AN APPLICATION OF QUEUING THEORY TO ORGANIZATION GROWTH Thesis for the Degree cf Ph. D. MICHIGAN STATE UNIVERSITY Lysle I. Beniamen I966 infll' IWWImnwmwflmfigflnll 7 v 3;);1; ' in: Ui‘fiv {mgr}, 5! v .- . h “fr-k $?‘ ~. '5 .~ This is to certify that the thesis entitled AN APPLICATION OF QUEUING THEORY TO ORGANIZATION GROWTH presented by Lysle I. Benjamen has been accepted towards fulfillment of the requirements for degree in DOCTOR OF PHILOSOPHY Major gofessor Date %X€/7€6 0-169 LC.- ' ""o _ pap.‘ ”the ABSTRACT AN APPLICATION OF QUEUING THEORY TO ORGANIZATION GROWTH by Lysle I. Benjamen The question of the proper time to add or subtract employees, and the question of the types of skills that these employees should have, are fundamental to problems of organization growth. The purpose of this study is to examine a use of queuing theory as a guide to under— standing this growth process. The use of queuing theory formulae is examined to determine whether queuing theory analysis is a valid predictive tool when applied to organization growth problems, or whether it instead may be a theoretical explanation of the growth process, or both or neither. The analytical formulae of queuing theory are used in the thesis to find the "size" of organization at which employees for various functions are typically added to an organization in specific, defined situations. Seventeen samples of eight firms of the American pleasure boat industry were taken to obtain data for evaluation of five hypotheses. These hypotheses were: 1. That there are regular and consequently predictable patterns in which personnel are added to various functional areas in Lysle I. Benjamen a marine organization as it grows in size as measured by certain stated parameters. 2. That these patterns are based on the princi- ple of adding personnel when the queue of demands for service in a functional area reaches a given length. 3. That demands for service in some, at least, of the functional areas arrive in a random manner making queuing theory a suitable tool for analysis. A. That queuing theory provides a good theo- retical explanation of the consistency in the historical data as to when personnel are added in certain functional areas. 5. That queuing theory is a practical tool for predicting at what point in a marine organization's growth, as measured by stated parameters additional personnel will be hired in various functional areas. Personnel changes for the below listed three functions were studied in the eight firms to find typical industry adjustments of number of employees to average work wait- ing line length of the function. A full personnel history of the three functions was obtained for one of the eight firms; the history of the (a) production, (b) purchasing, and (c) selling functions for this firm is compared with Lysle I. Benjamen personnel growth patterns developed by solution of queuing theory models based on typical industry personnel adjust- ments determined in the 1? firm samples. This comparison of actual personnel changes for the three functions with the function model solutions is used to evaluate the hypotheses. Propositions l, 2, 3 and A were consistently supported by the data. This data was limited in two ways beyond the necessary sampling limitations. (a) Samples were taken from eight firms in the marine industry only. Therefore, the re— sults will not necessarily apply to other industries, but no reason is seen to indi— cate this industry is peculiar in this re- spect. (b) Only one and two channel servicing agencies were included in the functional areas in- vestigated. The results may not be appli— cable to function areas arranged with a larger number of channels. Proposition 5 was not supported. The data showed queuing theory could, indeed, be used to predict when personnel would be added, but for this purpose the queuing theory proved to be only an added step in the analysis which contributed no prediction that was not inherent in the data without the queuing theory analysis. Lysle I. Benjamen The queuing theory analysis affords an explanation as to why, but contributed nothing as to what in the prediction of personnel additions in one and two channel functional operations. To illustrate, the historical data shows that when the rate of engine production increases a given amount, if no new personnel have been hired for purchasing, orders awaiting servicing will reach a given mean length. At this point an additional person will be hired for the purchasing department. This will result in shortening the mean queue length back to what apparently is accepta- ble in the industry. Queuing theory affords an explanation of why the mean queue length elongates as it does and why the added personnel reduces its length in the observed manner . AN APPLICATION OF QUEUING THEORY TO ORGANIZATION GROWTH By Lysle I. Benjamen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Personnel and Production Administration 1966 ACKNOWLEDGMENTS To Dr. Rollin Simonds the writer would like to offer his sincere thanks and appreciation for providing the critical encouragement and guidance so necessary to an understanding in depth of the vagaries of the American business system in general, and of business organization and management in particular. From Dr. David G. Moore the author obtained a comprehensive understanding of the place that behavior— alistic techniques and knowledge have within planning and controlling organizations. The late Dr. Karl Boedecker provided the writer with a disciplined understanding of the relationship theory has to empirical knowledge in the science of management. An eXpression of gratitude and thanks is due the entire graduate school faculty and the writer's fellow students for their patience and their capacity as intellectual whetstones. Finally, to his ex—colleagues at Dearborn Marine Engines, Inc., the writer expresses thanks for their aid and interest. ii VITA The writer of this thesis was born on July 20, 1927, in Detroit, Michigan. In 19AM, he was graduated from Redford High School, Detroit, and spent an academic year as a student in the College of Liberal Arts of Wayne University. In l9A5 he enlisted in the United States Coast Guard and was assigned to a year at the United States Coast Guard Academy Preparatory School at Groton, Connecticut. He was tendered an appointment as Cadet at the United States Coast Guard Academy in 19A6, and was graduated in 1950 with a Bachelor of Science degree in Mechanical (Marine Option) Engineering. Upon release from active military duty in 1954, the writer entered Rensselaer Polytechnic Institute, Troy, New York; and was graduated in 1955 with the degree of Master of Science in Management Engineering. Between 1955 and 1957 he was employed by the Eaton Manufacturing Company, Valve Division, Battle Creek, Michigan as Engineer of its Lawton Plant; and by the Prex Corporation, subsidiary of Wyman—Gordon Company, in Franklin Park, Illinois, as Chief Engineer. In September 1957, the writer was granted an educational leave from the Wyman-Gordon Company and was accepted iii as a candidate for the degree of Doctor of Philosophy by the Graduate School of Business Administration at Michigan State University, where he completed his resi— dence in August 1958. In September 1958, the writer Joined the faculty of Ferris Institute, Big Rapids, Michigan, as Assistant Professor of Commerce and Finance. He joined Dearborn Marine Engines, Inc., Madison Heights, Michigan, in June 1959, as Chief Engineer where he continued in capacity until May 1962. Concurrently, in September 1960, the writer joined the faculty of the University of Detroit as part—time lecturer in Industrial Management for the academic year 1960 to 1961. He has resided in Birmingham and Bloomfield Hills, Michigan for the past five years where the research and writing of the thesis were based. He is presently employed as Executive Vice President of Midwest Machine Company of Marysville, Michigan. iv TABLE OF CONTENTS ACKNOWLEDGMENTS. VITA LIST OF TABLES LIST OF FIGURES. LIST OF APPENDICES. Chapter I. INTRODUCTION Hypotheses Framework of the Hypotheses Conceptual Definitions Symbolic Definitions. II. HISTORICAL BACKGROUND AND QUEUING THEORY APPLICATION. . . . . . . History Application. III. APPLICATION OF QUEUING THEORY TECHNIQUES TO THE ORGANIC FUNCTIONS . . Accomplishment of Organizational Purpose Applicability of Queuing Theory Selling Service Problem. Job Factors in Organization Functions IV. RELATING SERVICING OF INDIVIDUAL FUNCTIONS TO A MEASURE OF ORGANIZATION SIZE Organizational Evolution . Development of the Organization Model ii iii viii ix Page (IDU'I Jon 10 IO 13 16 16 18 21 23 29 3O 32 Chapter Page V. THE RESEARCH PLAN. . . . . . . . . 3“ Research Sampling Basis . . . . . . 34 Research Methodology . . . . . . . 36 The Sample . . . . . . . . 37 Publications Research. . . . . . . 39 VI. RESEARCH DATA . . . . . . . . . . ”0 Determination of Queue Length and Queue Distribution . . . . . . . Al Organizational Case: Dearborn Marine Engines, Inc. . . . . . . . . A8 Evaluation . . . . . . . . . . A9 VII. QUEUING THEORY FORMULAE AS FUNCTION MODELS. . . . . . . . . . . . 61 Purpose of the Theory. . . . . . . 61 The Organization Model . . . . . . 61 The Infinite Universe. . . . 63 Analytical Tools——Sing1e Channel, Infinite Universe . . 64 Analytical Tools——Multiple Channels, Infinite Universe . . 65 Analytical Tools--Finite Universe Model. 67 Conclusion . . . . . . . . . . 67 VIII. SOLUTION OF FUNCTION MODELS . . . . . 69 Assumptions and Procedures . . . . . 69 Resident Sales Function . . . . . . 71 Purchasing Function . . . . 78 Production Supervision Function . . . 79 Monte Carlo Development . . . . . . 82 Conclusion . . . . . . . . . . 88 IX. COMPARISON: ACTUAL ORGANIZATION AND THE MODEL SOLUTIONS. . . . . . . . . 89 The Resident Sales Function. . . . . 89 Purchasing Function . . . . . 91 Production Supervision Function . . . 92 Summary of Comparison. . . . . . . 93 vi Chapter X. CONCLUSION. . . . . Evaluation of Hypotheses APPENDIX . . . . . . . . BIBLIOGRAPHY. Page 95 95 101 166 LIST OF TABLES Table I. In—plant Sales Function Model Solutions II. Purchasing Function Model Solutions III. Production Supervision Function Model Solutions . . . . . . . I viii Page 77 80 82 Figure LIST OF FIGURES Hypothetical Plot of Four Organic Functions: Function Service Demands Versus Firm Size. . . . . . Frequency Plots of Appendix A Data: Function Queue Data of Dearborn Marine Engines, Inc., and Willowby Boat Co. Frequency Plots of Appendix A Data: Function Queue Data of Graymarine Co., Renshaw Boat 00., and Raycraft . . Functional Growth of Dearborn Marine Engines, Inc.: Purchasing Function Functional Growth of Dearborn Marine Engines, Inc.: In—plant Sales Function . . . . . . Functional Growth of Dearborn Marine Engines, Inc.: Production Function Organizational Evolution of Dearborn Marine Engines, Inc. . . . Plot of Organic Functional Activity Versus Production Activity . Comparative Growth Chart: Queuing Theory Model Solutions Versus Actual Organ— ization Growth . . . . . Page 31 A6 “7 50 51 52 53 57 96 LIST OF APPENDICES Appendix A. Primary Data. B. Queuing Theory Model of In-plant Sales Department Based on the Monte Carlo C. Monte Carlo Sales Model Waiting— —Line Summary: Two Employees . D. Monte Carlo Sales Model Waiting— —Line Summary: Three Employees. . E. Data Sheet: Growth of Dearborn Marine Engines, Inc. . . . . . . F. Prediction of Personnel Needs Directly From Size Parameters Versus Prediction Based on Queuing Theory Model Solutions G. Statistical Background H. Monte Carlo-—A Simulation Method I. The Finite Universe Factor Page 101 119 128 138 1A7 15A 159 162 1611 CHAPTER I INTRODUCTION The question of the time at which employees are added to or subtracted from a business organization and the question of the types of skills that these employees have, are fundamental to problems of organization growth. The purpose of this study is to examine a use of queuing theory as a guide to understanding this growth process. The use of queuing theory formulae is examined to deter— mine whether queuing theory analysis is a valid pre— dictive tool when applied to organization growth problems, or whether it instead may be a theoretical explanation of the growth process, or both or neither. The ana- lytical formulae of queuing theory are used in the thesis to find the "size" of organization at which employees for various functions are typically added to an organization in specific, defined situations. To find this size, the thesis defines an organ— ization as being composed of a multiple combination of Job functions. It establishes three parametric names to identify the activity of three job functions that are examined in the thesis. Demands for service are made on the function from within or without the firm, a queue of these service demands develops, and the demands are serviced. The greater the number of employees in a given function and the more efficient their organization, the more rapid will be the servicing and the "better" the performance of this function. This desirable condition must be compared with employee cost considerations. This employee performance cost is non-determinate in this study since no attempt is made to determine the optimum cost of employee activity versus quality of service of the function. Instead, patterns of functional activity typical of suc— cessful firms in the American pleasure boat industry will be found, and there is the presumption that such patterns of successful firms likely represent reasonable adjust— ments of employee cost to function performance. Queuing theory seems possibly well suited to analysis of organizations, if we conceive of (1) an organization's customer, (2) a member of the organization, (3) a function of the organization, or (A) one of the organization's mechanical tools as a unit requiring "service"; and of (1) some other member of the organization, (2) another function of the organization, (3) another useful tool, or (A) a person or group external to the organization as the unit that renders the service. A waiting—line of demands for service may then develop. A waiting—time problem arises when either units requiring service or the facilities that are available for providing service stand idle, i.e., wait. Problems involving waiting-time fall into two different types depending on their structure. The first type of problem involves arrivals which are randomly spaced and/or service time of random duration. This class of problems includes situations requiring either determination of the optimal number of service facilities or the mini- mum arrival rate (or times of arrival), or both. The class of models applicable to the solution of these facility and scheduling problems is called waitingjline theory or (by the British) queuing theory.I A second type of waiting—time problem is concerned with the sequence (or order) in which a series of "servicing centers" provide service to units demanding it, and is called the sequencing problem. This problem is of no concern for thesis purposes. Through use of the systematic tools provided by queuing theory, the study will examine organization growth in both theory and practice. Hypotheses Five hypotheses provide the framework for the investigations. These hypotheses are: 1. That there are regular and consequently pre- dictable patterns in which personnel are added to various functional areas in a marine 1 Charles W. Churchman, Robert L. Ackoff, and Edward T Arnoff, Introduction to Operations Research (New York: John Wiley and Sons, i957), pp. 389—390. organization as it grows in size as measured by certain stated parameters. 2. That these patterns are based on the principle of adding personnel when the queue of demands for service in a functional area reaches a given length. 3. That demands for service in some, at least, of the functional areas arrive in a random manner making queuing theory a suitable tool for analysis. A. That queuing theory provides a good theoretical explanation of the consistency in the historical data as to when personnel are added in certain functional areas. 5. That queuing theory is a practical tool for predicting at what point in a marine organ— ization's growth, as measured by stated parameters, additional personnel will be hired in various functional areas- Framework of the Hypotheses It seems logical to believe that the delay en— countered in the performance of a work-task service by an organization function is a measure of functional and organizational effectiveness. That is, the more speed with which a demand for service is satisfied within a business organization the more effective that organ— ization is. The longer the delay before service begins, the more ineffective is the organization. Balanced with costs, there logically exists an optimum level of ser— vice for business organizations. A supermarket shOpper does not wish to wait excessive time before checking out with his groceries, but neither does he necessarily wish to pay the higher prices for his groceries required to employ sufficient checkout clerks to insure him that he would never have any delay. This condition exists with regard to all functional activities of business organizations. An organization's employee is a ser- vicing center analogous to the checkout counter of the supermarket. Demands are made on him for job performance and these demands are serviced by him. If these demands arrive at random intervals queuing theory may be used to measure this job performance. Through application of queuing theory to this problem we are seeking an understanding of the time at which personnel are added to an organization. Conceptual Definitions Throughout the thesis certain words and expressions will be used. It is necessary that their meaning be clear to the reader. Accordingly, the following defi— nitions are given to aid clarity and preciseness of the presentation. Ratio-Delay Study This expression describes a technique in use by industrial engineers which employs random sampling techniques of statistical methods to develop the work task "elements" of a total job, and the average time required for certain work elements. The technique is generally applicable to a work task that is not regular and time regulated in its functioning. It is used to develop constant and variable job factors. Constant Job Factor This expression denotes the fraction of a total specific work task that results from factors independent of demands caused by the amount of organization activity. It is defined as that fraction of a specific job that must be performed whether the level of activity of the organization is high, low, or none. It is analogous to accounting fixed costs. Variable Job Factor This expression denotes the fraction of a specific work task that results from factors dependent on the amount of activity of an organization. It is defined as the fraction of the job task that varies directly with the level of organizational activity. It is analogous to accounting variable costs. Evolution This word indicates a process of develOpment for a specific purpose or toward a specific goal. M9291 A model is a symbolic description or representation (in this case a mathematical formula or equation) of a work task about which decisions are to be made. A model describes the relationship between variables in a process such as a job.2 We may have (1) work task models, (2) function models, and (3) organization models. With re— gard to the organization, work task models are the elemental model form. Models of organization functions are compounded of function models and/or work task models. Organic Function The thesis will make use of the concept of the organic function. This is necessary to develop parameters for investigation of organization changes. Organic functions are defined as activities within an area of organization endeavor that are so necessary that the activity of the organization will be brought to a stop by a failure to perform them, somehow, somewhere, at 2Robert W. Crawford, ”Operations Research and Its Role in Business Decisions," Planning for Efficient Production (New York: American Management Association, 1953), p. 5. some time, by someone, in the minimum degree required for the satisfactory achievement of primary service needs. Symbolic Definitions This paragraph presents symbols, and their applicable definitions, that will be used in succeeding sections of the thesis. They are given here to aid in the clarity of the presentation of the analytical formulae of queuing theory. D = mean rate of service of demands—for— service by an organic function. A = mean rate of arriving demands-for—service from an organic function. n = number of service demands in a functional queue at a specific time (including the demands being serviced). P(n) = probability of "n" items in a functional queue at a specific time. E(n) = expected mean queue length of "n” size in a functional queue. E(w) = expected time of wait in a functional queue. M = number of functional service channels. N = number of total service demands in the demand for service universe. new state of a functional queue as it IT H 0 changes from n probability of an occurrence at functional queue. CHAPTER II HISTORICAL BACKGROUND AND QUEUING THEORY APPLICATION History The queuing theory has long been used to solve the problem of scheduling working times of telephone operators. It has made it possible to procure better service at a reduced cost by apprOpriately staggering employee working times in the manner indicated by the theory. As a matter of fact, queuing theory had its beginnings in its application to operational problems of telephone systems by the Danish engineer Erlang and others. For many years it remained largely re— stricted to the problems of telephone systems. In these cases the theory was highly successful in optimiz- ing service with frequent reduction in the cost of the service supplied. Since the end of World War II, the activity in many areas of management theory has been focused on in— creasing application of all facets of operations re- search. One of the leading exponents of queuing theory is professor of physics, Philip Morse, of Massachusetts Institute of Technology. In January of 195A Professor lO 11 Morse delivered a speech before the Society for the Advancement of Management.1 In this speech he described the application of operations research techniques to management problems and he also had much to say regarding queuing theory that is of interest to this study. As he shows, the effort in operations research activity is toward creation of a mathematical (quantitative) ”model" of the problem, the mathematical model is a "unifying" influence. A model ties together operational variables and is, therefore, widely used in the physical sciences. Queuing theory is a case in point. A mathematical model based on the theory can aid the problem of understanding any problem where ”units" arrive at an irregular rate at some point for service. The units are serviced and dispatched from this point at a different rate. Morse points out in his speech:2 Unless the mean service rate is greater than the mean arrival rate, a waiting-line of units to be serviced will be formed and will increase in length continually. But even if the mean service rate is larger than the mean arrival rate, the waiting—line is not abolished unless both arrival and service operations are regularized, not random. A central problem of waiting—line theory is to calculate the relationship between the mean length of waiting—line and the degree of randomness of arrival and disposal. On it can be based lPhilip Morse, ”Operations Research, Past, Present, and Future," Advanced Management, November 195A, p. 10. 2 Ibid., p. 10. 12 estimates of the optimum capacity of the servicing facilities when one balances the cost of letting the unit wait in line against the cost of increas— ing the service rate. These calculations are not very difficult if both arrivals and servicing take place in a purely random manner, as in the case of people coming into a store from the street. Having made the calculations we can proceed to apply them to a specific operational problem by first determining whether arrival and service are in fact random Next, one has to compare the cost of having end units waiting in line against the cost of increasing the service rate. If the ratio of arrival to service rates is small there is no great amount of time loss. But if the facility gets popular and arrival rate begins to approach service rate, the waiting-line increases rapidly in length and money must either be spent to in— crease service or effort must be spent to reduce or to schedule arrivals. Administrators not familiar with waiting-line theory can make wrong decisions with serious consequences in such cases. In the case of aircraft stacking over airports in bad weather when landings take time, it was hoped that careful scheduling of aircraft arrivals would materially reduce the line. Computing the effect on the waiting—line of changes of random- ness in arrivals and landings is a difficult mathematical problem. Preliminary analysis indi— cates that if arrivals are completely regular, each in equal time after the next previous, but if servicing is still purely random then the mean queue length is about half that predicted for the case of both arrivals and landings, or partially random in the cases where scheduling of flights and landings is attempted. No matter how hard they try to stick to schedules, some randomness creeps in. To solve this problem, a procedure known as the Monte Carlo was evolved. By use of a table of random numbers and of empirically determined probability distributions for scheduled and non— scheduled arrivals and landings, a whole series of virtual arrivals and landings could be worked out on a high speed computing machine which would have the same statistical properties as actual arrivals and landings. . . . The results do not have the elegance of generality of an analytical solution, but they have the advantage of being numerical answers corresponding to the case of interest. 13 As Morse says, there are two possible ways to solve a particular queuing problem. The first is through use of analytical techniques and mathematical formulae solutions. The second is through use of the Monte Carlo simulation. Through one or a combination of these pro— cedures the mathematical model may be built. Application Based on the idea that job function models can be developed from the analytical formulae of queuing theory, the application of the models to the problem of function growth is based on necessary rules of algebra. The steps in which these model: will be brought to bear on the problem is accomplished in the following way: 1. A suitable measure of the quantity of activity of the function to be studied is to be deter— mined. This measure is taken to be the average number of demands for service per unit time. 2. A suitable measure of the ”size” of the organ— ization to which the function belongs must be selected. This will be taken to be production of engines per unit time. 3. A quantitive relationship between these two parameters is to be found. That is, an algebraic relationship must be develOped. A. The statistical form in which servicing demands made upon the function distribute themselves, 1A in terms of queue length, must be determined (queuing theory requires expotential distri- butions). 5. The queue length mean at which function personnel additions are typically made is to be found through empirical investigation of eight similar successful organizations. 6. The mean rate of demands-for—service that a given function would handle will be found through ratio-delay study. 7. The queue discipline, the universe size and the number of service channels of the function are to be determined. 8. The applicable queuing model for the maximum allowable service demand arrivals at the function when personnel change typically occurs must be solved. 9. This measure of function activity will be related to a specific ”size" of the organ— ization at which function personnel changes normally occur. Thus, level of production per unit time will be compared with number of employees in the specific function. Personnel changes for three functions will be studied in eight firms to find typical industry adjustments of number of employees to average work waiting-line length 15 of the function. A full personnel history of the three functions will be evaluated from one of the eight firms; the history of the (a) production, (b) purchasing, and (0) selling functions for this firm is to be compared with personnel growth patterns develOped by solution of queuing theory models based on typical industry personnel adjustments determined in 17 firm samples. This comparison of actual personnel changes for the three functions with the function model solutions will be used to evaluate the hypotheses. CHAPTER III APPLICATION OF QUEUING THEORY TECHNIQUES TO THE ORGANIC FUNCTIONS Accomplishment of Organizational Purpose This paper is concerned with organization form and with a possible method for investigation of organ- ization growth. This is to be accomplished empirically through: (a) application if the "organic function" concept, (b) application of queuing theory to the concept, and (c) establishment of the scope of the organic function. Let us then conceive of a job or job task which is organic in its make—up and its operation. It is absolutely vital to the continued operation of an organ- ization. Let us further conceive of a person——a member of the organization—-who is primarily responsible for the performance of this job task. Accomplishment of this task may be considered to be a "service" rendered to the organization. There is a disciplinary demand made by the organization on the 16 l7 responsible organization member (or employee) for per— formance of the service. If the service is not rendered, or is unsatisfactorily rendered, the organization must procure a substitute source for this service. Otherwise the organization will——both by definition and in practice —-fail. The "other source" may be achieved by replace— ment of the delinquent employee. An economic decision problem occurs whenever there is a necessary demand for organic service. If arrival of service requirements is in any way irregular, the economic decision is further complicated, for the manager of the organization must make a decision concerning the legel at which he is to provide the service. If he pro— vides too little service capacity (e.g., not enough personnel), the job task will not be completed. This is one extreme. If he provides too much service, the cost to the organization of the service will be eco- nomically untenable; this is the opposite extreme. In short: As the organization grows larger, the problems of an organization for grouping, supervisin , and serving operations become more complex. Despite the complexity of the problem, there logically exists a solution between the two extremes; lLaurence Bethel, Frank Atwater, et el., Essentials of Industrial Management (New York: McGFaw—Hill, 1959), p. 2. 18 if there were none, the organization would cease to exist. Empirically, we know there is a solution. Solution to the problem of service level may be largely left up to eXperience and the "art". Tech— niques of Industrial Engineering have been unsuccessful at establishing predictability procedures for functions in which job services are not performed on a regular, repetitive basis. John Maynard Keynes discussed the "art" in the following way: . investment depended on a sufficient supply of individuals of . . . constructive impulses who embarked on business as a way of life, not really relying on a precise calculation of the prospective profit. The affair was partly a lottery, though, with the ultimate result largely governed by whether the abilities and character of the managers were above or below the average. Some would fail and some would succeed. . . . Businessmen play a mixed game of skill and chance, the average results of which to the players are not known by those who take a hand. Applicability of Queuing Theory If queuing theory is applicable, it may help our understanding of actual cases. Let us examine the problem of an in-plant Resident Sales Representative whose (organic) function is to process sales orders arriving from the field, through to the production 2John M. Keynes, General Theory of Employment, Interest and Money (New York: Harcourt, Brace and Co., 1935), pp- 150-151. 19 activity. These orders arrive by telephone. The in- plant salesman is thus supplying sales service to a group of customers who make random demands for service on him, such as order placing, product information, order follow—up, and so on. The manager must decide the level of sales service i.e., number of in—plant sales representatives to hire, that will be "best” for the organization. Effects of the level chosen may not be felt immediately by the organization; rapid trial and error techniques may not operate. The manager must look elsewhere for planning help and ordinarily may fall back on ordinary, unsupported experience for his answer. When the servicing capacity of this salesman is incapable of keeping up with the demand for sales ser— vice, a queue of unserviced customers will develop. A waiting customer is an iefuifilied customer whose experience with the conpacy he is demanding service of is negative, related in some way to the length of the wait. Empirically we knew that this adverse experience will adversely affect sales. On the other hand, if too many sales persons are hired the idle salesmen's time will be excessive. The effectiveness of the level of sales service )1 may depend on the manager's ability at preticting what L portion of the total customer waiting time should be 2O reduced. A manager, by increasing the number of sales persons or their effectiveness, can avoid loosing some business--but it costs excessively to increase the service level beyond some point. If an organic function is definable and measureable we may draw an analogy between it and the kind of service problem that exists between machines and service for these machines. If the machines may make demands for various kinds 9: service, then the analogy is complete. Compare the sales order-taking problem-just described, and a series of operator-tended automatic lathes. The customer requires various types of service; the lathe also does e.g., loading, unloading, setup, adjusting, etc. This service may be required at random intervals in both cases. If an operator is required to service an increased number of lathes, he may not be able to keep abreast of the machine demands for service as they occur. ”Down time" for some machines may then happen——the machines enter a waiting-line——with reduced productivity of the plant. If a Sales Department person is required to service an average number of calls per time period, and the number of calls, on average, increase, he may not be able to keep abreast of the calls as they occur. Customers that are waiting for service will become the rule. There will be lost sales and a resultant decrease in organizational effectiveness. 21 This same problem exists when materials handling service must be provided in the face of some random demand, such as movement of in-process materials from department to department via fork- lift trucks or overhead cranes. Here the problem is one of determining the optimum number of pieces of equipment to provide. The number of machines to provide in a job shOp production center pre- sents a similar problem when the arrival of orders is essentially random. Selling Service Problem The time required to satisfy service demands in such situations has two parts: (a) the time that the unit or customer is wait— ing for service, and (b) the time that the unit or customer is receiving the service that he requires. The latter time period may be required and unavoidable, but not so the first. If it were possible to determine, on average, how many customers or units would be found in the line waiting for service it would be possible to predetermine the effects of an increase or decrease in service availability to the customer or unit demanding the service. This predictability depends on two factors: (a) the probability that a salesman (or servicing unit) is occupied with an earlier demand. 3Edward Bowman and Robert Fetter, Analysis for Production Management (Homewood, Illinois: R. D. Irwin, 1957), p- 259. 22 (b) the probability that a customer will make a call for, or demand, service. In order to determine these probabilities, and thus be able to predict the results of organizational activities, there must be available information necessary to deter- mine the following factors: (a) What statistical form does the time required for servicing customer demands for service distribute itself in? (b) What statistical form does the time at which customer demands for service distribute itself in? (c) What the status of "queue discipline" is—- that is, are customers serviced on a first- come-first—served basis, or on some other priority basis? (d) The size of the customer population; is the number of customers finite or infinite in number? (e) Does the servicing facility operate in parallel (i.e.,multiple numbers of salesmen taking calls as they occur), in series (i.e., every call passes through each salesman as a stage in the total servicing process), or in a complex combination of the two? 23 If these facts are known, ordinarily we can develop waiting-line information from formulae solution; and barring this we can usually make use of simulation techniques of the Monte Carlo to determine the waiting- line information required. Imbalance between a facility and its need makes queuing situations in many places in a manufacturing company. . . . You can simulate queuing problems on paper either by formula or by Monte Carlo methods A . . . And you can use actual or hypothetical data. Job Factors in Organization Functions Thus it is possible to determine waiting-line data concerning the relationships existing between resident sales—order personnel and the customers of an organ— ization. If the costs are available for the Operation of this sales function, and the effeeee of the customer level of service activity (either real or assumed) are '~ 4-. ~.. ,,—. L 'i . .,. .... C. .1 ,. ‘ T- —. .\ .7.“ g; J. _,... ”1...; ._ .~ ;. 1.1.2- .: .-. ,., known 3 l L, Lilia) L 13*: DUO;- ,_!.,' I g L' ,L I; 1.. '. 11:: i’m,i_.'._'c: --I1L‘:‘ UL," L.- 11“.”.le serv1ce level and BC, by ex;ens1on; predetermine the ‘ w. " 1 , ,... v. .- 1‘ _; .V t. 4.- 1 - opt1mum form oi the Ciganization. Let us assume a small business organization in which no product develcpmeht activity is taking place, or is needed. This assumes away the existence of an Engineering Department or function. Accounting is .1...— . .... -_._ -_..-._......_._ ___. .-—» “Franklin G. Moore, Manufacturing Management (Homewood, Illinois: R. D. Irwin, 1961), p. 552. 2A done on contract. The organic functions of the organ— ization are specified as three: (a) The "raw material" procurement function—— purchasieg. (b) The production supervision function-- production (c) The resident selling function—-sales In addition, let us use the idea that the "service activity" of the organization's members to both those within and without the organization is composed of a fixed or constant component part of the servicing activity, and the complementary idea of a variable component of the job activity. The existence of analogous similarity between the constant and variable components of employee labor time, and the accounting concept of fixed and variable costs, highlights the meaning of this idea. The costs for which the plant——the fixed factor units——are responsible are known as fixed costs, while those arising from the use of the variable factors are known as variable costs. More pre- cisely, fixed costs may be defined as those which are the same in total amount regardless of the volume of the output, even if the latter is zero. Even if a firm produces nothing at all, the fixed costs continue unchanged . . . Variable costs may be defined as those which are eliminated if production is not carried on, and which vary with the rate of output.5 5John Due, Intermediate Economic Analysis (Homewood, Illinois: R. D. Irwin, 1956), pp. 152—15A. 25 The constant job factor has been defined as that portion of a specific job that must be performed whether the level of activity of the organization is high or low or zero. The variable job factor has been defined as the portion of the Job task that varies with the level of activity of the organization. There is probably no such thing as an absolute instance of a constant or a variable job factor. The concept of a constant or variable Job factor may be placed on a polar continuum. However, report writing, "meetings" with other members of the organization, personnel activity, etc., may be unrelated to organ— ization size. In general, the variable job factor of an organic function is the organic portion of that function, and so is the factor of concern for model purposes. The purchasing function has its function peculi— 6 arities. Report writing is a constant factor and procurement of plant services and facilities may be constant. In the small organizations that were 6"Organization and procedure are so closely related it is difficult to tell which determines the other. The procedures stipulate who is to do each of the steps in— volved. These assignments become a part of the job content which is considered in organization analysis.” From William H. Newman and James P. Logan, Business Policies and Management (Cincinnati: Southwestern Pub— lishing Co., 1959), p. 83A. 26 investigated for this study, activity of purchasing generally varied directly with production activity. The production supervision function also has its own constant factor in the form of plant maintenance activities and report making. The level of production supervisory activity seems mostly related to production demands. In the sales service illustration, if pure in— plant sales planning rests with the manager, then resident sales activity may be a 100 per cent variable factor within the total selling function. Statistical job time study analysis as proposed by Professor Harold W. Martin of Rensselaer Polytechnic Institute allows a level of determination of constant and variable Job factors adequate to the needs of waiting-line analysis. Pro- fessor Martin's technique, which he terms, Operations Analysis, is an effective extension and development of the techniques of ratio—delay analysis familiar to the field of Industrial Engineering. He employs the tech— nique of randomly sampling the job activity of a person over time; and so developing a statistical explanation of the job itself. Within the scope of our definition of organic function, it is apparent that managerial activity is vital to the life of the organization and so is ”organic". Demands on a managerial activity for service may be made 27 usually as demands for planning. Since planning activity within an organization is an activity whose primary characteristic is regularity, it will be assumed that the managerial activity is wholly composed of the constant job factor. The activity level of the constant job factor is definitionally determined by conditions independent of the level of organizational activity. In practice there must be a certain component of variable job factor in the managerial function. There are other functions, that may or may not be organic, which have this characteristic also. Product (development) Engineering activities and Research activities—~for example--may be organic. However, the level of activity of this function is determined by considerations largely independent of the organizational level of activity. The job activity is composed almost entirely of a constant job factor. So, the production function provides service, in the organic sense, to the sales function by providing the product on demand that the sales function has required. The purchasing function provides service to the production function in the form of the raw materials necessary to production; this provision is made on demand of the production function. The organic description of the organization matches the definition: 28 An organization basically is a group of indi- viduals who are cooperating to a common end under the guidance of leadership. This, then, is the usual organization form. 7Ralph C. Davis, Fundamentals of Top Management (New York: Harper and Bros., 1951), p. 787. CHAPTER IV RELATING SERVICING OF INDIVIDUAL FUNCTIONS TO A MEASURE OF ORGANIZATION SIZE Adequate framework has been developed now to move into theoretical formulation. It will be useful first, though, to summarize the theoretical position that has been discussed in the first three chapters. First, the formulation of hypotheses was ac— complished, directed at using queuing theory for an examination of the growth processes of organization change. Second, a concept of measuring function activity, was set forth. The measure is able to quantify the function activity within a reasonable confidence level. This chapter is concerned with developing function models for a small company organization structure. The structure of this model may be applicable to organ— izations of any size, however. If the line type of formal organization structure is the fundamental form of organization structures, primary concern will be for this type. Thesis discussion has been leading toward expla— nation of organization growth based on production level. 29 30 Such growth must be related to guality levels of function service as understood in queuing theory application. Organizational Evolution Steps must be taken toward examing the evolution of an organization as it grows from a single person operation. The reasoning that has been outlined will be used to develop an organization model for an evolving organization. The organization model is to be a quanti- tatively structured one. The method (the management consultants) intro— duced consisted of observing what was common in certain executive—type problems and ana- lyzing proposed solutions. It was only natural that efforts should eventually be made to try to find a common structure ('model') in these solutions and the bases on which such structures could be tested. These efforts amounted to the use of science in the study of executive—type problems. Possible functions are shown in graphical form in Figure l. The constant factor is defined as constant over changes in absolute organization size, and re— solves quantitatively to a measurement of the number of demands for service per unit time. The specific quantity is shown by the vertical distance from the abscissa, measured on the ordinate, that the function begins. These functions are assumed to be concave downward due to the economic principle of ”economies lChurchman, Ackoff and Arnoff, op. cit., p. 6. No. of demands for service per unit time No. of demands for service per unit time Figure l-—Hypothetical plots of four organic service demands versus measure 31 MARKETING FUNCTION f(m) I I l I zone | "A" I I I I constand factor I 1 .Production per unit time, or firm size PRODUCTION FUNCTION P(p) zone "A” Production per unit time, or firm size No. of demands for 'service per unit time No. of demands for service per unit time constant factor line ACCOUNTING FUNCTION f(a) zone "A" constant qactor line Production per unit time, or firm size PURCHASING FUNCTION Production per unit time, or firm size functions: functional of firm size. 32 of scale" applied to the variable job factor concept, but this may not be necessarily so. At first glance it would appear that an increase in inputs of all factors by the same proportion would necessarily result in an increase in output by the same proportion. If the total quantities of all factors employed are doubled, it would appear that the output would double as well. Actually this result may not follow, and ap— parently does not, at least over portions of ranges of possible variation. The behavior regarded as typical is indicated by the Principle of Returns to Scale: as a firm increases the quantities of all factors employed, output is likely to rise initially at a more rapid rate than the rate of increase in inputs, but ulti— mately at a less rapid rate. Development of the Organization Model Regardless of the quantity of production activity, and the size of each of the three organic functions, all constant and variable job factors within each of the four functions must be performed. If we know the following factors, we can build a waiting—line model for each of the organic functions under any particular set of condi- tions and so build a theoretical organization to be used for empirical comparisons. The factors are: (a) the mean time required to render a completed service, and (b) the mean number of demands for service, that may be expected in a functional service waiting—line. 2Due, op. cit., p. 1A0. 33 By knowing the statistical distribution of these two factors, the number of employees at any production level can be determined either by use of queuing formula or by use of simulation methods. The only remaining variable concerns the Job task efficiency of personnel performing these organic functions. It is obvious that work efficiency of the employees will vary. The remuneration of the employee might be taken as a measure of employee efficiency. If this were so then factor (b), above, could be modified to suit the index provided by the employee salary or wage. When employee efficiency differences within a function is of concern, a suitable index for modification of factor (a) must be found. CHAPTER V THE RESEARCH PLAN Research Sampling Basis The empirical research that was undertaken in an effort to evaluate the validity of the hypotheses of this thesis, was done in the American power boating industry. Due to the author's prior history of associ— ation with, and in, the industry, the opportunity for increasing the quality and completeness of data which otherwise would have been available was substantially enhanced. It was necessary to concentrate the research effort in one industry in order to negate the possi— bility that data would not be cross transferable from industry to industry. The pleasure boating industry in the United States is composed of three essential specialty fields: (a) the boat hull manufacturer, (b) the engine or propulsive system manufacturer, and (c) the ”hardware” manufacturer, with products such as propellers, cleats, line, steering apparatus, etc. 3A 35 The research was concentrated in firms of types (a) and (b). This was due to the fact that the third field of the industry has more of the operational characteristics of small volume retailers than do the first two. The first two possess, in general, the same organizational structure purpose. Thus, a degree of industry homogeneity was insured in the research. In addition, the boating industry possesses a degree of growth potential that is matched by few in- dustries at this time. This growth potential allows an evaluation of the organizational evolutions of firms in the industry over time periods that are not excessive. Uncontrolled variables may unknowingly get out of hand in a study made over a long period of time. Evidence of dynamic growth is given in the following authori- tative evaluation. The pleasure boating market tripled in size over the decade ending in 1959. This explosive growth brought about far—reaching changes in the character of the market and the distribution structure of the industry, especially over the past few years. In particular, during the years 1959 and 1960, we witnessed an accelerated growth and a diffusion of outlets, many of which were under capitalized and under experienced in the field they were entering. The year 1960 was marked by low profits and disillusionment for many of the newer entrants, and for some of the longer established businesses affected by price compe- tition.l 1Dunn and Bradstreet 1962 Boating Directory (New York: Dunn and Bradstreet, Inc., 1962), p. iv. 36 The purpose of the thesis is, therefore, served by using organizational samples from an industry of this type. Research Methodology2 Three research techniques were used to gather the data necessary to the investigation. The case study method in depth was used to provide detailed data not normally available to the researcher regarding the growth of an actual firm. The case study was made of a firm in the engine—propulsion system field of the industry. Material developed by the study that is pertinent to the thesis is brought together in the chapters that follow. The personal survey technique was also used in the investigation to provide data regarding the queue length acceptable to organic functions of industry organizations. It was necessary to determine whether the concept of an acceptable queue length either explicitly or implicitly existed within those firms; and to find out what the queue length and charactertistics might be. To do this 12 departments in eight firms were surveyed. Finally, the semi-directive interview technique was used to examine reasons and purposes, as they were understood by the functional managers, for organizational 2Pauline Young, Scientific Social Surveys and Re— search (Englewood Cliffs, N. J.: Prentice—Hall, 1956). 37 techniques and devices in use that might affect functional evolution. Persons responsible for the type or timing of organizational changes were inter— viewed concerning those situations of interest to the study in order to seek out unknown variables and evalu— ate the known ones. The Sample Each of the persons responsible for the three functional activities of the company that was eeee studied (Dearborn Marine Engines, Inc.) was interviewed regarding his function. These functions are the same as the three functions described in the last chapter. A total of seven management level persons were contacted for this purpose. It was necessary to use this technique in conjunction with ratio-delay studies for determining values of D, the mean servicing rate for the organic activity of each organization department. In all three departments ratio—delay studies of one week were used to check any subjective opinions of department heads regarding the rate of service activity of their depart- ments. Somewhat surprisingly the mean service rates found by ratio—delay techniques were within 10 per cent of the subjective values suggested by the department heads before the ratio-delay study. 38 Data for the case study, for the years 1955 to 1960, was provided by the records of Dearborn Marine. The following sources were the primary references for development of the data: (a) Resident Sales function—~Sales Department "Day Book" record. (b) Production Supervision function-—Monthly Production Control Summary. (c) Purchasing function——Purchase Order Register. From these sources, the data summarized in Appendix E was obtained which relates directly to that organization's growth. To develOp data regarding the mean length of the functional queue, 10 firms were approached by the author for the purpose of generating such information. Eight firms responded affirmatively, allowing a survey of 12 separate functions; in two firms, two departments were studied. In one firm, three functions were studied. The word "function" is used interchangeably here with "department" because those departments that were studied were carefully checked to insure that functions extraneous to the organic function concept of this thesis were eliminated from the data or were included as part of the constant job factor. Every attempt was made to insure that the expected mean queue length, E(n), met the definitional conditions of an organic function. 39 Publications Research Research in the field of organization theory, and its application to management science practice; and research in the area of practical application of manage- ment science to the boating industry were undertaken in a library study and through investigation of publications available through boating industry trade organizations. The library research was done in the library of the Michigan State University at East Lansing, and in the library of the University of Michigan at Ann Arbor. Some reference use was made of the Detroit, Michigan, Public Library. The library research was primarily directed toward generating information of interest in the field of organization theory. With regard to the boating industry, three industry trade organizations were helpful in avoiding pitfalls in the research, as well as providing aid through infor- mation which they possessed in published or unpublished form. These were the Outboard Boating Club of America located in Chicago, Illinois, the National Association of Engine and Boat Manufacturers of New York City, and the American Boat and Yacht Council of New York. All were helpful in directing the research toward COOperative firms; the Outboard Boating Club was most generous with industry-wide surveys which it had conducted. CHAPTER VI RESEARCH DATA The empirical investigation had two phases, which were designed to provide the two types of data necessary for the evaluation of the hypotheses. The first type of data needed was data supporting or destroying the idea that implicit organic queues do exist in organization functions, that the queue length is determinate, and that the queue is of such form to be useable with queuing theory formulae. The second type of data was concerned with information pertinent to the queuing formulae and the growth of an actual organization. The entire investigation was accomplished in the spirit of Dr. L. J. Hendersen's expressions in Three Lectures on Concrete Sociology. In the complex business of living . . . both theory and practice are necessary conditions of understanding, and the method of Hippocrates is the only method that has ever succeeded widely and generally. The first element of that method is hard, persistent, intelligent, responsible, unremitting labor . . . The second element of that method is accurate observation of things and events, selection, guided by judgment born of familiarity and experience, of the salient and the recurrent phenomena, and their classi- fication and methodical exploitation. The third element of that method is the judicious con- struction of a theory--not a philosophical theory, nor a grand effort of the imagination, nor a quasi-religious dogma, but a modest pedestrian AO A1 affair or perhaps I had better say, a useful walk— ing-stick to help on the way——and the use thereof. All this may be summed up in a word: (we) must have, first, intimate, habitual, intuitive familiarity with things; secondly, systematic knowledge of things; and thirdly, an effective way of thinking about things.1 Determination of Critical Queue Length and Queue Distribution The first step in the investigation was to develop information regarding expected mean queue lengths of organic functions at the time of addition of personnel to the function. The data is summarized in Appendix A. With four exceptions, the data is not listed under the actual company name; some firms did not care to be identified. Four organic functions were investigated in this beginning phase of the investigation, but the accounting function investigation was dropped for lack of an adequate parameter.2 The hypotheses require parametric measures of organic functions; for example that the number of pur— chase orders issued per unit time is a valid measure of the job activity of the purchasing function. Optimum parametric measures in this investigation were developed lFritz Jules Roethlisberger, Management and Morale (Cambridge, Massachusetts: Harvard University Press, 1959), p. 116 2Author's Note: Actual identification of the firms is,of course, available on a restricted basis to the interested reader. 1:2 by trial-and-error as an initial phase of the empirical investigation. "Number of Purchase Orders", and "Funds Spent" per unit time are two that were evaluated, with the elimination of the latter, in the case of the pur- chasing function. "Number of Productive Units Processed" was the only parameter considered for measuring the production function; but the in—plant sales function was also checked by two parameters: the "Number of Calls" made to the Sales Department and the "Gross Dollar Sales" per unit time. The latter was a measure of organic departmental activity, but was eliminated in favor of sales "calls”. Three functions were investigated at Dearborn Marine Engines for queue length, as part of the first phase of the study. There were two full—time, resident sales function employees, two full-time purchasing function employees at the time of the data gathering, and two production supervisors. A three person in-plant sales function was in— vestigated at the Francona Boat Company. The data from this company covered a three week period, which was adequate. Two functional departments were investigated at the Willowby Boat Company-—purchasing and production. The purchasing department had two employees, and the production department was checked at seven and eight supervisors. A3 Purchasing functions were sampled for data at the Renshaw and Graymarine Companies. Johnson and Raycraft Boat Companies provided added data on the in-plant sales function, and Crusader Marine on the production function. This investigation provided waiting-line data in these quantities: (a) Purchasing function--four firms (b) In-plant Sales function—~five firms (c) Production function-—three firms Data was gathered in the following way. The pur— chasing department queue length was taken to be the total number of purchase orders being processed, or waiting to be processed, at the end of days on which no overtime was worked. The in—plant sales function waiting-line data was developed by the company's switch- board operator and/or mail clerk. The switchboard operator, at a random time each day, recorded the number of "sales calls" in progress or waiting to be answered; the random times were supplied to the operator from a table of random numbers. The production function wait- ing-line was measured by the number of shop—order—cards in process or waiting to begin the manufacturing process at the end of the work day. The number of shop order cards in process was essentially constant in the firms sampled. The waiting-line data at the time of personnel additions as shown in Appendix A, is summarized below. AA (a) Purchase orders Dearborn Marine H = A.2 Willowby Boat H = 3.A Graymarine H = 5.A Renshaw Boat H = A.3 (b) Sales calls Dearborn Marine H = 3.72 and 1.73 Crusader Marine E = 1.57 Francona Boat H = 2.19 Johnson Boat E = 3.09 Raycraft H = 2.08 (0) Production orders Dearborn Marine H = 5.12 Willowby Boat 5 = 3.65 and 3.18 Crusader Marine H = 3.90 Three function mean waiting—line lengths were checked immediately 22223 personnel additions were made. One sample check was made for each of the three functions. The before-and—after data is as follows: (a) Purchase orders H before 5 after Graymarine 5.A0 1.1 (b) Sales calls Crusader Marine 1.57 0.11 (c) Production orders Dearborn Marine 5.12 1.9 “5 Based on the above mean waiting—line data, the following mean queue values were developed for use in the models for expected waiting—line lengths, E(n), and will be used in subsequent computations for solution of work task models. The mean of the E values for each function is taken to be the expected queue length, E(n). (a) Purchase Orders E(n) = A.3 (b) In—plant Sales "Calls" E(n) = 2.A (0) Production Orders E(n) = A.0 The next fundamental question is the statistical distribution types that the organic function queues fall into. In Figures 2 and 3 are shown the daily data of Appendix A, that resulted from the empirical investi— gation, arranged in a frequency distribution. In each case it is apparent that the data is distributed in an exponential manner. It is therefore accepted that the organic function queues investigated reasonably fill the Poisson-exponential distribution requirements for proper usage of analytical formulae.3 Since they are 3The derivation of the analytical formulae that will be presented in Chapter VIII requires that the arrivals for demands—for—service and that the time of servicing such a demand must be random in origin and the probability distributions of demand arrivals and functional servicing times be constant over time. The reader is referred to Churchman, Ackoff, and Arnoff, op. cit., for a discussion of this requirement. The requirement means that the demand arrivals will form an exponential distribution (generally based on the Poisson distribution in the case of natural phenomena). The servicing times will be governed by the binomial or Poisson distributions. Appendix H gives a short background to these statistical distributions. Number of Queued Calls 20 l8 16 1A 12 10 8 6 u 2 0 Number of Queued Production Orders A6 DEARBORN MARINE ENGINES IN-PLANT SALES DEPARTMENT Number of Queued _ Prod. Orders x xx —x xx x xx xxxxx xxxxxxxxxxxx WILLOWBY BOAT COMPANY PRO- DUCTION DEPARTMENT Number of Queued Purchase Orders —xx x xx xxx xxxxx xxxxxxx Frequency 22 2O 18 16 12 10 OIU.E<%