The Order Selection and Lot Sizing Problem in the Make-to-Order Environment

Transcription

1 Florda Internatonal Unversty FIU Dgtal Commons FIU Electronc Theses and Dssertatons Unversty Graduate School The Order Selecton and Lot Szng Problem n the Make-to-Order Envronment Zhongpng Zha Florda Internatonal Unversty, zhazhongpng@hotmal.com DOI: /etd.FI Follow ths and addtonal works at: Recommended Ctaton Zha, Zhongpng, "The Order Selecton and Lot Szng Problem n the Make-to-Order Envronment" (2011). FIU Electronc Theses and Dssertatons Ths work s brought to you for free and open access by the Unversty Graduate School at FIU Dgtal Commons. It has been accepted for ncluson n FIU Electronc Theses and Dssertatons by an authorzed admnstrator of FIU Dgtal Commons. For more nformaton, please contact dcc@fu.edu.

2 FLORIDA INTERNATIONAL UNIVERSITY Mam, Florda THE ORDER SELECTION AND LOT SIZING PROBLEM IN THE MAKE-TO-ORDER ENVIRONMENT A dssertaton submtted n partal fulfllment of the requrements for the degree of DOCTOR OF PHILOSOPHY n INDUSTRIAL AND SYSTEMS ENGINEERING by Zhongpng Zha 2011

3 To: Dean Amr Mrmran College of Engneerng and Computng Ths dssertaton, wrtten by Zhongpng Zha, and enttled The Order Selecton and Lot Szng Problem n the Make-to-Order Envronment, havng been approved n respect to style and ntellectual content, s referred to you for judgment. We have read ths dssertaton and recommend that t be approved. Purushothaman Damodaran Tao L Shh-Mng Lee Chn-Sheng Chen, Major Professor Date of Defense: March 4, 2011 The dssertaton of Zhongpng Zha s approved. Dean Amr Mrmran College of Engneerng and Computng Interm Dean Kevn O Shea Unversty Graduate School Florda Internatonal Unversty, 2011

4 ACKNOWLEDGMENTS I would lke to thank my dssertaton commttee and the ISE Department for ts gudance and support. In partcular, I would lke to express my deepest grattude to my major advsor, Dr. Chn-Sheng Chen, for hs excellent gudance, practcal nsghts and opportunty of partcpatng n hs research project, from whch ths research work was developed. I would lke to thank Dr. Purushothaman Damodaran for hs help n the areas of optmzaton, mathematcal modelng, and schedulng algorthms. I would also lke to thank Dr. Sh-Mng Lee for hs help on data analyss. I am also apprecatve of Dr. Tao L s valuable nputs n the research process. I also wsh to thank Dr. Qansan Shen for helpng me wth numerous experments conducted n the Enterprse Systems Lab. Lastly, my specal thanks go to Dr. Lxang Jang, for hs encouragement and help throughout my graduate study at FIU.

5 ABSTRACT OF THE DISSERTATION THE ORDER SELECTION AND LOT SIZING PROBLEM IN THE MAKE-TO-ORDER ENVIRONMENT by Zhongpng Zha Florda Internatonal Unversty, 2011 Mam, Florda Professor Chn-Sheng Chen, Major Professor Ths research s motvated by the need for consderng lot szng whle acceptng customer orders n a make-to-order (MTO) envronment, n whch each customer order must be delvered by ts due date. Job shop s the typcal operaton model used n an MTO operaton, where the producton planner must make three concurrent decsons; they are order selecton, lot sze, and job schedule. These decsons are usually treated separately n the lterature and are mostly led to heurstc solutons. The frst phase of the study s focused on a formal defnton of the problem. Mathematcal programmng technques are appled to modelng ths problem n terms of ts objectve, decson varables, and constrants. A commercal solver, CPLEX s appled to solve the resultng mxed-nteger lnear programmng model wth small nstances to valdate the mathematcal formulaton. The computatonal result shows t s not practcal for solvng problems of ndustral sze, usng a commercal solver. The second phase of ths study s focused on development of an effectve soluton approach to ths problem of large scale. The proposed soluton approach s an teratve process nvolvng three sequental decson steps of order selecton, lot szng, and lot v

6 schedulng. A range of smple sequencng rules are dentfed for each of the three subproblems. Usng computer smulaton as the tool, an experment s desgned to evaluate ther performance aganst a set of system parameters. For order selecton, the proposed weghted most proft rule performs the best. The shftng bottleneck and the earlest operaton fnsh tme both are the best schedulng rules. For lot szng, the proposed mnmum cost ncrease heurstc, based on the Dxon- Slver method performs the best, when the demand-to-capacty rato at the bottleneck machne s hgh. The proposed mnmum cost heurstc, based on the Wagner-Whtn algorthm s the best lot-szng heurstc for shops of a low demand-to-capacty rato. The proposed heurstc s appled to an ndustral case to further evaluate ts performance. The result shows t can mprove an average of total proft by 16.62%. Ths research contrbutes to the producton plannng research communty wth a complete mathematcal defnton of the problem and an effectve soluton approach to solvng the problem of ndustry scale. v

7 TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION Background and Motvaton Problem Descrpton Research Objectve and Methodology Sgnfcance and Contrbutons Dssertaton Outlne LITERATURE REVIEW Order Selecton Problems Lot Szng Job Shop Schedulng Lot Szng wth Order Selecton Order Selecton under Job Shop Envronments Integraton of Lot Szng and Job Shop Schedulng Summary A MATHEMATICAL FORMULATION Mathematcal Model Model Verfcaton What-If Analyss Usng the Proposed Model Run Tme Analyss HEURISTICS Overvew of Heurstc Approaches Order Selecton Approaches Deselectng Orders Schedulng Producton Lots Mnmum Cost Heurstc (LS1) for Lot Szng Mnmum Cost Increase Heurstc (LS2) for Lot Szng EXPERIEMENTATION Expermental Desgn Result Analyss on Heurstc Methods Comparson between Heurstc Solutons and CPLEX Results A CASE STUDY Descrpton of a Industral Case Implementaton Result Analyss v

8 7 CONCLUSIONS AND FUTURE RESEARCH Summary Future Research LIST OF REFERENCES VITA v

9 LIST OF TABLES TABLE PAGE Table 1-1 Example data... 5 Table 2-1 Order selecton lterature classfcaton Table 2-2 Summary of smlar research problems Table 3-1 Number of decson varables for the proposed model Table 3-2 Customer order related data for the small-sze problem nstance Table 3-3 Job shop related settngs for the small-sze problem nstance Table 3-4 Base case for large-scale problem nstances Table 4-1 Job shop data for dsjunctve graph example Table 4-2 PRDs used for schedulng Table 4-3 Lst of lot sze change methods Table 4-4 Decsons on ncludng future demands Table 5-1 Desgn for basc problem settngs Table 5-2 Desgn of parameters Table 5-3 Lst of heurstc methods Table 5-4 Summary statstcs on soluton qualty (LS2 to LS1) Table 5-5 Statstcs on the rato of heurstc soluton to CPLEX Table 5-6 Sgn test on the rato of heurstc soluton to CPLEX feasble soluton Table 5-7 Rato of run tme (CPLEX to Heurstc) Table 6-1 System parameters for each nstance v

10 LIST OF FIGURES FIGURE PAGE Fgure 1-1 Interrelatons among order selecton, lot szng and job shop schedulng 4 Fgure 1-2 An ntutve feasble soluton to the example nstance 5 Fgure 1-3 An nfeasble schedule of schedulng the most crtcal operaton frst 6 Fgure 1-4 An mproved soluton to the example nstance 6 Fgure 1-5 A feasble schedule for sngle order wth two producton lots 7 Fgure 1-6 An nfeasble schedule for both order A and B 7 Fgure 1-7 A feasble schedule for both order A and B 7 Fgure 2-1 Venn dagram of related lterature 11 Fgure 3-1 Illustraton of sequencng varables and dsjunctve constrants 29 Fgure 3-2 Illustraton of the ncreasng of number of bnary varables 31 Fgure 3-3 AMPL/CPLEX Soluton for the small-sze problem nstance 32 Fgure 3-4 Optmal soluton after ncreasng setup cost 34 Fgure 3-5 Optmal soluton after change of engneerng 35 Fgure 3-6 Optmal soluton after decreasng order ntal gross proft 36 Fgure 3-7 Soluton for the case wth an earler due date 37 Fgure 3-8 Soluton for the case wth earlest delvery 37 Fgure 3-9 CPLEX run tme for dfferent problem szes 38 Fgure 4-1 Order process drecton among sets of orders 40 Fgure 4-2 Illustraton of recourse utlzaton n deselectng order process 41 Fgure 4-3 A network presentaton of lot szng 43 Fgure 4-4 Illustraton of deselectng multple orders to acheve feasblty 48 x

11 Fgure 4-5 An example of the dsjunctve graph 50 Fgure 4-6 Illustraton of the dsjunctve graph for schedulng producton lots 53 Fgure 4-7 Illustraton of constrants among operatons 57 Fgure 4-8 Illustraton of shftng schedule of an operaton when changng lot szes 59 Fgure 4-9 Two drectons of lot sze change 59 Fgure 4-10 Effect on resource consumpton by changng lot szes 61 Fgure 4-11 Determnaton on lot sze change method 64 Fgure 4-12 Procedure of achevng feasblty by lot sze change 66 Fgure 4-13 Flow chart of LS2 heurstc at one tme perod 73 Fgure 5-1 Tree dagram on soluton qualty wth ncludng problem settng factors 81 Fgure 5-2 Soluton qualty of dfferent schedulng rules 84 Fgure 5-3 Run tme comparson for dfferent schedulng rules 84 Fgure 5-4 Box-plot for proft and tme rato (LS2 to LS1) 85 Fgure 5-5 Tree dagram of run tme (n mnutes) 86 Fgure 5-6 Run tme vs. Number of products for large-scale problem nstances 88 Fgure 5-7 Comparson on order selecton rules 89 Fgure 5-8 The proposed heurstc procedure 90 Fgure 5-9 Categores of CPLEX solutons 91 Fgure 5-10 Factors affectng run tme of CPLEX 92 Fgure 5-11 Relatve frequency of relmpgap n LP relaxaton solutons 93 Fgure 6-1 An exemplary process of work order 97 Fgure 6-2 Result of mplementng proposed heurstc approach 99 Fgure 7-1 Rato of heurstc soluton to CPLEX soluton 102 x

12 1 INTRODUCTION 1.1 Background and Motvaton The operaton modes of manufacturng enterprses are classfed nto two man categores: make-to-stock (MTS) and make-to-order (MTO). The basc dstncton between them s tmng of producton for customer orders. In the MTS mode, producton plan s based on demand forecasts. In the MTO mode, producton starts only after a customer order s receved. In the market place of rapdly ncreasng global competton, MTO s ganng ts popularty because t addresses ndvdual needs by mass-customzng each product (Chen 2006). MTO orders may vary sgnfcantly on ther routngs, materal requrements, and engneerng toolng, etc. Due to the producton nature of wde product/process varety and small quantty, job shop s the typcal operaton model used n an MTO operaton. To promptly respond to customer demands, detaled producton schedulng s mportant to MTO operatons to meet rgd delvery commtment. (Drexl and Kmms 1997) consder t as a concurrent szng and schedulng problem, assumng acceptng all orders. Therefore two questons to be answered are when and how many products to be produced over the plannng horzon of multple tme perods. Typcal lot szng problems consder setup cost and holdng cost (Jans and Degraeve 2005). Setup cost s assocated wth preparng the machne for processng. Holdng cost s the expense spent on mantanng goods n stock. The total setup cost decreases as the lot sze goes up and the number of setups goes down. However, the holdng cost goes up along wth the nventory level. It ncurs no holdng cost f the exact amount s produced that satsfes every delvery commtment (lot-for-lot); however, the total setup cost may ncrease as more 1

13 setups are needed. The prmary objectve for the lot szng problem s thus to balance between the setup cost and the holdng cost. Asde from cost-savng, lot szng can also be used to mprove order feasblty (Low et al. 2004). For example, lot szng can be appled to splttng a large producton order nto smaller producton lots, nestng them n a schedule such that the overall lead tme s reduced. In the MTO envronment, there s an ncrease demand for on-tme delvery. Ontme delvery helps the customer reduce nventory and ensure an effectve supply chan. Consequently on-tme delvery of orders has become mportant to MTO customers (Charnsrsakskul et al. 2004). In an MTO operaton, ncomng orders are revewed perodcally (per day or week). When ncomng orders exceed shop capacty, rejecton of ncomng has to come to play; or expected delvery commtment needs to be renegotated. In the meantme, lot szng and detaled schedulng need to be exercsed to ensure schedule feasblty. As a result, the MTO producton planner needs to make three decsons n concurrence: (1) whch ncomng customer orders to select, (2) how to splt each order (f selected) nto producton lots, and (3) how to schedule each producton lot n a job shop. 1.2 Problem Descrpton Ths study focuses on the problem of selectng a subset of ncomng customer orders to maxmze the total proft, whle meetng the deadlne of each selected order. Each customer order comes wth one delvery (product) tem only. The product routng s known and fxed. Each order may prescrbe more than one delvery date for a fxed quantty. Each commtment s vewed as a delvery deadlne; no late delvery s allowed. Job shop s used as the producton mode. Each operaton n the routng requres a setup n 2

14 addton to ts processng tme, whch s proportonal to the producton lot sze. Setup cost s usually defned by setup tme and unt setup cost. In ths problem, both setup cost and tme are fxed for each machne type and order type. Therefore, the tme on machne requred for a producton lot conssts of ts setup tme and processng tme. A producton lot may be scheduled over multple tme ntervals n the plannng horzon. Each producton lot travels down ts routng as a whole. It cannot be splt. It can be one machne at a tme, and one machne can process one lot at a tme. Ths problem also consders nventory, whose cost s defned by nventory level and length of holdng. The WIP s not consdered as nventory untl t becomes fnshed goods when the last operaton s completed. The objectve of ths problem s to maxmze the total proft, whch s defned as the ntal proft of selected orders mnus lot szng cost, whch conssts of setup and holdng costs. The ntal gross proft for a customer order s defned as the prce commtted by the customer for the order mnus the fxed manufacturng costs as defned n the routng, whch does not nclude setup and holdng costs. Ths problem assumes (1) the producton system s stable and there s no machne breakdown; (2) the nventory cost for work n process (WIP) s neglgble; (3) the buffer space between dfferent stages s assumed to be nfnte; (4) each producton lot s processed as a batch, whch moves n a lot; (5) preempton and re-crculaton are not allowed; and (6) all processng and setup tmes are determnstc. Ths research s ntended to study the ntegrated problem of order selecton, lot szng and job shop schedulng. Fgure 1-1 demonstrates the nature of the decson problem, whose nput s a set of ncomng orders. Each order comes wth a specfc 3

15 delvery commtment (.e., due dates and quantty for each due date), customer commtted prce, and producton routng. The decsons are order selecton, lot sze and a detaled producton schedule. The decson starts wth order selecton. The ntally selected orders are fed to lot szng decson, whch are n turn fed for detaled schedulng. The decsons are looped back to mprove feasblty and total proft. Adjustng lot sze may change lot szng cost and thus order selecton decson. Fgure 1-1 The order selecton, lot szng and job shop schedulng process The followng example llustrates the dynamcs among order selecton, lot szng and job shop schedulng decsons. Ths problem consders three orders n a plannng horzon of three tme perods. Each tme perod s 10 hours. Other data for ths problem are summarzed n Table 1-1. The order quantty for order A s 10; the due date s n 20th hour. The routng s machne 1 frst and then machne 2. The holdng cost s $5 per unt (tem) per perod of tme. The unt processng tme for each product on each machne s fxed for one hour. Its setup tmes and costs for both machnes 1 and 2 are 1 hour and $15. For order B, there are two delveres. The frst one s n 20th hour for 2 unts and n 30th hour for another 5 unts. 4

16 Order Intal Gross Proft Delvery 1 Delvery 2 QTY Due date Table 1-1 Example data QTY Due date Route Holdng cost Setup tme Setup cost m1 m2 m1 m2 A m1m B m2m C m If order A s processed as one lot (lot sze=10), the processng tme on each machne wll be 11 hours ncludng one hour of setup. Therefore, the flow tme wll be 22 hours, whch apparently exceeds ts due date. Thus, t s nfeasble to complete order A by ts due date. Therefore, only the other two orders can be further consdered. Fgure 1-2 shows a feasble schedule of orders B and C. Both meet ther due dates. The notaton B3, for example, denotes the producton lot of order B to be completed n the thrd perod. There are two producton lots of order B and one lot of order C. There ncurs no nventory and thus no holdng cost n ths producton schedule. The total lot szng cost n ths example conssts of only the setup cost whch s $110. The optmal total proft s $590. Fgure 1-2 An ntutve feasble soluton to the example nstance One way to mprove the soluton n Fgure 1-2 s to combne the two demands for order B nto one producton lot (lot sze of 7). When the producton lot s to be completed at the second perod, the total lot szng cost wll be reduced by $25. By applyng the 5

17 crtcal rato rule, B2 s scheduled before C2, for both are due n the second perod and yet B2 needs longer processng tme than C2. If so, the earlest fnsh tme for order C s n the 25th hour as shown n Fgure 1-3. The schedule s nfeasble because order C s late for 5 hours. If order C s scheduled before order B, then a feasble schedule exsts as shown n Fgure 1-4. The total proft for ths case s $615. Fgure 1-3 An nfeasble schedule of schedulng the most crtcal operaton frst Fgure 1-4 An mproved soluton to the example nstance Both feasble schedules do not nclude order A due to ts long flow tme. However, ts flow tme may be shortened f the order s dvded nto smaller producton lots and scheduled n parallel (Lxang and Gachett 2008). Fgure 1-5 shows order A s dvded nto two producton lots of A1 and A2 wth x 1 and x 2 as ther lot sze. A feasble schedule s shown n the fgure wth x 1= 3 and x 2 = 7. A2 s scheduled to complete at 20 th hour. In prncple, x 1 should be as smaller as possble, to mnmze ts holdng cost. Fgure 1-6 consders addng orders B and C to the schedule, after order A has been scheduled as two producton lots. Under ths stuaton, only order C can be 6

18 scheduled. Though order B s more proftable, t s not feasble, assumng lot-for-lot schedulng s practced. The fgure shows the completon for B3 at 34th hour, a delay of 4 hours. Fgure 1-5 A feasble schedule for sngle order wth two producton lots Fgure 1-6 An nfeasble schedule for both order A and B The nfeasblty problem can be solved by movng 2 unts from B3 to B2 and concurrently swappng the schedule for B2 and A1. Fgure 1-7 shows a feasble for orders A and B. The total proft s $750, whch s the best soluton so far. Fgure 1-7 A feasble schedule for both order A and B 7

19 The example demonstrates that each decson among order selecton, lot szng and schedulng can affect other decsons and the objectve of optmal proft. Any ndvdual decson on order selecton, lot szng and schedulng may not lead to the optmum. Wth n mnd that the tral-and-error approach s neffcent especally when dealng wth large sze problems, ths research ams at a thorough study of ths concurrent order selecton, lot szng and schedulng problem. 1.3 Research Objectve and Methodology The prmary objectve of ths research s to formally defne ths problem and develop an effectve soluton technque for solvng ths problem of large sze. The frst phase of ths study focuses on an analytcal defnton of ths job shop problem n concurrence wth lot szng and order selecton consderaton. The problem s modeled as a mxed nteger lnear program (MILP) wth ts objectve, decson varables, and constrants. The proposed model s solved wth a commercal solver, CPLEX. The model s solved to optmum for small problems to evaluate ts behavor and performance. The lot szng part of the problem s consdered an NP hard problem by (Chen and Thzy 1990). The job shop schedulng component s dentfed as a NP hard problem by (Blazewcz et al. 1996). Therefore, the problem that ntegrates order selecton, lot szng and job shop schedulng s also an NP-hard problem. It s nfeasble to solve ths problem analytcally when the problem sze s large. The second phase of ths study thus s focused on development of an effectve soluton technque for ths problem of large sze. Heurstcs are commonly appled to solve complex problems. They generally lead to good solutons wthn lmted computatonal tme, though they may not be optmal. Ths study proposes an teratve process for ths decson problem of concurrent order 8

20 selecton, lot szng and lot schedulng. As part of ths study, an experment s desred to examne the characterstcs (ncludng soluton qualty and run tme) of each decson under varous smple heurstcs and rules. The results are compared to optmal solutons and/or upper bounds, generated from the commercal solver CPLEX. They are used as benchmarks to evaluate the qualty of the proposed soluton method. In addton, an ndustral case s used to further valdate the proposed method and ts applcablty of solvng ndustral problems. 1.4 Sgnfcance and Contrbutons Even though there s volumnous research lterature n the producton plannng area, ths partcular problem of nterest ntegratng order selecton, lot szng, and job shop schedulng has not been studed. Related studes consder at most two of the three decson problems. Ths research s the frst attempt to address ths concurrent decson problem. The proposed mathematcal model formally defnes order selecton, lot szng, and job shop schedulng decsons n concurrence. The mathematcal formulaton s nnovatve n modelng ts dsjunctve constrants as lnear constrants, such that the model can be solved wth a commercal solver. In addton, the constrant for ensurng a producton lot to be completed n a desgnated tme nterval s unque, as t relates lot szng to job shop schedulng decsons. The proposed heurstc soluton method s effcent for solvng large-scale problems. It s bult on an experment desgned to evaluate performance of smple heurstc rules that are commonly used for these three decsons. The soluton approach makes use of these heurstcs and rules to mprove the decson process. Ths research also leads to dscovery of new rules for solvng ths problem. Among them, the proposed 9

21 weghted most proft rule s the best for order selecton and the earlest operaton fnsh tme and shftng bottleneck rules are best for job shop schedulng, whle the proposed mnmum cost-ncrease rule performs better for lot szng n a heavly loaded shop. These heurstcs alone could help MTO managers to make better order selecton, lot szng and schedulng decsons. In summary, the two major contrbutons by ths research are: (1) the formal defnton of the order selecton and lot szng problem n the job shop envronment, and (2) an effectve soluton technque for solvng large scale problems. 1.5 Dssertaton Outlne The rest of ths dssertaton s organzed as follows. Chapter 2 s a lterature revew of related research n the publc doman. Chapter 3 presents the MILP model wth experments conducted to valdate the proposed model. Chapter 4 presents the framework desgn for the teratve soluton approach. It also summarzes a study of varous heurstcs (both exstng and proposed ones) applcable to each of the three decson problems. Chapter 5 s an experment desgn and analyss for performance evaluaton of the above heurstcs under the proposed soluton framework. Chapter 6 presents a real-lfe case used to assess the applcablty of ths proposed soluton approach. Fnally, conclusons and future research are summarzed n Chapter 7. 10

22 2 LITERATURE REVIEW Ths research ams to unfy three decson problems order selecton, lot szng and job shop schedulng. The exstng lterature s dvded nto three levels, accordng to the number of decson problems consdered. The frst level contans the lterature that only studes one decson problem. The second level contans the lterature that consders two of the three decson problems. Lterature consderng all of them belongs to the thrd level. Although there s a large body of lterature n the frst and the second levels, there s no lterature found for the thrd level. The Venn dagram n Fgure 2-1 llustrates correspondng categores of lterature based on the levels. The abbrevatons for each category wll be used throughout ths dssertaton. The revew on the frst level s presented n Sectons , then followed by the dscusson of the second level n Sectons Secton 2.7 presents a summary of related research. Order Selecton (OS) OS-LS OS-JSS OS-LS-JSS Lot Szng (LS) LS-JSS Job Shop Schedulng (JSS) Fgure 2-1 Venn dagram of related lterature 11

23 2.1 Order Selecton Problems Order selecton has been a topc of growng nterests snce Mller (1969), who studes a queung system wth the objectve of maxmzng the expected value of customer orders. In ths revew, relevant order selecton researches are classfed by three crtera: order arrvals, resource settng and selecton crtera, as shown n Table 2-1. For dynamc order arrvals, customer demands are descrbed wth random dstrbuton; whle for statc arrvals, customer demands are determnstc. Resource settng refers to the producton envronment n whch selected orders are processed. Selecton crtera are a set of objectve functons, nformed by the tradeoff between rewards obtaned from selected orders and the cost of fulfllng them. Heren, meetng latest due date (LDD) refers to that an accepted order must be completed by latest due date; otherwse, t s rejected. Table 2-1 ndcates that order selecton problems wth sngle resource and tardness objectve attracted most research attentons. Table 2-1 Order selecton lterature classfcaton Crtera Character Prevous Research Order arrvals Resource settng Selecton crtera Dynamc Mller (1969), Wester (1992), Jalora (2006) statc Kern (1990), Slotnck (1996) Kern (1990), Wester (1992), Ten Kate (1995), Akkan (1997), Sngle resource Slotnck (1996), Charnsrsakskul (2004), Jalora (2006), Slotnck (2007), Blgnturk (2007) Multple resources Hans (2001), Ebben (2005), Roundy (2005) Earlness Akkan (1997), Charnsrsakskul (2004), Jalora (2006) Guerrero (1988), Kern (1990), Wester (1992), Slotnck Tardness (1996), Akkan (1997), Ghosh (1997), Hans (2001), Lews (2002), Charnsrsakskul (2004), Blgnturk (2007), Slotnck (2007) Meetng LDD Akkan (1997), Blgnturk (2007), Roundy (2005) 12

24 In terms of soluton approaches, although queung theory (Mller 1969), decson theory (Nagraj Balakrshnan 1996), and smulaton (Ten Kate 1995) are proposed n lterature, formulatng mathematcal models and applyng optmzaton technques are more common n order selecton lterature. In addton, heurstcs are developed to solve specfc problems n large scales. Kern and Guerrero (1990) present a conceptual model for demand management n the assemble-to-order envronment. They also formulate a MILP model wth the objectve functon of mnmzng total cost of lateness, nventory and setup. Slotnck and Morton (1996) explore order selecton wth weghted lateness penalty. They propose a branch-and-bound method for small-sze problems and heurstcs for large-sze ones. Ths research s further extended by Lews (2002) to mult-perod schedulng; an optmal dynamc programmng algorthm s devsed. To acheve overall schedulng feasblty for exstng orders and a newly arrved order, Akkan (1997) suggests several practcal methods, ncludng backward schedulng, forward schedulng, what-f analyss, mnmzng fragment cost and compacton. Charnsrsakskul et al. (2004) develop a mxed nteger programmng formulaton, and use numercal analyss to examne order acceptance, schedulng and due-date settng decsons. In ths research, the manufacturer has the flexblty to choose lead-tmes. Roundy et al. (2005) model a job nserton problem (selected orders are nserted nto a set of orders already scheduled) usng MILP. They also propose meta-heurstcs for ths problem, ncludng genetc algorthm (GA), smulated annealng (SA) and Tabu search. Order acceptance wth mnmzng weghted tardness s wdely dscussed. Slotnck and Morton (2007) examne order acceptance wth weghted tardness penalty. 13

25 They present straghtforward separaton of sequencng and job acceptance, together wth a branch-and-bound procedure. Smlar problems on order selecton wth tardness penalty are solved wth SA (Blgnturk et al. 2007) and GA (Rom and Slotnck 2009). 2.2 Lot Szng The frst lot szng model s the renowned Economc Order Quantty (EOQ), developed by Harrs (1913). Comprehensve survey on modelng lot szng problem and soluton approaches can be found n Maes (1988), Karm (2003), Quadt (2008) and among others. Lot szng problems are classfed manly based upon product complexty and exstence of resource constrants. If the fnal product s smply beng produced from raw materals, t s referred as a sngle-level problem. If there exsts parent component relatonshp among the tems, t s regarded as a mult-level problem. When nfnte resource capacty s assumed, lot szng problem s sad to be Uncapactated Lot Szng Problem (ULSP). On the contrary, f capacty constrants are explctly stated, the problem s named as Capactated Lot Szng Problem (CLSP). Except for sngle-level ULSP, the other varants of lot szng problems are strongly NP-hard (Btran and Yanasse 1982; Chen and Thzy 1990). As the research under study only consders smple products, ths revew only focuses on the sngle-level lot szng problem Sngle-level ULSP The EOQ model assumes constant demand rate and nfnte tme horzon. As an extenson to EOQ, Wagner-Whtn (WW) algorthm (Wagner and Whtn 1958) apples to tme-varyng demands and fnte dscrete plannng horzon. It consders all possble alternatves of processng an order n the current or prevous perods. Selecton of 14

26 alternatves s based on a mnmum cost polcy, under whch each lot sze s exactly the sum of a set of future demands. To avod complcated computaton of the WW algorthm, some practcal heurstcs for ULSP are proposed, such as EOQ-MRP, Slver-Meal (S-M), Part-Perod Balancng (PPB), Least Unt Cost (LUC), etc. Detaled descrpton of these heurstcs can be found n Orlcky (1975), Slver (1985), Erc (1986) and Nahmas (1989). A few researchers also compared the performance among ULSP heurstcs. For nstance, Blackburn (1980) compares PPB, S-M and WW n a rollng-schedulng envronment and revealed that smpler Slver-Meal heurstc can provde a cost performance superor to that of the WW algorthm. Gelders and Wassenhove (1981) state that the pror choce of sutable heurstc depends on varablty of demands and partcular cost structure at hand. When the demand varablty s low, EOQ s sutable; otherwse, S-M heurstcs s recommended Sngle-level CLSP Compared to ULSP, CLSP attract more research nterest snce the frst work of Manne (1958). It s more practcal but much more dffcult to be solved. Specalzed heurstcs and mathematcal programmng based approaches are commonly used to solve CLSP. (1) Specalzed heurstcs Specalzed heurstcs generally encompass three steps. The frst step s lot szng, whch s often based on the ULSP heurstcs. For example, Dogramac (1981) and Gunther (1987) smply employ Lot-for-Lot to generate ntal soluton. Dxon and Slver (1981) apply S-M method to ntalze lot szes. If the ntal lot szes for all tems are constructed from the frst perod to the last perod, t s named as perod-by-perod 15

27 method. Ths method can be found n the work of Lambrecht (1979), Dxon (1981), and Maes (1986). On the other hand, tem-by-tem heurstc s proposed by Krca (1994). The ntal plan starts from selectng a sngle tem and plannng the tem over the entre the plannng horzon. The second step s so-called feasblty routne. It s to ensure that all demands are satsfed wthout backloggng and capacty constrants are not volated. It s conducted wth feedback mechansm or look-ahead mechansm. In the feedback mechansm, excess demands are pushed back to an earler perod wth leftover capacty, gven that the savng n setup cost can make up the extra holdng cost (Lambrecht and Vanderveken 1979). Whle n the look-ahead method, the mnmum requred nventory s computed a pror n order to avod capacty volaton n later perods (Dxon and Slver 1981; Maes and Van Wassenhove 1986). The thrd step s to mprove the exstng soluton by adjustng lot sze. For example, Dxon and Slver (1981) ntroduce lot elmnaton, lot mergng, lot nterchange and use of optmal lot sze. Dogramac et al. (1981) propose left-shft procedure that searches for shfts wth the largest reducton n overall cost. Karn and Roll (1982) ntroduce 10 types of shfts wth calculatng cost-savng coeffcent whch s based on the tradeoff between setup and holdng costs. Tabu search s also appled to mprove CLSP soluton; see examples n Hnd (1996) and Karm et al. (2006). The mprovement step nvolves a large number of shfts and feasblty checkng; therefore, t s generally the most tme-consumng step n the lot szng procedure. 16

28 (2) Mathematcal programmng based approaches The revew by Karm et al. (2003) covers the most mportant results n exact and approxmaton algorthms untl They survey commonly used technques, such as branch and bound, LP-relaxaton and network flow algorthm. Recently, Heuvel and Wagelmans (2006) study CLSP wth lnear costs and present a dynamc programmng algorthm that solves the CLSP wth specal cost functon n polynomal tme. Abs and Kedad-Sdhoum (2009) consder safety stock and demand shortage n CLSP. They develop a Lagrangan relaxaton of the capacty constrants to obtan lower and upper bounds. The resultant uncapactated problem s modeled as a fxed-charge network and solved wth a dynamc programmng algorthm. Verma and Sharma (2010) desgn two Lagrangan relaxatons for CLSP wth consderng backloggng and setup tme. In the frst relaxaton, CLSP s relaxed to a mult-tem ULSP. In the second relaxaton, the nventory flow-balance constrant s relaxed; the problem s reduced to a sngle constrant contnuous knapsack problem wth an upper bound on the quantty produced. Compared to specalzed heurstcs, mathematcal programmng based methods usually produce solutons wth better qualty. However, they need more computatonal efforts, so that they are less applcable to real-lfe problems. 2.3 Job Shop Schedulng There are many varants of the JSS problem, accordng to dfferent schedulng objectves and constrants. Asde from job shop problems wth two machnes, or wth the processng tme of operaton s ether 0 or 1 can be solved n polynomal tme, other JSS problems are notorously NP-hard (Blazewcz et al. 1996; Pnedo 2002). In the problem 17

29 under study, lateness s not allowed; therefore, mnmzng makespan of a producton lot s mportant for feasblty of schedulng. If a producton lot s produced earler than demanded, holdng cost wll be ncurred. Mnmzng earlness cost may contrbute to maxmzng overall proft. In addton, rejectng an order causes loss of the correspondng revenue. Mnmzng ths loss s equvalent to mnmzng weghted number of tardy jobs (WNTJ) n a job shop schedulng problem. Therefore, the revew manly concentrates on schedulng problems wth mnmzng makespan, earlness and WNTJ. For makespan mnmzaton, shftng bottleneck method, developed by Adams (1988), s the most notable approxmaton algorthm for JSS. Wth consderng job nterdependency, Dauzere-Peres and Lasserre (1993) modfy Adams heurstcs and obtaned better computatonal performance. More recently, meta-heurstcs are appled nto JSS problems. For example, Huang and Lao (2008) employ Ant Colony Optmzaton (ACO) to generate ntal soluton, whch s further mproved by applyng tabu search teratvely. Zhang et al. (2008) apply a hybrd genetc algorthm for JSS. In ther research, genetc algorthm s used for global exploraton among the populaton; local search served as local explotaton around operaton-based chromosomes. Owng to the popularty of Just-n-Tme (JIT) concept, schedulng problems nvolvng earlness and tardness penaltes have receved consderable research attentons. Early research manly focuses on sngle machne systems; see Baker and Scudder (1990) for a survey. Recently, job shop envronment s consdered. Beck and Refalo (2003) apply a hybrd technque usng constrant programmng and lnear programmng to the earlness/tardness problem n the job shop. Thagarajan (2005) studes JSS wth multlevel jobs, wth the objectve of mnmzng the sum of weghted earlness, weghted 18

30 tardness and weghted flow tme of jobs. A set of dspatchng rules are presented by ncorporatng the relatve costs of earlness, tardness and holdng of jobs n the form of scalar weghts. Phlppe (2008) proposes two Lagrangan relaxatons of JIT schedulng wth relaxaton on precedence constrants and machne constrants, respectvely. For WNTJ problem, Karp (1972) establshes the NP-hardness for the sngle machne system. The specal case wth common due dates can be vewed as knapsack problem so that the weghted shortest processng tme (WSPT) heurstcs can be appled (Pnedo 2002). Other than sngle machne, WNTJ problem are also consdered n parallel machne (Ng et al. 2003; M'Hallah and Bulfn 2005), open shop (Brucker et al. 1993; Galambos and Woegnger 1995; Svetlana 2000; Baptste 2003) and flow shop (Charnsrsakskul et al. 2004). Heurstc solutons are mostly consdered n those problems. Lmted reserach n job shop can be found n Józefowska et. al (1994), who develop dynamc programmng algorthm applyng Jackson s ndexng method (James 1956). In ther work, only job shop wth two machnes s consdered. These objectves aforementoned are generally consdered ndependently. An excepton s Lee (1991), who studes mnmzng weghted number of tardy jobs and weghted earlness/tardness penaltes. Ther research s under a common due date assumpton and agreeable rato condton (f job s relatvely more mportant than job j, the weght of earlness and tardness wll be greater than that of job j ). They proved that the problem s NP-complete n the strong sense, and hence cannot be solved by usng any pseudo-polynomal tme algorthm. 19

31 2.4 Lot Szng wth Order Selecton Although extensve studes have been conducted n the order selecton or the lot szng, extant work ntegratng them s very lmted. Wester et al. (1992) study dfferent order selecton heurstcs n a sngle machne system, wth order arrvals followng Posson dstrbuton. They consder setup tme savng between smlar product groups when makng a schedule. In the monolthc approach, a new schedule s constructed for orders not yet n producton and the new order. The schedule s constructed wth a heurstc that mnmzes maxmum lateness and total setup tme sequentally. In the herarchc approach, re-schedulng of all avalable orders s based on operaton tmes of scheduled orders, the order to be scheduled, and a work content level chosen from smulaton experments. They also propose prorty rules based heurstc named myopc heurstc. It s to select the order that mposes mnmum lateness to exstng orders. The experments showed that monolthc approach performs better than the others. Geunes et al. (2002) study order selecton wth producton plannng problem, n whch dfferent customers order same product wth dfferent prces over tme perods. They provde a shortest path based soluton for uncapactated order selecton problem. Wth takng account of lot sze lmt, a Lagrangan relaxaton algorthm s proposed for the capactated order selecton problem. In terms of lot szng, ths research only consders sngle tem. 2.5 Order Selecton under Job Shop Envronments In the studes on the order selecton, job shop s frst consdered by Ebben et al. (2005), who employ resource loadng methods to support the order acceptance decson. They compared four resource loadng methods: aggregate resource loadng, resource 20

32 loadng per resource, EDD based order acceptance and Branch-and-Prce resource loadng. Ther experments ndcate that sophstcated approaches sgnfcantly outperform the straghtforward approaches when there are tght due dates. More recently, Chen et al. (2009) propose a MILP model for capacty plannng and order selecton problem n the MTO envronment. In ths research, overtme and outsourcng are consdered. The proposed model s further solved by Mestry et al. (2011) wth a branch and prce approach. 2.6 Integraton of Lot Szng and Job Shop Schedulng In the classcal CLSP, resources used by dfferent products can be smply added up. However, n a job shop envronment, workload on each machne depends on detaled schedule of products, whch s subject to precedence constrants. Dauzere-Peres and Lasserre (1994) consder an ntegrated model for lot szng and schedulng n the job shop. They propose a mult-pass decomposton approach that alternatvely solves the ntegrated problem at two levels. One level s lot szng for a gven sequence of jobs on each machne; the other level s sequencng lots wth fxed lot szes. Ther experments also ndcated that a modfed shftng bottleneck heurstc can provde a better soluton than prorty rule-based dspatchng methods. Anwar and Nag (1997) address the ntegrated schedulng and lot-szng problem wth complex assembles. The objectve s to mnmze the cumulatve lead tme of the producton and reduce setup and nventory costs. They propose a two-phase heurstc that addresses both precedence and capacty constrants. Jeong et al. (1999) study a batch splttng method for a job shop schedulng. They employ a modfed shftng bottleneck procedure to generate ntal schedule and then splt a batch to shorten makespan. 21

33 2.7 Summary These three decson problems (order selecton, lot szng and job shop schedulng) are usually treated separately n the lterature. The research that ntegrates two of them s very lmted; there s no research that addresses three decson problems smultaneously. Table 2-2 compares smlar problems n lterature and the problem under study. Table 2-2 Summary of smlar research problems Research Objectve Order Lot szng Job On-tme Soluton selecton shop delvery (Mestry et al. 2011) Maxmze proft Yes No Yes Yes Branch and prce (Dauzere-Peres and Mnmze No Multple tems Yes No Heurstcs Lasserre 1994) cost Setup cost Setup tme (Ebben et al. 2005) Mnmze lateness Yes No Yes Yes Heurstcs/ Branch and prce (Jeong et al. 1999) Mnmze No Multple tems Yes No Heurstcs makespan Setup tme (Wester et al. 1992) Mnmze No Setup tme No Yes Heurstcs lateness (Geunes et al. 2002) Maxmze proft Yes Sngle tem Setup cost No Yes Network flow/ LP relaxaton Proposed research Maxmze proft Yes Mult-tems Setup cost Setup tme Yes Yes Heurstcs In terms of order selecton, the problem under study consders statc order arrvals, multple resources and meetng latest due date of customer demands. There s no lateness cost; but an earlness cost s ncurred f a producton lot s completed earler than 22

34 demanded. From lot szng vew, the problem under study belongs to the class of snglelevel CLSP; the capacty s characterzed by job shop constrants. Both setup cost and tme are explctly defned; however, they are less addressed jontly n lterature. Maxmzng total proft relates to on-tme completon of producton lots and mnmzng weghted number of tardy jobs; but there s no research addressng these ssues concurrently n job shop schedulng lterature. Accordng to ths revew, mathematcal modelng and developng heurstcs are promsng n solvng complcated order selecton and producton plannng problem. However, most exstng models and heurstcs strongly depend on correspondng problem defntons. If extant research fndngs are adopted, modfcatons are needed to accommodate the characterstcs of ths problem. In the next chapter, mathematcal modelng wll be frst dscussed. 23

35 3 A MATHEMATICAL FORMULATION Ths chapter proposes a mathematcal formulaton for the order selecton and lot szng problem n the MTO envronment. Secton 3.1 presents the mathematcal model for the problem under study. The proposed model s verfed n Secton 3.2. Then the usefulness of ths model s llustrated n Secton 3.3. Fnally, the performance of solvng the proposed model wth a commercal solver s evaluated wth a set of numercal experments. 3.1 Mathematcal Model The nput of the problem under study s a set P of customer orders. Each order P only ncludes sngle product. The ntal gross proft ( r ), s the prce of the order excluded by some fxed producton costs (such as labor, utlty, and overhead). The producton plan s needed only for selected orders over a set of N plannng perods. For each perod t N, each demand for customer order ( d t ) must be satsfed wthout delay. Ths on-tme fulfllment s prescrbed by fully schedulng producton lots on a job shop wth a set M of machnes. For each machne ( k k M, the setup tme for each order ) s gven accordng to process plan. For any producton lot, t must be fully processed; therefore the total setup cost ncurred from dfferent machnes on the route A s always fxed. When the last operaton of a producton lot s completed, nventory s ncurred. For each order, the lot szng cost s the sum of setup cost (setup per lot ) and holdng cost (cost rate h ). As there s no machne breakdown, the capacty of each machne n each plannng perod always equals to the length of the plannng perod c. 24

36 The objectve functon of the problem under study s to maxmze total proft, whch s the ntal gross proft excluded by lot szng cost. More detaled nomenclatures used for the problem formulaton are shown as follows. Indces and sets : Order P { 1,2..., p} set of orders t : Tme perod N { 1,2..., n} set of tme perods k : Machne M { 1,2..., m} set of machnes Parameters L t Producton lot of producng fnal product for order durng perod t O tk Operaton of lot L t on machne k h Unt holdng cost of the product for order from one perod to the next Total setup cost for order over ts producton route r Intal gross proft obtaned from acceptng order d t Quantty of demand at the end of perod t from order k Unt processng tme of the product for order on machne k Setup tme for order on machne k k c Length of each tme perod M Set of machnes that can process order ; a M denotes the sze of M A Set of pars of machnes presentng precedence relatons for order 25

37 m Last machne on the route of order P k Set of orders that machne k can process; bk Pk denotes the sze of P k q t Producton quantty lmt for L t Unt processng of order on all machnes over ts route; k k Setup tme of for order on all machnes over ts route; k km M Decson varables Z X t 1, f order s selected 0, otherwse Lot sze of L t I t Inventory level of the product for order at the end of tme perod t Y t 1, f setup for product exsts at perod t 0, otherwse W ' ' t t k 1,f the sequence s 0,otherwse O ' ' t k to O tk S tk Start tme of operaton O tk F tk Fnsh tme of operaton O tk The mathematcal formulaton for the problem under study s presented below. Maxmze I Z r Y P tt t P tt h I t (3.1) 26

38 Subject to X t I, t1 Zdt It P, t N (3.2) X q Y P, t N (3.3) t t t F tk Stk k X t kyt P, t N, k M (3.4) S P, t N, k ' k A (3.5) tk F tk ' S ' ' t k Ftk ctw ' ' P t t k k, t N, ' Pk, t ' N ' t t wth ' or (3.6) W ' ' W ' ' 1 Pk, t N, ' Pk, t ' N t t k t tk ' t t or ' wth ' and (3.7) F Y ( t 1 c P, t N tm t ) (3.8) F nc Y ( n t c P, t N (3.9) tm t ) I 0 =0 I n =0 P (3.10) P (3.11) X t, X t, I t, tk S, F 0 (3.12) tk I t are nteger (3.13) 1 Y t 0 f product s setup for producng otherwse at perod t (3.14) 1 Z 0 f order s selected otherwse (3.15) 1 f the sequence s O ' ' to O t k tk W ' ' t t k (3.16) 0 otherwse 27

39 Expresson (3.1) shows the objectve functon, n whch total ntal gross proft, setup cost and holdng cost are denoted wth P Z r, P tn Y t and h I t, P tn respectvely. Constrants (3.2) defne nventory balance. They express that f an order s selected, the enterng nventory ( I,t1 ) added the current perod producton ( X t ) are used to satsfy the demand ( d t ); what remans s the nventory at the end of current perod ( I t ). Constrants (3.2) and Constrants (3.12) (no nventory shortage) ensure that all accepted orders are satsfed on tme. The couplng between setup and producton s descrbed n Constrants (3.3). If there s a producton lot, a setup s needed. The lot sze lmt q t s set n two ways. Frst, X t cannot exceed the total demand of order,.e., n X t d j j1. Second, for L t, the largest unt processng tme on machnes forces that the maxmum allowable lot sze s n ct ct. Therefore, qt mn{, dj}. max{ } max{ } k k k k j1 Constrants (3.4) transfer lot szng nto operatonal level. A setup tme s requred when a producton lot s processed; the processng tme s proportonal to lot sze. These constrants also ndcate no preempton s allowed. Constrants (3.5) state the precedence constrants. For each order, operatons follow a predefned route wth ' k as the precedent of k. The succeedng operaton can only start after ts precedent s completed. Constrants (3.6) and (3.7) are dsjunctve constrants, whch ensure that any two operatons cannot be processed smultaneously on the same machne. They are derved from the followng constrants: s or s ' ftk tk f ' ' t k ' ' t k or t t' (3.17) 28

40 Constrants (3.17) cannot be handled by most of commercal programs, so they are reformulated by ntroducng auxlary varable W t ' t ' k, whch ndcates the sequence of two operatons O tk and O t ' k '. If the sequencng varable W ' ' 0, Constrants (3.6) wll t t k result n S ' F,.e., O ' ' t ' s processed after O k tk. If W ' ' 1, Constrants (3.6) turn t k tk t t k out to be redundant, because the rght hand sde s non-postve. However, Constrants (3.7) wll enforce W ' ' 0, whch n turn mpose S tk F t, accordng to Constrants k t tk (3.6). These two stuatons are llustrated n Fgure 3-1. ' ' Fgure 3-1 Illustraton of sequencng varables and dsjunctve constrants Constrants (3.8) and (3.9) reflect the lnk between lot szng and schedulng. A producton lot s completed only when the last operaton s completed. For producton lot L t, the completon tme should fall nsde perod t. Constrant (3.10) and (3.11) show that there s no nventory at the begnnng or end of the plannng horzon. They reflect the basc MTO characterstc that producton s only trggered by customer orders and produced products are all used to satsfy customer demands. Constrants (3.12) mpose non-negatvty constrants for the lot sze, nventory and schedulng varables. Constrants (3.13) further enforce that lot sze and nventory varables are ntegers only. Fnally, Constrants (3.14) ~ (3.16) mpose the bnary restrctons on decson varables Y, Z and W, respectvely. 29