions of rod - p m: tion and price, the maximization of profit from the large eggs: 57 on the one hand, and from the small and medium eggs on the other, results in fixing long laying periods. (20 to 22 months) with dates of hatch in the fall. Because of the floor space limitations, several enterprises of the type which has just been described will be chosen and carried at varied levels of intensity in order to use the available floor space as near to full capacity as possible. The opportunity cost of floor space in the months in which the competition among enter- prises is greatest will fix these levels of intensity. But the use of floor space in one month is not independent from the use of floor space in any other month. The floor space inputs are perfect complements. The allocation of floor space made on the basis of the competition in the most limitational months will fix, _ip_s_o_ £3933, the use of floor space in the other month. For example the allocation based here in the cBOII'ljpetition of enterprises in months such as April or November rigorously fixes the use of December floor space at 9,111, and no more. Perfect complementarity of the inputs added to the consideration 0f o79130?tunity costs between the enterprises explain the persistence of Unused floor space in the optimum plan. In other words, to make use Of the floor space remining unused, it would be necessary to add new enterprises to the present combination, but the computation having been conducted up to the point where all the cJ. - are negative, 23. , s the Opportunity cost of any enterprise not in the plan is such that its addition would create a loss of revenue greater than the additional profit it would yield. Thus no new enterprise can be added nor can the levels of the enterprises in the present plan be altered without caus- ing a. I‘ed'llction in income. 58 B. Comparison with the current recommendations. In the first part of the discussion, the optimum plan has been compared revenue-wise to the old system of hatching in the spring and keeping the layers for 12 months. The possibility of making a net profit of more than 3,000 dollars instead of less than 1,000 need not be further discussed. The systems discussed by Marble and Jeffrey and presented at the beginning of this study (see pp. 1, ff.) will now be compared to the optimum plan. The possibility of using four hatches a year, as in the optimum plan, was said by Marble and Jeffrey to be quite common. The seven-months lay system advised for heavy breeds is here rejected . This is natural as the present study deals with a leghorn flOCk . , The keeping of leghorns for 15 or 16 months of lay is not used either but the optimum plan can be considered as a modification of this W of doing. Marble and Jeffi'ey advise to start the chicks in mid- winter ( December to February) in order to reach good egg size by the time egg prices reach their peak. "The next summer the birds are moved to t'E’I'Trkxleary quarters and carried as long as they remain in production. They are culled constantly but many will still be in heavy production at the end of 15 to 16 months."1 Here the farmer uses this ability of the leghorn breed to produce "me thJan-n 16 months and it has beenshown that he couples it with the influenC’e of favorable weather by advancing the date of hatch from December to July or October. Thus, he creates the possibility of'laying e p “was which reach 20 to 22 months. \ 1 Marble and Jeffrey, _1>_. cit. , p. 201;. 59 Under present conditions of production and prices, the profit which could be derived from a lS-month lay period with chicks hatched in December is easy to compute. Because of floor space limitation, this activity (X207), even carried alone, could not group more than 1,867 layers, five-months-old. The profit would be:1 765.29 x 1.867 = ' ti? l,h29 dollars or less than half the one derived from X142. With only one flock of birds born in December and kept laying for 20 months the 2 maximum profitawould be 1507.51; x 2% = 2558.11; dollars. This ’ indicates that with birds producing eggs as indicated in Appendix A, (Table 5, p. 70) a 20-month lay period would always be better than a lS-month one. The $300 difference between x210 and x142 shows also that, 88 it has been assumed, the profit from higher prices realized with a December hatch is more than balanced by the dimunition in the volume produced . Finally, Marble and Jeffrey also consider the possibility of a two- year thng period. This possibility is discarded here because limita- tion on floor space would allow any flock kept laying for 2).; months to count. only 1,535 layers, five months of age. This is much smaller than the intensity which can be reached with a 20-month or a lS'iHOnth lay System. Even when the revenue from a 2h-month lay system is higher than the one from a 20-month lagr system, the limitation introduced by the f1°0r space condition decreases the profit in such a way that the ___ - 1° 207 a 765.29 . 2 n c210 = 150751.. 6O 20—month lay system must be preferred. For example, with birds hatched on November 1, c198 = 1580.70 profit at the unit level of activity from at 2h—rnonth lay system is greater than 0194 = 1510.13 profit at the unit level of activity from the corresponding 20-month-lay system. However, the decrease in the level of activity from 1.6969 to 1.5359 is u“ enough for the profit from a 20-month lay system carried alone, to be smaller than the one from a 2h-month lay system carried alone: ' profit from X194 1510.13 3: 1.6969 = 2562.514 ll profit from x198 1580.70 1: 1.5356 = 2h27.32 0. Limitations of the study. A first set of limitations stems from the difficulties of applying the plan to the real world. The presence of 8 to 10 different age groups of birds on the farm in every month of the year is one of these Irate-1310115. These different age groups have to be separated. It would be imprac tical for the farmer to be continuously reallocating the floor t0 adjust to the slow but continuous changes in each flock due to mortality - The necessity of using the same floor space for brooders and for layers also will be difficult to meet. This suggests that as a develoI>rt1ent of this study it would be useful to consider the same OperatiOn but in distinguishing two kinds of floor space available each month (brooding space and layer space) 2).; limitations instead of 12 would then have to be respected. Finally, the time at which the major part of the brooding activities oc ~ cur (between August and December) renders necessary the organization 61 of well protected brooding quarters. No temporary arrangements would be satisfactory . 6 As capital has been considered available in any necessary quantity, no problem will here arise from this obligation. However, the need for a more complete study including the effects of capital limitations appears at this point. Even more important than the shortcomings is the fact that the results can be used only if the price situation corresponds to the one which has been chosen to compute the 03‘s and if thelayers considered have a ‘curve of egg production which follows the one used here (See Appendix: A, Table 13, p. 80). Any-other strain, with a different pattern of production would need to be managed along a different plan. W Charlge in prices also changes the'results. A110 ther set of limitations stem from the restrictive assumptions which have been made. The only factor of production studied was the £106]? Space and it was even taken as a fixed input for the farm as a whole - What about possible extension of the area available during the summer months through the use of some kind of temporary organization? What about the other inputs? The question of capital limitation has all‘ea.dy been raised. The problem raised by labor limitations are also Very important ones. No consideration was given to feeding prac- tices Whose influence on the cost. of any enterprise is of prime impartfiance. Finally, the present analysis can only be meaningful for special- ized commercial farms. If it were applied to a diversified farm, where 62 egg production cannot be considered independent from other activities, the study would have to be more comprehensive. Here nothing is known about alternative occupations which would bring the farmer the same profit . In the case of a diversified farm it would be necessary to know how the size of the layer enterprise is determined in relation to the feed grown on the farm or to the amount of capital which can be invested in different branches of the farm business: Fertilization, improvement of pastures, etc. The conclusions of this study remain valuable but it must not be forgotten that they deal with a very limited case of the laying flock mamgement problem . 63 CONCLUSIONS Linear programming was; used to find the optimal utilization of 10,000 square feet of floor space in a laying enterprise. The plan selected consists of seven different activities. These activities are [WET grouped on four different dates of hatch which range from July to : November . In addition to recommending a fall hatch, the optimum plan F fixes the length of the laying period at approximately 20 months. * These recommendations are very different from the formerly common Lil. management system of April hatching with a twelve-month lay period. The difference ‘in profits is also substantial. The increase is mainly due '00 the egg production pattern resulting from the combination cf ‘ enterprises selected. A comparison with the price curves for large, medium, and small eggs shows that the production of small and medium eggs very closely follows the movements in their respective prices and that the production peak of small and medium eggs occurs when their prices are at a maximum. Since there isno my of substantially increasing the production of small and medium eggs, this is the only method 01' maximizing the returns. Seasonal variations and length of lay, on the other hand, can greatly increase large egg production. This increase occurs in the present case. As a result, the importance 0f the discrepancy between the production peak and the peak in prices or large eggs is minimized. Despite a three-month difference between high prices and high production, the fall hatches with a twenty-month la y IEma-in the most profitable. 6h Because of the perfect complementary of inputs, this system does not use the available floor space in S of the 12 months of the year. Variations in the number of brooders and layers and the change in the use of floor space they require will create some practical difficulties. This points to the need for further studies. Among them, a study main- taining the distinction between laying space and brooding space through- .3 out the year might be suggested. It would also be interesting to know what is the influence of the introduction of other supplementary limitatienal inputs such as labor and capital. What would be the result 3. of introducing in the present model alternative management processes (buying young layers instead of rearing them, for example)? This, also, should be studied. snuff Lamas. APPENDIX A o e mo, cams coax coax «max «max cmax moax omx mes vow was max 4m mHmM coax meax Hoax mvflx mmax SHN max Hex max max sax MN vamx coax coax coax mvHN vaN coax max osx «out vmx was NN mHmN coax soax coax HvHN mmaN moaH sex sex How max max Hm «flux vmax coax coax ovflx «max scam cox sex com max max 0m HHmN mmax msax smflx anew Haas mosN mmN hex ovM me max . ma oamx «max vsflx coax mmflx omax «OHM vex cox ova omx max ma momN Hoax mess mmHN smHM mflax floax max new sex max HHN NH momN coax moax vaN mmax mass coax «ox vex ovx mmx oHN 0H comN coax HsHN mmflx mmax oHHx max Hex mmx new sax mN ma comN mmflx oeHN mmax vaN mass max omN moN vex mmx ox 4H momN omax coax Hmax mmaN mafia sex sex Hex mvx max ex ma momN coax coax omHN mmfix «Hex sex sex cox mvN «ax ox NH momN mmfix hoax mvHM Hmax mHHx max new max Hex max ax HH momx coax coax mvHM omfix NHHN vex mew mew ovx mmx ex OH Hoax hoax coax svax mmfix Haas mow new sex max me mm m oomN «max Voax mvHN swam oHHM new sex max new omN NM . m coax swam mmax mvHN smax coax Hex new mum emu max fix 5 a z 1 J m J Jim: me lie J was coeds mo specs mmapasaeoa.mo puss .H wands P 823 32.3 No npmnmq ll 66' Table 1: List of Activities - Concluded Rank of Limitation.Factor Process Non Used X217 Floor space in January x218 Floor space in February x219 ' Floor space in March x220 Floor space in April x221 Floor space in May x222 Floor space in June x223 ' Floor space in July. x224 Floor space in.August x225 Floor space in September x226 Floor space in Octdber x227 Floor space in November x228 Floor space 1n.December Table 2: Rates of Decrease in the Number of Birds Month Rate of Decrease Cause (Per cent) - let to the 2nd 5 Death 2nd to the 3rd 2 Death 3rd to the hth 2 Death Ltth to the 5th 2 Death 51:11 to the 6th 10 Culling 6th to the 7th 2 Death 7th to the 8th 2 Death 8th to the 9th 2 Death 9th to the 10th 2 ‘Death 10th to the 11th 2 Death 13.1211 to the 12th 2 Death 1 i i m i m m i m 29 th to the 30th 2 Death 68 Table 3: Number of Birds in the Flock According to the Month of Life Month of Life —5 Number of Birds Month of Life Number of Birds \0 (D -J (D \n if t» is J4 t4 £1 +4 t4 +4 #4 U1 U) A) F’ C) 12h3 1181 1157 113k 1111 1000 980 960 9h1 922 90h 886 868 851 83b 16 17 18 19 _ 20 21 22 23 2h 25 26 27 28 29 817 801 785 7 69 75h 739 72h 710 696 ~ 682 668 655 ohz 629 3.. -w.‘ v i.’ 69 e:s: e:s: e:s: e:s: e:e: e:s: emem emem emem emem emem emem emem emem emem emem emem emem emea m:e: m:e: m:e: m:e: m:e: m:e: m:e: meem meem meem mesm mesm mesm mesm mHsm masm mesm mesm ease ems: ees:. ems: ems: ems: ems: ems: ems: eesm eesm eesm eesm eesm eesm eesm eesm eesm eesm noes o:om o:om o:em o:om o:om o:om o:om o:om o:om emem emem emem emem emem emem emem emem emem esa m:em m:em m:em .m:em m:em m:em m:Hm m:Hm m:em m:em oeem oeem oeem oeem oeem oeem oeem oeem mes s:mm s:mm s:mm s:mm s:mm s:mm s:mm s:mm s:mm e:mm s:mm o:sm o:sm .o:sm o:sm o:sm o:sm o:sm est seem seem meme seem seem seem mmem seem seem seem seem seem oooe oooe coon ooom oooe .oooe nee meme mme: mme: mme: mme: mmo: mme: mmo: mme: mmo: mme: mme: mmo: mmmm mmmm mmmm mmmm mmmm ems s:ee s:ee see: sea: see: see: see: see: see: sea: see: see: see: see: eemm eemm eemm eemm e:a :mem :mem :mem smem smom smem seem smem smem seem seem smem seem smem smem emee smee smee ems emem emem emsm. emem m:sm m:sm m:sm m:sm m:sm m:sm m:sm m:sm m:sm m:sm m:sm m:sm Heee Heed ems. eses eses mses ssem sses See See See See See See See See See See See See Ema 2a sex sex sex max vex new new Hex oHH ex ex ex ex mm «M ma fix fix max 09 an to.“ mesoeoHMMooo pause psaMHilM: tents gnu“ 70 1 Table 5: Total Egg Production Per Bird Per Month in Dozens 1 Month of Date of Hatch Production Dec. Jan. PE. March Apr. May June July Augfiept. Oct. EW. 1. May . 1.5 . 2. .June 1.6 1.5 . 3. July 1.6 1.6 1.5 1;. August 1.5 1.6 1.6 .5 5. September 1.3 1.5 1.5 1.1 .5 6. October 1.1 1.3 1.3 1.6 1.1 .5 7. November 1.1 1.1 1.1 1.6 1.6 1.1 l.O 8. December 1.1 1.1 1.1 1.6 1.6 1.6 1.6 1.0 9. January 1.3 1.1 1.1 1.6 1.6 1.6 1.8 1.6 1.0 10. February 1.5 1.3 1.1 1.6 1.6 1.6 1.7 1.8 1.6 1.5 .37" ' 11. March 1.5 1.5 1.3 1.6 1.6 1.6 1.6 1.7 1.8 1.6 1.5 _ 12. April 1.1. 1.5 1.5 1.5 1.6 1.6 1.6 1.6 1.7 1.7 1.6 1.5 13. May 1.3 1.). 1.5 1.1; 1.5 1.6 1.5 1.61.6 1.7 1.7 1.6 114. June 1.3 1.3 1.1. 1.3 1.1. 1.5 1.1. 1.5 1.6 1.6 1.7 1.7 15. July 1.3 1.3 1.3 1.1 1.3 1.1; 1.3 1.1; 1.5 1.6 1.6 1.7 16. August 1.3 1.3 1.3 0.9 1.1 1.3 1.2 1.3 1.1m 1.5 1.6 1.6 17.5eptemher 1.1 1.3 1.3 0.8 0.9 1.1 1.0 1.2 1.3 1.3 1.5 1.6 18. October 1.0 1.1 1.3 0.6 0.8 0.9 0.8 1.0 1.2 1.1 1.3 1.5 l9.Noveanber 0.8 1.0 1.1 0.5 0.6 0.8 0.7 0.8 1.0 0.9 1.1 1.3 20. December 0.7 0.8 1.0 1.5 0.5 0.6 1.3 0.7 0.8 1.0 0.9 1.1 21. January 0.7 0.7 0.8 1.1; 1.5 0.5 1.5 1.3 0.7 1.0 1.0 0.9 22.Febr1mry 1.2 0.7 0.7 1.1. 1.1; 1.5.1.1. 1.5 1.3 1.2 1.0 1.0 23.March 1.11 1.2 0.7 1.3 1.3 1.h 1.3 1J4 1.5 1.3 1-2 1-0 21i-Apx-il 1.1. 1.1. 1.2 1.3 1.3 1.3 1.3 1.3 1.1; 1.14 1.3 1.2 25.May 1.1. 1.11 1.3 1.3 1.3 1.3 1.3 1.3 1.1; 1.14 1.3 26- June 1.11 1.3 1.3 1.3 1.2 1.3 1.3 1.3 1.11 1.14 27- July 1.3 1.3 1.3 1.1 1.2 1.3 1.3 1.3 l-h 28.11.11glm.o 1.3 1.3 0.9 1.1 1.2 1.2 1.3 1.3 29.3eptember . 1.3 0.8 0.9 1.1 1.0 1.2 1.3 30.03t0ber 0.7 0.8 0.9 0.9 1.0 1.2 31.NoVember , 0.7 0.8 0.8 0.9 1.0 32'De°ember 0.7 0.7 9.8 0.9 33. Janu-ary ' 1.0 0.7 0.8 314. February 1.0 0.7 35'M8chh 1.0 N E__sti"E.teg nnannrnri 'hv hr- Whnn'l gr- Unmrhnnn‘h nf‘ Arr-'1 mfl hlra'I Economics. 71 .mpemambesb 33m damaged: . moHsodoom Headpazoenwe. mo 0.559.3me $9323 :5 a3 3.9.36.5 mmpflfipmmH o.e schemata oe. om. H.H oo.e acesosoz so. ss. Ho. sm. m.e Ho. .so.H asoopoo He. ss. OH. 0». Ho. sm.e om. oe.H statesman ea. ms. :e. es. 0H. cm.e mm. mo.e sneeze sm. 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J Jill season not taste .mooHsm maHssm acsaoooeriamxsmz pHoapmn seem cpHgs s cease we «seem "a oHnaa mic 75 Table 8: Estimates for the Market Price of Snell Eggs January February March April May 1956 .081 38.1.9 (1)2 35.91; (D.M)3 37.17 (2)2 32. 21 rem... 1955 22.101 36.51: (1.--10)4 39.5 (D.M)3 31.98 (L-ll)4 27.89 (M2 , 19514 h3.b0 (6)2 13.59 (1)2 36.95 (D.M+l)3 32.57 (3)2 22.811 2 ' 2 a 2 2 . 1953 hl.h5 (5) 10.91 (2) 1.3.52 (D.M+l) hh.19 (2) 143.51. (7) ' 1'1'1'161‘ underlined numbers are reproduced from Table 7. ‘ We.-. 2The numbers in the parentheses indicate the number of data from the USDA market report which were available to compute the monthly average 0f the particular month. 3The abbreviations D.M. in the parentheses indicate that for this Particular month the figure used comes from one detailed weekly report fI‘om the Federal-State Mrket News Service, Detroit, Michigan. ‘The abbreviations L-lO and L-ll (February and April, 1955) indicate that for these two months no data were available and the price of Small eggs has been derived from the calculations by interpolation 0f the difference between the price of large and small eggs for each month between January 1955 and May 1955. . 76 assesses: Base 5&3on smoH20soom.HwaspHdoHnm<.mo pamspammom .mpmHmuan .hq.hn omnmasnsm some mean msoepepm mmmnp Scam apnonma was .oemo .somHommz new sameness aqoeHHsdm mo msoepmpm msenmsw map hp ewes mnewnms onP.Mo mmwdnwbew s.e s.e s.s s.s .s.s e e e s.e s.e s.e m.e essH me me m.m m.m m.m m.m m.m m.m m m m m6 :msH N N m m m w m m w o o m.o mmmH N N m.N m.N m.N N N N N N N N , ommH meme HHaae s.e s.e s.s, s.s . s.s _ s.s s.s s.s s s s s sssH o o m m m o o o m m m m.o m 2mmH me- me .mé m; mN e e e e e e e 33 N N m.N m.N m.N N N N N N N m.o cmmH . mmmm Edens: a s s s s e e e e e e e essH m6 m6 m; m; mN e e e e e . e s :msH s.e s.e e e e a.a a.e e.e s.s . s.s s.s s.e sssH o o m m m N N N m.® m.m m.w m.o ommH . _ mmmm owan .eem .eoz amoo .emee lemme sHee eeee ‘ as: HHmmeu. ease: .eee ‘ .ame memos new memwm Hermon one so mmmm Ho noomhm one c.2590 3 Hummb mfifimhg «m OHQQB 77 .suHmambens opmpm qmmenofiz .merosoom HmpspH50pre.mo pcmspumgmo .mpmHoman .mo mo mocmpmemmm one ape: oommHHnmpmo smog mam: mmpdsepmo muons .s eds e .s aeHoms as mass some ecseasma ee.mm He.mm H0.ee 0H.:e 0snee se.em. es.se :m.se se.se m0.se H:.:e ss.:e emsH 0e.Hm em.Hm m0.mH ee.eH :e.mm se.:m 0m.em He.:m sm.sm es.Hm sm.ee 0s.ee :msH ss.:e He.sm sH.em em.mm es.em ee.mm es.Hm se.sH es.mm m.em :m.0e 0e.mm mmsH mm.mm mm.mm H©.Hm No.mm mmJHm mm.mm mm.:m m:.mm NH.om mm.mm m:.Hm wo.om ommH mmmm HHmsm e:.o: em.ss em.e: as.0s ee.mm ee.mm sm.e: He.e: se.e: s:.s: se.H: .ee.H: mssH e:.mm 0s.em ss.mm me.em se.ee ee.mm me.0e mm.He s0.se. s0.se eH.:: m0.m: :msH e:.m: es.me em.:e e0.m: s:.se eH.ee HH.0e es.sm :e.ee 00.0: ms.sm 0s.sm mmsH Nw.om mm.mm o:.om ee.mm ee.Hm mm.mm mN.Hm :H.mm NN.mm mm.om mH.mm ee.Hm ommH mmmm Edens: :H.e: m0.mm H:.sm Hs.me :m.em m0.em :e.0m :0.e: :s.0m mm.s: ee.e: ms.:: emsH mm.Hm oN.mm mm.sm m:.J: mm.m: ee.mm 0N.:m mm.mm MN.mm Nm.H: mo.o: MN.m: 2mmH e:.m: :w.N: mm.N: 2N.mm mm.m: mm.sm mm.mm 0N.mm m:.Nm oo.m: :0.H: :0.Hm mmmH ss.:e 0e.se mm.m: ee.m: ss.0: me.0: se.ee as.ee m:.se e0.ee ss.ee :0.e: emsH . mmmm mwpsH . coo . >02 . poo . pmom . $2 .33. 9.86 mm: dean: no.3: . pom . use. 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HeN 0mg QVN 0e.0:0H mm.HNe .0m.s0s ee.Hm: 0e.ses sm.:He Hs.ooN 0e.sme ee.Hem m0.Hm: mm.e:e os.NOm :0.s: ss.:mH. eseHm. ee.msmH mmN NmN HmN omx QNN wNN sax wNN mmN #Nx mNN NNN HNN CNN QfiN mHN mm.s0mH see mm.somH 0H.” APPENDIX B .1," 82 APPENDIX B ‘ A. Computation of the an, 3 Number of birds in any of the 30 months . As previously mentioned (page 31), the basis for this computation is the fixed number of birds at 5 months old (beginning to lay) = 1000. The number of birds'in each of the other months is deduced from this number with the help of the death and culling rates. To simplify the computations, the change in the number of birds between the first and the second month as well as between every other two months, except for the (fifth and the sixth month (beginning of the laying period), have been considered as due to mortality; culling is ignored. All the Small changes it would introduce were it not ignored, therefore, have no effect on the total profit. The mortality rates adopted are the following: From the first to the second month 5%; from any one month to the following (except fmm the fifth to the sixth month) = 2%. To partially compensate for the minimization of culling, the de- crease on the number of birds in the month preceding the beaming 0f the hying period is considered as entirely due to culling and it is fixed at 10% (decrease in birds from the fifth to the sixth month).1 As a result of the adoption of these rates of decrease, the number of birds in 9&0h month is fixed at the following level: A lsee the table of the rates of decrease in the number of birds, in ppendix A, Table 2, page 67. 83 Number of birds in the sixth month ll :3 O) l I'-’ O O 0 Number of birds in the fifth month ll :3 01 l H O :3 0' ll :1 O) lOO 9On5 = 1000 100 2n4 £13... Number of birds in the fourth month ,= n4 - ----~ = 115 lOO 98n4 ' = 1111 n4 = 1131, :f 100 V in the same manner, 98113 L “100 = 1131; n3 = 1157 and 9811,,3 T56 =.- 1131; n2 = 1181 because of the 5% mortality during the first month: 95n1 z 1187 150 and in the same way, n, = 98n6 n, = 980 100 he a 98“? ns = 960 etc. 100 The complete table of number of birds constituting the flocks from the with of hatch to the 30th month of life (2hth month of lay) is given in Appendix A, Table 3, page 68. Computations of Several aij a) Computation of a1,1 - floor space requirements for the month of J “Diary for a flock hatched in January and which will lay for seven mnthg . 811— 'Number of birds: 12173. Floor space requirements per bird, 1 sq. ft. al’1 = 1 x l2h3 a1,1 = 112113 Because, in the first process (x1) the birds are sold" at the end of the first year, the computations of coefficients from a1,1 to 31,12 will follow this simple pattern. b) Computation of 8.1,; - floor space requirements in January for , flocks hatched in January and laying for eight months. I“ “(WI The birds are hatched in January, they begin to lay in June, and they are sold at the end of the following January. Thus, during the .fi“-_:_+‘._ua..... - .__._, 2.75, o . 0 month of January there are simultaneously two flocks in the hen house: The newly hatched chicks which require 1 x 12113 = 121:3 sq. ft. and the old hens which will be sold at the end of the month. This laying flock is 12 months old and only 868 hens remain, taking into account mortality. Each of these birds requires 3 square feet. The old flocks need 3 x 868 = 2601; square feet so, 611,2 = 12143 + 26011 = 381:7. Note that az’g is equal to 82,1 as it is the floor space require- ments in February for flocks hatched in January and kept laying for 8 months (sold at the end'of the following January). The only flock which remains in the hen house is the newly hatched one and it requires as in an, 1 x ll8l= 1181 square feet per unit of the process. For all the processes from x2 to x12 inclusive there will be two types of co- efficients. Some refer to months during which the newly hatched flock is in the hen house. These are computed as a1,1. Others refer to months during which the two flocks are in the hen house as in the case. of 31,2" They are computed in the same way as 811,2- 85 For x13 (flock hatched in January and sold in December of the following year after 19 months of lay) all the coefficients (a1,13) are computed, in the same way as 31’2- c) Computation of a1’14 - floor space requirements in January for flocks hatched in January and kept laying for 20 months (sold at the end of the month of January of the second year after the hatching date). During the month of January, three different flocks are in the hen house: The newly hatched flock which requires 1 x 12h3 = 12h3 square feet. The flock hatched the previous year which requires 3 x 868 = 260h square feet. The flock of old hens in their 20th month of lay - only 682 birds are remaining.1 Then this flock requires 3 x 682 = 20h6 square feet. As a result: a1’14 = 12h3 + 26Gb = 5893. One should note that a2’14 (floor space requirements in February for the same flock) is equal to 32,3 j = 3, b, . . 13. In all the corresponding processes (x3, x4, . . . x13) only two flocks are in the hen house during the month of February; the newly hatched flock, and the onefiyear-old flock. For all the processes from x14 to x18 inclusive there will be two types of coefficients. Some refer to months during which only two flocks are in the hen house and these are computed like a1,2. Others refer to months during which three flocks are in the hen house as in the case of a1,14. They are computed in the same way as 1,1h. B. Determination of the cj's: 'a) Questions of the quantities of products which can be sold. 1See the table of number of birds per month in.Appendix.A, Table 3, infige:68. 86 . . 1 1. Production of eggs. The data giving total egg production per bird in any month of the laying period are given in.Appendix.A, Table 5, page 70. An.inquiry to the poultry department of the states of New YOrk, Tennessee, Missouri, Florida, California and Wisconsin did not bring results as their organizations sent back only reports on random sample tests which dealt with flocks born in April or May and kept laying for 12 to 15 months. The computations have, in fact, been made by using a series of data which distinguish among large, medium and small eggs. These detailed egg production tables are in .Appendix A, Table 6, pages 71-73.' If, for any given month, large eggs aloneare produced, production in the succeeding months is also considered as being 100% large eggs. b) The question of the price of the output. 1. Price of eggs. The monthly average2 prices for small, medium and large grade A white eggs on the Detroit market have been utilized to compute the monthly average over January 1953 to December 1956. These prices in cents per dozen are given in .Appendix.A, Table 7, page 7h. The first problem which had to be solved arises from the fact that no prices for small eggs are available for the months of February, March, . 1The monthly egg production.per bird in.a 2h-month laying period for flocks hatched in any month of the year was furnished by Dr. Wheeler of the Agricultural Economics Department. Dr. D. A. Dawson of the Poultry Department was kind enough to check Dr. Wheeler's data and con- sidered it acceptable. 2These data were obtained from Dr. Larzelere of the.Agricultural Economics Department of Michigan State University, East Lansing, Michigan. 87 and April in any year as well as for the months of January 1953 and 195b, and May 1953 and 1955. In an effort to estimate the missing data several possibilities considered were: 1) Interpolation on the basis of the differences between prices of large and small eggs, on the one hand, and medium and small eggs on the other; 2) use of the data only from the grading stations of Hamilton, Michigan and Napoleon, Ohio-data .available for one week in March and interpolation to fill in the other months-- 3) use of every available datum whether it came from the Detroit market, from the daily United States Department of.Agriculture dairy and poultry news, or from the Hamilton.and Napoleon grading ,stations. In order to use as many real data as possible, this last solution was selected and led to an estimate of the market price of small eggs for the missing months. These estimates and their sources appear in. .Appendix.A, Table 8, page 75. I The second problem consisted in transforming these market prices into prices paid to the farmers. An estimate of the margin which had to be deduced from the market price to obtain the farm price was calculated as follows: The margin was computed from reports from the Hamilton and Napoleon grading stations for the months of March, June, September, and December of each of the four years. For each of these months the average of the margins at Hamilton and Napoleon was calculated, and this average was used for the month preceding, and the month following the one for 1 which it had been computed. .A table of these margins is in.Appendix A Table 9, page 76. 8 _For each month of the four years, the margin was deduced from the market price and the table of monthly farm prices for large, medium, 88 and small white Grade A eggs(Appendix A, Table 10, page 77) was thus obtained . ' As a last step, the monthly price averages which are to be used in the computation of the cj are derived from the four numbers avail- able for each month. (Appendix A, Table 12, page 79.) 26 Price of culled birds and old hens: Because the flocks are of leghorn stock, the birds culled in their fifth month (just before laying) are assumed to have been sold at the same price as the old hens. The only difference between them is the previously indicated one, namely, the weight. In order to compute a fourfiyear average for these prices, the prices of hens per pound on the farm in Indiana were taken from the daily market reports of the United States Department of Agriculture (Dairy and Poultry_ Eggs). Because prices tend to remain unchanged for several weeks, only one price each week (generally on wednesday's market report) was retained. From these weekly prices, a monthly average was computed for each of the h8 months. Then, for each month, an average was computed from the data available for 1953, l95h, 1955 and 1956. A difficulty arose from the fact that the United States Department of.Agriculture Dairy and Poultry;News service began to report Indiana farm prices,only in March l95h. Fourteen months from January 1953 to IFebruary'l95h inclusive, were missing. For these 1h months, estimates 'were established by referring to the prices of the following year and correcting the figures thus obtained according to information. about the poultry market. Table 11, page 78 in.Appendix.A shows all the prices 89 which were used in computing the prices of hens and culled birds necessary for the computation of cj's The final prices used in computing the cj's are indicated in Appendix A, Table 12, page 79. c) Questions of costs. For the costs of sexed chicks, the only data available from the Agricultural Prices Bulletins of the United States Department of.Agri- culture were the yearly prices of 100 sexed pullets in the East North Central Region. These prices were: 1953 - $30.2 l95h -v 32.2 . 1955 - 32.7 The fourfiyear average is 1956 - 31:3 $32 .35. The opportunities of using an average might be open to question since the succession of prices from 1953 to 1956 shows a definite upward trend in the price of sexed pullets. But in the computation of the'cj,s this price becomes a constant and thus has no influence on the results of the analysis. The costs of layer and chicken feed are also fourfiyear average prices derived on a monthly basis from the prices given in the.Agri- cultural Price Bulletin of the United States Department of.Agriculture 'under the heading: East North Central Region. These data are in Appendix A, Table 12, page 79.. i The problem here was to find an estimate of the averages for chicken .feed during the months of January, February, August, September, Octdber, and December for which no prices were reported. For the two kinds of 9O feed, for the month in which the prices are known, the ratio for two consecutive months of the price of layer feed is very close to the ratio for the same consecutive months of the price of chicken feed. The estimates of the price of chicken feed for the missing months have, therefore, been established as follows: The procedure is explained for the month of August. Price of layer feed for August = h.6 Price offilayer feed for July 5.6% Price of chicken feed for August = b.6g It is known that It is assumed that'Priee of chickefiwfeed’for July The price of chicken feed for July is known. Thus, the price of chicken feed for August = %;%%' x 5.09 = 5.06 Knowing the price of chicken feed for August and the ratio Price of the layer feed for September, Price of the—layer feed for.August " August is established by applying the same method. .A complete table of the price of chicken feed for the different monthly prices used in the computation of the cj‘s is given in Appendix A, Table 12, page 79. The preceding page shows how the shortage of data becomes the major jplague of every linear programming application. Each one of the esti- mates which were necessary because of the lack of data distorts the results of the analysis and diminishes the very advantages of the tech- nique, that is, to furnish a rigorous answer. However, a decision is necessary, and it is better to provide approximate solutions from not too accurate data than to wait fer perfect data and not study the problem at all. The preliminary questions have been resolved. One can.now'compute the Cj's' Two examples of computation will be given. The table of the 91 c is in Appendix A, Table 13, pages 80 and 81. 3'3 Computation of egg: i.e., computation of the net profit from a flock hatched on April 1, beginning to lay on September 1, then number- ing 1000 birds, and sold on the following April 1. According to the definition and the answers to the preceding questions of price and ‘ volume of production, the profits in 055 come from the selling of large, medium, and smll eggs during the seven months of the culling period, the culling of 111 birds each weighing h pounds at the end of August (before the beginning of the laying period) and the selling of 886 old hens each weighing 5 pounds at the end of March. The costs are: The cost of buying 1,2h3 sexed pullets at the. price” of $32.35 for 1003 the cost of chicken feed and cost of layer feed. 1. Receipts from egg production.. v—w—vv—v ' v f v v t.— fi_— v7 r 7 Number Dozen Eggs Price of * of Birds per Bird Eggs Profit (cents/doz.) (dollars) September Large 1000 . 03 51 .h3 15 .173 Medium 1000 .12 37 .03 hh .173 Small 1000 .35 21; .175 85 .57 October Large 980 .ll h7.2o 50.88 Medium 980- ' .117 . 32 .69 150 .57 small 980 .52 . 23.96 122.10 --and so on for each of the (months'of the laying periods. Total profit from egg production -- 33,3147.1:l.. _ 92 2. Receipts from culled birds. Price of hens in August times weight of a bird times number of birds $11.35 x h x 111 = $50.39 3. Receipts from selling the old hens. Price of hens in March times weight of one bird times number of birds $111.70 x 5 x 886 = $651.21 h. Cost of sexed pallets. 32.35 x 12h3 = $h02.11 5. Cost of chicken feed. Average price for chicken feed for the months of April to August inclusive, times the consumption per bird (25 pounds) times the number of birds during the fifth month (111): 5.10 x 277.75 ‘= $1h16.52. 6. Cost of laying feed. September + October + etc. _Number of layers times 8 times price of layer feed 1000 x 8 x h.59 = 367.20 + 356.72 + etc. and so on for each month of the laying period. Total cost of the layers' feed: 2320.03. So, c55 = 33h7.h1 + 50.39 + ~ 651.21 - h02.11 - lhl6.52 - 2h20.03 = ~189.65. Computation of 955: The 056 are the profit derived from a flock of the same type as the one used inx55 but with a laying period of 8 :months. .As a result, the cost of the sexed pullets (constant throughout the study), the cost of the chicken feed, and the profit from the culled 'birds remain equal to what they were in c55. This constant element is equa1_to 3 ~1768.2h. It is constant for all cj's related to flocks 93 hatched on April 1 (constant for C55 to c72 inclusive). .Another value, composed of the same data, will be constant for all flecks hatched on May 1 (C73 to 090) etc. The receipts from egg production is the one computed for c55 plus the receipts made during the eight months of production (in this case, April). So-- 33h7.h1 + (868 x 1.17 x hl.39) + (868 x .h3 x 38.83) receipts from receipts from medium eggs large eggs Total receipts: 3912.6h. The receipts from old hens sold in April is 1h x 5 x 868 = 607.60. The cost of layer feedis the one computed for 055 increased by the layer consumption in the eighth month of the laying period: 2h20.03 + (b.68 x 5 x 868) = 275h.01 So 056 = 3912.6h + 607.70 - 1768.2h - 275h.01 = $ +6.99. The same simplifications are used in the computation of all c from 056 to 072 inclusive. Any of the c 3'8 °dealing with flocks kept for a seven-month lay period are computed as 3'5 C55. All other cj's have a relationship to the immediately preceding . one in a sequence of 18 cj's’ a relationship comparable to the one- linking 056 to 055. ROOM USE ONLY Demco-293