Earliness tardiness production planning and scheduling in flexible flowshop systems under finite planning horizon

Applied Mathematics and Computation 184 (2007) 950–964 www.elsevier.com/locate/amc

Earliness tardiness production planning and scheduling in flexible flowshop systems under finite planning horizon M. Iranpoor, S.M.T. Fatemi Ghomi *, A. Mohamadinia Department of Industrial Engineering, Amirkabir University of Technology, Hafez Ave. No. 424, 15916-34311 Tehran, Iran

Abstract This paper studies a general flexible flowshop scheduling problem to minimize the earliness and tardiness penalties. We consider a finite planning horizon including some equal periods. There is a known demand for each product in each period. The objective in the all periods, except the last period is minimizing the total penalties of E/T which are originated from the less or excess quantity produced as compared with the cumulative undelivered demands. It is assumed that total demands of each product should be delivered at the end of the planning horizon, so the objective of last period is changed to minimizing the total penalties of E/T which are originated from the difference between completion time and final due date. Because of dividing the problem to some subproblems, we can use optimal algorithms (e.g., B&B) to solve the problem and find suboptimal solution in real-size problems. In the solution procedure, using the unused capacities of the past periods for producing the demand of each period, the solution is improved. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Scheduling; Earliness/tardiness measure; Flexible flowshop

1. Introduction Flowshop manufacturing systems exist in many real world situations. In this system, it is assumed that the products are processed at some stages (or work centers) such that all of the products pass all of the stages in a unique sequence. Flexible flowshop manufacturing system (FFMS) is a modification of flowshop manufacturing system in which some products do not require to be processed at some stages. This system is more general and more realistic than the flowshop system. With popularity of the just-in-time philosophy, production managers have started to utilize this approach in their manufacturing systems. In fact earliness/tardiness production scheduling and planning (ETPSP) tries to implement JIT for a mass manufacturing system as much as possible. So every job (or product) completed earlier than its due date could cause opportunity cost for its carrying inventory until the due date, while any *

Corresponding author. E-mail address: [email protected] (S.M.T. Fatemi Ghomi).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.188

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job not completed until its due date could cause trouble to its customer. Therefore it has been desired to get all the jobs finished as exactly as possible on their assigned due dates. Many scheduling researches with earliness/tardiness (E/T) measure have been reported in the literature. This problem was originally called the minimum weighted absolute deviation problem. The early researches of such scheduling problem are traced back to the late 1970s and early 1980s by Sidney [9] and Kanet [5]. Since about 1990 Ahmed and Sundararaghavan [1] have referred this problem to as the ET problem. Panwalkar et al. [8] examined the single machine scheduling problem to minimize E/T costs for a set of independent jobs with common linear penalty rates a and b, in the context of common due date assignment in which the location of due date is a decision variable. Bagchi et al. [2] extended the results of Panwalker et al. for the equivalent scheduling problem with a given common due date (as a parameter). Ip et al. [4] developed a model to address the multi-product production environment and used genetic algorithm for finding its near-optimal solution. Cheng et al. [3] applied genetic algorithm to solve the identical parallel machine earliness/tardiness scheduling problem, with the objective of minimizing the maximum weighted absolute lateness. Wang [12] addressed a multi-product multi-process ETPP problem from an aggregate planning perspective with consideration of capacity limitations in processing stages of all periods. Li et al. [6] addressed the ETPP problem dealt by Wang [12] using goal programming (GP). Rasika and Wirth [11] used E/T measure for scheduling a single machine to process a number of jobs that are classified into some families using group technology (GP). They considered sequence-independent setup times, when processing switches from a job of one family to a job of another. Min and Cheng [7] discussed E/T scheduling problem for parallel machines including determining the location of common due date as a decision variable by using the genetic algorithm. Zhu and Heady [14] developed E/T formulation in a non-identical parallel machine problem. In their problem each job should be processed on one machine. The processing time of each job depends on the job-machine combination. Further, they considered sequence-dependent setup times. Sung and Min [10] concluded that the vast majority of the published E/T articles have considered only single or parallel processing machines. However, they, in turn, considered a two-machine flowshop scheduling problem with at least one batch processing machine (BPM) incorporated, where BPM can process a number of jobs jointly in a batch. Yeung et al. [13] discussed two-stage flowshop earliness/tardiness scheduling involving a common due window. In this paper we developed an extensive planning approach in flexible flowshop systems in general sense. There are some stages and some jobs (or products). A total demand of each job should be delivered at the end of the planning horizon, which includes some equal periods. In other words, there is a common due date at the end of each period. The demand of each product in each period is known, so for all periods, except last period, ETPP is determined by the weighted summation of all excess production (as the early completed products) and stockouts (as the tardy to be completed products). But, because of the total demands of each job (or product) should be delivered at the end of the finite planning horizon; in the last period ETPSP is determined by the weighted summation of earliness and tardiness time of completing all of demands of each product as compared with the final due date (the end of the last period). The paper is organized as follows. Section 2 presents the general formulation of the problem. Section 3 introduces the solution procedure. A real size numerical example is completely analyzed in Section 4. Finally, Section 5 is devoted to conclusion remarks. 2. General formulation of the problem The flexible flowshop ET problem for delivering the total demand of each product during a finite planning horizon in some periods can be formulated using the following notation: M di(k) pi(k) T L Lek

a large number demand of product i in period k production quantity of product i in period k number of periods within planning horizon length of each demand period length of expanded period k

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ai aTi Ei bi bTi Ti bijk wij si lik Iik Bik N m

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earliness cost per unit of product i (used for periods 1 to T  1) earliness cost per unit of time for product i (used for period T) earliness time of product i in the last period tardiness cost per unit of product i (used for periods 1 to T  1) tardiness cost per unit of time for product i (used for period T) tardiness time of product i in the last period beginning time of processing product i at stage (or machine) j in period k processing time of unit of product i at stage j lot-size of product i inventory of product i at the beginning of period k inventory of product i at the end of period k (early produced quantity) backorder of product i at the end of period k (quantity to be produced with tardiness) number of products number of stages

The objective function expressed as follows: #þ ( " " #þ ) N X T 1 k k k k X X X X X Minimize ai lik þ pi ðtÞ  d i ðtÞ þ bi d i ðtÞ  pi ðtÞ  lik i¼1

þ

N X

k¼1

t¼1

t¼1

t¼1

 T þ þ ai ½LT  bimT  wim pi ðT Þ þ bTi ½bimT þ wim pi ðT Þ  LT 

t¼1

ð1Þ

i¼1

which [x]+ equals zero if x < 0 and equals x otherwise. As shown, expression (1) adds the penalties of producing more and less than cumulative not responded demands until each period for periods 1 to T  1 and penalties originated from the difference between final due date (end of the last period) and completion time of producing not responded demands in the last period. 0 To convert the objective function to a linear expression, we replace the expression (1) by expression (1 ) and append eight sets of constraints ((2)–(9)) Minimize

N X T 1 N X X  T  ðai I ik þ bi Bik Þ þ ai Ei þ bTi T i ; i¼1

k¼1

ð10 Þ

i¼1

s:t:: I ik P lik þ

k X

pi ðtÞ 

t¼1

Bik P

k X t¼1

d i ðtÞ 

k X

d i ðtÞ;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

ð2Þ

pi ðtÞ  lik ;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

ð3Þ

t¼1 k X t¼1

I ik P 0;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

ð4Þ

Bik P 0;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

ð5Þ

Ei P LT  bimT  wim pi ðT Þ;

i ¼ 1; . . . ; N ;

ð6Þ

T i P bimT þ wim pi ðT Þ  LT ;

i ¼ 1; . . . ; N ;

ð7Þ

Ei P 0;

i ¼ 1; . . . ; N ;

ð8Þ

T i P 0;

i ¼ 1; . . . ; N :

ð9Þ

In flowshop manufacturing systems there is a unique sequence for all of the products from first stage to last one (mth stage in our notation). In flexible manufacturing systems, however, some products may not require some stages. This specification does not affect our formulation at all. Just, we should let the processing time of these stages equal to zero for such products. So, the constraints of the priority are as follows:

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bimk þ wim pi ðkÞ 6 kL;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

bimk P bi;m1;k þ wi;m1 pi ðkÞ;

bi2k P bi1k þ wi1 pi ðkÞ; bi1k P 0;

ð10Þ

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T ;

bi;m1;k P bi;m2;k þ wi;m2 pi ðkÞ; .. .

953

ð11:1Þ

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T ;

ð11:2Þ

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T ;

ð11:m-1Þ

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T :

ð11:mÞ

In the above constraints we assumed that each product batch must be processed by only one machine at any time. In addition, we assume that each machine can process at most one product at any time. Consider two products i and i 0 are to be processed at stage (or machine) j. According to the above assumption the process of product i on machine j should be started after completing the process of product i 0 on machine j or vice versa. To guarantee that our solution space has such specification we add three sets of constraints (12)–(14) i ¼ 1; . . . ; N  1; i0 ¼ i þ 1; . . . ; N ;

bi0 jk P bijk þ wij pi ðkÞ  My ii0 jk ;

j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; k ¼ 1; . . . ; T ; bijk P bi0 jk þ wi0 j pi0 ðkÞ  Mð1  y ii0 jk Þ;

ð12Þ 0

i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ;

j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; k ¼ 1; . . . ; T ; 0

y ii0 jk 2 f0; 1g;

ð13Þ

i ¼ 1; . . . ; N  1; i ¼ 1; . . . ; N j ¼ 1; . . . ; mjwij 6¼ 0 & w 6¼ 0; k ¼ 1; . . . ; T : i0 j

ð14Þ

It is obvious that the production of each period on each machine should be started after that the production of its last period is finished, so we append below constraints to the model: bijk P bi0 j;k1 þ wi0 j pi0 ðk  1Þ;

i ¼ 1; . . . ; N ; i0 ¼ 1; . . . ; N ;

j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; k ¼ 2; . . . ; T :

ð15Þ

In many practical environments, operation managers consider lot-size for each product. The production quantity of each item should be a product of its lot-size. we consider this characteristic in our problem by adding the sets of constraints (16) and (17) pi ðkÞ ¼ ri si ;

i ¼ 1; . . . ; N ; k ¼ 1; . . . ; T  1;

ri 2 f0; 1; 2; . . .g;

i ¼ 1; . . . ; N :

ð16Þ ð17Þ

li1 is known as parameter, but lik for periods 2 to T  1 is calculated by set of constraints (18) lik ¼ li;k1 þ pi ðk  1Þ  d i ðk  1Þ;

i ¼ 1; . . . ; N ; k ¼ 2; . . . ; T  1:

ð18Þ

However, since we should deliver the total demands of all periods of the planning horizon; the production quantity of each product is not a decision variable anymore in the last period. In fact, the quantity of production for product i in the last period is calculated by Eq. (19) as a set of constraint pi ðT Þ ¼ d iT  liT ;

i ¼ 1; . . . ; N :

ð19Þ

3. Solution procedure As shown in last section the number of binary variables is not reasonable, so we cannot use optimal solution of above model for real-size scheduling problems. So, we introduce an optimal-based heuristic solution procedure in this section to reach a suboptimal solution. In this approach, we make a model for each period and solve it to reach a suboptimal solution. For improving this solution we permit the production of each demand period be carried out in its past periods. In other words, we expand the length of producing periods backward. For example in Fig. 1, point S is considered as the start point of producing period 2. But machine B and machine C are busy at the beginning of this expanded period (Fig. 1).

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Fig. 1. Using unused capacity of past period.

As shown, a single product is processed at stage A, then at stage B and then C. It is assumed that the processing of each product at each stage cannot be started until its process is completed at the last stage; for example in Fig. 1, the processing of product at stage B cannot be started till the whole of items of that product are processed at stage A (and similarly C after B). The objective function in period k (k from 1 to T  1) expressed as follows: ( " " #þ ) #þ N k k k k X X X X X : ð20Þ Minimize ai lik þ pi ðtÞ  d i ðtÞ þ bi d i ðtÞ  pi ðtÞ  lik i¼1

t¼1

t¼1

t¼1

t¼1

As shown, expression (20) adds the penalties of producing more and less than cumulative not responded demands until each period. To convert the objective function to a linear expression, we replace the expression 0 (20) by expression (20 ) and append four sets of constraints (21)–(24). Minimize

N X ðai I ik þ bi Bik Þ;

ð200 Þ

i¼1

s:t:: I ik P lik þ

k X

pi ðtÞ 

t¼1

Bik P

k X t¼1

d i ðtÞ 

k X

d i ðtÞ;

i ¼ 1; . . . ; N ;

ð21Þ

pi ðtÞ  lik ;

i ¼ 1; . . . ; N ;

ð22Þ

t¼1 k X t¼1

I ik P 0;

i ¼ 1; . . . ; N ;

ð23Þ

Bik P 0;

i ¼ 1; . . . ; N :

ð24Þ

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The constraints of due date and priority for period k are as follows: bimk þ wim pi ðkÞ 6 Le;k ;

i ¼ 1; . . . ; N ;

bimk P bi;m1;k þ wi;m1 pi ðkÞ;

i ¼ 1; . . . ; N ;

bi;m1;k P bi;m2;k þ wi;m2 pi ðkÞ; .. . bi2k P bi1k þ wi1 pi ðkÞ; bi1k P 0;

ð25Þ ð26:1Þ

i ¼ 1; . . . ; N ;

ð26:2Þ

i ¼ 1; . . . ; N ;

ð26:m-1Þ

i ¼ 1; . . . ; N :

ð26:mÞ

In the above constraints we assume that each product batch must be processed by only one machine at a time. This assumption also was shown in Fig. 1. Since each machine can process at most one product at a time we add three sets of constraints (27)–(29) bi0 jk P bijk þ wij pi ðkÞ  My ii0 j ;

i ¼ 1; . . . ; N  1; i0 ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0;

bijk P bi0 jk þ wi0 j pi0 ðkÞ  Mð1  y ii0 j Þ;

ð27Þ

0

i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; ð28Þ

y ii0 j 2 f0; 1g;

0

i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0:

We consider lot-size in our problem by adding the sets of constraints (30) and (31) pi ðkÞ ¼ ri si ; i ¼ 1; . . . ; N ; ri 2 f0; 1; 2; . . .g; i ¼ 1; . . . ; N :

ð29Þ ð30Þ ð31Þ

li1 is known as parameter. After solving the above model (expressions (20 0 )–(31)) for period k  1, lik should be calculated at step k by using Eq. (32) and replaced in the model lik ¼ li;k1 þ pi ðk  1Þ  d i ðk  1Þ;

i ¼ 1; . . . ; N ; k ¼ 2; . . . ; T  1:

ð32Þ

In addition, for periods 2 to k  1 the right hand side (RHS) of constraint (25) may be greater than the length of each period (that is L). For example in Fig. 2; suppose the processing unit time is minute and there are 480 min per each workday. In this example Le2 equals L + 5 * 480. But since machine B is busy in 2 days of 0 these added 5 days, we add constraint (25 a); also since machine C is busy in the whole of these added 5 days, 0 we add constraint (25 b)

Fig. 2. Expanding of the length of periods using unused capacity of past periods.

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biC2 þ wiC pi ð2Þ 6 Lþ 5  480;

ð25 Þ

biB2 P 2  480;

ð250a Þ

biA2 P 5  480;

ð250b Þ

Since we should deliver the total demands of all periods during the planning horizon; the production quantity of all of the products is not a decision variable anymore in the last period. In fact, the quantity of production for product i in the last period is calculated by Eq. (33) and replaced in the model of last period pi ðT Þ ¼ d iT  liT ;

i ¼ 1; . . . ; N :

ð33Þ

The objective function for period T is as follows: Minimize

N X  T  ai ½L  bimT  wim pi ðT Þþ þ bTi ½bimT þ wim pi ðT Þ  Lþ :

ð34Þ

i¼1 0

To convert the objective function to a linear expression; we replace expression (34) by (34 ) and append four sets of constraints ((35)–(38)) Minimize

N X  T  ai Ei þ bTi T i ;

ð340 Þ

i¼1

s:t:: Ei P L  bimT  wim pi ðT Þ;

i ¼ 1; . . . ; N ;

ð35Þ

T i P bimT þ wim pi ðT Þ  L;

i ¼ 1; . . . ; N ;

ð36Þ

Ei P 0;

i ¼ 1; . . . ; N ;

ð37Þ

T i P 0;

i ¼ 1; . . . ; N :

ð38Þ

Similar to period 2 through T  1, some machines are unavailable during the added time to period T. Let uj be the duration time that machine j is unavailable during this added time. We express this unavailability by set of constraint (39) bijT P uj ;

i ¼ 1; . . . ; N :

ð39Þ

The other constraints are similar to the model of past periods: bimT P bi;m1;T þ wi;m1 pi ðT Þ;

i ¼ 1; . . . ; N ;

bi;m1;T P bi;m2;T þ wi;m2 pi ðT Þ;

ð40:1Þ

i ¼ 1; . . . ; N ;

ð40:2Þ

.. . bi2T P bi1T þ wi1 pi ðT Þ; bi1T P 0;

i ¼ 1; . . . ; N ;

ð40:m-1Þ

i ¼ 1; . . . ; N ;

ð40:mÞ 0

bi0 jk P bijT þ wij pi ðT Þ  My ii0 j ; i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; bijT P

0 bþ i0 jT wi j pi0 ðT Þ  Mð1  y ii0 j Þ;

ð41Þ

0

i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0; ð42Þ

0

y ii0 j 2 f0; 1g; i ¼ 1; . . . ; N  1; i ¼ i þ 1; . . . ; N ; j ¼ 1; . . . ; mjwij 6¼ 0 & wi0 j 6¼ 0:

ð43Þ

Note that as compared with the formulation for period 1 through T  1, the model does not include set of constraint (25) because we do not have to deliver the demand of last period at its due date; instead, we need to deliver all of the undelivered demands in this period. So we changed the E/T measure from a quantity-based to a time-based function. Also, since pi(T) is parameter, we cannot limit its values to a product of si (lot size of product i), so the model does not include sets of constraints (30) and (31).

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4. Numerical example An small firm has one production line including four stages (Fig. 3). This firm has contracted to supply four parts for an assembling company within next six weeks. The manufacturing process of each of four parts is as follows in Fig. 4. The processing time of each part on each machine expressed as follows in Table 1. Lot sizes of the parts 1–4 are respectively 50, 100, 50, 100. The demand of each part at the end of each week expressed in Table 2.

Fig. 3. Production line diagram.

Fig. 4. Process diagram of each product (part).

Table 1 Processing time of each unit part on each machine (min) Product

Operation 1

Operation 2

Operation 3

Operation 4

1 2 3 4

2.1 – – 1.8

1.8 1.7 2.1 –

– 1.5 2.3 1.7

1.6 1.8 – 2.0

Table 2 Demand of each product (part) to be delivered at the end of each week Part

1 2 3 4

Period 1

2

3

4

5

6

Total

400 450 550 350

350 300 450 400

600 350 400 600

300 450 400 300

600 300 600 500

600 550 600 550

2850 2400 3000 2700

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Table 3 E/T penalties of each part ($) Part (i]

E/T penalty

1 2 3 4

ai

aTi

bi

bTi

2 3 2 3

10 10 10 10

4 5 3 4

10 15 10 12

Each week includes 6 workdays and each workday, in turn, includes 480 minutes, that is 2880 min (excluding idle times). There is no on hand inventory of each part at the beginning of the planning horizon. The earliness/tardiness penalties of each part is expressed in Table 3. After solving the model which involves expressions (20 0 )–(31) for period 1, the optimal production quantities for this subproblem are p1 ð1Þ ¼ 400;

p2 ð1Þ ¼ 400;

p3 ð1Þ ¼ 400;

p4 ð1Þ ¼ 300:

The optimal value of the objective function in this period equals 900$. The scheduling Gantt chart in this period is as follows: We can conclude from Fig. 5 that in period 2 L ¼ 2880 þ ð2880  1520Þ ¼ 4240 min; bi12 P 0; i ¼ 1; . . . ; 4; bi22 P 2240  1520; bi32 P 2570  1520;

i ¼ 1; . . . ; 4; i ¼ 1; . . . ; 4;

bi42 P 2880  1520;

i ¼ 1; . . . ; 4:

Fig. 5. Scheduling Gantt chart of period 1.

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Furthermore, initial inventory of each part in period 2 is l12 ¼ 0;

l22 ¼ 50;

l32 ¼ 150;

l42 ¼ 50:

After solving the model which involves expressions (20 0 )–(31) for period 2, the optimal production quantities for this subproblem are p1 ð2Þ ¼ 350;

p2 ð2Þ ¼ 400;

p3 ð2Þ ¼ 600;

p4 ð2Þ ¼ 500:

The optimal value of the objective function in this period equals 300$. The scheduling Gantt chart in this period is as follows: We can conclude from Fig. 6 that in period 3 L ¼ 2880 þ ð2880  1690Þ ¼ 4070 min; bi13 P 0; i ¼ 1; . . . ; 4; bi23 P 2320  1690; i ¼ 1; . . . ; 4; bi33 P 2880  1690; bi42 P 2880  1690;

i ¼ 1; . . . ; 4; i ¼ 1; . . . ; 4:

Furthermore, initial inventory of each part in period 3 is l13 ¼ 0;

l23 ¼ 50;

l33 ¼ 0;

l43 ¼ 50:

So, the optimal production quantities for period 3 using model (20 0 )–(31) are p1 ð3Þ ¼ 500;

p2 ð3Þ ¼ 400;

p3 ð3Þ ¼ 400;

p4 ð3Þ ¼ 500:

The optimal value of the objective function in this period equals 1000$.

Fig. 6. Scheduling Gantt chart of period 2.

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The scheduling Gantt chart in this period is as follows: We can conclude from Fig. 7 that in period 4 L ¼ 2880 þ ð2880  1010Þ ¼ 4750 min; bi14 P 0;

i ¼ 1; . . . ; 4;

bi24 P 1860  1010; bi34 P 2780  1010;

i ¼ 1; . . . ; 4; i ¼ 1; . . . ; 4;

bi44 P 2860  1010;

i ¼ 1; . . . ; 4:

Furthermore, initial inventory of each part in period 4 is l14 ¼ 100;

l24 ¼ 0;

l34 ¼ 0;

l44 ¼ 150:

So, the optimal production quantities for period 4, using model (20 0 )–(31) are p1 ð4Þ ¼ 400;

p2 ð4Þ ¼ 500;

p3 ð4Þ ¼ 400;

p4 ð4Þ ¼ 500:

The optimal value of the objective function in this period equals 300$. The scheduling Gantt chart in this period is as follows in Fig. 8. So, for period 5 L ¼ 2880 þ ð2880  0Þ ¼ 5760 min; bi15 P 0;

i ¼ 1; . . . ; 4;

bi25 P 1750  0;

i ¼ 1; . . . ; 4;

bi35 P 2670  0;

i ¼ 1; . . . ; 4;

bi44 P 2790  0;

i ¼ 1; . . . ; 4:

Fig. 7. Scheduling Gantt chart of period 3.

M. Iranpoor et al. / Applied Mathematics and Computation 184 (2007) 950–964

Fig. 8. Scheduling Gantt chart of period 4.

Fig. 9. Scheduling Gantt chart of period 5.

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Fig. 10. Scheduling Gantt chart of period 6.

Furthermore, initial inventory of each part in period 5 is l15 ¼ 0;

l25 ¼ 50;

l35 ¼ 0;

l45 ¼ 50:

So, the optimal production quantities for period 5, using model (20 0 )–(31) are p1 ð5Þ ¼ 600;

p2 ð5Þ ¼ 300;

p3 ð5Þ ¼ 600;

p4 ð5Þ ¼ 500:

The optimal value of the objective function in this period equals 300$.

Fig. 11. Scheduling Gantt chart of the planning horizon (includes all of the periods).

M. Iranpoor et al. / Applied Mathematics and Computation 184 (2007) 950–964

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The scheduling Gantt chart in this period is as follows: We can conclude from Fig. 9 that in the last period L ¼ 2880 þ ð2880  640Þ ¼ 5120 min; bi16 P 0;

i ¼ 1; . . . ; 4;

bi26 P 1720  640; bi36 P 2880  640; bi46 P 2880  2840;

i ¼ 1; . . . ; 4; i ¼ 1; . . . ; 4; i ¼ 1; . . . ; 4:

Initial inventory of each part in the last period (period 6) is l16 ¼ 0;

l26 ¼ 50;

l36 ¼ 0;

l46 ¼ 50:

Now, the production quantity of each product to satisfy the total undelivered demands is calculated by expression (44) pi ð6Þ ¼ d i ð6Þ  li ð6Þ;

i ¼ 1; . . . ; 4:

ð44Þ

So p1 ð6Þ ¼ 600;

p2 ð6Þ ¼ 500;

p3 ð6Þ ¼ 600;

p4 ð6Þ ¼ 500:

Now the optimal scheduling solution for the last period is calculated, using the model which includes expressions (34 0 )–(43). The Scheduling Gantt chart of this period is as follows in Fig. 10. The optimal value of the objective function in the last period equals 256.67$. Finally, the summary of the above results is shown in Fig. 11. 5. Concluding remarks In this paper, we presented an extensive method for the production planning and scheduling with E/T measure in flexible manufacturing systems. We considered a finite planning horizon that is divided to some equal periods. The E/T penalty was considered quantity-based during all periods except the last one.In the last period the E/T measure is changed to a time-based function, because the production quantities were not decision variable in this period anymore. References [1] M.U. Ahmed, P.S. Sundararaghavan, Minimizing the weighted sum of late and early completion penalties in a single machine, IIE Transactions 22 (3) (1990) 288–290. [2] U. Bagchi, Y.L. Cheng, R.S. Sullivan, Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date, Naval Research Logistics 34 (1987) 739–751. [3] R.W. Cheng, M.S. Gen, T. Tozawa, Minmax earliness/tardiness scheduling in identical parallel machine system using genetic algorithms, Computers and Industrial Engineering 29 (1–4) (1995) 513–517. [4] W.H. Ip, Y. Li, K.F. Man, K.S. Tang, Multi-product planning and scheduling using genetic algorithm approach, Computers and Industrial Engineering 38 (2000) 283–296. [5] J. Kanet, Minimizing the average deviation of job completion times about a common due date, Naval Research Logistics 28 (4) (1981) 643–651. [6] Lei Li, Daniel J. Fonseca, Der-San Chen, Earliness-tardiness production planning for just-in-time manufacturing: a unifying approach by goal programming, European Journal of Operation Research, in press. [7] Liu Min, Wu Cheng, Genetic algorithms for the optimal common due date assignment and the optimal scheduling policy in parallel machine earliness/tardiness scheduling problems, Robotic and Computer-Integrated Manufacturing 22 (4) (2006) 279–287. [8] S.S. Panwalkar, M.L. Smith, A. Seidmann, Common due date assignment to minimize total penalty for one machine scheduling problem, Operation Research 30 (2) (1982) 391–399. [9] J. Sidney, Optimal single machine scheduling with earliness and tardiness penalties, Operation Research 25 (1) (1977) 62–69. [10] C.S. Sung, J.I. Min, Theory and methodology: scheduling in a two-machine flowshop with batch processing machine(s) for earliness/ tardiness measure under a common due date, European Journal of Operation Research 131 (2001) 95–106. [11] Rasika H. Suriyaarachchi, Andrew Wirth, Earliness/tardiness scheduling with a common due date and family setups, Computers and Industrial Engineering 47 (2004) 275–288.

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