An Extension of the Multi-machine Multi-level Proportional Lot Sizing and Scheduling Model for Product-Life-Cycle Demand

Transcription

1 International Journal of Operations Research Vol., No., (00) An Extension of the Multi-machine Multi-level Proportional Lot Sizing an Scheuling Moel for Prouct-Life-Cycle Ping-eng Chang *, Ching-Hsiang Chang, an Chia-sung Chen Department of Inustrial Engineering an Enterprise Information, Box 98, unghai University, aichung 07, aiwan, ROC Abstract Often lot sizing an scheuling moels have neglecte the consieration of influence of prouct life cycles (PLCs). But, accoringly as esirable in a firm s prouction strategy, ecisions of lot sizing scheuling may have to respon to the stage changes in a prouct s life cycle. In this paper, we propose an extension of the lot sizing scheuling, the multi-item, multi-level prouct-structure an multi-machine proportional lot sizing scheuling moel by Kimms (999), by the concept of PLC. he avantages have been realize an inclue both the setup an holing cost improvements an an improve lot sizing scheuling that better matches the proucts emans in the PLCs. In aition, the combine effects of the setup (cost) learning effect an cash flow are examine in the moel an the genetic algorithm is aopte as the solution ai tool. A numerical example is provie an emonstrates the avantages of this moel an the effects of the factors consiere. Keywors Proportional lot sizing an scheuling problem; Prouct life cycle; Cash flow, Learning curve; Genetic algorithms. INRODUCION Lot sizing an scheuling or etermining which quantities of what items have to be prouce at what time to meet the emans of the proucts is two of the most important problems in prouction planning. Many researchers have evelope moels an approaches to these problems (e.g., see Drexl an Kimms, 997 for a goo review). Four types of problems have been reporte an investigate an inclue the capacitate lot sizing problems (e.g. Barany et al., 98; Chen an hizy, 990; Haase, 99), iscrete lot sizing an scheuling problems (e.g. Fleschman, 99), continuous setup lot sizing problems (e.g. Bitran an Matsuo, 98;Karnarkar an Schrage, 98), an proportional lot sizing an scheuling problems (PLSP) (e.g. Drexl an Haase, 99; Kimms, 99). In this paper, we inten to propose an extension of the PLSP with multi-item multi-level-prouct-structure an multi-machines (PLSP-MM) (Kimms, 999) for the concern of prouct-life-cycle (PLC) emans. As the PLC concern can be an important factor, but it has been neglecte in the existing moel. In this PLSP-MM problem, we make the same assumptions as those one by Kimms (999) besies the PLC emans. We assume a finite horizon an which is ivie into iscrete perios with ynamic an eterministic external emans for the items. No shortage or backlog is allowe. Items may consist of other items that are to be prouce before the former items can be prouce. he intermeiate items may in turn cause internal emans for other items an so on. Also, we assume that the capacity per perio of each machine is constraine. he prouction of one item consumes an item-specific amount of machine capacity. o prouce an item the machine has to be set up for the item. Every setup causes item-specific setup costs. Setup time an sequence epenent setup cost are ignore. If a remaining capacity occurs in a perio it can be use to scheule another item. An ile perios between lots of a same item o not cause aitional setup costs. Items that are prouce within a perio without a eman in that perio are store in inventory to meet future emans, while inventory is assume to be uncapacitate but have holing costs. herefore, the obective is to fin the prouction plan feasible an cheap, i.e., the sum of setup an holing costs must be as low as possible. Furthermore, uner prouct life cycles (Polli an Cook, 99; Rink an Swan, 979), a firm often has to change its market strategies several times (Lambkin an Day, 989), (Levitt, 9), which is not only ue to the competitors challenges but also that the chance of the prouct s market economic state an its role as acting in the market may have change (Aaker, 99). herefore, a firm often has to prepare a series of strategies for ifferent stages of PLCs (Kurawarwala an Matsou, 998) an the strategies for prouction to meet the ifferent stages of PLCs. Existing lot sizing an scheuling moels often have focuse on the problems without the consieration of PLC stages an changes of emans in the PLCs. hough Hill (99) has propose a moel with the PLC concern, but his assumption was mae on a single item, single-prouct-level moel an cannot solve the multi-item multi-level prouct structure problems. Especially it is not consiere that the lot sizes of the proucts therefore may have to respon to * Corresponing author, ptchang@ie.thu.eu.tw.

2 IJOR Vol., No., (00) the stage changes of the PLC. It is only a pure inventory moel instea of multi-item scheuling an inventory moel. herefore, in this paper the PLSP moel will be consiere with the PLC concept. Besies that, the effects of the cash flow an learning curve (Argote an Epple, 990) will also be examine. he technique of genetic algorithms (GAs) will be aopte as the solution tool for the current moel. he remainer of this paper is organize as: Section provies the moel evelopment. Section provies the solution proceure by incorporating the genetic algorithm. Section provies a numerical example for emonstration. In section we raw conclusions.. MODEL DEVELOPMEN. he multi-level PLSP with multiple machines (PLSP-MM) In this subsection the PLSP-MM moel by Kimms (999) will be reviewe. Aitional assumptions in aition to those given in the last section will be provie as follows. In a multi-level prouct structure, the preceence relationships among the items can be efine as an acyclic gozinto-structure of a general type. Several resources (or machines) can be available for manufacturing the items an each item is prouce on item-specific machines (or resources). Also, positive lea-times of the items can be efine ue to the technological restrictions such as cooling or transportation for instance. Moreover, items may share common machines (or resources), some of them may be scarce, an capacities of these resources may vary over time. For this problem, able efines the ecision variables. able provies the problem parameters neee. Using these symbols, the PLSP-MM may be moelle as a mixe integer program (999) as follows: able. Decision variables Symbol Definition I t Inventory for item at the en of perio t. q t Prouction quantity of item in perio t. x t y t J = i= Binary variable for inicating whether a setup for item occurs in perio t x = ) or not (x t = 0). Binary variable which inicates whether machine m is set up for item at the en of perio t (y t = ) or not (y t = 0). Min ( sx + hi ), () t t t I able. Parameters for the PLSP-MM Symbol Definition J Number of items. M Number of machines. Number of perios. a i Gozinto -factor. Its value is zero if item i is not an immeiate successor of item. Otherwise, it is the quantity of item to prouce one item i. C mt Available capacity of machine m in perio t. t External eman for item in perio t. b Non-negative holing cost for having one unit of item one perio in inventory. m Machines on which items is prouce. p Capacity nees for proucing one unit of item. s Non-negative setup cost for item. v Positive an integral lea time of item. I 0 Initial inventory for item. y 0 Unique initial setup state. Set of all items that share the machine m, i.e., α m β { {,,..., } } α m = J m = m. Set of immeiate successors of item, i.e., { {,,..., } i 0} β = i J a >. { t+ v } a q min, t i iτ i β τ= t+,0 =,,, J, t =,,,, J α m y t m =,,, M, t =,,,, x y y t t ( t ) =,,, J, t =,,,, ( t m t ( t ) t ) pq C y y =,,, J, t =,,,, α m pq C t mt m =,,, M, t =,,, { }, y t 0,, i =,,, I, =,,, J, t =,,,, () () () () (7) (8) subect to I = I + q a q, t ( t ) t t i it i β =,,, J, t =,,,, () I, q, x t t t 0, =,,, J, t =,,,, i =,,, I, =,,, J, he obective () is to minimize the sum of setup an holing costs. Eqs. () are the inventory balances. At the (9)

3 IJOR Vol., No., (00) en of a perio t there shoul be in inventory what was in there at the en of perio t plus what is prouce minus external an internal emans. Constraints () guarantee a lower boun of inventory of item at the en of a perio for fulfillment of the internal emans for the amount of items require to prouce other items but within the next v perios of its positive lea time. Constraints () make sure that the setup state of each machine is uniquely efine at the en of each perio. hose perios in which setups of a machine m for item happen are spotte by Eqs. () that if the machine is not set up for the particular item at the beginning of a perio, y (t-) = 0, but it is at the en of that perio (y t = ), then a setup must have occurre in perio t, i.e., x t 0 =. Ile perios may occur in orer to save setup costs. Due to Eqs. (), prouction can take place only if there is a proper setup state either at the beginning or at the en of a particular perio. Note that if the machine is not set up for an item at the beginning of a perio but it is at the en, the setup takes place in that perio, which means that prouction may take place in that perio. As the consequence of Eqs. (), at most two items can be manufacture on each machine per perio. Capacity constraints are formulate in Eqs. (7). Eqs. (8) efine the binary-value setup state variables, while Eqs. (9) are simply non-negativity conitions. One shoul easily convince himself that ue to Eqs. () in combination with Eq. () setup variables x t are inee zero-one value. Hence non-negativity conitions are sufficient for these variables. By letting inventory variables I t be nonnegative, backlogs cannot occur. he following will introuce the present extene moel.. he propose PLC-base PLSP moel his subsection introuces the propose PLC-base PLSP-MM moel. he aitional parameters ue to this consieration will be first introuce. hen, the extension of the PLSP-MM moel will be given... Notations for the PLC concerns In applications of PLCs, important steps relevant to this research inclue etermining the transitional points between the consecutive stages, such as the introuction, growth, maturity, an ecline stages an forecasting the emans of the PLCs. Useful technique has been foun in (Chang an Chang, 000), (Chang an Chang, 00) an the reviews therein. able provies the aitional parameters to the PLC-concerne moelling ue to the concern of emans of the PLCs. Here we shoul exten the original PLSP-MM moel by extening it to allow for ifferent aitional capacity an lot sizing constraints for the various stages of PLCs. he moelling resultant may have a better scheuling for the PLC emans... Aitional capacity-allocation constraints propose In lot sizing scheuling moels, often, the capacity constraints have been treate equally for all items competing for it. No consierations have been place on the proucts eman-stage changes. As a result when the PLC stages of the proucts exist strongly, the allocation of capacities for ifferent stages of proucts may occur inaequately for fulfillment of the emans. Higher inventories have to be use in some perios. However, a reality can be observe as well that in practice a company usually can have ifferent strategies for capacity allocation an ifferent proucts in ifferent stages for any preictable market changes. herefore, if PLC is not consiere, the solutions of the conventional lot sizing scheuling moels may become unsuitable or suboptimal. o overcome this problem, here a terme stage capacity-allocation weight for each stage of the proucts base on their PLCs may be introuce. It may be efine by Stage capacity-allocation weight Stage total eman PLC total time length = PLC total eman Stage total time length, (0) able. Aitional parameters for the PLC base PLSP moelling Symbol Definition B Starting perio of the PLC of item. ransitional perio between the introuction an In growth stages of item. ransitional perio between the growth an Gr maturity stages of item. ransitional perio between the maturity an Ma ecline stages of item. E En perio of PLC of item. In Number of perios in the introuction stage of item. Gr Number of perios in the growth stage of item. Ma Number of perios in the maturity stage of item. De Number of perios in the ecline stage of item. Number of perios of the PLC of item. n Number of immeiate successors of item. an taking into consieration the PLC stage emans an lengths of the proucts in capacity allocation of the lot-sizing scheuling. herefore, by the stage capacity-allocation weights, the aitional constraints of capacity allocations for each item in each perio can be efine (will be enote as C mt) an also replace their original C m t in Eqs. (). Also, for consieration of the original constraints C mt, C mt can be efine by multiplying the original C mt by the respective stage capacity-allocation weight of the items at each stage. Hence, for introuction stage,

4 IJOR Vol., No., (00) In t B t Cmt Cmt E In t B for =,,, J, t =, +,, B B In () number of immeiate successors of an item into consieration too, to avoi shortages for the emans. he aitional require lot-sizing constraints may be efine by Aitional lot sizing constraint Stage total eman = No. of immeiate successors Stage total time length () For growth stage, Gr t In + t Cmt Cmt E Gr t B for =,,, J, t = +,, In Gr () Hence For introuction stage: In B 0 q t n, In for =,,, J, t =, +,, For growth stage: t B B In () For maturity stage, Ma t Gr + t Cmt Cmt E Ma t B for =,,, J, t = +,, Gr Ma () Gr In + 0 q t n, Gr for =,,..., J, t = +,...,, For maturity stage: t In Gr (7) For ecline stage, E t Ma + t Cmt Cmt E De t B for =,,, J, t = +,, Ma E () In Eqs. ()-(), the minimum is taken to the purpose of maintaining the original capacity constraints un-exceee. An the results fulfill the strategic reassignment of capacity for each item in each perio at each stage. herefore, the stage capacity-allocation weight will reflect the capacity assignment requirement (constraints) of the items in ifferent stages... Lot sizing constraints propose Analogous to the arguments in capacity allocation constraints, PLC emans vary with stages. he lot-sizing ecisions of the proucts shoul also be given with the consieration of the stage variations too not only the capacity availability. o supply this iea into the lot sizing ecision of ifferent stage prouction requirements of the items, aitional lot-sizing constraints may be efine. Note that an item may be require for manufacturing other item(s). For these lot-sizing constraints to be feasible, the lot sizing an thus resultant inventories shoul take the Ma Gr + 0 q t n, Ma for =,,..., J, t = +,...,, For ecline stage: E Ma + t Gr Ma 0 q t n, De for =,,..., J, t = +,...,. t Ma.. Cash flow an learning curve consierations E (8) (9) Practically time value of money is consiere necessitate in any investment, revenue an expenses. Some lot-sizing moels have investigate a cash-flow-oriente moel to replace the non-cash-flow one (e.g., see Hofmann, 998; Ram an homas, 99). he moel propose here will also examine the cash flow effects for a further investigation. Moreover, as an item s prouction usually exhibits the learning curve, which is important in PLCs, the effects of learning curve cost will also be examine in the current moel. he setup costs may be consiere with the learning curve effect. Following these results (Argote an Epple, 990; Ram an homas, 99; Yelle, 979), the setup cost

5 IJOR Vol., No., (00) for an item may be moelle in a power function: (Setup cost) S = S t t b, { t min} for =,,..., J, t S S (0) where the learning rate b = 0.98 is use in the numerical example. For the effects of cash flows, iscounting of the costs at the various perios with a rate of return i per perio to the equivalent present worth or proecting them into the future worth or equivalent annual worth performs the task (William et al., 997) he equivalent present worth is aopte here. hey inclue: (Setup costs at each perio) α m pq C t mt, m =,,, M, t =,,,, y t { 0, }, i =,,, I, =,,, J, t =,,,, 0.98 S t = S t, for =,,..., J, In t B t Cmt Cmt E In t B for =,,, J, t =, +,,, B B In J = S t x t t, for t =,,..., ( + i ) (Holing costs at each perio) J = hi t t, for t =,,..., ( + i ) () () he entire moel propose, PLC-base PLSP-MM, can be efine as, by collecting the above formulations: Min J = i= subect to s t x h I t + t t + t + ( i) ( i) I I q a q t = ( t ) + t t i it, i β =,,, J, t =,,,, I t i β { t+ v } a q min, τ =+ t =,,, J, t =,,,, J α m y t, i iτ m =,,, M, t =,,,, x y y, t t ( t ) =,,, J, t =,,,, ( t m t ( t ) t ) pq C y y =,,, J, t =,,,,,, () Gr t In + t Cmt Cmt E Gr t B for =,,, J, t = +,,, In Gr Ma t Gr + mt = mt mt E Ma t B C min C, C, for =,,, J, t = +,,, Gr Ma E t Ma + t Cmt Cmt E De t B for =,,, J, t = +,,, Gr In + Ma E t 0 q t n, Gr for =,,..., J, t = +,...,, E Ma + In Gr 0 q t n, De for =,,..., J, t = +,...,, t Ma E I, q, x 0, =,,, J, t =,,,. t t t his moel will also be enote as PLC-PLSP-MM-CL, where CL stans for the cash flow ( C ) an learning curve ( L ) effect. Also a similar enotation will be use for

6 IJOR Vol., No., (00) the original moel, PLSP-MM, an therefore, PLSP-MM-CL enotes the original PLSP-MM moel with the cash flow an learning curve effects. Also, for the comparison purpose, PLC-PLSP-MM will enote the current moel without the cash flow an learning curve effects. he following introuces the solution proceure of the moel.. HE SOLUION PROCEDURE For optimization problems, genetic algorithms have been prove powerful an general for any type of problem an particularly for those of complex nature an moelling. With the vast literature an application cases, here the reaer is referre to the references (Disney et al., 000), (Hyun et al., 998), (Ip et al., 000), (Khoua et al., 998), (Kim an Kim, 99), (Kimms, 999), (Chang an Lo, 00), (S ik )provie for the basic iea. he esign of the GA for aiing in the solution of the moel above is provie as the solution proceure as follows. Step. GA representation/coing: Let S M enote a chromosome consisting of M binary genes s mt, m =,,, M, t =,,. hese genes are arrange in the orer of machines an time perios as Figure. Each gene represents whether a set up of the corresponing machine occurs or not in the particular perio t. Step. Initial population of chromosomes: An initial population of chromosomes S mt of a esirable size (PSIZE) is generate ranomly an each S M represents a feasible solution. PSIZE = 0 has been use in the numerical example. Step. Selection rules for setting up for items: For the above moel, we have use a backwar proceure for eciing the lot size scheuling, i.e., starting from the perio an then working backwars towars perio. Meanwhile, for each s mt = (t =,,,, m =,,, M) of each SM, selecting the proper item that requires the machine to be set up has been the central of this step. Here five rules are use an applie in orer of rule (δ ) to rule (δ ). If with a selection rule there are more-than-one items that can be selecte, the next rule is applie. able gives the parameters to be use in these selection rules. Rule (δ ) Base on minimizing the holing cost: Step (δ -): Fin the set α mt of items, which require the machine m to prouce an have unfulfille eman at time t. αmt = { αm CDt ( + ) > 0} () αm Nr q τ > 0. τ =+ t able. Parameters for the selection rules Symbol Definition CD t Cumulative eman for item that has not been met yet in perio t, i.e., = ( ) CD t q an CD = 0. τ= t τ τ = t+ τ + Nr Net requirement of item, i.e., = τ = τ Nr. pat Path of an item (with the meaning to be explaine in rule ). ep Depth of an item (with the meaning to be explaine in rule ). IS k Set of items selecte by selection rule k. Step (δ -): Evaluate the holing cost that can result from each of the items in α mt. he iea is that in orer to let all immeiate successors i of an item can be prouce in the next perio (t = t + ), the prouction of item at machine m must be complete in perio t, or otherwise it will cause holing cost of the item. herefore, the holing cost that an item α mt coul cause may be evaluate as h CD (t+). his rule is to select the item that can result in the possible maximal holing cost an to scheule it first in orer to avoi high holing costs. Namely, IS δ an ( CDl( t+ ) ) max{ αmt h ( CD ( t+ ) )} h l = l αmt, (a) =, if ISδ, S mt = (b) 0, otherwise, where S mt inicates the prouction an setup status of item in perio t whether it is scheule on machine m or not. If IS δ, apply the next rule δ. Rule (δ ) Base on minimizing the setup cost: Step (δ -): he iea for this rule is that for the same machine if an item can be prouce in consecutive perios aitional setup can be avoie. For this rule, one woul like to scheule an item first if it has alreay been scheule for prouction on machine m in the next perio (t + ). Namely, this rule will select the item(s) as: resetting α mt for δ, Machine m M Perio t Gene s mt Figure. Chromosome representation.

7 IJOR Vol., No., (00) α mt = { δ } m( t ) IS S = α, (a) + mt with the four items. herefore, this rule realizes that 7 an { αm ( t+ ) t 0} αmt = CD + > αm Nr q τ > 0. τ =+ t (b) he purpose of (b) is to ensure that the item selecte has an unfulfille eman incluing that for perio t. If in (a) { ISδ S + = } = m( t ), let ISδ = IS δ, omit the next step an go to the next rule. Otherwise, continue to the next step. Step (δ -): Select the item from α mt with the minimal setup cost. if ISδ =, S mt = { l αmt sl = min { αmt s }}, (7) 0, otherwise. Again if IS δ, apply the next rule δ. Rule (δ ) Base on the maximal epth of the items in the prouct structure: Assume a prouct manufacturing structure is as shown in Figure. he epths of the items can be etermine as ep =, ep =, ep =, ep =. In orer to reuce the generating of inferior or even infeasible solutions, we woul look for the eeper-epth or eeper-associate item to scheule first. his rule realizes that if ISδ =, S mt = { l ISδ ep = max { δ } l IS ep }, (8) 0, otherwise. Again if IS δ, apply the next rule δ. Figure. An example of prouct structure. Rule (δ ) Base on the longest path of the items: By a similar iea to rule, rule focuses on the entire path of the items in the prouct structure for avoiing inferior solutions ue to long influence of an item to the inventory an setup costs if it is not scheule first. Again by referring to Figure, the path lengths of the items as shown for instance can be etermine as pat = pat = pat = pat = as they are all in a path if ISδ =, S mt = { l ISδ pat = max{ δ } l IS pat } (9) 0, otherwise. If again IS δ, apply the final rule δ. Rule (δ ) Ranom selection: If applying the above rules still cannot settle the choice, a ranom selection may be applie. Step. Carry out the program () an etermine all other ecision variables values an the obective function value. Step. GA reprouction/selection: he GA reprouction is esigne an performe here with the roulette wheel selection base on the fitness function values. In GA operation, this requires the evaluation of the fitness (obective) function value of each chromosome an which is for maximization. he fitness function is efine as the reciprocal of the obective function above for each chromosome. Step. GA crossover: he crossover operation is esigne with a crossover probability P c for actually performing the crossover operation or otherwise the two ranomly selecte chromosomes enter irectly the next generation. It is performe in two ranomly selecte chromosomes with two ranomly selecte gene-points. hen exchange of the genes between the crossover points of the two chromosomes generates two offspring chromosomes. Step 7. GA mutation: Mutation operation is carrie out with a mutation probability P m on a ranomly selecte gene of a chromosome ranomly selecte here an mutates it from 0 to or vice versa. After completing this operation, a generation of the GA is reache. hen, if the termination conition is met, the GA process terminates. Or otherwise it returns to step. In aition, in the above proceure we also esigne it with a floating crossover an mutation probability mechanism for escaping from entrapment of local optimum possible. his is with that, when the process continues to fin the same value for a number of generations n, the GA increases graually the probabilities P c an P m with (P cu P cd )/GEN an (P mu P md )/GEN, respectively, where P cu an P cd enote the upper limit an efault, respectively, of P c an similarly for P m, an GEN enotes the number of generations. ill the solution changes, they are then reset to the efault value. If the number of generations GEN is reache an still the same solution is foun, the process terminates. After a number of empirical trial runs with all moels, these probability parameters, (P cu P cd )/GEN = (0.9 0.)/00, (P mu P md )/GEN = (0. 0.)/00, GEN = 00, an n = 0 were use.. NUMERICAL EXAMPLE Assume a firm is planning for prouction of two

8 IJOR Vol., No., (00) 8 proucts X or an Y or for the next ten perios for emans as shown in able. Figure. shows the manufacturing structure of the two proucts accoring to the bills of materials, where the numbers along the arrows inicate the quantities of the items require to make a unit of another item. able shows the manufacturing parameters of all these items. here are three machines available an each machine has 80 units of capacity per perio of time. able 7 shows the manufacturing relationships between these machines an items. It shows that which item can be prouce on what machine(s). For example, item can be prouce on machine or. able 8 shows the optimal chromosomes of machine setups S M with each moel introuce after 0 runs of the solution proceures. Other results obtaine by these moels are provie an analyze in the next sections. able. s for proucts X an Y Prouct Perio X Y Figure. he gozinto-structure for manufacturing. able. Manufacturing parameters Parameter Item S h P V I y able 7. Relationships between machines an items Relationship between the machine an item Machine Item 0 a 0 0 a a inicates that an item can be prouce on a machine an 0 inicates that it cannot.. he optimal lot sizing an scheuling he outputs are from the PLC-PLSP-MM-CL moel an the original PLSP-MM, PLC-PLSP-MM, an PLSP-MM-CL moels too. able 9 shows the optimal setup statuses of the machines for the items at the ens of perios, y t, with the applie selection rule by the PLC-PLSP-MM-CL moel. able 8. Optimal chromosomes of machine setups S mt with each moel S mt Machine Original PLSP-MM PLC-PLSP-MM PLSP-MM-CL PLC-PLSP-MM-CL able 9. Optimal setup statuses y t by the PLC-PLSP-MM-CL moel y t Machine Item t = (δ )(δ )(δ )(δ ) (δ ) (δ ) (δ ) (δ ) (δ )(δ ) (δ ) (δ ) 0 (δ ) (δ ) (δ ) (δ ) (δ ) (δ ) (δ ) (δ ) (δ ) (δ ) (δ )(δ ) 0 0 (δ ) 0 (δ ) (δ ) (δ ) (δ ) (δ )(δ ) (δ ) 0 (δ ) (δ ) 0 (δ )(δ ) (δ ) (δ ) (δ ) (δ )(δ )(δ ) (δ ) (δ ) 0 (δ ) (δ )(δ )(δ ) herefore, able 0 shows the optimal machine setups that are to be occurring in the ifferent perios for items, x t of the PLC-PLSP-MM-CL. Furthermore, the optimal lot sizing q t an optimal inventory ecisions I t of the various items by the PLC-PLSP-MM-CL moel are shown in ables an. he results of the original PLSP-MM, PLC-PLSP-MM, an PLSP-MM-CL moels are also obtaine; but for space limitations, they will not be provie here in etail. But, Figures -7 provie the final lot sizing an scheuling of the four moels in respective Gantt charts. In the next subsections, comparisons of these moels shall be mae.

9 IJOR Vol., No., (00) 9 able 0. Optimal machine setups to occur in ifferent perios for the items, x t, of the PLC-PLSP-MM-CL x t Machine Item t = able. Optimal lot-sizing ecisions q t of the items by the PLC-PLSP-MM-CL moel q t Item t = able. Optimal inventory ecisions I t of the items by the PLC-PLSP-MM-CL I t Item t = Machine Perio : Unuse Capacity : Unuse Perio Figure. Optimal lot sizing an scheuling by the PLC-PLSP-MM-CL.

10 IJOR Vol., No., (00) 0 Machine : Unuse Capacity Unuse Perio Figure. Lot sizing an scheuling by the original PLSP-MM. Perio Machine Perio : Unuse Capacity : Unuse Perio Figure. Optimal lot sizing an scheuling by the PLC-PLSP-MM. Machine Perio : Unuse Capacity : Unuse Perio Figure 7. Optimal lot sizing an scheuling by the PLSP-MM-CL.

11 IJOR Vol., No., (00). Comparison of the lot sizing scheuling costs of the moels able shows the setup cost, holing cost, an their total resulting with each moel. he following important remarks can be mae. () he effect of PLC concern: Examining the two pairs of moel s costs, original PLSP-MM moel with the PLC- PLSP-MM an PLSP-MM-CL with the PLC-PLSP-MM- CL moel (able ) inicates that both pairs iffer in whether the PLC is consiere or not an that not only the holing cost but also the setup cost are improve by the PLC consierations. Particularly a greater saving can be even given in the holing cost. he pairs of comparison of moels costs are summarize in able. () he combine effect of cash flow an setup-cost learning: Although the two moels, original PLSP-MM moel an PLSP-MM-CL (an similarly PLC-PLSP-MM moel an PLC-PLSP-MM-CL) may not be irectly compare ue to the reason of ifferent cost bases, one shoul still observe that the improvement of the total cost ue to the PLC consieration combine with the cash flow an setup cost learning (i.e., PLSP-MM-CL PLC-PLSP- MM-CL) is greater than that with only the PLC factor (i.e., original PLSP-MM PLC-PLSP-MM) in able. his inicates that the CL has also an effect on the moel an therefore shoul be taken into consieration in the moel. able. Optimal cost of the moels Moel Setup Holing Cost Cost otal Cost Original PLSP-MM PLC-PLSP-MM 00 8 PLSP-MM-CL PLC-PLSP-MM-CL 7 7 able. Comparison of the optimal cost improvement of the moels Reuction Setup Cost HolingCost otal Cost Original PLSP-MM PLC-PLSP-MM (-.%) (-.%) (-7.0%) PLSP-MM-CL PLC-PLSP-MM-CL 8 7 (-.%) 8 7 (-0.%) 8 (-.%). Comparison of eviations between scheule outputs of the moels an emans his subsection further compares the eviations between the scheule output an items emans by the moels. his task can be one either in a single-perio-by-single-perio manner or by a cumulate manner. For space limitations, only the cumulate results are provie here. In able, first the emans of the items in each perio are summarize. herefore, the cumulate emans an cumulate scheule outputs for all items (in units of capacity) up to each perio by the four moels are shown in able. he sum of square ifferences, SSD, between the cumulate emans an outputs of each moel can be shown in able 7. Similar remarks to that in Section. can be mae. he SSDs by the PLC-PLSP-MM-CL moel an PLC-PLSP-MM moel are much smaller than that by the PLSP-MM-CL an PLSP-MM, respectively. Furthermore, although the SSD by the PLSP-MM-CL is higher than that by the PLSP-MM, the SSD by the PLC-PLSP-MM-CL is smaller than that by PLC-PLSP-MM. So, again we observe that the PLC-PLSP- MM moel prouces a lot-sizing scheule that is better matching the proucts emans in the PLCs than that by the PLSP-MM moel. An the combine effect of the cash flow an setup cost learning curve (i.e., the PLC- PLSP-MM-CL moel) again reinforces this result. he extension of the PLSP-MM by the PLC-PLSP-MM-CL moel provies benefits to the PLSP problem with the prouct-life-cycle consierations. able. s of the items for each perio Item t = able. Cumulate emans an scheule outputs of the moels Cumulate s Cumulate Outputs: Original PLSP-MM PLC-PLSP-MM PLSP-MM-CL PLC-PLSP-MM-CL able 7. Sum of square ifferences (SSD) between cumulate emans an scheule outputs by each moel Original PLC-PLSP-MM PLSP-MM-CL PLC-PLSP-MM-CL PLSP-MM SSD CONCLUSIONS his paper has presente an extene PLSP-MM moel base on the prouct life cycle concept. It has introuce the proper PLC prouction capacity-allocation constraints an lot sizing constraints for the proucts for ifferent stages as the aitional constraints. he results of the example showe that the benefits of the extene moel provie not only reuce both the setup an holing costs but also that the lot sizing scheuling was significantly much better matching the proucts emans in the PLCs than the PLSP-MM moel. Also, in the moel

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