Identical Parallel Machine Scheduling Problem with Additional Resources and Partial Confirmed Orders in Make-to-Stock Strategy

Abstract

This research deals with the parallel machine scheduling problem for identical machines that need additional operational resources during the changeover. The production strategy is mainly based on make-to-stock. When the current inventory is less than the quantity of the new order, the corresponding products will be scheduled for production in predetermined production batches that are larger than the quantity of the order. Because the additional resources are limited, batch splitting, which increases the number of changeovers, is not permitted. The objective is to minimize both the makespan and total tardiness. A two-phase methodology is proposed. In the first phase, a mixed-integer program is developed to minimize the makespan. The resulting minimal makespan becomes the constraint in the second phase. An extended mixed-integer program is then developed to minimize the total tardiness. A case study of a plastic pallet manufacturing company is introduced. The experimental results show that the proposed methodology can minimize the makespan and total tardiness efficiently. Moreover, it also shows the promise of the proposed methodology for solving practical applications.

1. Introduction

Parallel machine scheduling (PMS) is a classic problem in the literature. In general, PMS can be classified into three categories according to the types of machines, namely, identical machines, uniform machines, and unrelated machines [1]. This research deals with the PMS problem for identical machines in which the processing time of a job is the same for all machines [2]. The objective is to minimize the makespan and total tardiness.

The motivation for this study is a practical problem at a plastic pallet manufacturing company in Taiwan. The main manufacturing process for the plastic pallets is injection molding. At the company, there are several identical injection molding machines. Each plastic pallet only needs to be processed by one of these machines. When a machine completes one batch of a product, additional resources are required for the changeover if a different product is to be produced subsequently. The main purpose of a changeover is to change the molds. The weights of the molds are between 10 and 20 tons, and the heaviest can be as much as 24 tons. A thirty-five-ton crane is used to move the molds. For reasons of safety, the molds are moved slowly and at least two operators must be involved. While one worker is operating the crane, the other has to watch and ensure all situations are safe. After the molds are fixed in the machines, the machines require a series of adjustments. All operations are executed slowly, so it takes 6 to 8 h to exchange a mold. There is only one crane, and only one team can undertake the mold exchange operation, so only one machine can be changed at a time. Moreover, in order to avoid accidents, changeovers can only happen on a day shift, so there can only be one changeover per day.

The company has a make-to-stock strategy. Because of the lack of inventory space, no safety stocks are kept for any products. Once an order for a product is received, if there is insufficient inventory to fill the order, the company will arrange to produce the product. Because of the resources needed to change molds, production batch sizes are always larger than the quantity of the received order and determined on the basis of historical demand data; a product with higher demand is produced in larger batches. Additional product is stored in the warehouse for future orders. The last day on which the mold must be exchanged in order to produce product i, LS_i, can be estimated from the due date of the most recent order for which there is insufficient inventory to satisfy the order. In order to satisfy requests from customers, the day that machine exchanges the mold for product i, DS_i, should be earlier than LS_i. Figure 1 illustrates the case in which the inventory of product i is too low for the new order and DS_i < LS_i.

In Figure 1, the quantity of the order for product i is A_i, which is larger than the current inventory. In order to satisfy the order, the machine mold is changed on day DS_i. After the 8 h changeover, the machine starts to produce a batch and the inventory increases. There is more inventory than Ai on the due date, when quantity Ai is delivered to the customer and the inventory is reduced. The machine continues to produce the product and the inventory increases again. When the total current production output equals the production batch, the machine stops producing product i. The total production time for product i is PT_i.

If DS_i > LS_i, the inventory will be less than A_i on the due date. The whole order will be late, and it will be delivered to the customer once the inventory is equal to A_i. The tardiness of the order can be calculated by DS_i − LS_i, and the total tardiness for the order is A_i (DS_i − LS_i). The inventory status for delayed orders is illustrated in Figure 2.

Typically, the identical PMS problem is an NP-hard problem [3]. Therefore, a manual system is not efficient in providing a practical solution when considering both the limitations of the changeover and the due date of all received orders simultaneously. This research aims to minimize the makespan and total tardiness for the proposed identical PMS problem. A two-phase methodology is proposed. The remainder of this paper is organized as follows: Section 2 reviews the pertinent literature. Section 3 presents the methodology proposed to solve the problem. Details of the empirical observations are discussed in Section 4. Conclusions and future research opportunities are addressed in the final section.

2. Literature Review

The study of PMS in which additional resources are required for the changeover is a significant area of research. Edis et al. [4] surveyed the related literature and classified it into an efficient framework. The present study deals with additional renewable resources, also called servers, which can only be deployed within certain constraints. The recent related literature is reviewed in the following.

Torabi et al. [5] proposed a multi-objective model for a PMS problem with unrelated machines in which the setup times are sequence and machine dependent, processing times and due dates of jobs are uncertain, and jobs may have different ready times. Assignment of a job to a machine is permitted if the required resources (e.g., tool and die) are available, and the number of required resources is limited. An effective multi-objective particle swarm optimization algorithm was proposed to find a good approximation of the Pareto frontier, for which the total weighted flow time, total weighted tardiness, and total machine load variation are to be minimized simultaneously. Afzalirad and Rezaeian [6] addressed a PMS problem for unrelated machines with resource constrains, sequence-dependent setup times, different release dates, machine eligibility, and precedence constraints. Integer mathematical modeling was proposed to first minimize the makespan. Since the problem is strongly NP-hard, two new meta-heuristic algorithms, genetic algorithm and artificial immune system, were then deployed to find the optimal or near-optimal solutions. Fanjul-Peyro et al. [7] analyzed a PMS problem for unrelated machines in which the processing of jobs on the machines required a number of limited resources, which were fixed throughout the production horizon and depend both on the job and on the machine. Two integer linear programs were used to minimize the makespan. Hamzadayi and Yildiz [8] dealt with the PMS problem for identical machines with a common server and sequence-dependent setup times. They proposed a mixed-integer linear programming (MILP) model to minimize the makespan. In order to solve the proposed problem efficiently, they used simulated annealing and genetic algorithm approaches. Zheng and Wang [9] proposed a PMS problem for unrelated machines in which a common shared renewable resource is considered, and the speed of a machine depends on the energy the machine consumes. A collaborative multi-objective fruit fly optimization algorithm was proposed to minimize the makespan and the total carbon emission. Liu et al. [10] dealt with a PMS problem with a single server. A branch-and-bound algorithm was used to minimize the total weighted job completion time. Akyol Ozer and Sarac [11] dealt with a PMS problem for identical machines with sequence-dependent setup times, machine eligibility restrictions and multiple copies of shared resources. Mixed-integer programming models and a model-based genetic algorithm were proposed to minimize the total weighted completion time. Lee et al. [12] dealt with a PMS problem in which jobs can be split into multiple sections and the number of setups that can be performed simultaneously is restricted because of the limited setup operators. A mathematical programming model was presented first and then an iterative job splitting algorithm was proposed to minimize the makespan. Lopez-Esteve et al. [13] dealt with a PMS problem for unrelated machines with the need of additional resources during the processing of the jobs, as well as during the setups. Several heuristics and metaheuristics were proposed to minimize the makespan.

Regarding the objectives of PMS in the literature, the makespan is the most widely used [13]. A minimum makespan usually implies high utilization of the machines [14]. However, tardiness is another important criterion when the due dates of the orders are considered. The literature related to PMS that deals with minimizing both makespan and total tardiness is rare.

Chyu and Chang [15] dealt with a PMS problem for unrelated machines in which the setup times are job-sequence- and machine-dependent, and each job has its own due date. A measure based on fuzzy set theory was used to assess the satisfaction level with both the makespan and average tardiness. Then, simulated annealing and a greedy randomized adaptive search procedure were presented for solving the problem. Zarandi and Kayvanfar [16] dealt with an identical PMS problem in which each job could be processed in a shorter or longer period of time depending on its efficacy in the objective function by reducing or increasing the available resources. The are two objectives, as follows: the first is minimizing the total cost of tardiness and earliness, as well as compression and expansion costs, of the job processing times; the second is minimizing the tardiness. Two multi-objective evolutionary algorithms were employed to solve the proposed bi-objective problem. Manupati et al. [17] dealt with an unrelated PMS problem in which the setup times were job-sequence- and machine-dependent; the jobs could have different ready times; and the processing times and due dates of the jobs were formulated as fuzzy numbers. A multi-objective-based evolutionary artificial immune nondominated sorting genetic algorithm was used to minimize the makespan, flow time, tardiness and machine load variation simultaneously. Yu et al. [18] dealt with a PMS problem for two identical machines where jobs could be split among the machines. A multi-agent system that can be applied to dynamic and static environments was designed for minimizing the total tardiness and makespan.

In this research, an identical PMS problem with renewable additional resources are considered to deal with the make-to-stock production strategy. The objectives is to minimize both the total tardiness and the makespan. To the best of the author’s knowledge, no previous research has addressed the same conditions as in the presented study. Moreover, a real case study of a plastic pallet manufacturing company is introduced. Thus, the present study may not only contribute practical solutions but also develop academic perspectives.

3. Solution Methodology

A two-phase methodology is proposed to minimize the makespan and total tardiness for a practical identical PMS problem, as shown in Figure 3. Only the products that receive orders and with corresponding inventories that are less than the quantity of the orders are considered. In the first phase, an integer program was developed. When given the production times of product i for its corresponding production batches, PTi, it aims to minimize the makespan without considering the due dates of the orders that have been received and their corresponding quantities. Although one schedule may result from Phase 1, multi-optimal solutions exist. In Phase 2, an integer programming approach, modified from Phase 1, is developed. It aims to minimize the total tardiness by considering the resulting minimal makespan from Phase 1 for those among the received order, Ai, and the last day that the molds have to be exchanged for the corresponding products, LSi. A detailed explanation of both phases is given in the following sections.

3.1. Phase 1: Minimizing Makespan

The main purpose of the first phase is to minimize the makespan for the products that have received orders. The limited number of changeovers per day is also considered. For the development of the Phase 1 model, the required notation is defined as follows:

• Indices:
  • i  Index of products;
  • j  Index of machines;
  • k  Index of sequences;
  • d  Index of working days.
• Input variables:
  • I  Number of products;
  • J  Number of machines;
  • K  Number of sequences;
  • D  Scheduling period (working days);
  • ${PT}_i$  Production time of product i (hours).
• Decision variable:
  • X_ijk $=\left\{\begin{array}{l}1, \mathrm{If} \mathrm{product} i \mathrm{is} \mathrm{produced} \mathrm{by} \mathrm{Machine} j \mathrm{with} \mathrm{Sequence}k\\ 0, \mathrm{Otherwise} \end{array}\right.$
  • Y_jk $=\left\{\begin{array}{l}1, \mathrm{If} \mathrm{any} \mathrm{product} \mathrm{is} \mathrm{produced} \mathrm{by} \mathrm{Machine} j \mathrm{with} \mathrm{Sequence}k\\ 0, \mathrm{Otherwise} \end{array}\right.$
  • C_jkd  $=\left\{\begin{array}{l}1, \mathrm{If} \mathrm{mold} \mathrm{exchanged} \mathrm{for} \mathrm{Machine}j \mathrm{with} \mathrm{Sequence}k \mathrm{is} \mathrm{in} \mathrm{day} d\\ 0, \mathrm{Otherwise} \end{array}\right.$
  • WD_jk  Working days required by Machine j with Sequence k (integer)
  • S_jk  The day that Machine j with Sequence k exchanges mode;
  • F_jk  The day that Machine j with Sequence k completed;
  • M  Makespan.

The mathematical model in the first phase aims to minimize the makespan as given in Equation (1). Equation (2) ensures that batch splitting is not allowed for any product, and all products have to be scheduled. Equation (3) is used to judge if any product is produced by Machine j with Sequence k. Equation (4) ensures, at most, one product is produced by Machine j with Sequence k. In the case company, the work time is 24 h per day. Because molds can only be exchanged on the day shift, it is assumed that 8 h are required. Equation (5) is used to calculate the number of working days required for each product including the changeover. Equation (6) ensures that the mold in a machine cannot be exchanged before day 1. Equation (7) defines the relationship between the starting and completion days of Sequence k in Machine j. Equation (8) defines the relationship between the completion day of Sequence k and the starting day of Sequence k + 1 in Machine j. Equation (9) is used to judge the date for the mold exchange for Machine j with Sequence k. Equation (10) ensures that, at most, one mold is exchanged per day. The completion time for all of the sequences on all machines cannot exceed the makespan stated in Equation (11).

$$\mathrm{Minimize} M$$

(1)

$${\displaystyle \sum _{j=1}^J}{\displaystyle \sum _{k=1}^K}{X}_{ijk}=1 \forall i$$

(2)

$${\displaystyle \sum _{i=1}^I}{X}_{ijk}=Y_{jk} \forall j,k$$

(3)

$$Y_{jk}\le 1 \forall j,k$$

(4)

$${\displaystyle \sum _{i=1}^I}\left({PT}_i+8\right)X_{ijk}/24\le {WD}_{jk} \forall j,k$$

(5)

$$S_{jk}\ge 1 \forall j,k$$

(6)

$$S_{jk}+{WD}_{jk}-1=F_{jk} \forall j,k$$

(7)

$$F_{jk}+1\le S_{jk+1} \forall j,k=1,\dots ,K-1$$

(8)

$${\displaystyle \sum _{d=1}^D}{d\ast C}_{jkd}=S_{jk}{Y}_{jk} \forall j,k$$

(9)

$${\displaystyle \sum _{j=1}^J}{\displaystyle \sum _{K=1}^K}{C}_{jkd}\le 1 \forall d$$

(10)

$$F_{jk}\le M \forall j,k$$

(11)

Figure 4 shows an example result from the integer program in Phase 1. In this example, for Machine 1, Product 1 is assigned to Sequence 2 and Product 2 is assigned to Sequence 4. No product is assigned to Sequences 1, 3, and 5. Therefore, Machine 1 is arranged to exchange the mold on Day 1 and produce Product 1 until Day 3. Then, Machine 1 exchanges its mold on Day 4, and produces Product 2 until Day 5. For the two machines, at most one changeover per day is allowed, and the makespan is 5 days.

3.2. Phase 2: Minimizing Total Tardiness

Based on the makespan found in Phase 1, the second phase aims to minimize the total tardiness. An integer program modified from Phase 1 is developed. The main difference between Phase 1 and Phase 2 is that the minimal makespan generated by Phase 1 becomes the constraint in Phase 2. Therefore, the makespan, M, in Equation (11) is given in the model of Phase 2. Moreover, the quantities of received orders, and the last day that the model has to be changed for the corresponding products are both taken into consideration. For the development of the Phase 2 model, the required notation is defined as follows:

• Input variables:
  • ${LS}_i$  The last day that the model has to be exchanged to produce product I;
  • $A_i$  The quantity of product i in the received order.
• Decision variables:
  • ${DS}_i$  the day that the mold is exchanged (and starts production) for product i;
  • $L_i$ $=\left\{\begin{array}{l}1, \mathrm{if} \mathrm{product} i \mathrm{is} \mathrm{produced} \mathrm{after} \mathrm{the} \mathrm{last} \mathrm{starting} \mathrm{day}\\ 0, \mathrm{otherwise} \end{array}\right.$
  • T Total tardiness;

Based on the integer program in Phase 1, the objective function is replaced with Equation (12) in Phase 2, aiming to minimize the total tardiness. Moreover, Equations (13) to (15) are involved. Equation (13) is used to calculate the day that the mold is exchanged for all products. Equation (14) is used to judge if a product is late. The total tardiness for any product cannot be negative, as stated by Equation (15).

$$\mathrm{Minimize} T=\sum _{i=1}^I{L}_i\left({DS}_i-{LS}_i\right)A_i$$

(12)

$${\displaystyle \sum _{j=1}^J}{\displaystyle \sum _{k=1}^K}{X}_{ijk}\times S_{jk}={DS}_i \forall i$$

(13)

$$\left({DS}_i-{LS}_i\right)/D\le L_i \forall i$$

(14)

$$L_i\left({DS}_i-{LS}_i\right)\ge 0 \forall i$$

(15)

From the example in Figure 4, it can been found for Phase 1 that the minimum makespan is 5 days. Based on the minimum makespan, the result of Phase 2 is illustrated in Figure 5. The scheduling in Figure 4 and Figure 5 are different. In the example shown in Figure 5, for Machine 1, Product 2 is assigned to Sequence 3, Product 3 is assigned to Sequence 4, and Product 5 is assigned to Sequence 5. No product is assigned to Sequences 1 and 2. Therefore, it is arranged that Machine 1 will exchange its mold on Day 1, and produce Product 2 until Day 2. Then, Machine 1 will exchange its mold on Day 3, and produce Product 3 on the same day. After, Machine 1 will exchange its mold on Day 4, and produce Product 5 until Day 5. As in Figure 4, at most one mold exchange can be accomplished per day.

In Figure 5, Products 1 and 4 are manufactured late for their corresponding orders. The tardiness of these two products are both 1 day (2-1 and 5-1). Thus, the total tardiness is 250 (50 × 1 + 200 × 1). In Figure 4, Products 2 and 3 are manufactured late for their corresponding orders. The tardiness of these two products are both 2 days (4-2 and 5-3). The total tardiness is 500 (100 × 2 + 150 × 2). Therefore, the proposed integer program in Phase 2 can further reduce the total tardiness with the same minimal makespan.

Since a typical identical PMS problem is an NP-hard problem, more constraints are taken into consideration in the proposed problem. LINGO is a comprehensive tool designed to make building and solving mathematical optimization models easier and more efficient [19]. Therefore, this research adopted LINGO 17 to optimize both the integer programming problems in Phase 1 and Phase 2.

4. Empirical Illustrations

In the case company, there are four identical parallel injection molding machines that produce plastic pallets. On average, each machine produces 45 pallets per hour. The production manager always schedules the products whose due dates are within about 2 weeks (i.e., 10 working days). Because, at most, one changeover is allowed per day, this research generated nine products randomly for testing. Descriptions of the generation processes are shown in

Table 1. The generation of test data.

No.	Product Data	Random Generation	Description
1	PTi	[8 h, 110 h]	The number of working days required by a product is between 1 and 5 days. It can also be found that the production batch sizes of different products are quite different.
2	Ai	PTi × [10, 40] × 0.01 × 45	The quantity of received orders are 10% to 40% of the production batches.
3	LSi	[Day 1, Day 9]	The latest starting day is Day 9, and the maximum production time is 5 days. Thus, all products can be completed within 15 days.

[a, b] Random integers between a and b generated by uniform distribution.

.

This research generated 20 sets of data for testing. Both integer program in Phases 1 and 2 are optimized by the commercial software LINGO 17. According to a prior test using a computer with Intel Core i7-9700 3.00 GHz, the maximum run time was set at 2 min for the integer program in Phase 1. As the integer program of Phase 2 is constrained by a makespan generated by Phase 1, longer run times are required. The maximal run time for the integer program of Phase 2 is set as 20 min. In order to reduce the required run time, $S_{j1}=j$ for all machines are set. The results are shown in

Table 2. The experimental results.

No.	Phase 1		Phase 1 + Phase 2
	M	T	M	T
1	10	8052	10	1250
2	10	6892	10	1239
3	10	14,890	10	4897
4	9	66,054	9	300
5	10	5813	10	1380
6	9	39,252	9	2040
7	10	5987	10	899
8	9	6129	9	4232
9	10	18,240	10	1087
10	9	5268	9	2144
11	9	4618	9	1119
12	9	5726	9	1097
13	10	13,349	10	1471
14	9	5241	9	2114
15	10	29,982	10	2759
16	11	11,208	11	1588
17	10	11,360	10	2267
18	11	15,645	11	4603
19	10	17,610	10	1743
20	9	2841	9	2841

.

In Table 2, Phase 1 indicates the results generated by the integer program in Phase 1, and Phase 1 + Phase 2 indicates the results generated by the integer program in Phase 2, based on the makespan results from Phase 1. According to the results shown in Table 2, the total tardiness resulting from Phase 1 + Phase 2 are always much shorter than Phase 1, except in Case 20. Therefore, the proposed two-phase methodology can minimize the makespan and total tardiness sequentially.

The present study then compared the proposed methodology with the following two heuristics: longest processing time (LPT) and largest weighted slack time (LWST). The LPT is a heuristics that was developed for PMS to minimize the makespan. The unscheduled job with the longest processing time has the highest priority in being assigned to a machine whenever one is free [14]. Regarding the LWST, it incorporates the following two criteria: weight and slack time. For a product with a larger quantity of orders, a 1 day delay will increase the total tardiness more than a product with a smaller quantity of orders. Thus, the products with larger quantities of orders should have higher priority. In this research, the weight of a product was set as the quantity of the received orders, A_i. The slack time means the remaining spare time at the current time [20]. The less slack time, the higher the priority of the product. By definition, the slack time is equal to the last day that the model has to be exchanged for producing the product, LS_i. Accordingly, LWST arranges all products in decreasing order of A_i/LS_i.

The results for the LPT and LWST are shown in

Table 3. Comparison between the proposed methodology and other methods.

No.	Proposed Methodology		LPT		LWST
	M	T	M	T	M	T
1	10	1250	10	5998	10	1680
2	10	1239	10	6845	10	2345
3	10	4897 *	11	9188	11	4560
4	9	300	10	2252	10	1311
5	10	1380	10	3480	10	2796
6	9	2040	10	3775	9	2399
7	10	899	11	9492	11	1747
8	9	4232	10	7822	11	6322
9	10	1087	11	7487	11	2916
10	9	2144	10	4705	10	2976
11	9	1119	10	5059	9	1533
12	9	1097	9	4915	9	1817
13	10	1471	11	6403	11	2181
14	9	2114	9	5040	10	3017
15	10	2759	10	6485	10	2935
16	11	1588	11	14,884	11	1588
17	10	2267	10	8510	10	3197
18	11	4603	11	12,690	11	5197
19	10	1743	10	6450	10	2621
20	9	2841 *	10	4856	11	470

* Indicates the performances by the proposed methodology that were worse than those of other heuristics.

. It can be seen from the results that with the proposed methodology the makespans are shorter or equal to the LPT and LWST in all cases. For the total tardiness, the proposed methodology had worse results than the LWST in two cases, and the LWST results had longer makespans. For these two cases, it is not easy to say which methodology is better. This study then used Phase 2 to minimize the total tardiness for Cases 3 and 20 with 11 days of makespan, which is equal to the results of LWST. The resulting total tardiness values are 4320 and 138 days, which are both better than that of the LWST. Therefore, the proposed methodology can outperform the LPT and LWST for the proposed identical PMS problem. In Table 3, it can also be seen that LWST outperformed the LPT in all 20 cases.

The company in the case study wants to relax the constraints imposed by the mold exchange. Adding another team that can perform a mold exchange is one option (although the two teams would still have to share the crane). This research then modified Equation (10) to be Equation (16). The results of the proposed methodology are shown in

Table 4. Comparison among teams of different numbers for the mold exchanges.

No.	# Team of Mold Exchange				Improvement (%)
	1		2
	M	T	M	T	M	T
1	10	1250	9	0	10.00	100.00
2	10	1239	8	69	20.00	94.43
3	10	4897	8	2603	20.00	46.85
4	9	300	7	0	22.22	100.00
5	10	1380	8	1186	20.00	14.06
6	9	2040	7	0	22.22	100.00
7	10	899	9	0	10.00	100.00
8	9	4232	8	774	11.11	81.71
9	10	1087	7	0	30.00	100.00
10	9	2144	8	0	11.11	100.00
11	9	1119	7	0	22.22	100.00
12	9	1097	8	0	11.11	100.00
13	10	1471	8	189	20.00	87.15
14	9	2114	7	0	22.22	100.00
15	10	2759	7	1562	30.00	43.36
16	11	1588	10	154	9.09	90.30
17	10	2267	8	0	20.00	100.00
18	11	4603	10	888	9.09	80.71
19	10	1743	8	1467	20.00	15.83
20	9	2841	9	0	0.00	100.00

.

$${\displaystyle \sum _{j=1}^J}{\displaystyle \sum _{K=1}^K}{C}_{jkd}\le 2 \forall d$$

(16)

In Table 4, it can be seen that adding another team that can perform a mold exchange can reduce the makespan for all cases, except Case 20 (in which the results are the same). The total tardiness can also be improved in all cases, and the tardiness can even be reduced to zero in 11 cases.

The total computing times were highly dependent on the mathematical model of Phase 2. According to observations during LINGO 17’s run, the times required to result in the minimal total tardiness are illustrated in Figure 6. It can be found that the results were within 2 min for nine cases. However, more than 10 min are required in five cases (1, 12, 13, 17, and 18).

5. Discussion

This research deals with an identical PMS problem, which was motivated by an examination of a company that manufactures plastic pallets. Because of the limitation of additional resources, one mold exchange is allowed per day. The production strategy of the company is MTS. Once the inventory of a product is less than the quantity of the received order, the production manager will schedule the product to be manufactured in a predetermined batch size, which is larger than the quantity of the order. Then, the remaining amount will be stored in the warehouse. If the order is not completed by the due date, the whole order will be delayed for delivery.

This research proposed a two-phase methodology for the identical PMS problem. In Phase 1, an integer program that considers the limitation of the mold exchange is developed to minimize the makespan. The makespan results from Phase 1 then become the upper bound. Another integer program that takes the upper bound of the makespan and information about the orders into consideration is then developed in Phase 2 to minimize the total tardiness.

This research randomly generated 20 sets of data based on the manufacturing environment of the company case study. The proposed methodology was then compared with two heuristics, as follows: LPT and LWST. The results show the effectiveness of the proposed methodology for a practical application. Identical parallel machines with additional resources for MTS are common in industry, and scheduling decisions always play an important role in the performance of whole manufacturing systems. Therefore, the proposed methodology can be applied to other industries, in addition to the plastic pallet manufacturing company.

Based on observations during the experiments, it is believed that longer run times will be required when the number of machines or products increases. Fortunately, these 20 cases were all generated based on actual conditions in which the working days required for all products ranged from 1 to 5 days. Therefore, the run times were sufficient for the proposed case. However, in order to solve different practical problems, an efficient heuristic, such as genetic algorithms, Tabu search or simulated annealing, might be proposed. Then, in addition to experiments on larger problems, more analyses and comparisons can be conducted. This may be an opportunity for future research.

The results also show the improvements that can be achieved by relaxing the constraint of the mold exchange by adding one more team capable of performing a mold exchange. Reducing the time required for a mold exchange and making it possible to exchange two molds per day by one team is another way reduce the constraint. For example, if the changeover times can be reduced to 5 h, then two molds can be exchanged in 1 day with 2 h of overtime. However, adding one more team and reducing the time required for a mold exchange have different implications for developing the mathematical models. Reducing the time required for a mold exchange does not add additional resources and cannot be modeled by simply modifying Equation (11). Therefore, developing a mathematical model that considers a shorter length of time for a mold exchange and analyses of the effect on the makespan and total tardiness could also be opportunities for future research.