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A new model for aggregate output planning
O M E G A . The Int. JI of M g m t Sci.. Vol. 6. No. 3. pp. 267 272 © Pergamon Press Ltd 1978. Printed in Great Britain
0305-114~;~ 7~ 0701-0267S02 01) t)
A New Model for Aggregate Output Planning THOMAS
J HINDELANG
JOHN
L HILL
Drexel University, Philadelphia {Receiled March 1976: in reci~,,d Jbrm January 197,~1
Aggregate production planning has been aided in the recent past by the advent of mathematical programming models and the improvement in computer solution techniques. Such models have only considered single objectives; this paper formulates a multi-objective production planning model as a goal program and discusses the problems involved in its implementation.
INTRODUCTION THE FOCUS of this paper is to formulate an aggregate output planning approach which capitalizes on the strengths of goal programming (GP) in incorporating multiple behavioral and economic considerations into the analysis. The new model will be formulated for a single plant with several departments each having the capability to produce one or more products. The model allows readily for disaggregation in several ways. The decision variables incorporated in the model are the aggregate production rate, the work force level, the aggregate amount of overtime worked, and the total amount of subcontracting. These four controllable factors are the major strategies available to management which enable the adjustment of inventory levels so as to smooth out demand fluctuations and achieve an hierarchy of relevant goals. The next section of the paper surveys the aggregate output planning problem and relevant research to date. The new formulation is then developed, and finally, implementation of the model is discussed. STATEMENT OF THE PROBLEM Aggregate output planning (AOP) is the essence of a manufacturing firm's intermediate production planning phase, having a one to a
twelve-month planning horizon wherein decisions are made relative to the production rate, workforce level, capacity utilization, amount of subcontracting to be undertaken, inventory levels to be maintained, and the procurement of raw materials. AOP concentrates on determining which combination of these decision variables should be utilized in order to optimally adjust to demand fluctuations within the constraints imposed by long-range plans. Without question, the economic, goal-oriented, and behavioral significance of the AOP problem requires careful balance and coordination between robust models and managerial talent in all of the firm's sub-systems which contribute to the decision process. Smooth integration of these subsystems is required in order to achieve effective and efficient performance of the overall firm as it interacts with the environment. Research to date in the AOP area has been rather extensive; surveys can be found in Groff and Muth [3, Chapter 13] and Buffa and Taubert [1, Chapters 5-7]. To date, two models have been suggested which apply GP to the AOP problem setting. The former was by Jaaskelainen [-4] which merely has four goals: one for meeting demand requirements and one for each of the three major costs (stockout cost, costs associated with changing employment levels, and inventory costs) which are relevant in AoP. He does not incorporate any of the
267
268
Hindelang, Hill--A New Model for Aggregate Output Planning
other relevant dimensions of the problem such as: interactions among various departments and sub-systems of the firm; management policies or goals relative to stockouts, frequency of changes in employment levels, and the public image of the firm; and worker motivation, satisfaction, and performance. The more recent GP model was proposed by Goodman [2] who utilized GP as a method of linearizing the quadratic cost terms of the AOP problem and considering higher order cost terms. His formulation also ignores the relevant dimensions of the problem just cited and also does not make use of GP's ability to penalize different cost components on various priority levels in the objective function. The new formulation recommended below extends the above GP models and capitalizes on the following aspects ignored by them. First, the overall cost function is segregated into its three major components--work-force size, production rate, and inventory costs--so that management has additional flexibility in penalizing deviations from the various types of costs depending on the firm's characteristics (e.g., its size, industry, level of mechanization, stability of demand, cost of products, financial viability, etc.) and management's perceptions of tradeoffs among the cost components. Second, behavioral objectives, managerial insights and policy considerations are incorporated. Third, the model optimizes the aggregate production variables as well as determining the optimal product mix given the multidimensional problem setting described above. We now turn to a discussion of the new formulation.
THE GOAL PROGRAMMING FORMULATION The following specific goals are incorporated in the new formulation: 1. To come as close as possible to a specified worker productivity goal; 2. To promote worker motivation through job rotation and labor force stability; 3. To limit the dollar cost associated with work force level to a specified amount or to minimize this cost;
4. To limit the dollar cost associated with production rate and subcontracting to a specified amount or to minimize this cost; 5. To limit the dollar cost associated with inventory levels to a specified amount or to minimize this cost; 6. To achieve a minimum level of total dollar contribution to cover joint fixed costs and earn profits for the firm or to maximize dollar contribution. The rationale supporting each of these goals will be discussed. In addition, the mathematical representation for each goal will be given in the usual GP framework. Readers unfamiliar with the structure of GP constraints are encouraged to consult Lee [5].
Manpower level and productivity goals On a departmental as well as a plant wide basis, the efficient utilization of manpower is an important consideration and criterion for evaluating performance. The following important decision areas will be optimized by the model so as to achieve several goals in management's hierarchy: hiring/firing decisions, rotating workers between departments, and the use of overtime. The following definitional constraint will be needed in several goals described below: Lk.t-1 + (Nlkt -- NDkt) = L~,.
(1)
This recursive relationship shows the labor force size (in number of workers) in department k during period t will equal the labor force size of the previous period plus the net change in workers during period t. Equation (2) quantifies management's goal for the number of productive hours available in each department k during each time period t:
r~ (L,.t-, - ND,,) + T~ Nlk, +
T~ 0,,
+ Di-k, -- D~-k, = Y~ T ~ P~k,.
(2)
Management's goal for the production of each unit of product i in department k is given by the input parameter T~ which multiplied by the number of units to be produced (Pik,--a decision variable in the model) and summed over all products, i, gives a target number of hours to be used in the department. The other
Ome(la, Vol. 6, No. 3
three input parameters Tk~, T 2, and T~ in this goal take learning curve effects into account by attaching a greater productivity coefficient for experienced workers (T~,) than for newly hired or transferred workers (T2); similarly, possible differences in productivity of overtime hours (compared to regular time) are shown by multiplying the coefficient Tk3 by the decision variable, Okt, which gives the number of overtime hours to be worked in department k. It should be noticed that the above goal assumes for simplicity that the learning curve is such that following one period of employment in a particular department, a new worker has attained the same level of proficiency as the more seasoned employees; of course, more complex assumptions could be incorporated into the model but the above illustrates a general two-stage learning effect. The deviational v a r i a b l e s D l k t and D ~ k t show respectively a deficiency and a surplus in available productive hours. Job rotation and labor force stability goals Considerable research cited in [3, Chapters 5 and 16] suggests that employee motivation, performance on the job, and satisfaction derived by workers are all enhanced when job enlargement or job rotation are implemented by the firm and when workers perceive a stable employment environment. The following two goals are incorporated into the model in order to provide at least a surrogate for job rotation and work-force stability. The basic rationale for goal (3) is that management establishes a desired number of workers to be rotated among departments during period t(M,). Of course, discretion must be exercised in setting the level M, so that workers and first-line supervisors do not view such rotation as disruptive rather than beneficial. Now, the number of transfers between departments will equal the smaller of either the total net increases or the total of the net decreases in the work force of all departments because it is assumed that all hiring and firing is done by a centralized personnel office within the plant, i.e., A l t = Min (~'~k Nit,,, Y~k NDkt). Thus, workers not needed in one department will be transferred to another department where they are needed and will be released from the firm only if no department is in need of additional
269
laborers. Similarly, new employees will be hired from the outside work force only if the number of workers transferred out of other departments is insufficient to meet the requirements of departments in need of additional workers. The goal is represented as: A , , + D3, - D~, = M ,
(3)
where D3t and D~t are, respectively, underachieving or overachieving the established goal. The number of workers transferred between all pairs of departments is determined by the above goal, equation (1), the demand for various products, and subsequent goals which determine the optimal product mix and the optimal allocation of resources among departments so as to achieve the profitability goal as well as various cost control goals. In addition to the above goal, workers also perceive greater job security if the firm does not make frequent and significant changes in the overall work force level. Further, the firm may feel that its image in the community and local labor force is enhanced through a conscious effort to maintain work force stability. Due to these considerations, which are relevant even apart from the economic costs involved in changes in aggregate work force level, the following goal is proposed: A2, + O~t - O~, = Qt
(4)
where Qf is the maximum desired fluctuation in aggregate work force level established by management; D,~t and D2, are, respectively, the number of workers less than or in excess of the desired maximum; and Azt = ]Z k N l , t - ~'k NDktk
Due to centralized hiring and firing discussed above, the change in aggregate work force (AEt) is the difference between the net increases and net decreases among all departments. As with goal (3), the present goal enters into the determination of the optimal values of the model's decision variables for the number of workers to be hired or fired in each period based on optimal product mix, departmental requirements, and cost considerations. Cost minimization goals Overall cost minimization is one of the most overriding criteria upon which aggregate production decisions are based. However~ it is not
Hindelang. Hill--A New Model for Aggregate Output Planning
270
the sole criterion in the AOP problem and, more importantly, the various cost components of the overall cost function (i.e., production rate, work force level, and inventory) may be assigned significantly different importance by various firms. Thus, the new model has a distinct cost goal for each of the three major AOP cost components. This segregation provides flexibility to management in penalizing deviations from the various types of costs in that the cost goals can be placed on different priority levels which enables the consideration of a fuller range of tradeoffs among the cost components in model solution and through sensitivity analysis. If the firm desires to consider all costs simultaneously, they can be placed on the same priority level and, even here, they can be given different relative weights to show varying importance among the costs. The cost minimization goals shown below seek to find the optimal values of the decision variables by minimizing all relevant costs (both out-ofpocket and opportunity) according to a desired hierarchy and weighting scheme. The following three cost minimization goals are established: 1. Costs related to changes in work force size. One set of relevant costs in AOP are those related to changing the size of the aggregate work force and inter-departmental transfers of workers. The following goal quantifies these costs: C 1 m3t -+- C 2 mat + C 3 Air + D~] - D~-t = W F C ,
(5)
where WFC, is a target level of dollar cost incurred in period t in changing work force size and transferring workers between departments and D~t and D~-t are respectively a shortfall or excess of dollars spent relative to the target. The input parameters C 1, C 2, and C 3 are respectively the costs involved in hiring, firing and transferring an employee. It should be noted that A3t shows the net number of workers hired and is found:
{ ~ Nlkt -- ~. NDkt if positive A3 t =
k
,~
0 otherwise
On the other hand, A4t is the net number of workers fired in a given period and is equal to:
Aa, = f~ ~ NDkt - ~k Nlkt
(
0 otherwise
if positive
Thus, goal (5) is the major cost vehicle by which the decision variables related to changes in work force levels (Alt thru A,t) are optimized. 2. Costs related to production rate. Several relevant costs must be taken into account in determining the optimal production rate for the firm. These costs are incorporated in the following goal:
~k { ~ (C~tPik, + C~kHik, + C~(SC. + Xm) + C9El,k, + CTk,Ok,+ C~,O,q,}
(6)
+ Dg, - Dg, = PRC, where the necessary cost parameter inputs are the standard variable cost of producing one unit of product i in department k during period t (C4,), the cost on one unit of shrinkage of product i in department k (C~k), the cost of subcontracting out one unit of product i (C6), the cost per overtime hour in department k (CkTt) and the cost per idle-time hour in department k (Ckat). The value PRCt is management's target level for production rate costs and the deviational variables are interpreted in the usual way. The variable H~kt is an input which represents the shrinkage of work-in-process inventory of product i in department k. The remaining values in goal (6) are decision variables which are optimized by the model for each period; i.e., the number of units of product i to be produced in department k (Pikt), the number of units of product i to be subcontracted out to increase finished good inventory (SC,) or the work-in-process inventory in department k (Xikt) and the number of overtime hours to be worked in department k (Okt). 3. Costs related to inventory level. Inventory costs are another important component of total AOP costs and for finished goods include carrying costs, costs of shrinkage, and backorder costs. A target level for this cost is established for each time period t:
E,(C'~ FG. + C~° R,, + C~' BO,,) + D(, - D~, = IC, (7) where the cost parameter inputs required for this goal are the cost incurred for carrying one unit of product i (C9), the shrinkage cost for one unit of product i (C~°), and the cost incurred for one unit of product i backordered per period (C]1); these costs should include
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both out-of-pocket and opportunity costs. IC t is management's target level for inventory cost and the values FGit and BOit are the number of units in finished good inventory of product i and the number of units of product i backordered respectively. These values are computed from the inventory balance equation based on beginning inventory level, demand, production, and subcontracting. The amount of finished goods inventory shrinkage (R~t) can be a constant or a fixed percentage of the inventory level. Of course, there are definite interrelationships among these three cost functions, and between each one and other goals formulated above. However, separating them provides greater flexibility to the firm in directing the focus of the optimization process.
Contribution margin 9oal Besides the cost control goals, departments or overall facilities are often evaluated as profit centers wherein the total amount contributed to cover joint fixed costs and earn profits in each period is established as a target and used for performance evaluation. In addition, in the multi-product setting of this model, the dollar contribution per unit of each product will be used to arrive at the optimal product mix. The following goal incorporates this dimension: E,~(SPit - VCit)Sit + Dst - Da~ = C M ,
(8)
where S P i t is the selling price per unit of product i; VCi, is the standard variable cost per unit associated with the production and sale of product i in period t; S. is the number of units of product i to be sold and shipped to customers in period t; C M t is the dollar contribution goal with the deviational variables being interpreted in the usual manner. The values SPit, VCI,, and C M t a r e input parameters whereas Sit is a decision variable optimized by the model.
Economic constraints Besides the managerial goals and policies described above using soft constraints, there are several relevant economic constraints which define relationships and resource limitations that the firm faces in its AOP process. These latter constraints determine the feasible alternatives available to the firm as it strives to achieve its multiple goals discussed above.
271
The major economic constraints incorporated in the model are: 1. A budget constraint quantifying a strict limitation on cash outflows on a departmental or plant-wide basis; 2. A constraint which shows other limited productive resources required to produce various products; 3. An inventory balance equation for work-inprocess units; 4. An inventory balance equation for all finished products; and 5. Constraints showing upper and/or lower limits on any decision variable in the model. Because these constraints are somewhat common in usage, they will not be specifically formulated here. (See [5].) IMPLEMENTATION AND CONCLUSIONS During the implementation stage, all GP models become unique to the subject firm. Each company has its own hierarchy of objectives and preferences concerning tradeoffs among goals. These different preferences and objectives are reflected in the magnitude of the managerial goals established, the priority level selection as well as the relative weights assigned to deviational variables within those priority levels. The first question to be resolved in order to implement the proposed GP model is the scope or focus which the model is to take. For example, at the facility level, AOP is usually related to a single economic sector; whereas, at the corporate level, fluctuations in demand in a single economic sector are insulated from exposure as a result of the corporation's product mix. The time element of this GP AOP model is of considerable importance during the initial phases of implementation. Model administrators should consider learning curve effects, production lead-time, shrinkage rates, and subcontract procurement lead-time as the primary limiting factors in selection of the proper time
272
Hindelan9, HilI--A New Model for Aggregate Output Planning
period duration. Furthermore, a planning horizon based on factors limited primarily by capital equipment acquisition lead time (the amount of time required to increase facilities capacity) must be selected. However, more horizons tend to be annual corresponding to budgeting and related financial decisions. Another consideration of the model is time references as limited from the short duration side. Since there is a period of time over which all factors of production are fixed, it would be meaningless to design a model with reference less than that period. Frequently multiproduct manufacturing operations utilize intermediate products as common input for the production of many finished goods. During the implementation stage of the model, it will become necessary to identify those stages of manufacture where products become unique (i.e., the split-off points). Continuous process manufacturing concerns in particular must devote considerable effort to this stage. On the other hand, job shops tend typically to manufacture goods which require different sequences of operations. Identification of this sequence for each product is mandatory as a result of the model's accounting for interdepartmental transfers. Certain conversion constants are required in the model. These constants should reflect direct (out-of-pocket), indirect (ramification), and opportunity costs. Indeed great care must be exercised in the accurate evaluation of these factors. Each management team has its own set of corporate or plant objectives in terms of longrun plans, productive capacities, product lines, etc. Further, each manufacturing system operates subject to a cost structure which results
from those plans, capacities, product lines and so forth. As a consequence the model needs to be specifically tailored to reflect these aspects. Certainly the time chain effects of those constraints are obvious to the professional manager. But, the priority levels for the various goals, the quantification of certain among those goals and the interrelationships among those goals warrant the most serious consideration affordable. It is in this area of implementation that the most time must be spent in order for the final model to accurately replicate the reality of the situation. ACKNOWLEDGEMENTS The authors gratefully acknowledge the helpful comments of Dr John F Muth, Indiana University; Dr Gene K Groff, Georgia State University; Dr Michael F Pohlen, University of Delaware; Dr William Berry, Indiana University; Dr Elwood S Buffa, UCLA; and Dr Laurence J Moore, VPI. Of course, any remaining errors of omission or commission are the exclusive responsibility of the authors.
REFERENCES 1. BUFFA ES ,~ TAUBERT WH (1972) Production--Inventory Systems: Planning and Control (Rev. Edn.) Richard D Irwin, Homewood, Illinois. 2. GOODMANDA (1974) Goal programming approach to aggregate planning of production and work force. Mgmt Sci. 20, 1569-1575. 3. GROFF GK & MUTH JF 0972) Operations Management: Analysis for Decisions. Richard D Irwin, Homewood, Illinois. 4. JAAKELAINENV (1969) A goal programming model of aggregate production planning. Swedish J. of Econ. 2, 14-27. 5. LEE SM (1972) Goal Programmingfor Decision Analysis. Auerbach, Philadelphia, Pennsylvania. ADDRESSFOR CORRESPONDENCE: Thomas J Hindelang, Esq, Department of Finance and Statistics, College of Business and Administration, Drexel University, Philadelphia, Pennsylvania 19104, USA.
0305-114~;~ 7~ 0701-0267S02 01) t)
A New Model for Aggregate Output Planning THOMAS
J HINDELANG
JOHN
L HILL
Drexel University, Philadelphia {Receiled March 1976: in reci~,,d Jbrm January 197,~1
Aggregate production planning has been aided in the recent past by the advent of mathematical programming models and the improvement in computer solution techniques. Such models have only considered single objectives; this paper formulates a multi-objective production planning model as a goal program and discusses the problems involved in its implementation.
INTRODUCTION THE FOCUS of this paper is to formulate an aggregate output planning approach which capitalizes on the strengths of goal programming (GP) in incorporating multiple behavioral and economic considerations into the analysis. The new model will be formulated for a single plant with several departments each having the capability to produce one or more products. The model allows readily for disaggregation in several ways. The decision variables incorporated in the model are the aggregate production rate, the work force level, the aggregate amount of overtime worked, and the total amount of subcontracting. These four controllable factors are the major strategies available to management which enable the adjustment of inventory levels so as to smooth out demand fluctuations and achieve an hierarchy of relevant goals. The next section of the paper surveys the aggregate output planning problem and relevant research to date. The new formulation is then developed, and finally, implementation of the model is discussed. STATEMENT OF THE PROBLEM Aggregate output planning (AOP) is the essence of a manufacturing firm's intermediate production planning phase, having a one to a
twelve-month planning horizon wherein decisions are made relative to the production rate, workforce level, capacity utilization, amount of subcontracting to be undertaken, inventory levels to be maintained, and the procurement of raw materials. AOP concentrates on determining which combination of these decision variables should be utilized in order to optimally adjust to demand fluctuations within the constraints imposed by long-range plans. Without question, the economic, goal-oriented, and behavioral significance of the AOP problem requires careful balance and coordination between robust models and managerial talent in all of the firm's sub-systems which contribute to the decision process. Smooth integration of these subsystems is required in order to achieve effective and efficient performance of the overall firm as it interacts with the environment. Research to date in the AOP area has been rather extensive; surveys can be found in Groff and Muth [3, Chapter 13] and Buffa and Taubert [1, Chapters 5-7]. To date, two models have been suggested which apply GP to the AOP problem setting. The former was by Jaaskelainen [-4] which merely has four goals: one for meeting demand requirements and one for each of the three major costs (stockout cost, costs associated with changing employment levels, and inventory costs) which are relevant in AoP. He does not incorporate any of the
267
268
Hindelang, Hill--A New Model for Aggregate Output Planning
other relevant dimensions of the problem such as: interactions among various departments and sub-systems of the firm; management policies or goals relative to stockouts, frequency of changes in employment levels, and the public image of the firm; and worker motivation, satisfaction, and performance. The more recent GP model was proposed by Goodman [2] who utilized GP as a method of linearizing the quadratic cost terms of the AOP problem and considering higher order cost terms. His formulation also ignores the relevant dimensions of the problem just cited and also does not make use of GP's ability to penalize different cost components on various priority levels in the objective function. The new formulation recommended below extends the above GP models and capitalizes on the following aspects ignored by them. First, the overall cost function is segregated into its three major components--work-force size, production rate, and inventory costs--so that management has additional flexibility in penalizing deviations from the various types of costs depending on the firm's characteristics (e.g., its size, industry, level of mechanization, stability of demand, cost of products, financial viability, etc.) and management's perceptions of tradeoffs among the cost components. Second, behavioral objectives, managerial insights and policy considerations are incorporated. Third, the model optimizes the aggregate production variables as well as determining the optimal product mix given the multidimensional problem setting described above. We now turn to a discussion of the new formulation.
THE GOAL PROGRAMMING FORMULATION The following specific goals are incorporated in the new formulation: 1. To come as close as possible to a specified worker productivity goal; 2. To promote worker motivation through job rotation and labor force stability; 3. To limit the dollar cost associated with work force level to a specified amount or to minimize this cost;
4. To limit the dollar cost associated with production rate and subcontracting to a specified amount or to minimize this cost; 5. To limit the dollar cost associated with inventory levels to a specified amount or to minimize this cost; 6. To achieve a minimum level of total dollar contribution to cover joint fixed costs and earn profits for the firm or to maximize dollar contribution. The rationale supporting each of these goals will be discussed. In addition, the mathematical representation for each goal will be given in the usual GP framework. Readers unfamiliar with the structure of GP constraints are encouraged to consult Lee [5].
Manpower level and productivity goals On a departmental as well as a plant wide basis, the efficient utilization of manpower is an important consideration and criterion for evaluating performance. The following important decision areas will be optimized by the model so as to achieve several goals in management's hierarchy: hiring/firing decisions, rotating workers between departments, and the use of overtime. The following definitional constraint will be needed in several goals described below: Lk.t-1 + (Nlkt -- NDkt) = L~,.
(1)
This recursive relationship shows the labor force size (in number of workers) in department k during period t will equal the labor force size of the previous period plus the net change in workers during period t. Equation (2) quantifies management's goal for the number of productive hours available in each department k during each time period t:
r~ (L,.t-, - ND,,) + T~ Nlk, +
T~ 0,,
+ Di-k, -- D~-k, = Y~ T ~ P~k,.
(2)
Management's goal for the production of each unit of product i in department k is given by the input parameter T~ which multiplied by the number of units to be produced (Pik,--a decision variable in the model) and summed over all products, i, gives a target number of hours to be used in the department. The other
Ome(la, Vol. 6, No. 3
three input parameters Tk~, T 2, and T~ in this goal take learning curve effects into account by attaching a greater productivity coefficient for experienced workers (T~,) than for newly hired or transferred workers (T2); similarly, possible differences in productivity of overtime hours (compared to regular time) are shown by multiplying the coefficient Tk3 by the decision variable, Okt, which gives the number of overtime hours to be worked in department k. It should be noticed that the above goal assumes for simplicity that the learning curve is such that following one period of employment in a particular department, a new worker has attained the same level of proficiency as the more seasoned employees; of course, more complex assumptions could be incorporated into the model but the above illustrates a general two-stage learning effect. The deviational v a r i a b l e s D l k t and D ~ k t show respectively a deficiency and a surplus in available productive hours. Job rotation and labor force stability goals Considerable research cited in [3, Chapters 5 and 16] suggests that employee motivation, performance on the job, and satisfaction derived by workers are all enhanced when job enlargement or job rotation are implemented by the firm and when workers perceive a stable employment environment. The following two goals are incorporated into the model in order to provide at least a surrogate for job rotation and work-force stability. The basic rationale for goal (3) is that management establishes a desired number of workers to be rotated among departments during period t(M,). Of course, discretion must be exercised in setting the level M, so that workers and first-line supervisors do not view such rotation as disruptive rather than beneficial. Now, the number of transfers between departments will equal the smaller of either the total net increases or the total of the net decreases in the work force of all departments because it is assumed that all hiring and firing is done by a centralized personnel office within the plant, i.e., A l t = Min (~'~k Nit,,, Y~k NDkt). Thus, workers not needed in one department will be transferred to another department where they are needed and will be released from the firm only if no department is in need of additional
269
laborers. Similarly, new employees will be hired from the outside work force only if the number of workers transferred out of other departments is insufficient to meet the requirements of departments in need of additional workers. The goal is represented as: A , , + D3, - D~, = M ,
(3)
where D3t and D~t are, respectively, underachieving or overachieving the established goal. The number of workers transferred between all pairs of departments is determined by the above goal, equation (1), the demand for various products, and subsequent goals which determine the optimal product mix and the optimal allocation of resources among departments so as to achieve the profitability goal as well as various cost control goals. In addition to the above goal, workers also perceive greater job security if the firm does not make frequent and significant changes in the overall work force level. Further, the firm may feel that its image in the community and local labor force is enhanced through a conscious effort to maintain work force stability. Due to these considerations, which are relevant even apart from the economic costs involved in changes in aggregate work force level, the following goal is proposed: A2, + O~t - O~, = Qt
(4)
where Qf is the maximum desired fluctuation in aggregate work force level established by management; D,~t and D2, are, respectively, the number of workers less than or in excess of the desired maximum; and Azt = ]Z k N l , t - ~'k NDktk
Due to centralized hiring and firing discussed above, the change in aggregate work force (AEt) is the difference between the net increases and net decreases among all departments. As with goal (3), the present goal enters into the determination of the optimal values of the model's decision variables for the number of workers to be hired or fired in each period based on optimal product mix, departmental requirements, and cost considerations. Cost minimization goals Overall cost minimization is one of the most overriding criteria upon which aggregate production decisions are based. However~ it is not
Hindelang. Hill--A New Model for Aggregate Output Planning
270
the sole criterion in the AOP problem and, more importantly, the various cost components of the overall cost function (i.e., production rate, work force level, and inventory) may be assigned significantly different importance by various firms. Thus, the new model has a distinct cost goal for each of the three major AOP cost components. This segregation provides flexibility to management in penalizing deviations from the various types of costs in that the cost goals can be placed on different priority levels which enables the consideration of a fuller range of tradeoffs among the cost components in model solution and through sensitivity analysis. If the firm desires to consider all costs simultaneously, they can be placed on the same priority level and, even here, they can be given different relative weights to show varying importance among the costs. The cost minimization goals shown below seek to find the optimal values of the decision variables by minimizing all relevant costs (both out-ofpocket and opportunity) according to a desired hierarchy and weighting scheme. The following three cost minimization goals are established: 1. Costs related to changes in work force size. One set of relevant costs in AOP are those related to changing the size of the aggregate work force and inter-departmental transfers of workers. The following goal quantifies these costs: C 1 m3t -+- C 2 mat + C 3 Air + D~] - D~-t = W F C ,
(5)
where WFC, is a target level of dollar cost incurred in period t in changing work force size and transferring workers between departments and D~t and D~-t are respectively a shortfall or excess of dollars spent relative to the target. The input parameters C 1, C 2, and C 3 are respectively the costs involved in hiring, firing and transferring an employee. It should be noted that A3t shows the net number of workers hired and is found:
{ ~ Nlkt -- ~. NDkt if positive A3 t =
k
,~
0 otherwise
On the other hand, A4t is the net number of workers fired in a given period and is equal to:
Aa, = f~ ~ NDkt - ~k Nlkt
(
0 otherwise
if positive
Thus, goal (5) is the major cost vehicle by which the decision variables related to changes in work force levels (Alt thru A,t) are optimized. 2. Costs related to production rate. Several relevant costs must be taken into account in determining the optimal production rate for the firm. These costs are incorporated in the following goal:
~k { ~ (C~tPik, + C~kHik, + C~(SC. + Xm) + C9El,k, + CTk,Ok,+ C~,O,q,}
(6)
+ Dg, - Dg, = PRC, where the necessary cost parameter inputs are the standard variable cost of producing one unit of product i in department k during period t (C4,), the cost on one unit of shrinkage of product i in department k (C~k), the cost of subcontracting out one unit of product i (C6), the cost per overtime hour in department k (CkTt) and the cost per idle-time hour in department k (Ckat). The value PRCt is management's target level for production rate costs and the deviational variables are interpreted in the usual way. The variable H~kt is an input which represents the shrinkage of work-in-process inventory of product i in department k. The remaining values in goal (6) are decision variables which are optimized by the model for each period; i.e., the number of units of product i to be produced in department k (Pikt), the number of units of product i to be subcontracted out to increase finished good inventory (SC,) or the work-in-process inventory in department k (Xikt) and the number of overtime hours to be worked in department k (Okt). 3. Costs related to inventory level. Inventory costs are another important component of total AOP costs and for finished goods include carrying costs, costs of shrinkage, and backorder costs. A target level for this cost is established for each time period t:
E,(C'~ FG. + C~° R,, + C~' BO,,) + D(, - D~, = IC, (7) where the cost parameter inputs required for this goal are the cost incurred for carrying one unit of product i (C9), the shrinkage cost for one unit of product i (C~°), and the cost incurred for one unit of product i backordered per period (C]1); these costs should include
Omega, Vol. 6, No. 3
both out-of-pocket and opportunity costs. IC t is management's target level for inventory cost and the values FGit and BOit are the number of units in finished good inventory of product i and the number of units of product i backordered respectively. These values are computed from the inventory balance equation based on beginning inventory level, demand, production, and subcontracting. The amount of finished goods inventory shrinkage (R~t) can be a constant or a fixed percentage of the inventory level. Of course, there are definite interrelationships among these three cost functions, and between each one and other goals formulated above. However, separating them provides greater flexibility to the firm in directing the focus of the optimization process.
Contribution margin 9oal Besides the cost control goals, departments or overall facilities are often evaluated as profit centers wherein the total amount contributed to cover joint fixed costs and earn profits in each period is established as a target and used for performance evaluation. In addition, in the multi-product setting of this model, the dollar contribution per unit of each product will be used to arrive at the optimal product mix. The following goal incorporates this dimension: E,~(SPit - VCit)Sit + Dst - Da~ = C M ,
(8)
where S P i t is the selling price per unit of product i; VCi, is the standard variable cost per unit associated with the production and sale of product i in period t; S. is the number of units of product i to be sold and shipped to customers in period t; C M t is the dollar contribution goal with the deviational variables being interpreted in the usual manner. The values SPit, VCI,, and C M t a r e input parameters whereas Sit is a decision variable optimized by the model.
Economic constraints Besides the managerial goals and policies described above using soft constraints, there are several relevant economic constraints which define relationships and resource limitations that the firm faces in its AOP process. These latter constraints determine the feasible alternatives available to the firm as it strives to achieve its multiple goals discussed above.
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The major economic constraints incorporated in the model are: 1. A budget constraint quantifying a strict limitation on cash outflows on a departmental or plant-wide basis; 2. A constraint which shows other limited productive resources required to produce various products; 3. An inventory balance equation for work-inprocess units; 4. An inventory balance equation for all finished products; and 5. Constraints showing upper and/or lower limits on any decision variable in the model. Because these constraints are somewhat common in usage, they will not be specifically formulated here. (See [5].) IMPLEMENTATION AND CONCLUSIONS During the implementation stage, all GP models become unique to the subject firm. Each company has its own hierarchy of objectives and preferences concerning tradeoffs among goals. These different preferences and objectives are reflected in the magnitude of the managerial goals established, the priority level selection as well as the relative weights assigned to deviational variables within those priority levels. The first question to be resolved in order to implement the proposed GP model is the scope or focus which the model is to take. For example, at the facility level, AOP is usually related to a single economic sector; whereas, at the corporate level, fluctuations in demand in a single economic sector are insulated from exposure as a result of the corporation's product mix. The time element of this GP AOP model is of considerable importance during the initial phases of implementation. Model administrators should consider learning curve effects, production lead-time, shrinkage rates, and subcontract procurement lead-time as the primary limiting factors in selection of the proper time
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Hindelan9, HilI--A New Model for Aggregate Output Planning
period duration. Furthermore, a planning horizon based on factors limited primarily by capital equipment acquisition lead time (the amount of time required to increase facilities capacity) must be selected. However, more horizons tend to be annual corresponding to budgeting and related financial decisions. Another consideration of the model is time references as limited from the short duration side. Since there is a period of time over which all factors of production are fixed, it would be meaningless to design a model with reference less than that period. Frequently multiproduct manufacturing operations utilize intermediate products as common input for the production of many finished goods. During the implementation stage of the model, it will become necessary to identify those stages of manufacture where products become unique (i.e., the split-off points). Continuous process manufacturing concerns in particular must devote considerable effort to this stage. On the other hand, job shops tend typically to manufacture goods which require different sequences of operations. Identification of this sequence for each product is mandatory as a result of the model's accounting for interdepartmental transfers. Certain conversion constants are required in the model. These constants should reflect direct (out-of-pocket), indirect (ramification), and opportunity costs. Indeed great care must be exercised in the accurate evaluation of these factors. Each management team has its own set of corporate or plant objectives in terms of longrun plans, productive capacities, product lines, etc. Further, each manufacturing system operates subject to a cost structure which results
from those plans, capacities, product lines and so forth. As a consequence the model needs to be specifically tailored to reflect these aspects. Certainly the time chain effects of those constraints are obvious to the professional manager. But, the priority levels for the various goals, the quantification of certain among those goals and the interrelationships among those goals warrant the most serious consideration affordable. It is in this area of implementation that the most time must be spent in order for the final model to accurately replicate the reality of the situation. ACKNOWLEDGEMENTS The authors gratefully acknowledge the helpful comments of Dr John F Muth, Indiana University; Dr Gene K Groff, Georgia State University; Dr Michael F Pohlen, University of Delaware; Dr William Berry, Indiana University; Dr Elwood S Buffa, UCLA; and Dr Laurence J Moore, VPI. Of course, any remaining errors of omission or commission are the exclusive responsibility of the authors.
REFERENCES 1. BUFFA ES ,~ TAUBERT WH (1972) Production--Inventory Systems: Planning and Control (Rev. Edn.) Richard D Irwin, Homewood, Illinois. 2. GOODMANDA (1974) Goal programming approach to aggregate planning of production and work force. Mgmt Sci. 20, 1569-1575. 3. GROFF GK & MUTH JF 0972) Operations Management: Analysis for Decisions. Richard D Irwin, Homewood, Illinois. 4. JAAKELAINENV (1969) A goal programming model of aggregate production planning. Swedish J. of Econ. 2, 14-27. 5. LEE SM (1972) Goal Programmingfor Decision Analysis. Auerbach, Philadelphia, Pennsylvania. ADDRESSFOR CORRESPONDENCE: Thomas J Hindelang, Esq, Department of Finance and Statistics, College of Business and Administration, Drexel University, Philadelphia, Pennsylvania 19104, USA.