A Mathematical Model for Profit Maximization General Lot-sizing and Scheduling Problem (PGLSP)

Transcription

1 Proceedings of he 2012 Inernaional Conference on Indusrial Engineering and Operaions Managemen Isanbul, Turkey, July 3 6, 2012 A Mahemaical Model for Profi Maximizaion General Lo-sizing and Scheduling Problem (PGLSP) Narges Sereshi, Mehdi Biari and Ghasem Moslehi Deparmen of Indusrial and Sysem Engineering Isfahan Universiy of Technology Isfahan, Iran Absrac This paper exends he general lo-sizing and scheduling problem (GLSP), o include demand choice flexibiliy. In Profi maximizaion general lo-sizing and scheduling problem wih demand choice flexibiliy which is named PGLSP in his paper, he acceped demand in each period, lo-sizing and scheduling are hree problems which are considered simulaneously. The proposed problem will resul in beer decisions especially when he mix of producs is decided in mid-erm planning and shor-erm decision is abou scheduling of differen machines. A mahemaical model is developed for he PGLSP. Keywords Lo-sizing, scheduling, profi maximizaion, demand choice flexibiliy 1. Inroducion Lo-sizing and scheduling are imporan problems in he field of producion planning. Mos of he ime, decisions for hese wo problems are made hierarchically. In his mehod, he lo-sizing problem is solved a firs and hen he resul is used for sequencing and scheduling problem [1]. GLSP considers lo-sizing and scheduling problems simulaneously due o heir dependency. In mos of he models for producion planning and especially in lo-sizing and scheduling problems, he obecive funcion is minimizing coss. In hese models, he assumpion is ha all he cusomers demands should be me. However, in a business wih a goal of maximizing profi saisfying all poenial demands may no always be an opimal soluion [2]. In his siuaion, appropriae selecion of demand (o be me) is an effecive sep for demand managemen. In his paper, he profi maximizaion general lo-sizing and scheduling problem wih demand choice flexibiliy is sudied. The acceped demand in each period can vary beween is upper and lower bounds. The upper bound could be he forecased demand and he lower bound could be he organizaion commimens owards cusomers or minimum producion level according o producion policy. In oher words, he amouns of acceped demands, losizing and scheduling are problems which are considered simulaneously. According o hese assumpions he obecive funcion of he problem is maximizing he revenue of sales minus he producion, invenory holding, and seups coss. The resul of his model is he amoun of acceped demand for each produc in each period, he size of producion los, and he sequence of hem. In his problem, he mix produc problem which perains o mid-erm planning and lo-sizing and scheduling problems which relae o shor-erm scheduling are considered in one problem. The proposed problem will resul in beer decisions especially when mix of producs is decided in mid-erm planning while shor erm decision is abou scheduling of differen machines. This model is presened based on he characerisic of indusries like moquee weaving indusry. In his ype of indusry seup imes and coss are considerable so i is wise o use he GLSP model. Besides ha selecing he demand wihou considering back order coss is also possible and predicing he exac amoun of demand is impossible and i is more reliable o define upper and lower bounds for i. Wholesalers accep his policy, so someimes he amoun of producs hey delivered is less han heir orders. For preserving he percenage of producs and enhancing he service level, he lower bounds are greaer han zero. 230

2 We have a few producs in his company in differen models and colors and he planning period of wo weeks and he planning horizon of 8 weeks are suiable for he company. 2. Lieraure Review Because PGLSP, which is a new problem, is a combinaion of GLSP and demand choice flexibiliy, we discuss he relaed lieraure of hese wo problems. GLSP was firs presened by Fleischmann and Meyr [3]. For modelling GLSP, he auhors use wo kinds of ime buckes, small buckes, and large ones. Small buckes or posiions are wihin he large buckes or periods. The posiions are used for sequencing. Meyr [4] exended GLSP o deal wih sequence-dependen seup imes. Kocklar [5] represened a new model for GLSP wih sequence-dependen seups by using he combinaion of Meyr [4] approach for modelling and ransporaion problem. The basic idea in his model is o disaggregae he producion variables by relaing each producion quaniy o he period a which i will be required. Gupa and Magnusson [6] represened a new model based on he capaciaed lo-sizing problem (CLSP) wih sequence-dependen seup imes. This problem is relaed o ravelling salesman problem (TSP) or he vehicle rouing problem (VRP) problem. Seup coss in GLSP are like disance cos in TSP. Because TSP is an NP-hard problem, CLSP wih sequence-dependen seup is also NP-hard [6]. Almada-lobo e al [7] have demonsraed ha he Gupa s model does no avoid disconneced sub ours. Almada-lobo e al [8] presened a more efficien model especially in comparison wih he models which use he Meyr [4] approach for scheduling. Simulaneous lo-sizing and scheduling is noiced in many pracical siuaions. Paloch e al [9] represened a heurisic algorihm for producion lo scheduling in obacco indusry. Quad [1] discussed he lo-sizing and producion scheduling for flexible flow-line and represened a case sudy in a semiconducor facory. This problem was modelled and solved for sof drink producion [10,11], seel casing facory [12] and animal food plan [13]. These sudies emphasized he imporance of considering lo-sizing and scheduling a he same ime due o heir ineracions. Companies ofen produce many producs, bu one of he mos imporan problems is selecing which producs o produce. This problem is known as he produc mix problem [14]. The problem is o find a producion plan ha can maximize he ne profi of he company by considering he resource consrain and each produc demand. This problem only calculaes he amoun of producion and doesn ake ino accoun lo-sizing and scheduling problem. Amoun of producion should be wihin is upper bound and lower bound. PGLSP is a combinaion of he mix produc problem and GLSP which are peraining o mid-erm planning and shor-erm scheduling. For mos producion planning problems wih he assumpion of responding o all demands, because he ne profi from selling all producs is fixed, he obecive is o minimize differen coss. Flexibiliy in demand choice resuls in changing his obecive o maximizing he sales revenues minus differen coss. Haugen e al [15] sudied profi maximizaion capaciaed lo-sizing problem (PCLSP) and represened a mahemaical model for i. In his model, demand is a funcion of price and price is a variable in he model. In his problem he demand is deermined indirecly by he model. The obecive funcion is maximizing sales revenue minus differen coss. In anoher paper, Haugen e al [16] sudied he large scale of he same problem. Geunes e al [17] represened a model ha implicily decides, hrough pricing decisions, he demand levels of he firm which should be saisfied in order o maximize he profi. Merzifonluoglo and Geunes [18] sudied a planning problem which deermined he opimal level of demand, producion, and invenory for every planning period when flexibiliy exiss in selecing demands and heir delivery iming. They use a variable in heir model which deermines he proporion of acceped demand wih he obecive of maximizing he profi. Shen [2] sudied he demand choice flexibiliy in a supply chain and wih he obecive of profi maximizaion. 231

3 3. PGLSP Definiion and Formulaion 3.1 PGLSP definiion PGLSP is described as follows: Having P producs and T planning periods, decision maker wans o define: (1) he acceped demand of each produc in each period which is beween upper and lower bounds, (2) he quaniy of los for each produc, and (3) he sequence of los. The obecive funcion is maximizing he revenue of sales minus producion, holding, and seup coss. Assumpions of he model are as follows: Backlog is no allowed. Seup imes and coss are sequence-dependen. The riangular inequaliy is exised beween seup imes. The model has he characerisic of seup preservaion. I means ha if we have an idle ime, se up sae would no change afer i. The breakdown of seup ime beween wo periods is no allowed and he seup is finished in he same period in which i begins. In his par we propose a model for PGLSP. Parameers for he model are represened in able 1 Table 1: Model parameers Parameers C Available capaciy in each period = 1,, T Ld Demand lower bound for produc in period = 1,, P, = 1,, T Ud Demand upper bound for produc in period = 1,, P, = 1,, T h Holding cos for one uni of produc = 1,, P r Sales revenue for one uni of produc in period = 1,, P, = 1,, T Cp Producion cos for one uni of produc = 1,, P p Processing ime for one uni of produc = 1,, P S i Seup cos for ransiion from produc i o produc i = = 1,, P S i Seup ime for ransiion from produc i o produc i = = 1,, P I 0 Iniial invenory level for produc = 1,, P 3.2 The model of PGLSP In his model lo-sizing variable is disaggregaed according o period which is used and scheduling mehod uses he concep of TSP. Scheduling mehod is based on Almada-lobo e al [8] model and uses he concep of TSP. This model and is decision variables are proposed as follows: Q k : Producion quaniy of Produc in period o fulfil he demand of period k D : The acceped demand of produc in period X i : Binary variable which is 1 when seup occur from produc i o produc in period A : Posiive variable which is 1 when machine is seup for produc a he beginning of period Y : Auxiliary variable ha assign produc o period Q 0 : Quaniy of iniial invenory which is used in period R 0 : Unused porion of iniial invenory a he end of he planning horizon M (1) 232

4 Subec o: (2) (3) (4) (5) (6) (7) (8) i (9) i (10) (11) (12) The obecive funcion of he model (1) is o maximize oal revenue minus producion and invenory cos of producion quaniies in differen periods, seup coss, iniial invenory holding cos unil he period which is used, and he unused iniial invenory holding cos. CQ is a parameer which combines producion cos and invenory cos of produc which will be used periods afer producion period. This parameer is calculaed by saemen (13). i (13) Consrains (2) show ha he acceped demand in period is fulfilled by producions for his period or iniial invenory. Consrains (3) guaranee ha he acceped demand in each period is beween is lower and upper bounds. Consrains (4) ensure ha he iniial invenory is equal o iniial invenory which used in differen periods and he unused porion of i. Consrains (5) guaranee ha he machine is seup for producion, and he producion upper bound in hese consrains is shown in saemen (14). 233

5 (14) Consrain (6) shows he capaciy limi for producion. Consrains (7) o (9) deermine he sequence of producs, changing he seup saes and ransiion of seup saes beween differen periods. Consrain (7) shows ha a he beginning of each period here is only one seup sae. Consrain (8) guaranees ha he seup saes nework is a conneced nework and consrain (9) avoids disconneced sub ours. Consrains (10) o (12) show he differen kinds of variables in he model. We can omi variable D and use he saemen k =0 Q k insead, consequenly, consrain (2) is omied from he model, and consrain (3) changed o consrain (15). Wih his change, a se of variables and consrains are omied from he model, bu because he model is MIP, his change should be well sudied. k Ld Q k Ud =0 = 1,, P, = 1,, T (15) Number of binary variables and coninuous non-negaive variables for each model are represened in able 2. Table 2: Number of variables Coninuous non-negaive variables Binary variables 2 P In he models, because producion los are disaggregaed according o he periods in which hey will be used, he number of producion variables and consrains is increased inensively by he increase in number of periods. 4. Numerical Experimens 4.1 Tes daa In his par, we discuss experimenal resuls. We use he mehod of Hass and Kimms [19] for generaing daa ses for GLSP wih some modificaions when needed. The mehod of producing hese daa is represened as follows. Demand upper bounds, Ud, have uniform disribuion beween [50, 70] and is lower bounds, Ld, have he same disribuion beween [30, 49]. Producion capaciy in each period is calculaed by saemen (16). (16) In saemen (16) capaciy in each period is calculaed based on he demands upper and lower bounds in he period. uil is a parameer which aduss he capaciy and if i is increased, he capaciy will be igher. In differen scenarios his muliplier is equal o 0.4, 0.6, or 0.8. Seup imes, S i, have uniform disribuion wih parameers [2,10] and seup coss are generaed by saemen (17). Producion cos of one uni of a produc is generaed based on is average seup coss and is calculaed by saemen (18). In his saemen Co is a muliplier for adusing producion cos. This parameer in differen scenarios is equal o 8 or 12. (17) 234

6 (18) Sale price for each produc is calculaed by relaion (19). Sales prices have relaion wih producion coss which is adused by Pr as a muliplier. This muliplier in a scenario wih high profi is equal o 1.25 and in a scenario wih low profi is equal o 1.1. (19) Wihou losing generaliy, we assume ha iniial invenory is zero and machine is seup for produc 1 a he beginning of planning horizon. According o differen mulipliers, 12 problem ses are proposed. These ses are shown in able 3. Table 3: Daa ses and heir mulipliers NO Uil Co Pr Compuaional ess were execued on a compuer wih a 2.8 GHZ CPU and 3.25 GB RAM. For solving mahemaical models we use GAMS 22.1 sofware and CPLEX solver. According o large amoun of ess, for organizing inpu parameers and oupu resuls, we link C++ and GAMS. 4.2 Numerical ess In his par he performance of he model and he effec of omiing variable D is sudied. As menioned before, we can omi his variable from models and subsiue he equal saemen. To sudy his change, problems wih differen sizes have been generaed. Number of producs and periods for hese problems are presened in able 4. Table 4: Number of produc and periods for Numerical ess No. P T No. P T No. P T No. P T Two saes of he Model were run in a ime limi of 3600 seconds. For comparison, he percenage difference beween compuaion ime of an sae and he bes compuaion ime is calculaed. The bes compuaion ime is he minimum of he wo saes compuaion imes for each problem. The average of his percenage (AvgDif%) for all problems in a se is represened as an evaluaor for he model in ha se. The higher his evaluaor is, he worse is he model performance. The sandard deviaions of hese percenages (DevDif) are also calculaed. According o ime limi, number of problems in each se in which he bes soluion is found (nbes) is also represened. Comparaive resuls are shown in able

7 Table 5: The resuls of models comparisons model wih variable D model wihou variable D No. Uil Co Pr nbes AvgDif% DevDif nbes AvgDif% DevDif As we can see in able 5 he performance of he model saes are differen for various ses. In oher words, alhough he second sae has a beer performance in comparison wih he firs we canno inroduce one sae as he bes one. For example he model wihou D variable in he se 1 has he wors performance; on he oher hand, he model has he bes performance in he 12h se. To sudy he effec of differen parameers on performance of he models, previous evaluaors are calculaed for differen inpu parameers. To do his, problems in differen daa ses wih similar parameers are pu in o one se and evaluaors are calculaed for each new se. Table 6 shows he effec of mulipliers uil, Co and Pr on model performance. NO. Table 6: Mulipliers effec on models performance model wih variable D model wihou variable D Muliplier nbes AvgDif% DevDif nbes AvgDif% DevDif Uil Co Pr Toal

8 Omiing D variables from he model has been invesigaed and we can say when hese variables can be easily defined, heir exisence resuls in beer performance. When he capaciy is no igh hese variables are more easily deermined. For example when uil is equal o 0.4, he capaciy is high so invenory coss are low and he D can be defined easily by he upper bounds of demands because higher producion means higher revenue. When he uil is equal o 0.8 and capaciy is igher, he D variables canno be defined easily and heir exisence increases he number of variables and consrains of he models. In he second par of able 6, he effec of muliplier Co is shown. In boh scenarios model wihou variable D has much beer performance; his means ha he effec of his muliplier is no he same as uil. When Co is equal o 12, he producion cos is higher. In his case, performance of boh model saes become worse in comparison wih he case ha Co is equal o 8. Because by increase in coss, he profi of each produc is decreased, he disincion beween producs is decreased and selecing producs becomes harder. The hird par shows he effec of muliplier Pr on he model performance. When his muliplier becomes higher, he price of each produc increases and producs become more disincive and Producion decisions easier. From able 6, he performance of boh saes becomes beer when his parameer is increased from 1.1 o When Pr is equal o 1.25 he performance of model wih D variables is beer han he model wihou D, because in his scenario high profi resul in more disincive producs so i is easier o define D. Models performances according o all solved problems are shown in able 6 in Toal secion. The model performance wihou D variables is beer han he model wih D. Highes number of problems wih bes soluions also perains o he firs sae. So in general, he model wihou D is beer in hese samples. 4.3 Solved problems size Since he model wihou D variables has he beer performance in general, o sudy he dimensions of problems which are solved opimally, we use his model sae wih a ime limi of 7200 seconds. In able 7, he maximum number of producs wih a definie number of periods in differen ses is represened. The dimensions of solved problem in ses 9, 11, and 12 are low in comparison wih oher ses. In hese ses, according o low disincion beween producs, he low dimensions are accepable. In a number of problems due o low memory space we couldn find any opimal soluions. NO. Uil Co Table 7: solved problems size Pr Number of periods * *Opimal soluion is no found in 7200 second Maximum number of producs 237

9 5. Conclusion and Fuure Research Direcions In his paper, profi maximizaion general lo-sizing and scheduling problem wih demand choice flexibiliy is sudied. In his problem, decisions abou lo-sizing, scheduling and demand selecion are made in order o maximize he oal revenue minus producion, holding, and seup coss. This new problem which is an exension of GLSP by adding he assumpion of flexibiliy in choosing demands, inegraes wo differen levels of producion planning, shor-erm planning and mid-erm scheduling. The resul of his model is he acceped demand in each period for each produc, he quaniy of producion los, and he sequence of hem. In his paper a mahemaical model represened for PGLSP. Represening exac mehods which ry o find an opimal soluion based on he model srucure like branch and bound algorihm is a good aspec for fuure research. Efficien heurisic and mea-heurisic algorihms are also appropriae mehods o solve PGLSP. Considering he effec of ransporaion coss in choosing demands, obaining he delivery ime of each produc as one of he oupu of he model and exension of he problem environmen o more complex one like flow shop and ob shop are suggesed for fuure researches. References 1. Quad, D., Lo-sizing and scheduling for flexible flow lines, Verlag: Springer. 2. Shen, Z.J.M., A profi-maximizing supply chain nework design model wih demand choice flexibiliyˮ. Operaions Research Leers, 34(6), Fleischmann, B., and Meyr, H., The general losizing and scheduling problemˮ. OR Specrum, 19 (1), Meyr, H., Simulaneous losizing and scheduling by combining local search wih dual reopimizaion ˮ. European Journal of Operaional Research, 120(2), KoÇLar, A., The general lo sizing and scheduling problem wih sequence dependen change oversˮ. Thesis (MSc) Middle Eas Technical Universiy. 6. Gupa, D., and Magnusson, T., The capaciaed lo-sizing and scheduling problem wih sequencedependen seup coss and seup imesˮ. Compuers and Operaions Research, 32(4), Almada-Lobo, B., Oliveira, J.F., and Anónia Carravilla, M., A noe on he capaciaed lo-sizing and scheduling problem wih sequence-dependen seup coss and seup imesˮ, Compuers and Operaions Research, 35 (4), Almada-Lobo, B., Klaban, D., Carravilla, M.A., and Oliveira, J.F., Single machine muli-produc capaciaed lo sizing wih sequence-dependen seupsˮ. Inernaional Journal of Producion Research, 45 (20), Paloch, M., Schmid, G., and Kovalyov, M.Y., Heurisic algorihms for losize scheduling wih applicaion in he obacco indusryˮ. Compuers & Indusrial Engineering, 39(3-4), Ferreira, D., Morabio, R., and Rangel, S., Soluion approaches for he sof drink inegraed producion lo sizing and scheduling problemˮ. European Journal of Operaional Research, 196(2), Toledo, C.F.M, França, P.M., Morabio, R. & Kimms, A., Muli-populaion geneic algorihm o solve he synchronized and inegraed wo-level lo sizing and scheduling problem ˮ. Inernaional Journal of Producion Research, 47(11), de Arauo, S.A., Arenales, M.N., and Clark, A.R., Lo sizing and furnace scheduling in small foundriesˮ, Compuers and Operaions Research, 35(3), Toso, E.A.V., Morabio, R., and Clark, A.R., Lo sizing and sequencing opimisaion a an animal-feed planˮ. Compuers and Indusrial Engineering, 57(3), Mongomery, D.C., and Johnson, L.A., Operaions Research in Producion Planning, Scheduling, and Invenory Conrolˮ, New York: Wiley. 15. Haugen, K.K., Olsad, A., and Peersen, B.I., The profi maximizing capaciaed lo-size (PCLSP) problemˮ. European Journal of Operaional Research, 176(1), Haugen, K.K., Olsad, A., and Peersen, B.I., Solving Large-scale Profi Maximizaion Capaciaed Losize Problems by Heurisic Mehodsˮ. Journal of Mahemaical Modelling and Algorihms, 6(1), Geunes, J., Romein, H.E., and Taaffe, K., Requiremens planning wih pricing and order selecion flexibiliyˮ. Operaions Research, 54(2), Merzifonluoglu, Y., and Geunes, J., Uncapaciaed producion and locaion planning models wih demand fulfillmen flexibiliyˮ. Inernaional Journal of Producion Economics, 102 (2), Haase, K., and Kimms, A., Lo sizing and scheduling wih sequence-dependen seup coss and imes and efficien rescheduling opporuniiesˮ. Inernaional Journal of Producion Economics, 66 (2),