hierarchical production planning: comparison of some heuristics

Engineering Costs and Production Economics, 11 (1987) 203-2 14 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

203

HIERARCHICAL PRODUCTION PLANNING: COMPARISON OF SOME HEURISTICS

R.P. Mohanty and R.V. Kulkarni National institute for Training in Industrial Engineering, Bombay 400 087 (India)

ABSTRACT

This paper presents a hierarchical production planning approach as a distinct alternative to the monolithic approach by highlighting the detailed attributes of each approach in both philosophical and functional terms. A hierarchical approach interlinks the various decision making levels in an organization and the solutions are obtainedfor each level depending upon

the characteristics of the decision problem at that level. An attempt is made here to study some heuristic approaches and apply them to a real-life problem situation. It is seen that the proposed heuristic approach performs better than some other available heuristics when the minimization of the number of backorders is considered to be a dominant objective.

1. INTRODUCTION

Contributions towards the development of theoretical models are abundant in the field of Production Planning and Control over the last several years. According to Svardson [ 11, the production planning includes philosophy, strategies and techniques. There also exists today a variety of models and solutions to practical problems in industries. Production systems are concerned with the effective management of systematic flows of materials from the acquisition stage to final production stage. The management of production system requires two types of basic knowledge inputs, namely about hardwares (Production Engi- neering) and softwares (Production Planning and Control). Hardwares and softwares have to be integrated rationally and synthetically, and that synthesis has to cover several organizational echelons forcing a great deal of inter-

facing coordination both vertically and among functional areas. Anthony [ 21 was the first to recognise the multiplicity of decisions in production planning and control. He proposed a framework for classifying the problems in three distinct categories: strategic planning, tactical planning and operations control. Buffa [ 31 has also proposed a generalised description model of production, the significance of which lies in integrating the physical flow within the system with the information flows and decision mak- ers. Bjorke [4] has recently remarked that technological and management systems were traditionally separated although they are simi- larly executed in sequence and highly influence each other. Doumeingts [ 51 has suggested that the general structure of a production management system should specify the basic function of the system, links between the functions, the decision systems and the structure of the physical system.

0167-188X/87/$03.50 0 1987 Elsevier Science Publishers B.V.

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The various approaches and models that have been in use can broadly be classified in two distinct categories: Monolithic planning approaches and Hierarchical planning approaches. Monolithic planning approaches try to view the multiple decision making field of production systems in its totality, although the selection of value goals of all production decision fields and their empirical analyses are not distinct from one another but are closely intertwined. Table 1 presents a summary of the monolithic models. Prominent among them are the Linear decision rule, Search decision rule, Dynamic programming, Management coefli- cient models etc. These monolithic approaches are based on the following rationale: (1) The text of a good production planning

strategy is typically that various analyses find themselves directly agreeing on a set of basic premises and assumptions about the nature of production systems.

(2) Policy formulation is not approached through means-ends analysis, rather first the ends are isolated, and then the means are searched.

Problems arise when such monolithic models are applied to real-life situations. It is generally observed that such models have to be trans- formed into prescriptive and predictive norms under very restricted conditions which are no exact match to realities. Furthermore, when a large-scale production problem is faced involving several thousand products, the data requirements and the computational complexity might become overwhelming and some- times unmanageable. In the recent years, as an alternative to the monolithic approaches, hierarchical planning approach has been proposed. In the subsequent sections we will be discussing such an approach. The purpose of the paper is to enrich the hierarchical planning concepts by way of proposing a heuristic method for solving a production planning problem.

Earlier, several heuristics have been proposed and utilised in solving real-life prob-

lems. The heuristic proposed in this paper will provide a comparison with the other available heuristics.

2. THE HIERARCHICAL PLANNING APPROACH

Early motivation for this approach can be found in the works of Holt et al [ 61 and in Winters [ 71. However Hax and Meal [ 81 have been the pioneers in formalising a hierarchical production planning structure and have also proposed a set of coordinated heuristics for implementation. The monolithic approach, as pointed out earlier considers the total production planning problems as a system defined by a set of structural relationships. These relationships may not be generic at a single level of the production system structure, but usually spill over to or from different levels. The monolithic approach establishes meta-control of the production system by aggregation. The hierarchical approach as opposed to monolithic approach can be defined as the formation of part systems, aspect systems, subsystems or phase systems. The decomposition of the total system essentially amounts to the subset of planning and control actions of the total system. Bertrand and Wortman k9] in their recent book have made it very explicit how to structure the complex production problem. The fundamental concept has been that the interactions between subsystems may be due not only to the production system under control but also to the decision procedures used. The decision procedures by themselves may have to take cognisance of local organizational cir- cumstances. Mohanty and Krishnaswamy [ lo] reveal in their assessment that the hierarchical approach has a conceptual advantage in the sense that an effective image of the response to the same input can be brought out more trans- parently because of explicit considerations of interactions according to managerial requirements than in the conventional monolithic approach. Many authors, such as Hax [ 111,

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TABLE 1

A summary of monolithic models used in production planning

Criteria Linear programming

Linear decision rule

Dynamic Management Search Multi- programming coefficient decision rule objective

programming

1 Number of products

2 Number of periods in the planning horizon

3 Number of production stages

4 Number of objectives for decision making

5 Natureof demand

6 Structure of cost

7 Solutions obtained in

8 Availability of computer package

9 Acceptability to decision maker

10 Situational applicability

11 Involvement of decision maker

12 cost of sensitivity analyses

13 Implementa- bility

Multiple Single

Multiple Multiple

Multiple Single

Single Single .

Deter- ministic Linear

Single run

Yes

Deter- ministic Non-linear

Single run

No

Good Limited

Small to medium production system Little

Small system

Less costly

Little

More costly

Simple Difficult

Single

Multiple

Single

Single


Single run

No

Limited

Small to medium

Little

More costly

Difftcult

Single

Multiple

Single

Single


Single run

No

Limited

Small to medium

Little

Very costly

Difftcult

Multiple

Multiple

Multiple

Single


Single run

Yes

Good

Small to large

Moderate

Very costly

Simple

Multiple

Multiple

Multiple

Multiple

Deter- ministic Linear and Non-linear Multiple runs

No

Good

Small to large

High

Very costly

Difftcult

Golovin [ 121, Ritzman et al [ 131, Dempster et al [ 141, have highlighted the computational advantages, adaptiveness to changing requirements, and easy implementation of the hierarchical approach. Rolstadas [ 151 has pointed out that production planning functions in future years will be more decentralised, which implies the search for a hierarchical management system. According to him, a hierarchical system will be more amenable to the use of new

techniques such as text processing and computer graphics with user-oriented languages at appropriate decision making levels in the organization. Huebner and Hoefer [ 161 have also mentioned that in striving for an overall optimum solution, it is necessary that the production planning expand to embrace the various subsystems through a coordinated optimization approach. They argue that such an approach is necessary because today’s indus-

206

trial production system is exposed to the influence of numerous environmental factors, some of which influence the production system structure directly and other which influence indirectly through their direct influence on other fields interfaced with the production system.

The concept of hierarchical planning is not new. A complex structured system consists of part systems and insight into them determines the insight into complexity [ 171. According to Simon [ 181, the essence lies in ordering the decision layers or levels. Findeison et al [ 191, Himmelblau [ 201, Milkiewies [ 211 and Mohanty et al [ 221 have used such a concept in other types of problem situations. The applicability of such a concept lies in the complex nature of the problem structure and decision making requirements for implementation. According to Haimes [ 231, hierarchy of a system can be described in various ways, in terms such as: temporal, physical, spatial, and functional. These descriptions facilitate analysis and comprehension of the behaviour of the subsystems at a lower level and provide evaluative feedbacks for the higher level. For example, decisions are made in a sequential manner, first making and imposing aggregate decisions. In turn, the detailed decisions provide the feedback to evaluate the quality and viability of the aggregate decision. Each level, in real life, has its own decision making requirements - for instance, the length of decision horizon, the level of detail of the required information, type of manager in charge of executing the’plan and the nature of interactions among subsystems, etc.

An important criterion in designing the hierarchical production planning system has been the product structure. Probably, the complexity of a production system has been recognised to be generic from the configuration of the product. Historically also, the production systems have been classified from this viewpoint [ 241. In fact, in industries today product structure is being recognised for developing

master production schedules, orders information, strategical information like inventory policies, subcontracting procedures, invest- ment in product lines, etc. [ 51. Hax and Meal [ 81 identify three different levels for developing a hierarchical approach. They are as follows: ( 1) Items are the final products to be delivered

to the customers. They represent the highest degree of specificity regarding the man- ufactured products. A given product may generate a large number of items differing in characteristics such as colour, packag- ing, labels, accessories, size, and so on.

(2) Families are group of items which share a common manufacturing set up cost. Eco- nomics of scales are accomplished by jointly replenishing items belonging to the same family.

(3) Types are groups of families whose production quantities are to be determined by an aggregate production plan. Families belonging to type normally have similar costs per unit of production time, and similar seasonal demand patterns.

These three levels hierarchically establish basic product structure in almost all batch- processing industrial situations. Figure 1 gives a conceptual overview of such a system [ 81.

This paper reports a comparative study of some methods used at family disaggregation and item disaggregation levels. Earlier studies con- ducted at these levels seem to be inappropriate for industries having severe capacity restric- tions to reduce the number of backorders. Here, we would like to propose an alternate heuristic, which can be compared with other heuristics: Hax and Meal [ 81, Winters [ 71, Mohanty and Krishnaswamy [ lo], for example.

3. THE METHODOLOGY

From Fig. 1, it can be seen that aggregate production planning is considered to be the highest level planning decision. Here, attempts

207

READ IN LAST PERIODS USAGE

UPDATE DEMAND FORECASTS, St\;;;; STOCKS, OVERSTOCK

, AND RUNOUT TIMES I

I I

DETERMINE EFFECTIVE DEMANDS FOR EACH PRODUCT TYPE

, 1

(1) 1 AGGREGATE PLAN FOR TYPES (AGGREGATE PLANNING REPORTS)

I

(2) FAMILY DISAGGREGATION NANAGENENT

(FAI*ILY PLANNING REPORTS) INTERACTION

I t I

Fig. 1. Conceptual overview of hierarchical planning system.

are made to allocate the total production capacity among product types, and normally the planning horizon considered is at least a full year, or a period which can fully reflect the fluctuating demand requirements. At this level, total cost, involving production, inventory, regular time and overtime costs, is minimised. Set-up costs are generally ignored at this level as they have no real impact in aggregate decisions.

At the family disaggregation level, the aggregate planning decisions of the product types are used as inputs. Due to uncertainty in the planning process, only the results of the first period of the planning horizon are disaggregated, thus the amount of data handling and computational efforts are reduced substantially. The main condition to be satisfied at this level for a coherent disaggregation is that the sum of the productions of the families in a product type equals the amount dictated by the higher level for this type. Thus, the consistency and feasi- bility among the type and family are assured. Furthermore, since family has to share a common set-up, it is at this level that set-up costs are minimised.

Item disaggregation divides the family production quantities among the items belonging to each family. For the current planning period, the problem considers to distribute the production quantity in such a manner that the time between the family set-ups is maximised for future periods so as to.maintain all items in the inventory. In order to save set-ups in the future periods, it seems reasonable to distribute the family run quantity among its items in such a way that the item run-out times coincide with the run-out time of the family. A direct conse- quence which can be realised at this level is that all items of a family will trigger simultane- ously, minimising remnant stock, the remain- ing inventory held in that family.

Winters [ 71 has proposed a model in which families are produced in economic order quantities, in order of their increasing run-out times, until the aggregate total is reached. The family run-out time is considered to be the minimum of item run-out time belonging to that family. Mohanty and Krishnaswamy [ lo] have suggested modification to this approach. They have proposed a simple heuristic which defines modified run-out time as follows:

R ROT --

mod - C . . . . .

where ROT = original run-out time of the family R mod = modified run-out time of the family C = triggering coefficient which is defined

as:

Number of items in the family triggered during the current planning period

Total number of items in the family An item is said to be triggered when its run out time is less than one period.

This heuristic seems to perform better in order to minimise the number of backorders under the restrictive conditions of plant capacity. In case the objecive is to minimise the total cost, Winter’s model performs better. Hax and

208

Meal [ 81 have suggested another heuristic approach which works as follows: (1) Schedule those families in each product

that must be run in the currect planning period in order to meet the item service requirements.

(2) Set the initial family run quantities so as to minimise the cycle inventory and set up costs.

(3) Adjust the family run quantities so as to use all the production time allocated to each product type by an aggregate planning model while observing items overstock limits.

Bitran and Hax [ 25 ] have formalised this heuristic into a convex knapsack problem which minimises the total set-up costs. Equal- isation of run out times is another heuristic suggested by Karmarker [ 261, whose main focus is to equalize the run-out times of the item belonging to each product type and thus skipping the family level disaggregation.

4. THE PROPOSED HEURISTICS

Notations

t = time index i = product type index

j = family index k = item index d = demand d = effective demand x = production quantity allocated to pro-

duct type Y = production quantity allocated to family Z = product quantity allocated to item s = family set-up cost AI = available inventory S‘S = safety stock limit T = number of total time periods in the

planning horizon I = number of products type J = number of families K = number of items

Formulation

From the aggregate planning level, we obtain the optimal production quantity XTr for prod-

_ uct type i= l,..., Iand t= l,..., T. The family-level subproblem can be solved

for each product type i and for each period t.

The problem can be stated as: Find the optimum allocation. yj(jei) for the problem

Min j4 Sj (4 -Yj) (1)

Subject to:

(2) ,’

The item-level sub-problem can be stated as: Find the optimum production quantity zk(kcj) such that; -

Min C (zk - &) ktj .* ,

Subject to: ’

1 ZkSYj W

(4)

& =effective demand for the kth item can be stated as ’ ’

d, = (dk -A& + S&J (5)

The algorithm repeats itself iteratively for every time period, type and family.

; - *

The Algorithm

We demonstrate here the solution procedure for a single time period and for a product type i. “‘. Step 1: Calculate the effective demand at the

item level: ilk= (dk-AIk+S&), fork= l,..., ,k

: .’ where K= total number of items in family j (j= l,..., J); where J=total number of families for product type i) .

Step 2: Calculate the effective demand for the family by considering the effective demand for all the items belonging to

Fig. 2. Product structure.

that family; “=z&, forj= l,...J

Step 3: Solve the family-level subproblem;

Min jf, S’<a’-Yj>

Subject to:

i YjS$,

j=l

Step 4: Solve the item-level subproblem for every familyj, Min (~~--a~) Subject to: zj zk S Yj

Step 5: Calculate the inventory for the period t+l; Alk= tzk-ak)

Step 6: Iterate to the next product type. Step 7: Iterate to the next time period.

5. AN ILLUSTRATIVE EXAMPLE

The product structure is shown in Fig. 2, and the relevant data used for analysis are shown in the appendices.

Table 2 presents the results of the aggregate level. Table 3 gives the comparison between the

TABLE 2

Aggregate level decisions

209

various methods for the family level disaggregation. Tables 4 and 5 present the result of sensitivity analyses with respect to the capacity changes. Tables 6 and 7 present the sensitivity to forecast errors. Table 8 gives the result of sensitivity to the changes in the set-up costs.

Discussion

The study reported above compares the various approaches under similar conditions. The following remarks are considered to be significant:

(1)

(2)

(3)

When set-up cost of a family is comparable to the inventory holding cost and the minimization of the number of backorders is an equally important objective for the management as compared to the total cost, the proposed heuristic is comparable with that of Hax and Meal. Hax and Meal’s model has an advantage over the proposed heuristic in terms of the total cost when the shop faces the tight capacity constraints. The proposed heuristic has advantages when both the tight and loose capacity conditions exist but the management has the dominant objective of minimising the backorders. All the approaches are equally well in terms of minimising the number of backorders when a positive error is found in demand forecasts. However, the proposed heuristic is comparable with Winter’s model in

Period Production quantity Inventory

Type P] Type PZ Type PI Type Pz

Time

Regular Overtime

1 3,660 11,950 0 2,540 2,000 756 2 13,940 9,280 0 0 2,000 1,250 3 17,890 14,875 0 5,375 1,789 0 4 15,610 2,195 0 0 2,000 0

Capacity: 2000 regular hours and 1250 overtime hours.

210

TABLE 3

Family level disaggregation decisions

Cost component Winters

Set-up cost 1,800 Holding cost 9,133 Overtime cost 19,057

Total cost 29,990 Number of backorders 4,313

Capacity: 2000 regular hours and 1250 overtime hours.

Mohanty and Krishnaswamy

1,800 9,243

19,057

30,100 2,560

Hax and Meal

2,160 4,965

19,057

26,181 0

Proposed model

1,800 9,133

19,057

29,990 0

TABLE 4

Aggregate level decisions

Period Production quantity Inventory Time

Type pI Type PZ Typepr Type PZ Regular Overtime

1 3,660 10,700 0 1,290 2,100 406 2 13,940 10,530 0 0 2,100 1,400 3 17,890 14,375 0 4,875 1,789 0 4 15,610 2,895 0 0 2,100 0

Capacity: 2 100 regular hours and 1400 overtime hours.

TABLE 5


Cost component Winters Mohanty and Hax and Meal Proposed Krishnaswamy model

Set-up cost 1,800 1,800 20,404 1,800 Holding cost 7,496 8,712 5,820 7,416 Overtime cost 17,517 17,157 17,157 17,157

Total cost 26,453 27,669 25,107 26,453 Number of backorders 3,549 4,073 340 0

Capacity: 2 100 regular hours and 1400 overtime hours.

(4)

terms of the total cost. Mohanty and Krishnaswamy’s model and Hax and Meal’s model are almost equal in perform- ance under this condition. When there is a negative error in the forecast, the proposed heuristic is far superior to other approaches in terms of minimising the number of backorders, although the total cost function value remains same for

Winter, and Mohanty and Krishnaswamy. Hax and Meal’s model gives a higher total cost.

(5) When set-up cost has been changed for type 2 from Rs 120 = 0 to Rs 500 = 00 (which is very high), the proposed model is superior to all other models and comparable with Hax and Meal so far as total cost and the number of backorders are concerned.

211

TABLE 6


Cost component Winters Mohanty and Krishnaswamy

Hax and Meal Proposed model

Set-up cost 1,800 1,800 1,800 1,800 Holding cost 16,332 14,215 13,398 12,853 Overtime cost 17,157 17,157 17,157 17,157

Total cost 35,289 33,172 32,355 31,810 Number of backorders 0 0 0 0

Forecast error f 10%.

TABLE 7


Cost component Winters

Set-up 2,160 Holding cost 2,146 Overtime 17,157

Total cost 21,463 Number of backorders 12,841

Mohanty and Krishnaswamy

2,160 2,146

17,157

21,463 12,841

Hax and Meal

2,160 2,570

17,157

22,887 11,988

Proposed model

2,160 2,993

17,157

22,310 11,068

Forecast error - 10%.

TABLE 8

Family level disaggregation decisions (sensitivity to changes in set-up costs)

Cost components Winters Mohanty and Krishnaswamy

Hax and Meal Proposed model

Set-up cost* 4,500 6,720 6,720 6,720 Holding cost 9,133 9,243 4,964 4,955 Overtime cost 19,057 19,057 19,057 19,057

Total cost 32,690 35,020 30,741 30,732 Number of backorders 6,423 3,248 0 0

*Family set-up cost for type 1: Rs 90 = 00 Family set-up cost for type 2: Rs = 500 = 00

(6) The distinct advantage of the proposed (7) The models have been compared under model is that it achieves the family level and item level solutions using the same

different conditions and the sensitivity analyses with respect to forecasting errors

algorithm and hence the computational and set-up costs have been carried out complexities and the data requirements are under a rolling horizon basis. The results low in comparison to Hax and Meal. indicate that the aggregation of the plan-

212

ning horizon could be useful in many situations since it can improve the forecasting accuracy in more distant time periods and will facilitate reduction in the computational effort of data processing.

6. CONCLUSION

In this paper, we have aimed at presenting the hierarchical production planning approach as a distinct alternative to monolithic planning in order to gain an insight into the different management levels of the production system. Through this approach, we have the advantage of breaking down the complex product structure into a set of families and items which includes the production and material control. Also, we can simplify the complex production system, reduce dimensionality of the problem at each level, establish interactions between various management levels, and tinally enable each level in the hierarchy to deal with an appropriate type of algorithm depending upon the nature of the problem at that level. In terms of our Table 1, it can be observed that a hierarchical planning approach can satisfy almost all the criteria.

In a real-life production situation, the basic functions as envisaged by managers are planning and monitoring. The planning includes the decision process. Through the decision process, it i’s necessary to decide how much, when and what to produce with the various meas- ures of efficiency and effectiveness. Monitor- ing establishes evaluative feedbacks. These two basic functions cut across several levels in the organization and the product configuration links these levels. Planning and monitoring essentially call for flexible systems that should cover the entire functions without loosing transparency. Monolithic production planning systems are generally centralised software systems. What we need today is the modular and transferable softwares which can facilitate optimization of resources at every level in the organization and can improve parameter con-

trol. Towards these ends, a hierarchical approach seems to match adequately. The research in this approach is still in the embry- onic stage. The purpose of this paper was merely to enrich the literature through the sug- gestion of an alternate heuristic approach with a limited objective in mind such as to minimise the backorders in a batch processing industrial environment. However, hierarchical approach is to be recognised not merely as a tool for production planning but as a philosophy to depict an adaptive, iterative and coordinated multi-functional decision process.

ACKNOWLEDGEMENTS

The authors express gratitude to the esteemed reviewers as well as to Prof. Grub- striim for their helpful comments and suggestions.

REFERENCES

Svardson, B., 198 1. The ABC-view on production planning. Proc. IFIP TC-5 Int. Workshop on Automation of Production Planning and Control. Trondheim, Norway, 3-5 September, pp. 2 1 l-2 19. Anthony, R.N., 1965. Planning and control systems: A framework for analysis. Grad. School Bus. Admin., Har- vard Univ., Boston, MA. Buffa, E., 1973. Modem Production Management, 4th edn. Wiley, New York, p. 34. Bjorke, O., 1980. Interconnection between technology and management systems. Proc. IFIP TC-5 Int. Work- shop on .Automation of Production Planning and Con- trol, Trondheim, Norway, 3-5 September, pp. 15-34. Doumeingts, G., 1984. State of the art in production management systems in France and U.S.A. In: G. Dou- meingts and W.A. Carter (Eds.), Advances in Produc- tion Management Systems. Elsevier Science Publishers B.V., Amstedam, pp. l-20. Holt, C.C., Modigliani, F., Muth, J.F. and Simon, H.A., 1960. Planning Production, Inventories and Workforce. Prentice-Hall, Englewood Cliffs, NJ. Winters, P.R., 1962. Constrained inventory rules for production smoothing. Manage. Sci., 8 (4): 470-48 1. Hax, A.C. and Meal, H.C., 1975. Hierarchical integra- tion of production planning and scheduling. Manage. Sci. 1. In: M.A. Geisler (Ed.), Logistics. North Hol- land-American Elsevier. Bertrand, J.W.M. and Worstman, J.C., 198 1. Produc- tion Control and Information Systems for Component Manufacturing Shops. Elsevier, Amsterdam.

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Mohanty, R.P. and Krishnaswamy, M.V.R., 1984. An assessment of some approaches to hierarchical production planning. Int. J. Oper. Prod. Manage. 4 (1): 48-60. Hax, A.C. and Golovin, J.J., 1978. Hierarchical production planning. In: Hax A.C. (Ed.), Operations Manage- ment. Elsevier, Amstedam. Golovin, J.J., 1975. Hierarchical integrationofplanning and control. HIT, O.R. Centre, Tech. Rep., p. 116. Ritzman, L.P., Krajewski, L.J., Berry, W.L., Goodman, S.H., Hardy, ST. and Vitt, W.D., 1979. Disaggregation Problems in Manufacturing and Service Organizations. Martinus Nijhoff, Boston. Dempster, M.A.H. Fisher, M.L., Jansen, L., Lageweg, B.J., Lenstra, J.K. and Rinooy Kan, A.H.G., 1981. Ana- lytical evaluation of hierarchical planning systems. Oper. Res. 29 (4): 707-716. Rolstadas, A., 1984. State of the art on production management. In: G. Doumeingts and W.A. Carter (Eds.), Advances in Production Management Systems. Else- vier, Amsterdam, pp. 37-47. Huebner, H. and Hoefer, H., 1984. Strategy oriented production management. In: G. Doumeingts and W.A. Carter (Eds.), Advances in Production Management Systems. Elsevier, Amsterdam, pp. 5 l-7 1. Mohanty, R.P. and Kulkami, R.V., 1986. Multilocation plant sizing and timing: A study of some decomposition procedures. J. Comput. Ind. Eng., 10 (3): 149-161. Simon, H.A., 1962. The architecture.of complexity. Proc. Amer. Philosophical Sot., 106: 467-482.

APPENDIXES

A. Cost data

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24

25

26

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Findeison, W., Bailey, F.N., Bady, M., Malinowski, K., Tatjewski, P. and Wozniak, A., 1980. Control and Coor- dination in Hierarchical Systems, II ASA. Int. Ser. Appl. Syst. Anal., John Wiley, N.Y. Himmelblau, D.M., 1975. Decomposition of Large Scale Problems. North Holland, Amsterdam. Milkiewies, P., 1977. Multi-horizon-multi-level opera- tive production and maintenance control. IFAC-IFORS -11 ASA Workshop on Syst. Anal. Application to Com- plex Programmes, Biola, Poland. Mohanty, R.P., Sahu, K.C. and Patnaik, N., 1975. Hier- archical system formulation for socio-economic problems. Proc. Indo-British Conf. Eng. Prod., New Delhi. Haimes, Y.Y., 1977. Hierarchical Analysis of Water Resources Systems. McGraw-Hill, New York. Bensoussan, A., Crouby, M. and Proth, J.M., 1983. Mathematical Theory of Production Planning. North Holland, Amsterdam. Bitran, G.R. and Hax, A.C., 1977. On the design of hierarchical production planning systems. Decision Sci., 8: 28-55. Karmarker, U.S., 1981. Equilisation of run-out times. Oper. Res., 29 (5): 757-762.

(Received May 29, 1985; accepted in revised form Decem- ber 1, 1986)

Product type p1 Product type p2

Family set-up cost 90 120 Holding cost 0.3 1 /unit/period 0.40/unit/period Overtime cost 9.5 lhr 9.5 /hr Unit time requirement 0.1 /hr/unit 0.2 /hr/unit Production lead time period 1 period

B: The demand forecasts

Period Forecast demand

Type p1 Type p2

12,660 12,410 15,440 12,020 17,890 9,500 15,610 7,500

214

C: Family Run Quantity

Family Economic order quantity

Proportion of total type demand

P#‘I 4,78 1 0.6 P#‘2 2,972 0.4 P2F1 4,4004 0.4 P2F2 4,897 0.3 P2F3 6,003 0.5

D: Inventory of items

Item Initial inventory

Proportion of family demand

PI F, II 2,27 1 0.40 P1F112 2,654 0.25 P,F,IJ 3,128 0.35 PI F21, 1,825 0.90 PI F212 935 0.10 PzF111 1,390 0.50 PzF, I2 224 0.50 P2F211 227 0.70 P2F212 1,324 0.30 f’2F3b 435 0.80 P2F312 156 0.20

hierarchical production planning: comparison of some heuristics

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