AN INERACIVE PROCEDURE FOR AGGREGAE PRODUCION PLANNING Maciej Nowak Universiy of Economics in Kaowice 40-287 Kaowice, ul. 1 Maja 50, Poland E-mail: maciej.nowak@ue.kaowice.pl Posal Address Absrac Minimizing producion cos over he planning period is usually assumed o be he objecive of aggregae planning. However, oher issues of sraegic ype may be even more imporan. Smoohing employmen levels, driving down invenory levels or meeing high level of service usually are also considered. hus, aggregae planning problem consiues a muliple crieria decision making problem. In he paper a new approach for producion aggregae planning problem is proposed. he procedure combines linear programming, simulaion, and ineracive approach. Linear programming models are used o generae iniial soluions. In order o check how he flucuaions in demand will affec he resuls obained under each of hese soluions simulaion experimens are performed. Finally, an ineracive procedure is used for idenifying he final soluion of he problem. Key words: Aggregae producion planning, Producion risk, Simulaion, Muliple crieria analysis, Ineracive approach 1. INRODUCION he objecive of any manufacuring sysem is o deliver producs in righ quaniies, on ime and a he appropriae cos. A rough axonomy of decisions affecing he producion sysem involves hree caegories: sraegic, acical, and operaional. Sraegic planning decisions are mosly focused on he developmen of resources o saisfy cusomers requiremens. he objecive of he acical decisions is he mos effecive use of hese resources (Biran and irupai, 1993). Finally, operaional decisions are concerned wih operaional and scheduling problems and require disaggregaion of informaion generaed on higher levels. In his paper, acical producion planning is considered. I deals wih deermining and iming of producion for he inermediae fuure (from 3 o 18 monhs). erms aggregae producion planning or sales and operaions planning are ofen used o describe his par of planning aciviies. he word 247
aggregae means, ha plans are prepared on a produc family basis. Produc families are defined as groupings of producs ha share common manufacuring faciliies and seup imes. Decisions made in he developmen of an aggregae plan include deermining he bes way of meeing forecased demand by adjusing producion raes, labor levels, invenory levels, overime work, subconracing raes, and oher conrollable variables (Heizer and Render, 2004). Various opimizaion echniques are employed in aggregae planning. Linear programming, mixed ineger programming, and dynamic programming are used mos ofen. he objecive is ypically o find he lowes-cos-plan (Vollmann, Berry, Whybark and Jacobs, 2005). However, oher issues may be even more imporan. Smoohing employmen levels, driving down invenory levels or meeing high level of service are usually under managers consideraion. As a resul, aggregae planning consiues a muliple crieria decision making problem. Alhough mahemaical programming approaches for aggregae planning are already subsanially sophisicaed, hey sill do no reflec he siuaion of mos firms sufficienly. he main problem is ha hey ignore uncerainies inheren in any managerial decisions. In fac, he fuure demand can be only roughly esimaed. Producion coss vary due o flucuaions in raw maerial prices. Finally, he volume of producion ha company would be able o subconrac canno be precisely evaluaed. As a resul, uncerainy and risk have o be aken ino accoun while consrucing a producion plan. In order o solve a muliple crieria problem, single-crierion evaluaions mus be aggregaed. Roy (1985) idenified hree main aggregaing conceps: a concep wih a single synheic crierion, ouranking concep, and dialog concep wih rial-and-error ieraions. he las idea, also known as ineracive approach, is ofen used for solving real-world problems. Iniially, ineracive approach was used for solving decision making problems under cerainy. In uncerain conex ineracive mehods are mainly used for solving muliobjecive linear programming problems (Novak and Ragsdale, 2003; Urli and Nadeau, 2004). On he oher hand, ineracive procedures are also proposed for discree problems, where he number of feasible soluions is moderae (Nowak, 2004; Nowak, 2006). In his paper an ineracive procedure for aggregae producion planning is proposed. he problem is formulaed as a muliple crieria decision making problem. Insead of cos minimizaion, hree oher crieria are considered: minimizaion of invenory, minimizaion of producion volume ousourced o a subconracor, and minimizaion of flucuaions in producion rae. he procedure combines linear programming, simulaion, and ineracive approach. Linear programming models are used o generae iniial soluions. In order o check how he flucuaions in demand affec he resuls obained under 248
each of hese soluions simulaion experimens are performed. Finally, an ineracive procedure is used for idenifying he final soluion of he problem. 2. MAHEMAICAL PROGRAMMING APPROACHES FOR AGGREGAE PRODUCION PLANNING None organizaion can operae wihou a planning sysem. Usually companies make several plans a differen levels of aggregaion, using differen planning horizons (homas and McClain, 1993). acical plans should be harmonized wih company s long-erm goals and work wihin he resources allocaed by earlier sraegic decisions. hey are also he saring poin for shor-erm producion scheduling. Decisions made on his level involve a medium-range planning horizon (ypically one year), and aggregaion of iems ino produc families (Biran and irupai, 1993). Some examples of quesions ha he aggregae plan should answer are as follows (Waers, 2002): Should he producion raes be consan, or should hey be adjused o mach he demand requiremens in successive planning periods? Should subconracors be used o overcome capaciy shorages in some periods? Should work-force levels be adjused by hiring or laying off employees? Is back ordering a viable alernaive? A good plan balances he conflicing objecives of minimizing producion cos, maximizing cusomer service, minimizing invenory invesmens, mainaining a sable workforce. Several opions can be used o absorb demand flucuaions, including changing invenory levels, subconracing, varying producion raes hrough overime and idle ime. Quaniaive approaches used in aggregae producion planning include, among ohers, linear programming (Shapiro, 1993), mixed ineger programming (Vollmann, Berry, Whybark and Jacobs, 2005) and dynamic programming (rzaskalik, 1990). Below, an example of a linear programming model for aggregae producion planning problem is presened. he basic assumpions are as follows: he forecass of he demand for he nex periods have been prepared for each monh probabiliy disribuion of he demand is available. he model should provide a plan ha minimizes he paricular objecive assuming ha in each monh he demand is equal o he mean of he probabiliy disribuion represening i s flucuaions. he company would like o mainain a high level of he cusomer service. herefore, he final invenory in monh should be high enough o guaranee a high probabiliy of meeing he demand in monh + 1. 249
he premier objecive is o minimize oal cos, which includes producion cos, invenory cos and he cos of he idle ime. As he model assumes ha shorages are no allowed, he cos of delays in deliveries is no aken ino accoun. he producion is measured in producion hours required for is compleion. he noaion used in he model is as follows: c R he cos per labor-hour of regular ime producion, c O he cos per labor-hour of overime producion, c S he cos per labor-hour of subconracor producion, c I he cos per monh of carrying one labor-hour of work, c U he cos per labor-hour of idle regular ime producion, x he regular ime producion hours scheduled in monh, o he overime ime producion hours scheduled in monh, s he subconracor ime producion hours scheduled in monh, i he number of working hours sored in invenory a he end of monh, u he number of idle ime regular producion hours in monh, d he expeced demand in monh (hours of producion), b he highes demand he company should o be able o saisfy in monh (hours of producion), m 1 he maximum number of regular ime hours in monh, m 2 he maximum number of overime hours in monh, m 3 he maximum number of subconracor hours in monh, r he reducion in he number of producion hours scheduled in monh compared o he number of producion hours scheduled in monh 1, g he increase in he number of producion hours scheduled in monh compared o he number of producion hours scheduled in monh 1, a 1 iniial invenory, he number of monhs in he planning horizon, he producion plan should saisfy he following consrains: i x o s i d 1 for = 1,..., (1) i x o s b 1 for = 1,..., (2) x u m1 for = 1,..., (3) x o s r g x o s for = 2,..., (4) 1 1 1 i a r 0, g 0 (5) 0 1, 1 1 0 x m 1, 0 o m 2, 0 s m 3 for = 1, 2,..., (6) 250
In his paper we minimize four conflicing crieria: f 1 he oal cos: r 0, g 0, u 0 for = 1, 2,..., (7) c R x coo cs s cii cu u 1 f 2 he oal number of he overime producion hours: o 1 f 3 he oal number of he subconracor producion hours: s 1 f 4 he oal flucuaions in producion raes: (8) (9) (10) r s 1 Linear programming is a good ool if he company is able o prepare precise forecass of he fuure demand. Unforunaely, mos companies operae in uncerain environmen. If he demand canno be prediced wih high precision, anoher ools should be used o analyze he resuls of he producion plan. his applies in paricular o crierion f 1, as he esimaes of he oher crieria arise direcly from he producion plan. Simulaion is a good ool for such analysis. In our procedure we use i for deermining probabiliy disribuions of he oal cos. I allows also o esimae he overall cusomer service level. In our simulaion model we will consider he case of delayed deliveries. We will assume, ha he cos he company has o cover in he case of delays is equal o c D per one producion hour. (11) 3. HE PROCEDURE FOR AGGREGAE PRODUCION PLANNING he soluions of single-crierion linear programming problems wih crieria funcions specified above provide soluions minimizing he objecives under consideraion assuming ha in each period sales are equal o he expeced demand. In his paper, however, we consider he siuaion of uncerain demand. In such case, we canno esimae he values of he crierion f 1 precisely. As we assume here ha he plan is implemened regardless of he acual demand, values of he oher crieria arise direcly from he producion plan. he echnique ha we propose combines linear programming, simulaion and ineracive approach. Firs, four single-crierion line programming problems are solved. Nex simulaion is used o analyze 251
how good are hese plans wih respec o crierion f 1. Finally, ineracive procedure is used o deermine he producion plan saisfying decision maker s requiremens. Iniially four soluions are proposed o he decision maker. If he/she acceps any of proposals, he procedure ends. Oherwise he decision maker is asked o indicae maximal accepable values of crieria f 2 f 4. he new proposal is idenified by solving linear programming problem in which he oal cos is minimized under addiional consrains on he values of he oher crieria. Again he simulaion model is used o analyze he cos of new producion plan and he resuls are presened o he decision maker. he following seps are performed o complee he procedure: 1. Solve four linear programming problems wih objecive funcions given by (8) (11) and consrains given by (1) (7). 2. Perform simulaion runs o evaluae he oal cos of producion plans deermined in sep 1. 3. Presen he soluions generaed in sep 1 o he decision maker and ask him/her wheher he/she acceps any of hem as a final soluion. If he answer is ÝES end he procedure. 4. Ask he decision maker o specify maximal accepable values of crieria f 2 f 4. 5. Generae a new proposal for he decision maker solving he linear programming problem in which he oal cos is minimized under addiional consrains on he values of he oher crieria. If he problem is infeasible noify he decision maker and go back o sep 4. 6. Perform simulaion runs o evaluae he oal cos of producion plan deermined in sep 5. 7. Presen he new proposal o he decision maker. If he/she is saisfied wih he proposal end he procedure, oherwise ask he decision maker o redefine his/her requiremens go back o sep 4. Commens: Sep 1: As alernae soluions of linear problems may exis, we recommend o use hierarchical approach. Once he minimum value of crierion f k is idenified, oher crieria are minimized preserving he minimal value of f k. Sep 2: For each soluion generaed in sep 1, a series of simulaion runs is performed aking ino accoun probabiliy disribuions of he demand. he resuls are used o consruc probabiliy disribuions describing how good are he soluions wih respec o he crierion f 1. Sep 4: We assume here, ha f 1 is he mos imporan crierion. In order o consider decision maker s preferences on he values of oher crieria addiional consrains are defined specifying maximum values of hem. 252
Sep 5: If he decision maker s requiremens on he crieria f 2 f 4 are o srong, he problem may be infeasible. In his case he decision maker is asked o redefine consrains on crieria oher han cos. Sep 7: he new soluion mees he decision maker s requiremens on he crieria f 2 f 4. However, he increase of he cos may be oo high for he decision maker. herefore, i is proposed o allow he decision maker o change he consrains defined previously. 4. NUMERICAL EXAMPLE o illusrae applicabiliy of he procedure le us consider following example. A company prepares an aggregae plan. ime horizon is 6 monhs. Basic daa are as follows: c R = 1,00, c O = 1,50, c S = 1,70, c I = 0,30, c U = 0,50, c D = 5,00. m 1 = 900, m 2 = 100, m 3 = 300 for = 1,, 6. Probabiliy disribuions of he demand are presened in able 1. able 1: Probabiliy disribuions of he demand Monh 1 Monh 2 Monh 3 Monh 4 Monh 5 Monh 6 Demand Prob. Demand Prob. Demand Prob. Demand Prob. Demand Prob. Demand Prob. 620 0,05 800 0,05 1020 0,05 900 0,05 740 0,05 620 0,05 640 0,1 820 0,05 1040 0,1 920 0,05 760 0,05 640 0,1 660 0,25 840 0,1 1060 0,1 940 0,1 780 0,1 660 0,1 680 0,2 860 0,25 1080 0,3 960 0,25 800 0,3 680 0,3 700 0,15 880 0,25 1100 0,25 980 0,25 820 0,25 700 0,25 720 0,1 900 0,15 1120 0,1 1000 0,15 940 0,15 720 0,1 740 0,1 920 0,1 1140 0,05 1020 0,1 960 0,05 740 0,05 760 0,05 940 0,05 1160 0,05 1040 0,05 980 0,05 760 0,05 Means of he demand are as follows: d 1 = 685, d 2 = 874, d 3 = 1087, d 4 = 974, d 5 = 836, d 6 = 687. he company assumes ha he iniial sock in each monh should be enough o provide 95% probabiliy of meeing he demand. hus, he highes demand ha he company should be able o saisfy is as follows: b 1 = 740, b 2 = 920, b 3 = 1140, b 4 = 1020, b 5 = 960, b 6 = 740. he iniial invenory is i 0 = 0. he procedure operaes as follows. Ieraion 1 253
Sep 1: Four single crierion linear programming problems wih consrains given by (1)-(7) are solved. Resuls are presened in able 2. able 2: Soluions of linear programming problems solved in ieraion 1. Monh Soluion 1 Soluion 2 Soluion 3 Soluion 4 x o s x o s x o s x o s 1 800 0 0 800 0 0 800 100 0 800 100 17 2 800 5 0 800 0 5 800 100 0 800 100 17 3 800 100 194 800 0 294 800 100 40 800 100 17 4 800 100 67 800 0 167 800 100 40 800 100 17 5 800 100 14 800 0 114 800 100 0 800 100 17 6 616 0 0 616 0 0 616 0 0 800 100 17 Sep 2: Simulaion experimens are performed o evaluae producion plans deermined in sep 1 wih respec o crierion f 1.. Sep 3: he soluions idenified in sep 1 are presened o he decision maker. able 3: Iniial soluions proposed o he decision maker. Soluion f 1 (mean) f 2 f 3 f 4 Service level 1 6027,5 305 275 772 99,06% 2 6080,5 0 580 772 99,09% 3 6061,2 500 80 364 99,33% 4 6303,4 600 102 0 99,69% he decision maker says, ha he is no fully saisfied wih any of he proposals. Sep 4: he decision maker specifies he maximum accepable values of crieria: f 2 300, f 3 300, f 4 50. Sep 5: he new linear programming problem is formulaed. Objecive funcion f 1 is minimized under he consrains (1)-(7) and addiionally: o 1 s 1 300 (12) 300 (13) 1 50 r (14) s As he problem is infeasible, he decision maker is asked o redefine his requiremens. 254
Sep 4: he decision maker decides o weaken he consrain on he value of he crierion f 4 he maximum value of his crierion is changed o 150. Sep 5: he linear programming problem wih new consrained is solved. he soluion is presened in able 4. able 4: he soluion no. 5 proposed o he decision maker. Monh Soluion 5 x o s 1 800 100 16,5 2 800 100 16,5 3 800 0 116,5 4 800 100 16,5 5 800 0 114 6 800 100 16,5 Sep 6: Simulaion runs are performed o evaluae he oal cos of producion plan defined by soluion no. 5. Sep 7: he resuls obained for he soluion no. 5 are presened o he decision maker (able 5). able 5: he resuls obained for he soluion no. 5. Soluion f 1 (mean) f 2 f 3 f 4 Service level 5 6104,6 300 280 150 99,65% According o he decision maker, he oal cos is oo high. herefore, he procedure goes back o sep 4. Sep 4: he decision maker decides o weaken he consrain on he value of he crierion f 4 he maximum value of his crierion is changed o 400. Sep 5: he linear programming problem wih new consrained is solved. he soluion is presened in able 6. Sep 6: Simulaion runs are performed o evaluae he oal cos of producion plan defined by soluion no. 6. Sep 7: he resuls obained for he soluion no. 6 are presened o he decision maker (able 7). 255
able 6: he soluion no.6 proposed o he decision maker. Monh Soluion 6 x o s 1 800 0 84 2 800 84 0 3 800 16 132 4 800 100 50 5 800 100 14 6 616 0 0 able 7: he resuls obained for he soluion no. 6. Soluion f 1 (mean) f 2 f 3 f 4 Service level 6 6065,9 300 280 400 99,19% he decision maker decides o accep curren soluion. 4. CONCLUSIONS Aggregae producion planning provides a key communicaion links for op managemen o coordinae he various planning aciviies in a company. From a manufacuring perspecive, i provides he basis o focus he deailed producion resources o achieve he firm s sraegic objecives. Various mahemaical programming formulaions are proposed for aggregae producion planning. he objecive is ypically o find he lowes-cos plan. However, oher crieria are also aken ino accoun by managers, including for example minimizing flucuaions in producion runs or minimizing he volume of producion ousourced o subconracors. As a resul, he aggregae planning can be considered as a muliple crieria decision making problem. In his paper he ineracive procedure for aggregae producion planning was proposed. I was assumed ha cos minimizaion is he mos imporan objecive. he dialog wih he decision maker is conduced in order o find a soluion ha is accepable wih respec o all crieria. he idea of he procedure is quie simple and can be easy implemened. REFERENCES Biran, G.R. and irupai, D. (1993), Hierarchical Producion Planning, in: Graves, S.C., Rinnooy Kan, A.H.G., and Zipkin, P. (Eds.), Handbooks in Operaions Research, Vol. 4., Logisics of Producion and Invenory, Elsevier Science Publishers B.V., Amserdam, pp.523-568. Heizer, J. and Render, B.(2004), Operaions Managemen, Pearson Educaion, Upper Saddle River. 256
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