Comparing two computer search models for aggregate production planning

Meij, J. T. Article Comparing two computer search models for aggregate production planning South African Journal of Business Management Provided in Cooperation with: University of Stellenbosch Business School (USB), Bellville, South Africa Suggested Citation: Meij, J. T. (1982) : Comparing two computer search models for aggregate production planning, South African Journal of Business Management, ISSN 2078-5976, African Online Scientific Information Systems (AOSIS), Cape Town, Vol. 13, Iss. 2, pp. 67-69, https://doi.org/10.4102/sajbm.v13i2.1175 This Version is available at: https://hdl.handle.net/10419/217795 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Comparing two computer search models for aggregate production planning J.T. Meij Department of Industrial Engineering, University of Stellenbosch In this paper a comparison is made between the results and the cost-effectiveness of two computer search models for aggregate production planning when applied to a very sensitive high-order cost structure. The Search Decision Rule (SOR) model developed by Taubert outperforms the Sectioning Search Model (SECT) of Goodman in both the areas of total optimum cost. and cost-effectiveness. S. Afr. J. Bus. Mgmt. 1982, 13: 67-69 In hierdie artikel word 'n vergelyking getref tussen die resultate en die koste-doeltreffendheid van twee rekenaarsoekmodelle vir geheelskedule-produksiebeplanning, soos toegepas op 'n baie sensitiewe hoe-orde kostestruktuur. Die 'Search Decision Rule'-model (SOR) ontwikkel deur Taubert lewer in albei areas, naamlik totale optimale koste en kostedoeltreffendheid, beter resultate as die 'Sectioning Search Model' (SECT) van Goodman. S.-Afr. Tydskr. Bedryfsl. 1982, 13: 6769 Second in a series of three articles Prof. J.T. Meij Head of Industrial Engineering, University of Stellenbosch, Stellenbosch 7600, Republic of South Africa Received October 1981; accepted November 1981 Introduction Many production managers are faced with the problem of planning production, inventory and work force under the constraint of limited recources to meet a seasonal demand. In those cases where linearity of the cost functions of an undertaking may reasonably be assumed, an ordinary linear programming model suffices. In many cases, however, this simple linear approach to certain essentially non-linear cost functions is unacceptable owing to the gross approximation made. Considerable research has been done on this planning problem and various models have been proposed. These models can be divided into three broad categories, namely heuristic models, mathematical optimization models and computer search models. In this paper a comparison is made between the results of two of the published computer search models on a high-order cost function. One of the following four basic strategies can be followed to meet the fluctuations in demand. 1. Work-force level and production rate are kept constant and inventory is used to absorb fluctuations in demand. 2. Work-force level and inventory are kept constant and demand fluctuations are handled by changing the production rate, i.e. working overtime or allowing idle time. 3. Production rate and inventory are kept constant and the work force is varied to suit the demand. 4. A combination of the three strategies given above. In most cases in industry the combination type of strategy (4) is usually the most appropriate. The extent to which the different strategies should be mixed to present an overall plan is dependent on the cost structure of the particular industry. Cost structures vary, and may have anything from linear or almost linear, to highly non-linear relationships. In many cases ordinary linear or piecewise linear functions may be adequate to describe the relationship between cost and one of the above-mentioned variables. On the other hand it may well be that for certain costs a linear approach is unrealistic and far removed from the real world situation. In the latter case it becomes extremely difficult to obtain a proven optimum solution. Various methods have been suggested to solve this problem. For a solution method to be practical it must comply 68 with the following primary properties: It must be cost effective. It must assure, with reasonable confidence, that a global optimum will be reached. It must be universally applicable. With the development of the high-speed digital computer, computer search methods have been developed and implemented to comply, in the field of aggregate production planning, with these properties. Taubert I compared various search algorithms and found the HookeJeeves algorithm2 particularly suitable for the solution of high-order functions. He made use of this algorithm in the development of his computer search system, Search Decision Rule (SOR). Goodman3 applied a modified Sectioning Search Model (SECT) to a high-order cost f unction. The author applied the SOR model to the high-order cost function used by Goodman and compared the results with those of the SECT. Description of the cost structure In order to test the Sectioning Search Model, Goodman developed a fourth-order cost model. The real world costs, of which this model is an approximation, are given in Table 1. The cost components considered in this model are: direct payroll, overtime and idle time, hiring and layoff, change of production rate and inventory holding and shortages. The objective cost function to be minimized is: C = 1 [340 W, + 0,2 (P, - 6W,) 4 + 64(W1 -W1 _.>4 + 0,1 (P, - P,_.)4 + 0, 1 (320 - 1,)1 Where W, is the work force in period t, P, is the production in period t, and /1 is the inventory in period t; subject to the following constraints: /1 = /1_ 1 + P, -D, 0 '- W,< 150 (t = 1 to N) (t = 1 to N) S.-Mr. I yJ,kr. Bcdryfsl. 1982, IJ(i) (t = 1 to N) where D, is the demand in period t. Work force and production quantity in each period are the independent variables. From these variables, as well as the given demand (D,), the other variable contributing to the cost, that is inventory, is calculated. Results In Table 2 the monthly production plans and corresponding costs given by SOR and SECT are compared for a 24-month planning horizon. SOR gave an improvement of nearly 6% on the total cost of $14 196 488 obtained by Goodman's SECT. It can also be seen that the cost model is very sensitive to small changes in any of the variables - compare for example the monthly costs for months 3 (SECT 53"7o higher than SOR), 4 (SECT 52% higher than SOR) and 8 (SECT 52% higher than SOR). There are no major differences between the two plans. From a practical point of view any one of the two plans could be adopted. It must thus be emphasized that for highlysensitive cost structures as the one used here, extreme care must be taken in the choice of an optimization method. To measure the cost-efficiency of the two search techniques, the computer time required per decision (independent variable) is compared. It should be kept in mind, however, that as Goodman states, ' ... computer time usage is a function of both the computer used and programming efficiency and method used'. 3 He states that on average the Sectioning Search Model uses 0, 75 s per decision. It was found that SOR used only 0,38 s per decision on a UNIV AC 1110 computer. By decreasing the number of search repetitions of the SOR procedure, a plan was obtained using only 0,24 s per decision (a 680/o saving in computer time). The total cost of this plan was only 0, 14% higher than the results previously obtained. Conclusion In this paper a comparison is made between the results of two well-known search models developed for aggregate production planning. For comparison purposes a highorder cost model has been used. The SOR-model of Taubert outperforms the SECT-model of Goodman when compared on the basis of total optimum cost and cost-effectiveness. Table 1 Real world cost on which the cost model is based (Rand) 1w,-w 1_11 Cost IP,-P,_,, Cost II, - 3201 Cost IP,-6W,I Cost 0 0 1 I I I 0 0 I 66 2 2 2 2 I 2 1001 4 24 3 9 2 4 3 5210 s 68 4 28 3 14 4 20100 7 22S 6 122 s 131 s 38120 10 1049 8 392 7 457 7 86300 16 6310 11 1370 1876 10 9 139200 22 26100 IS 5417 12 3780 12 224400 34 123400 21 18240 14 7795 14 279600 S2 487200 39 231200 18 20600 19 401100 87 1140000 SI 474400 22 34900 2S 698700 ISO 2224000 70 762500 30 58200 S. Afr. J. Bus. Mgmt 1982, 13(2) 69 Table 2 Production plans, Search Decision Rule (SOR) and Sectioning Search (SECT) Work force Production Inventory Period cost Period Demand SDR SECT SDR SECT SDR SECT SDR SECT (Rand) (Rand) I 430 73 75 447 431 337 301 36205 160572 2 447 70 72 431 440 321 294 36187 76836 3 440 67 69 406 426 287 280 182702 392632 4 316 64 65 376 392 347 356 163305 340082 5 397 62 62 362 374 312 333 29178 39620 6 375 60 59 352 348 289 306 109783 75042 7 292 62 60 364 348 361 362 305626 335780 8 458 64 63 395 386 298 290 151989 316936 9 400 63 64 383 391 281 281 244834 253710 10 350 61 61 353 355 284 286 273730 330447 II 284 63 63 359 356 359 358 278352 277808 12 400 68 68 399 399 358 357 532877 593728 13 483 73 73 442 444 317 318 414825 475143 14 509 78 78 477 481 285 290 362126 340645 15 500 83 83 491 488 276 278 432210 381629 16 475 88 88 510 508 311 311 103618 118576 17 500 94 94 553 552 364 363 853346 835740 18 600 101 IOI 608 607 372 370 1752367 1728065 19 700 107 107 662 662 334 332 1090724 1068460 20 700 112 112 698 699 332 331 315675 373248 21 725 107 107 658 659 265 265 1273906 1264146 22 600 IOI IOI 600 600 265 265 2171033 2244341 23 432 95 95 545 545 378 378 2222244 2240081 24 615 93 93 552 553 315 316 32637 33204 Total cost 13368479 14196458 References and statistical problems, J. Assoc. Comp., April 1961, pp. I. Taubert, W.H. A search decision rule for the aggregate scheduling 212-229. problem, Manage. Sci., Feb. 1968, 14(6), pp. 83438359. 3. Goodman, D.A. A Modified Sectioning Search Approach to Ag2. Hooke, R. & Jeeves, T.A. 'Direct Search' solution of numerical gregate Planning. Ph.D-dissertation, Yale University, 1972.