Separable programming for aggregate production planning: A high-order cost case

Meij, J. T. A icle Sepa able p og amming o agg ega e p oduc ion planning: A high-o de cos case Sou h A ican Jou nal o Business Managemen P o ided in Coope a ion wi h: Uni e si y o S ellenbosch Business School (USB), Bell ille, Sou h A ica Sugges ed Ci a ion: Meij, J. T. (1982) : Sepa able p og amming o agg ega e p oduc ion planning: A high-o de cos case, Sou h A ican Jou nal o Business Managemen , ISSN 2078-5976, A ican Online Scien i ic In o ma ion Sys ems (AOSIS), Cape Town, Vol. 13, Iss. 1, pp. 18-22, h ps://doi.o g/10.4102/sajbm. 13i1.1166 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217786 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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Meij Depa men o Mechanical Enginee ing, Uni e si y o S ellenbosch Many p oduc ion manage s a e aced wi h he p oblem o planning p o- duc ion, in en o y and wo k- o ce unde he cons ain o limi ed esou ces o mee a seasonal demand. Conside able esea ch has been done on his planning p oblem and a ious planning models ha e been in oduced. In hose cases whe e linea i y o he cos unc ions o an unde aking may easonably be assumed. an o dina y linea p og am- ming model su ices. In many cases. howe e , his simple linea ap- p oach o ce ain essen ially non-linea cos unc ions is unaccep able owing o he g oss app oxima ion made. Sepa able p og amming (SEP) is in oduced as a solu ion me hodology o his agg ega e p oduc ion planning p oblem in a com- plex. high-o de cos s uc u e case. The cos s uc u e was used by Goodman o he applica ion o goal p og amming (GP) in his ield. The Goodman GP model makes p o ision o posi i e o nega i e slack o he p oduc ion le el, wo k- o ce le el and in en o y le el wi h penal y cos s o hese slack-de ia ions. Goodman also made use o a 'sec ion- ing sea ch' model o his high-o de cos case o se e as a measu e o his GP model. A compa ison is made be ween he esul s o hese h ee app oaches. SEP o e ed an imp o emen o mo e han 4% in o al cos in compa ison wi h he sec ioning sea ch model, and pe - o ms 26% be e han he GP model. S. A . J. Bus. Mgm . 1982, 13: 18-22 Baie p oduksiebes uu de s wo d gekon on ee me die p obleem an die beplanning an p oduksiehoe eelheid, oo aad lak en a beidsmag me inagneming an die bepe k e b onne o die onde neming se beskikking i die be ediging an 'n seisoenale aan aag. Heelwa na o sing is al oo hie die beplanningsp obleem gedoen en 'n e - skeidenheid wiskundige en ande modelle is ge oe s. In die ge alle waa linee e kos e unksies by 'n onde neming me 'n g oo ma e an seke heid aan aa kan wo d, kan an gewone linee e p og amme- ingsmodelle geb uik gemaak wo d. In baie ande ge alle is die aan- name an 'n linee e kos es uk uu eg e onaan aa baa weens g owwe aannames wa gemaak wo d. Skeiba e p og amme ing (SEP) wo d oo ges el as 'n oplos- singsme odiek i die ak iese p oduksiebeplanningsp obleem in ·n komplekse, h<*o de, kos es uk uu ge al. Hie die kos es uk uu is deu Goodman geb uik i die oepassing an doelwi -p og amme ing (GP) in die gebied. Die Goodman doelwi -p og amme ingsmodel maak oo siening i posi iewe en nega iewe a wykings i die p oduksie lak a beidsmag lak en oo aad lak me boe e-kos e i die a wykings. ' Goodman geb uik ook 'n ' e deling-soek'-model, i hie dle ho -o de kos e-ge al, i e gelykingsdoeleindes een sy GP-model. ·n Ve gely- king wo d gemaak ussen die esul a e an hle die d ie benade ings. Die SEP-model loon 'n e be e ing an 4% op die o ale kos e an die e deling-soek-model en p es ee 26% be e as die GP-model. S.-A . Tydsk . Bed y sl. 1982, 13: 18 - 22 I.T. MelJ P o esso , Depa men o MC(:hanicaJ Enaince ina, Uni eni y o S ellenbosch S ellenbosch 7600, Republic o Sou h A ica ' Recei ed Sep embe 1981 ; accep ed No embe 1981 The agg ega e p oduc ion planning p oblem, in i s simples o m, may be s a ed as ollows: To de elop he lowes cos p oduc ion plan gi en a luc ua ing demand pa e n and limi ed p oduc ion esou ces. Va ious solu ion me hodologies o sol e his op 1m1za ion p oblem ha e been sugges ed. Hol , Modigliani, Mu h and Simon 1•2 de eloped he Linea Decision Rule (LOR) app oach o assumed quad a ic cos s uc u es. Thei published applica ion o LOR a a pain ac o y se ed as a ya ds ick o many di e en models. Hanssman and Hess3 used he LOR model as a basis o he de elopmen o hei p og amming model because ... 'i appea s, howe e , ha in he majo i y o p ac ical applica ions and heo e ical models he cos unc ions a e assumed o be linea '. Based on his app oach Goodman 4•5 p esen ed an al e - na i e linea iza ion me hod and used goal p og amming (GP) o sol e he pain ac o y p oblem. In compa ison wi h linea p og amming (LP) and LOR, goal p og am- ming ga e good esul s o he quad a ic cos unc ion case. Howe e , in he case o he highe -o de cos unc- ion used by Goodman, he GP app oach ailed in com- pa ison wi h a compu e sea ch me hod (sec ioning sea ch). In his pape he use o sepa able p og amming (SEP) o he case o high-o de cos unc ions is shown o gi e excellen esul s. The ad an age o his me hod lies in he ac ha o dina y linea p og amming algo i hms can be used o sol e he model. To de elop his ou h-o de cos unc ion Goodman made use o a hypo he ical eal wo ld se o da a, ep oduced in Table 1. SEP is also applied o his se o da a o es i s abili y o i he eal wo ld si ua- ion. Goal p og amming model The Goodman GP model was de eloped o i he hypo he ical his o ical cos da a-se shown in Table 1. In his able only he absolu e alues o changes a e shown. I is assumed ha he cos s a e symme ical abou he ze o cos poin o each a iable. The cos model is cons uc ed as ollows. De ine: P, = P oduc ion a e in pe iod D, = Demand in pe iod I, = In en o y le el a he end o pe iod W, = Wo k- o ce le el in pe iod By i ing cu ilinea segmen s o a se o he hypo he· S. A . J. Bus. Mgm 1982, 13(1) ical cos da a he ollowing cos unc ions we e ob ained: 340 w, Regula pay oll. 0,2 (P, - 6 W,>4 O e ime and idle ime 64( w, - W, _ / Hi ing and layo . O,l(P, - P,_ / P oduc ion le el inc ease and de- c ease. O, 1(/, -320)4 In en o y and sho ages. These cos unc ions lead o he ollowing o al cos model. Minimize he o al cos : ° (340 W, + 0,2(P, -6 W,)4 /= I + 0,1(/, - 320)4) Subjec o: o = 1,2,3 .... n Goodman sol ed his model wi h goal p og amming. I is based upon he no ion ha each o he ou h-o de cos e ms becomes ze o when he exp ession inside he pa en heses is ze o. Minimiza ion o each cos e m is ega ded as a goal and is o mula ed as a cons ain . I is necessa y o allow posi i e and nega i e slack in hese cons ain s because i is no possible o minimize simul a- neously all he cos e ms while a he same ime sa is y- ing he demand equi emen s. The esul ing goal con- s ain s can be exp essed as: P, - 6W, + Q,+ Q,- = 0 W- I W -I + R+ I R- I = 0 P, P,-1 + s+ I s- I =0 1 320 + + I - I =0 /1_1 + P, -D, = I, P, ;?I; 0 w, ;?I; 0 = 1,2 .... n, whe e Q, R, Sand Ta e slack a iables. Posi i e coe icien s a e assigned o he slack a iables in he objec i e unc ion. The e ec is o penalize de ia- ions om he desi ed goals. The objec i e unc ion is gi en by: Min J1 [340 W, + C Q/ + C,Q, -+ CzR/ + CzR,- + C3S,+ + c~,- + C4T,+ + c. ,-1 19 The coe icien s C1, C2, C3 and C4 mus be selec ed o gi e good cos app oxima ions o he cos e ms ha hey ep esen . This is done by app oxima ing he cos unc- ions by linea segmen s. The slopes o hese linea segmen s gi e he desi ed cos coe icien s. (See Figu e 1.) The slopes a e se so ha a ea A is equi alen o a ea B, o example: Figu e 1 App oxima ion o a cos unc ion by a linea segmen (Goodman). Sepa able p og amming o mula ion In sepa able p og amming he same p inciple is applied as in goal p og amming. The main di e ence is ha he cos e ms a e now app oxima ed by se e al linea segmen s. (See Figu e 2.) COST COST VARIABLE Fl11u e 2 App oxima ion o high-o de cos unc ion by se e al linea segmen s. 20 The ollowing model can hus be o mula ed. De ine: OT = O e ime OA = Idle ime HT = Inc ease in wo k- o ce (Numbe o people) HA = Dec ease in wo k- o ce (Numbe o people) PT = Inc ease in p oduc ion a e PA = Dec ease in p oduc ion a e VT = In en o y VA = Sho ages W = No mal wo k- o ce P = P oduc ion a e I = Ne in en o y le el D = Demand = Time pe iod T = Numbe o ime pe iods (Planning ho izon) b, c, d, e, J, g, u, s = Cons an s om piecewise app ox- ima ions o a iables B, C, D, E, F, G, U, S = Cos cons an s om piecewise app oxima ion i, j, k, I, m, n, q, = In e -subsc ip s de e mining hese- quence o he piecewise segmen s. G = Numbe o piecewise segmen s equi ed (kep cons an o simpli y he o mula ion) a = Regula wo k- o ce cos coe icien Z = Ra io o p oduc ion o wo k- o ce (p oduc i i- y cons an ). The sepa able p og amming model can be exp essed as: Minimize he objec i e unc ion: ;i [aw, + J1 B; OT;, + J1 cj OAj, + J1 DkHTkT+ I E1 HA 11 + J1 Fm PTm, + Ji GnPAn, + JI uq VTq, + IS, VA,,] Subjec o he cons ain s: W, - W,_ 1 - HT, + HA, = O P, - P,_ 1 - PT, + PT, = O I, - 320 -VT, + VA, = 0 P, -6 w, -OT, + OA 1 = 0 S.-A . Tydsk . Bed y sl. 1982, 13(1) P, -I, + I, -I = D, w, ;;i. 0 P, ;;i. 0 b;OT;, - OT,+ ZW, = 0 i=I J = I eiOAi, - OA, + ZW, = 0 JI d*HT*, - HT,= 0 i e1 HA 1 , - HA, = 0 m=I mPTm, - PT,= 0 ! n=I g n PA n - PA 1 = 0 ! q=I u VT - q q VT-'+ 320 = 0 I G :I: =I s VA - VA,+ 320 = 0 o all = 1,2 .... T. No e: G may a y o each a iable depending on he desi ed quali y o he i o he linea segmen s o he cos s uc u e. Applica ion o he sepa able p og amming model The sepa able p og amming model was applied o wo al e na i e cos s uc u es: The i s un was done using cos s as gi en by he ou h-o de cos unc ions de eloped by Goodman. Each cos unc ion was app oxima ed by six piece- wise linea segmen s. Thus G = 6 in he o mula- ion. This un will be called SEP (model applica ion). The second un was done by using di ec ly he hypo he ical his o ical cos da a shown in Table l. Six piecewise linea segmen s we e used o app oxi- ma e each a iable. This un will be called SEP (di ec applica ion). The cos e ms o bo h cos s uc u es we e g aphed o de e mine he unc ion alues o he linea segmen s and he associa ed cos coe icien s. I mus be emppasized ha he model de eloped when i ing he linea segmen s o he eal wo ld cos s uc u e gi en in Table l mus be Table 1 Hypo he ical-his o ical cos da a by Goodman 1 1w,-w,_ 1 1 Cos IP,-P,-11 Cos II, - l Cos IP,-ZW,I Cos 0 0 1 1 1 0 0 66 2 2 2 2 2 1 001 4 24 3 9 2 4 3 .S 210 .s 68 4 28 3 14 4 20 100 7 225 6 122 .s 131 .s 38 120 10 1 049 8 392 7 457 7 86 300 16 6 310 11 1 370 10 1 876 9 139 200 22 26 100 l.S .S 417 12 3780 12 224 400 34 123 400 21 18 240 14 7 79.S 14 279 600 52 487 200 39 231 200 19 401 100 18 20 600 87 1 140 000 .SI 474 400 22 34 900 2.S 698 700 I.SO 2 224 000 70 702 500 30 58 200 S. A . J. Bus. Mgm 1982, 13(1) 21 Table 2 Agg ega e p oduc ion plan using sepa able p og amming applied o he high-o de cos model and applied o he eal-wo ld cos s uc u e P oduc ion (Uni s) Wo k- o ce (Men) In en o y (Uni s) Pe iod Demand Model applica ion Di ec applica ion Model applica ion Di ec applica ion Model applica ion Di ec applica ion 0 4SO 4SO 7S 7S 320 320 I 430 446 441 73 73 336 331 2 447 431 424 71 70 319 308 3 440 409 402 67 67 289 271 4 316 378 380 64 6S 3SO 33S s 397 361 37S 62 63 314 313 6 37S 349 368 60 62 289 30S 7 292 364 368 62 62 360 381 8 4S8 39S 390 64 64 297 312 9 400 383 383 63 63 280 29S 10 3SO 3S2 361 60 61 282 30S 11 284 361 3S3 64 61 3S9 374 12 400 401 37S 69 64 360 348 13 483 441 464 74 7S 318 329 14 S09 478 486 79 79 288 30S IS soo 493 soo 84 82 280 30S 16 475 508 492 89 84 313 322 17 soo 548 Sl3 94 88 360 33S 18 600 600 629 99 104 360 364 19 700 668 663 108 109 328 327 20 700 708 68S 113 112 337 312 21 72S 668 663 108 109 280 2SO 22 600 600 600 103 102 280 2SO 23 432 S60 572 98 97 408 390 24 61S S49 S60 94 9S 342 33S To al cos -(Calcula ed by using cos s in Table I)} Model applica ion: R9 817 794 Di ec applica ion: R9 818 227 Table 3 Agg ega e p oduc ion plan using he sec ioning sea ch and goal p og amming models o Good- man P oduc ion (Uni s) Wo k- o ce (Men) In en o y (Uni s Pe iod Demand Sec ioning sea ch Goal p og amming Sec ioning sea ch Goal p og amming Sec ioning sea ch Goal p og amming I 430 431 4SO 7S 7S 301 320 2 447 440 447 72 74 294 320 3 440 426 403 69 67 280 283 4 316 392 37S 6S 63 3S6 342 s 397 374 37S 62 63 333 320 6 37S 348 37S S9 63 306 320 7 292 348 37S 60 63 362 403 8 4S8 386 37S 63 63 290 320 9 400 391 37S 64 63 281 29S 10 3SO 3SS 3S3 61 S9 286 298 11 284 3S6 3S3 63 59 3S8 367 12 400 399 353 68 S9 3S7 320 13 483 444 483 73 30 318 320 14 S09 481 496 78 83 290 307 IS soo 488 496 83 83 278 303 16 47S sos 496 88 83 311 324 17 soo SS2 496 94 83 363 320 18 600 607 600 101 100 370 320 19 700 662 700 107 117 332 320 20 700 699 700 112 117 331 320 21 72S 6S9 700 107 117 26S 29S 22 600 600 S57 101 93 26S 2S2 23 432 S4S SS7 9S 93 378 377 24 6IS SS3 SS7 93 93 316 319 To al cos -(Calcula ed by using cos s in Table I) Sec ioning sea ch: RIO 625 200 Goal p og amming: R12 237 846 22 S.-A . I yJ,k . UeJ yhl. 1982, 13(1) Table 4 A compa ison o he cos a eas and o al cos associa ed wi h each solu ion me hodology (All cos s calcula ed using Table 1) Sepa able p og amming: Sepa able p og amming: Sec ioning sea ch Goal p og amming Applica ion o model cos s Di ec cos applica ion Cos s (las 13 pe iods) (las 13 pe iods) (las 13 pe iods) (24 pe iods) (las 13 pe iods) (24 pe iods) R R R R R R Regula wo k- o ce 408 000 (100) 8 408 340 (100) 412 080 (IOI) 653 480 408 000 (100) 649 740 Change in p oduc ion a e 3 856 718 (100) 6 783 299 (176) 3 724 148 ( 97) 4 074 351 3 838 973 ( 99) 3 954 773 Changes in wo k- o ce 602 981 (100) I 859 610 (308) 578 620 ( 96) 635 312 749 782 (124) 765 339 O e ime and idle ime 179 565 (100) 46 ( 0) 270 616 (151) 320 716 44 941 ( 25) 53 406 In en o y and shonages 3 015 479 (100) 1 376 231 ( 46) 2 733 031 ( 91) 4 133 935 2 744 128 ( 91) 4 394 969 To al cos 8 062 743 (100) IO 427 520 ( 129) 7 718 495 ( 96) 9 817 794 7 785 824 ( 97) 9 818 227 "Numbe s in b acke s a e pe cen ages. ega ded as he be e model - he i o he Goodman model was done o con ol pu poses. Resul s and compa ison In o de o compa e he esul s o he wo SEP models wi h ha o he wo app oaches, goal p og amming and sec ioning sea ch used by Goodman, a wen y- ou - pe iod planning ho izon was used. The p oduc ion plans o bo h SEP models a e gi en in Table 2 and can, o all p ac ical pu poses, be ega ded as simila . The o al cos di e ence is negligible. The compa ison be ween he sec- ioning sea ch model and he goal p og amming model o his planning pe iod is gi en in Table 3. These esul s illus a ed clea ly ha he cos s uc u e used is e y sen- si i e, so ha small de ia ions om he global op imum plan in ol es la ge changes in he o al cos . Cos s a e calcula ed o each o he p oduc ion plans om he cos da a in Table 1 using in e pola ion whe e necessa y. A compa ison was made be ween he esul s o he ou models men ioned abo e, on he basis o cos s in he a- ious cos a eas. The s a ing condi ions used by Good- man we e unknown and he e o e only he las 13 pe iods ~e co_mpa ed o elimina e i s e ec . The compa ison is gi en m Table 4. SEP (model applica ion) o e ed an im- p o ei:nen o m~ e han 40/o in he o al cos in compa i- son wi h he sec ioning sea ch model and pe o ms 260Jo be e han he GP models. These imp o emen s a e s?mewha lowe in he case o he SEP (di ec applica- ion) - 3,40Jo and 2SOJo espec i ely. A pe cen age-wise compa ison is also made be ween he models in Table 4 The sec ioning sea ch esul s a e used as a basis o hi~ c~mpa i.son. Majo di e ences can be summa ised as in- dica ed m Table 5. Concluslon In conclusion he oil owing ad an ages o he SEP a _ p oach.mus be ~nde lined. This is a ma hema ical p ~- g ammmg ~hmque ha can be used in a e y lexible way. Mo e linea segmen s may be added in cases whe e Table 5 A compa ison be ween he a ious models, using sec ioning sea ch esul s as a basis Majo di e ences compa ed o he sec ioning Model sea ch esul s Goal Changes in he wo k- o ce is dominan (app oxima e- p og amming ly h ee imes mo e), no o e ime/unde ime and less in en o y. Highe o al cos . SEP (model applica ion) SEP (di ec applica ion) Mo e o e ime/unde ime and less in en o y. To al cos less. Mo e changes in he wo k- o ce, less o e ime/ unde ime and less in en o y. To al cos less. mo e accu acy is needed. In he case o mono onous cos inc eases (e.g. conca e cos s uc u es), he o dina y simplex algo i hm may be used o sol e he p oblem. The linea segmen s can be i ed g aphically o eal-wo ld cos da a wi hou de i ing, by means o labo ious cu e i ing me hods, a complex ma hema ical model. In many cases o dina y linea cu es, while in o he ins ances (e.g. ~ e ime pay as a linea unc ion o no mal ime pay) hnea segmen s, ep esen he eal wo ld. Re e ences I. ~OLT, C., MODIGLIANI, F. & SIMON, H.A. 'A Linea Deci- s10.n Rule o P oduc ion Employmen Scheduling', Manage. Sc,., Oc obe 1955. 2. ~O~T, ~-, MODIGLIANI, F., SIMON, H.A. & MUTH, J.F. De i a ion o a Linea Decision Rule o P oduc ion and Employmen ', Manage. Sci., Janua y 1956. 3. HANSSMANN, F. & HESS, W. 'A Linea P og amming Ap- p oach o P oduc ion and Employmen Scheduling', Manage. Technol., Janua y 1960, 1(1). 4. GOOD~AN, D.A. 'A New App oach o Scheduling Agg ega e P oduc ion and Wo k Fo ce', AIIE T ansac ions June 1973 Vol. S. ' ' 5. GOODM~N, D.A. 'A Goal P og amming App oach o Agg e- ga~e Planning o P oduc ion and Wo k Fo ce'' Manage. Sc1.,Augus 1974, 20(12), p.1569.