Modified production switching heuristics for aggregate production planning

Computers Ops Res. Vol. 22, No. 5, pp. 531-541, 1995

Pergamon

o.~m~-o~(94)ooo24-4

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0305-0548/95 $9.50+0.00

M O D I F I E D P R O D U C T I O N S W I T C H I N G HEURISTICS FOR AGGREGATE PRODUCTION PLANNING Sang-jin Namt and Rasaratnam Logendran:~§ Department of Industrial and Manufacturing Engineering, Oregon State University, Corvallis, OR 97331, U.S.A.

(Received May 1993; in revisedform April 1994) Scope and Purl~se--Aggregate production planning (APP) is probably the most critical area of planning in the design of production systems. Although numerous solution techniques for solving the APP problem have been proposed in the past, only a few are implementable in an industrial situation. In this paper, we present two new modified production switching heuristics for solving the APP problem, complete with a series of programs which can be run on a personal computer with relative ease. The advantage of these heuristics when compared with other published work dealing with switching is that they do not do so in the context of a pre-packaged program that might be adapted for use by practitioners in industry. Al~tract--The Production Switching Heuristic (PSH) developed by Mellichamp and Love [Management Science 24, 1242-1251 ] has been suggested as a more realistic, practical and intuitively appealing approach to aggregate production planning (APP). In this paper, the PSH has been modified to present a more sophisticated open grid search procedure for solving the APP problem. The effectiveness of this approach has been demonstrated by determining a near-optimal solution to the classic paint factory problem using a personal computer based application written in THINK PASCAL. The performance of the modified production switching heuristics is then compared in the context of the paint factory problem with results obtained by other prominent APP models including the linear decision rule, parametric production planning, and PSH to conclude that the modified PSH offers a better minimum cost solution than the original PSH model.

1. I N T R O D U C T I O N

Aggregate production planning (APP) involves the simultaneous determination of a company's production, inventory and employment levels over a finite time horizon. Its objective is to minimize the total relevant costs while meeting non-constant, time varying demand, assumed fixed sales and production capacity I-1-3]. Since the early 1950s, approaches for APP have varied from simplistic, graphical methods to more sophisticated optimizing, search, parametric and dynamic methods. These approaches may be classified broadly as those which guarantee a mathematically optimal solution and those that do not [4, 5]. Unfortunately, despite all of the approaches available to managers, APP methods have not had a significant impact on industry operating practices. Several reasons are cited for the lack of assimilation of aggregate planning techniques into management practice I-6,7]. One is that many tSang-jin Nam graduated from Dong-A University in Korea with a Bachelor of Science in industrial engineering. He completed a Master of Science in industrial engineering at Oregon State University where he pursued his academic interests in production planning and scheduling. Currently, he is working in the area of production planning and scheduling for Goldstar Research Center in Seoul, Korea. :~Rasaratnam Logendran is an Associate Professor in the Department of Industrial and Manufacturing Engineering at Oregon State University, Corvallis, Oregon, U.S.A. He previously taught at Southern Illinois University at Edwardsville, Illinois. He received his Bachelor of Science Degree in Mechanical Engineering with honors from the University of Sri Lanka, a Master of Engineering Degree in Industrial Engineering and Management from the Asian Institute of Technology, Thailand, and Ph.D. in Industrial Engineering and Management from Oklahoma State University. Dr Logendran's primary areas of interest are cellular manufacturing, production systems, group scheduling, and applications of operations research and simulation in industrial engineering. He has published articles in such journals as European Journal of Operational Research, Journal of Computers and Operations Research, International Journal

of Production Research, International Journal of Production Economics. International Journal of Systems Science, Journal of Computers and Industrial Engineering and Journal of Logistics and Transportation Review. §To whom all correspondence should be addressed. 531

532

Sang-jin Nam and Rasaratnam Logendran

of these models require simplifying assumptions such as linear or quadratic cost functions which limit their applicability [8-12]. Recent work on the APP problem has focused on aggregation and disaggregation issues [13-15] such as the incorporation of learning curves into linear decision rules [16], the inclusion of multiple products or more complex cost structures [17-19] and of marketing and/or financial variables [20,21]. Near optimal approaches, including search decision rule [5], management coefficients model [6] and parametric production planning [22], overcome some of the problems associated with optimal approaches. Complex cost functions which accurately describe actual costs may be embodied in most near-optimal models. Despite these improvements, however, the near optimal models suffer from a limitation that also applies to optimal models. That is, most of the models produce a different set of values for the decision variables--production rate (P,), work force level (W~), and inventory level (/t)---for each time period (t) in the planning horizon. In fact, these multiple decision variable values may very well be the single most important factor in limiting the application of all aggregate planning models [4, 7, 23]. The Production Switching Heuristic (PSH) was developed by Mellichamp and Love [7] to specifically address the problem of too many decision variable changes over the planning horizon. It was the authors' belief that PSH would thus have more appeal to practicing managers, who tended to favor less production and/or workforce changes over the planning horizon. PSH is based on the observation that managers seem to favor one large change in work force over a series of smaller and more frequent changes during the planning horizon. Thus, as long as demand is being met. i.e. stockouts do not occur too frequently and inventory levels do not increase drastically, managers are often inclined to maintain the same production and work force levels, making minor adjustments when necessary. It has subsequently been noted that a production policy requiring frequent hiring and firing of personnel might be impractical because of prior contract agreements. Frequent hiring and firing over the planning horizon has also been shown to be undesirable due to the potential negative effects on the firm's public image [24].

2. P R O B L E M S T A T E M E N T

Orr [25] and Elmaleh and Eilon [26] proposed a production policy which can only be carried out at discrete levels. The policy was later renamed as the PSH and was applied to a limited set of production problems by Mellichamp and Love [7]. The PSH approach is directed to situations where three-production levels, namely; high, normal and low, can only be changed in discrete increments or decrements, such as adding or removing a production shift. In their approach, Mellichamp and Love [7] also defined switching algorithms to determine the desirable fixed production levels over a planning horizon by analyzing alternative values of the various control parameters. Evaluation of these algorithms provided a set of production, work force, and inventory decisions which were directly related to cost performance. The objective of their production problems, therefore, was to find the best set of the control parameters that minimize cost over the planning horizon [27]. The PSH approach of Mellichamp and Love [7], however, limits grid search options in analyzing all sets of the control parameters. Consequently, their approach was hindered from determining a better solution. This paper modifies the PSH used by Mellichamp and Love [7] and incorporates a more elaborate grid search method which exhausts reasonable incremental values over the entire cost surface. This search method opens all grid options to evaluate a broader set of alternative parameters than the original PSH approach. Two different schemes are proposed as options of the grid search in the alternative approach. Each scheme evaluates the productivity function used in PSH to determine the optimum combination between production and work force sizes. The productivity function developed in PSH is modified in the proposed approach in order to provide a better balance between regular work force and overtime rates than the original PSH formulation. Furthermore, it is demonstrated that this modified productivity function yields better results for reducing regular payroll costs. To evaluate and validate the modified approach, we utilize the paint factory problem which was first described in Holt et al. [28]. Based on the two search schemes, which we refer to as MPSH1 and MPSH2, the paint factory problem is solved using THINK PASCAL software on an IBM PC compatible computer. These results are also compared with those obtained from the Parametric

Modified PSH for APP

533

Production Planning (PPP), PSH and Linear Decision Rule (LDR) reported in Mellichamp and Love [7]. The remaining sections of the paper are organized as follows. The determination of production levels followed by the determination of work force rates are presented in the next two sections. A summary of the program used for performing the elaborate grid search is presented next. The application of the proposed approach to the paint factory problem [28] is presented in the next section. The computational results followed by the analysis of results are then presented. Summary and conclusions are discussed in the last section.

3. DETERMINATION OF PRODUCTION LEVELS The production switching algorithm is used for the assignment of production levels to each planning period using a reasonable control mechanism. MeUichamp and Love 1"7] proposed a PSH using three production levels for period t, H (high), L (low), and N (normal) as follows:

L,

ifFt-It_ 1< L - C ,

Pt=' H, i f F ~ - I t _ I > H - A , N,

otherwise,

where

Pt = production level at period t, Ft = forecasted demand for period t, / t - 1 =ending inventory for period t - 1,

A = minimum acceptable target inventory, and C =maximum acceptable target inventory. The heuristic suggests that if the net production required after taking into account on-hand and target inventories is less than the low level of production, one should produce at the low level. If the net production required is greater than the high level of production, then one should produce at the high level. Finally, if required net production is between the low and high levels of production, one should produce at the normal level. Generally speaking, the PSH, based on the exsiting rationale, may be extended to incorporate more than three levels. With a greater number of levels, the PSH is expected to perform better since its ability to meet fluctuations in demand increases. On the other hand, as the number of levels increases, the frequency of switching and, therefore, the complexity of the production system also increases. The very advantage of a switching heuristic--less frequent rescheduling of production and work force, is lost gradually as more levels are added. This research is, therefore, directed to situations where the three production levels (H, N and L) can only be changed in discrete increments or decrements, such as adding or removing a production shift.

4. DETERMINATION OF WORK FORCE RATES Mellichamp and Love [7] determined the work force rates for a given production level by introducing a productivity function. They provided little or no explanation about this function's operation in determining an appropriate level of regular payroll versus overtime labor. Their productivity function, describing the work force level (Wt) after period t was given as:

Wt =f(P,, G), where G is the percentage increase or decrease in the work force required to achieve high or low levels of production. In order to determine the optimum combination between the production and the work force sizes, the following two schemes, labeled MPSH1 (Modified PSH1) for scheme 1, and MPSH2

534

Sang-jin Nam and Rasaratnam Logendran

(Modified PSH2) for scheme 2, are used in this research. Each of these can be run separately: Scheme 1 Wt = N/k,

= (H/k)*G, = (L/k)*G,

if Pt = N, if P , = H , if Pt = L.

Scheme 2 Wt = N / k , =N/k+(E/k)*G, =N/k-(E/k)*G,

if P, = N, if P , = H , if P,=L,.

where k = the productivity factor such that N / k equals the number of workers necessary to produce N units in regular time without incurring any overtime or undertime, and E = the difference between either the normal and high or low and normal production rates. The factor, G, as an option of grid search, controls the proportion of hiring (firing) and overtime in adjusting the work force size when production is switched from normal to high (low) levels. G ranges from 1 to 0 (decreased in steps of 0.1). For example, when G equals one in both of the schemes, the work force sizes for high and low production levels become H / k and L/k, respectively, resulting in no overtime costs but high hiring and firing costs. Under Scheme 1, some of the values of G result in impossible situations. For example, G=0.0 for high and low regular work force size essentially means that no workers should be employed and further implies that all production is accomplished using overtime labor, which is not possible if no workers are employed. Therefore, only a limited portion of the range of G needs to be evaluated to minimize the cost function. This is very useful in reducing the computer run time required to evaluate the problem using Scheme 1. On the other hand, using Scheme 2 seems more reasonable as there are only small incremental changes to the work force size for each change in G. Thus, the whole range needs to be examined to find the minimum cost. 5. PROGRAM SUMMARY The program seeks to minimize total production costs by looking at the best combinations of inventory, work force and overtime costs. It incorporates five parameters (N, E, B, D and G) using the relationships specified below: H=N+E, L=N-E, A=B-D, C=B+D,

where D = the difference between either the maximum acceptable inventory level (C) and the target inventory level (B) or the minimum acceptable inventory level (A) and the target inventory level. A fundamental assumption made is that productivity per worker is constant. Starting with demand forecasts, an initial production rate (N) is arbitrarily determined by looking at the demand over the time horizon and selecting an initial value for N that falls somewhere between the high and low demand values. An estimation of target inventory level (B) is made by looking at the current level of inventory, I o, and then selecting an inventory level small enough to include the initial inventory level in a grid search. With initial values for N and B the grid search can be iteraled by fixed increments to determine the best values of N and B that minimize the cost function.

M o d i f i e d P S H for A P P

535

The level of production is assigned as either High, Normal or Low, over the planning horizon, by using the appropriate switching algorithms. The differences E and D are also increased by fixed increments after each iteration. The search method is carried out in three steps to reduce the CPU time and to systematically search over the entire cost surface for the minimum cost point. In the first step, using a relatively large incremental value of 20 for the parameter N, the search method rapidly determines the zone for the lowest cost point. In the second step, the search method uses a medium incremental value of five to search a smaller surface around the previous point for a better solution. Finally, the small incremental value of one is used to determine the global minimum point--here "global" does not mean a mathematical optimum but the "best answer" for the entire set of incremental values. Clearly, a smaller incremental value will yield a better solution than that obtained with a larger incremental value. Given the demand forecasts for the next 12 months in the paint factory problem, the MPSH determines the levels of the best control parameters available. These parameters include shift settings, overtime levels, production levels, and inventory levels that minimize aggregate costs. The MPSH is interactive in nature. The firm exercises the option to view the total set of regular and overtime production settings and inventory target levels or any number thereof. Aggregate plans can be determined based on restricted production rate shift settings, with or without overtime, with various forecast series and with varying ranges for inventory targets. Figure 1 illustrates the decision tree and interactive nature of the two MPSH's. It shows the MPSH options compared to the original PSH. Table 1 presents the incremental values of each parameter, the range varied for the parameter, and the number of arrays (combinations) for each iteration. Once the starting conditions are provided, aggregate costs are then calculated for each planning component and are explicitly given.

Evaluate all possibilities of normal production rates (N)? MPSH input options

Modified PSH

Original PSH Fig. 1. Input options for grid search.

Table 1. Grid search for options Control parameter N Normal production level { High (H) production level E Low (L) production level

Incremental value

Range

Array

20, 5, 1

80, 40, 10

5 + 9 + 11

5

90

19

10

80

9

D [ Maximum acceptable inventory level (C) t Minimum acceptable inventory level (A)

5

80

17

G Overtime and regular production

0.1

I

B Target inventory level

Maximum number or evaluations

11 799,425

536

Sang-jin Nam and Rasaratnam Logendran 6. A P P L I C A T I O N - - T H E

PAINT

FACTORY

PROBLEM

In order to demonstrate the modified production switching heuristics (MPSH) described and to evaluate their performance relative to other aggregate production planning approaches, the proposed MPSH is applied to the paint factory problem originally described by Holt et al. [28]. The paint factory problem has been used in the context of introducing most new or modified APP models proposed by various authors to evaluate whether these newer models can perform as well as the LDR. The performance of the proposed methods is evaluated by a solution procedure that minimizes the costs using an LDR-type quadratic cost function.

The paint factory The cost relationships used for the paint factory were: Regular payroll cost (C1,)= 340W, Hiring and layoffs cost (C2t)=64.3(Wt-W,_ 1)2, Overtime cost (C3,)= 0.2(P,- 5.67 W~)2 + 51.2P,- 281 W, Inventory cost (C4t) = 0.0825(•, - 320)2, where Pt = production level during the period t, Wt = work force size during the period t, I, = ending inventory for period t, t = l , 2 . . . . . 12. The objective is to minimize total costs: 4

TC = ~ Cu. i=1

Other information given about the Paint Factory are: Beginning inventory (Io)= 263, Beginning work force (I4/o)--81. The cost of changing the work force (hiring and layoffs) is a squared function of the amount of increase or decrease in the work force. The overtime cost component yields negative overtime costs for certain values of P, and I4/,. Whenever this occurred in the calculations, Cat is set to zero. For this problem, when back orders occur they show up in the results as negative inventory. 7. C O M P U T A T I O N A L

RESULTS

The computer search routines were utilized to determine the best values for the control parameters with both MPSH schemes. With MPSH1, the computer run time was 52 min. With MPSH2, the run time was 92 min. In both cases the problems were written in THINK PASCAL and a MaclI, personal computer was used to perform the runs. The results for the period by period production plans using MPSH1 and MPSH2 as compared to those obtained from the LDR [12] are shown in Table 2. The reader should refer to the article by Mellichamp and Love [7] for the period by period production plans resulting from using PPP [22] and PSH [7] for the paint factory problem. It should, however, be noted that the LDR approach gives an optimal solution to the paint factory problem against which all other approaches are generally compared. If total production is considered between the various techniques, P P P yields the largest annual production with 4735 gallons, followed by LDR with 4700, and the PSH with 4659, while MPSH1 and MPSH2 yield 4590 and 4599, respectively. This production can be compared against the total annual demand of 4589 gallons which shows that MPSH1 and MPSH2 yield the least difference between the amount of total annual production and forecasted demand. This, however, does not imply that MPSH1 and MPSH2 provide better solutions (in terms of cost) when compared with LDR as discussed later in the paper.

Modified PSH for APP

537

Table 2. Aggregate plans with LDR and modified PSH LDR

Period

Forecast

0 1 2 3 4 5 6 7 8 9 10 11 12

-430 447 440 316 397 375 292 458 400 350 284 400

MPSHI

Production Work force (gallons) (people) -467.72 441.32 414.88 379.83 375.28 367.09 358.51 380.57 376.80 366.70 366.59 405.95

81.00 78.63 75.32 72.24 69.55 67.21 66.29 65.66 65.87 66.49 67.68 69.67 72.62

Inventory (gallons) 263.00 292.72 289.08 263.92 328.75 309.03 301.12 369.64 295.21 270.01 283.71 365.30 366.24

Production Work force (gallons) (people) -450 450 450 360 360 360 360 360 360 360 360 360

81.00 71.43 71.43 71.43 63.49 63.49 63.49 63.49 63.49 63.49 63.49 63.49 63.49

MPSH2 Inventory (gallons)

Production (gallons)

Work force (people)

Inventory (gallons)

263 283 286 296 340 303 288 356 258 218 228 304 264

-447 447 447 362 362 362 362 362 362 362 362 362

81.00 72.83 72.83 72.83 63.84 63.84 63.84 63.84 63.84 63.84 63.84 63.84 63.84

263 280 280 287 333 298 285 355 259 221 233 311 273

Table 3. Comparison of costs Cost component

MPSH1

MPSH2

PSH

PPP

LDR

Regular payroll Overtime Hiring/Layoff Inventory

$267,142 15,437 9941 2658

$269,661 13,295 9484 2538

$276,338 13,200 8863 1494

$285,141 7810 3229 1865

$282,642 8518 3514 1362

Total cost Adjusted cost

295,178 $301,2947

294,979 $300,555

299,895 $301,873

$298,405

$296,036

?$301,294 = 295,178 + 340 (102 gallons)/(5.67 gallons/man month).

In the paint factory problem, the inventory cost function is represented by the inventory costs accrued from the difference between the end of the month inventory and a target inventory value of 320 [i.e. C4, = 0.0825(It- 320)2]. This means that the penalty costs of holding inventory will be greatly reduced even when I, is large, due to the multiplying factor (0.0825). This can also be viewed as being equivalent to the per unit holding cost e~ in the evaluation of total inventory costs (IC) which is given by: IC = ei(Pt - Ft + It- 1). The best policy, therefore, is to continuously change the production level which results in a minimum inventory holding cost. As stated previously, the work force level changes at every period for both the optimal LDR approach and the PPP near-optimal approach. The PSH was developed in part to reduce frequent production and work force changes. As presented in Table 2, with the MPSH1 and MPSH2, these changes are further reduced to just two production and work force level changes over the planning horizon. Under the conditions calling for the original development of PSH by Mellichamp and Love [7], this should be even more appealing to practitioners. Table 3 provides a comparison for all approaches with respect to their total variable costs. It should be noted that, as in previous comparisons of APP approaches, the regular payroll costs are considered in analyzing the performance of the approaches. Vergin [29] pointed out that these payroll costs should be treated as essentially fixed or the various models should require some similar ending conditions if any comparative analysis is to be made. It means that, although the total costs of $295,178 for MPSH1 and $294,979 for MPSH2 are less than those of the PSH and those of other models, no direct comparison of the models can be made since the total costs evaluated for MPSH1 and MPSH2 in Table 3 include neither the "relevant" costs nor similar ending conditions. To develop similar ending conditions (i.e. inventories), it is known that MPSH1 and MPSH2 resulted in an ending inventory difference from the optimal LDR balance of 102 gallons (366-264) and 93 gallons (366-273), respectively. In order to make MPSH1 and MPSH2 costs comparable

538

Sang-jin Nam and Rasaratnam Logendran Multiple X - Y p l o t 45

--

x

41

--

~t

O Q C~ v

• + m t3 x

MPSH.MPSHI N MPSH.MPSH1E MPSH.MPSHI B MPSH.MPSItl D MPSH.MPSH1G

\ 37

- -

O

"x X

33 --

x % 'K, %.

29 --I 0

l

l

i

i

l

I

2

4

6

8

10

12

P a r a m e t e r s N, E, B, D, G MPSH.MPStt l Fig. 2. Total costs on five parameters for MPSH1.

we must consider the cost without overtime to make their ending inventories equal to that of the optimal LDR. The regular payroll cost associated with producing 102 and 93 gallons is determined from the cost function C12 = 340W,. This means that, at most, we must incur $6116 [ = 102(340/5.67)] and $5576 [=93(340/5.67)], respectively. These amounts are added to the total costs in Table 3 to arrive at the adjusted MPSH1 and MPSH2 cost figures. Comparing the results based on similar ending conditions we find that the modified PSH's--both MPSH1 and MPSH2, perform better than the original PSH by $579 and $1318, respectively. The total cost of $300,555 ($301,294) obtained with MPSH2 (MPSH1) is only 1.52 (1.77)% greater than the optimum value of $296,036 generated by LDR. The less frequent production changes and total costs should make both MPSH approaches more appealing to practitioners than the PSH approach. 8. A N A L Y S I S

OF RESULTS

This section addresses the analysis performed to determine the sensitity of the various control parameters with regard to the total cost function. To perform the analysis, the parameter whose sensitivity is of interest was fixed at a constant value while the others were varied according to the grid search routine. Thus, for each fixed value of the control parameters N, E, B, D and G, respectively, various combinations of the remaining parameters were evaluated to determine the local minimum associated with that value. Two composite graphs (Fig. 2t for MPSH1 and Fig. 3 for MPSH2) show the variation of the total cost vs each parameter. G, at values less than 0.8, shows a large variation in total costs compared to the other parameters as shown in Fig. 2. It is followed in minimum total cost sensitivity by parameters N and E. On the other hand, the minimum total cost is relatively insensitive to various values of parameters B and D. These results are reasonable since parameters N, E and G

t A s they are composite graphs, the same scale is used on the x-axis for all parameters. However, the true values of the parameters indicated by each of the 11 points on the graphs are: for N, m i n i m u m and m a x i m u m of 320 and 420, with a step size of 10; for E, m i n i m u m and m a x i m u m of 0 and 100, with a step size of 20; for B, m i n i m u m and m a x i m u m of 250 and 350, with a step size of 20; for D, m i n i m u m and m a x i m u m of 0 and 100, with a step size of 20; and for G, m i n i m u m and m a x i m u m of 0 and 1, with a step size of 0.2. O n the y-axis, the total cost is expressed in 10,000's of units.

539

Modified PSH for APP Multiple X - Y plot

31.4 --

\ 30.6

_

~

X.

C)

+ • 13 x

~ ~

g) ,

•

30.2

MPSH.MPSH2E MPSH.MPSH2B MPSH.MPSH2D MPSH.MPSH2G

\

\

u~ O

,"

al

[.-

• MPSH.MPSH2N

"l~"~O~o

31.0

29.8

-

/\

,~+~+--, "~.x,

- ." ...

~

/,

/to

x.., x

x • + .-~." ,-t"

.+

29.4

29.0

-I

I

I

I

I

I

I

0

2

4

6

8

10

12

Parameters N, E, B, D, G MPSH.MPSlt2 Fig. 3. Total costs on five parameters for MPSH2.

are directly related to the total cost function while parameters B and D are related to the production switching algorithm. As can be seen in Fig. 3 the minimum total cost in MPSH2 is most sensitive to various values of parameters N and E followed by G. Accordingly, in MPSH2 each level of G has a small but meaningful impact on the determination of Wt. This accounts for G being more significant in MPSH1 where work force size evaluated at low levels of G can dramatically affect the cost function but less significant in MPSH2 where it only has a small impact at each level of G. In Fig. 3, again the minimum total cost appears to be relatively insensitive to various fixed values of parameters B and D as it was in Fig. 2. A final point should be made with regard to the computer run times experienced in this research. It is important to point out that PSH, P P P and LDR results were obtained using a UNIVAC 1110 computer or its equivalent with run times, in the case of PSH, being in the vicinity of 30 s. In this research, run times of 52 min with MPSH1 and 92 min with MPSH2 were observed on a MaclI Personal Computer (PC). Two factors should be considered in evaluating two or more solution techniques for solving an APP problem. The first is the quality of the solution (total cost) obtained, and the second is the efficiency of the technique itself. Within the context of cost, however, PC time might be argued to be substantially cheaper for managers than time-sharing or other mini or main frame arrangements. The run time difference, between the models is the difference between searching over the entire range of G, in MPSH2, and only searching in the range of G=0.9, in MPSH1. The savings is approx. 40 min of PC time for a total cost solution that is within 1.52% of the optimal LDR solution with MPSH2 and within 1.77% with MPSH1. In light of this, it is clear that when two implementable solution techniques for solving an APP problem are compared, the deciding factor is the quality of the solution and not the efficiency of the technique. While no claim of efficiency of MPSH1 or MPSH2 over PSH can be made as run times on equivlent machines are not available, since the quality of the solution obtained with the modified production switching heuristics is better, both of them are superior to the production switching heuristic.

540

Sang-jin Nam and Rasaratnam Logendran 9. S U M M A R Y A N D C O N C L U S I O N S

The modified production switching heuristic approaches described in this paper offers several clear advantages over the other approaches for solving the APP problem. The principal advantage is that they produce production, work force, and inventory decisions which require a minimum amount of period to period adjustment--a characteristic that should be appealing to practicing managers in industry. In this research, the PSH has been modified with an improved search method, which exhaustively searches over the entire cost surface. These modifications are accomplished using two schemes which are labeled MPSH1 and MPSH2 for convenience. The computational requirements of either scheme are quite large, however, the search procedure is relatively straightforward. The modified PSH, applied to the paint factory problem, has shown that it can improve the total cost performance of the original PSH by Mellichamp and Love [7-1. It also offers benefits in simplicity and flexibility for a minimal sacrifice in cost--around 2% of optimal. This feature may appeal to the decision makers in industries who are not pursing an optimal scheduling policy. Analysis of the model showed the minimum cost function to be sensitive to values of the control parameters directly related to the cost function and insensitive to those only indirectly related. The modified approaches presented in this paper offer practitioners an alternative APP model, complete with a series of programs which can be run on a personal computer with relative ease. This is an advantage over other published work which offers switching heuristics but do not do so in the context of a pre-packaged program that might be adapted for use by practitioners in industry. Finally, further research might follow the lead of other APP investigations that point out that hiring and firing costs generally do not take into account the effects of the skill of the workers being laid off or hired. This investigation may be described as attempting to refine the MPSH models to account for learning curve effects and would entail developing some sophistication in the general cost function representing the cost of hiring or firing workers. Acknowledgements--The authors wish to thank three anonymous referees for their insightful comments which enhanced the clarity of the paper significantly.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

E• S. Bu•a and •. H. Taub•rt• Pr•du•ti•n-•nvent•ry Systems: P•annin• and C•ntr•l. H•mew••d• I••in•is• Irwin ( • 972). E.A. Elsayed and T. O. Boucher, Analysis and Control of Production Systems. Prentice-Hall, Englewood Cliffs, NJ (1994) E. A. Silver, A tutorial on production smoothing and work force balancing. Ops Res. 15, 985-1010 (1967). M. D. ••i• and G. K. Le•ng• A discrete pr•ducti•n switching ru•e f•r aggregate p•anning. Decisi•n Sci. •8• 582-596 ( • 987). W. H. Taubert, A search decision rule for the aggregate scheduling problem. Mgmt Sci. 14, 343-359 (1968). E. H. Bowman, Consistency and optimality in managerial decision making. Mgmt Sci. 9, 310-321 (1963). J. M. Mellichamp and R. M. Love, Production switching heuristics for the aggregate planning problem. Mgrnt Sci. 24, 1242-1251 (1978). F. Hanssmann and S. W. Hess, A linear programming approach to production and employment scheduling. Mgmt Technol. 1, 46-54 (1960). D. A. Goodman, A goal programming approach to aggregate planning of production and work force. Mgmt Sci. 20, 1569-1575 (1974). S. M. Lee and L. J. Moore, A practical approach to production scheduling. Prod. Inventory Mgmt 15, 79-92 (1974). E. H. B•wman• Pr•ducti•n scheduling by the transp•rtati•n meth•d •flinear pr•gramming. •ps Res. 4• •••-• •3 ( • 956). C. C. Holt, F. Modigliani, J. F. Muth and H. A. Simon, Planning Production, Inventories and Work Force. Prentice-Hall, Englewood Cliffs, NJ (1960). S. Axsater, Aggregation of product data for hierarchical production planning. Ops Res. 29, 744-756 (1981). G. R. Bitran and A. C. Hax• •n the design •fhierarchical pr•ducti•n planning systems. Decisi•n Sci. • 28-55 (1977). K. Zoeller, Optimal disaggregation of aggregate production plans. Mgmt Sci. 17, B533-549 (1971). R. J. Ebert, Aggregate planning with learning curve productivity. Mgmt Sci. 23, 171-182 (1976). G.L. Bergstrom and B. E. Smith, Multi-item production planning--an extension of the HMMS rule. Mgmt Sci. 16, B614-629 (1970). E.P. Newson, Multi-item lot size scheduling by heuristic Part 1: with fixed resources. Mgmt Sci. 21, 1186-1193 (1975a) E. •. News•n• Multi-it•m l•t siz• scheduling by heuristic Part 2: with variab•e res•urc•s. M gmt Sci. 2•• l l 94-12•3 ( • 97 5b) W. W. Damon and R. Schramm, A simultaneous decision model for production, marketing and finance. Mgmt Sci. 19, 161-172 (1972). R. A. Leitch, Marketing strategy and the optimal production schedule. Mymt Sci. 20, 302-312 (1974). C. H. Jones, Parametric production planning. Mgmt Sci. 13, 843-866 (1967). M. D. Oliff, H. S. Lewis and R. E. Markland, Aggregate planning in crew-loaded production environments. Comouters Ops Res. 16, 13-25 (1989).

Modified PSH for A P P 24. 25. 26. 27. 28.

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S. Nahmias, Production and Operations Analysis. Irwin, Boston, MA (1989) D. Orr, A random walk production-inventory policy: rationale and implementation. Mgmt Sci. 9, 108-122 (1962). J. Elmaleh and S. Eilon, A new approach to production switching. Prod. Res. 12, 6?3-684 (1974). S. Eilon, Five approaches to aggregate production planning. AIIE Trans. 7, 118-131 (1975). C. C. Holt, F. Modigliani and H. A. Simon, A linear decision rule for production and employment scheduling. Mgmt Sci. 2, 1-30 0955). 29. R.C. Vergin, A new look at production switching heuristics for the aggregate planning problem. Mgrat Sci. 26, 1185-1186 (1980).