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Applied System Innovation
, Volume 5 (4) – Jun 22, 2022

/lp/multidisciplinary-digital-publishing-institute/coordinating-production-and-delivery-schedule-of-multi-product-and-6ptDhQdvon

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Article Coordinating Production and Delivery Schedule of Multi-Product and Multi-Customer through Mathematical Programming The Jin Ai * and Ririn Diar Astanti Department of Industrial Engineering, Universitas Atma Jaya Yogyakarta, Yogyakarta 55281, Indonesia; ririn.astanti@uajy.ac.id * Correspondence: the.jinai@uajy.ac.id; Tel.: +62-274487711 Abstract: Due to the shortage and high cost of shipping containers, manufacturers that distribute products for worldwide customers must optimize their delivery. Commonly, they complete their delivery schedule ﬁrst before creating the production schedule. This article tries to address the scheduling problem arising in the process of creating the production schedule to meet the delivery schedule. The complexity of the situation is increased since there is concern about producing as few product types as possible in a single period and having a balance of production load over multiple periods. A mathematical programming model for addressing this production scheduling problem is proposed in the form of integer programming. This model can be solved using LINGO, a commercial mathematical programming software. Microsoft Excel spreadsheets are used to store both the data input and the output of the mathematical programming model. The advantages of using Excel are its easy-to-read format and familiarity with most users in various companies. Computational experiments in several cases show that the proposed method is effective to create the production and delivery schedule in various problem settings as expected. Keywords: production schedule; integrated production and delivery; mathematical programming; Citation: Ai, T.J.; Astanti, R.D. Coordinating Production and integer programming Delivery Schedule of Multi-Product and Multi-Customer through Mathematical Programming. Appl. Syst. Innov. 2022, 5, 59. https:// 1. Introduction doi.org/10.3390/asi5040059 This paper is motivated by the current situation in a manufacturing industry that Academic Editor: Abdelkader Sbihi produces a variety of products in response to fulﬁlling orders from a large number of customers from various countries. This manufacturing industry operates using a make- Received: 3 May 2022 to-order (MTO) strategy. Because shipping containers are currently very expensive, this Accepted: 17 June 2022 industry prioritizes that each container sent has the best load efﬁciency possible, which Published: 22 June 2022 is close to a full container load. In this manner, the production planning department will Publisher’s Note: MDPI stays neutral await the shipping department’s product delivery schedule in the form of a container with regard to jurisdictional claims in delivery schedule and the contents of each container within a speciﬁc planning period, published maps and institutional afﬁl- such as the next 4 to 8 weeks. iations. The production planning department is in charge of creating a production schedule in order to meet the delivery schedule established by the shipping department. Based on their experience, the company needs to minimize product variation in each period of the planning horizon. They also preferred to balance the production load across the period. Copyright: © 2022 by the authors. In the make-to-order environment, determining the production schedule is impor- Licensee MDPI, Basel, Switzerland. tant [1–10]. This decision is made for satisfying various purposes, with the most common This article is an open access article purpose being to meet the customer delivery due dates [1,4,10–15]. Frequently, this pro- distributed under the terms and duction schedule decision is made in conjunction with other decisions. Furthermore, this conditions of the Creative Commons production schedule decision frequently inﬂuences subsequent decisions. Examples of Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ related decisions to production schedule are supplier selection [2,16], price [4,14,17,18], and 4.0/). transportation [1,18–22]. Appl. Syst. Innov. 2022, 5, 59. https://doi.org/10.3390/asi5040059 https://www.mdpi.com/journal/asi Appl. Syst. Innov. 2022, 5, 59 2 of 14 Previous research has demonstrated that the production schedule can be prepared using a mathematical programming approach by considering a variety of settings, con- straints, and objectives. Galbraith and Miller [23] discussed a scheduling problem for a multi-objective, multi-period, multi-product manufacturing operation in terms of four objectives: capacity utilization, delivery schedule compliance, cost minimization, and overtime minimization. They compared the approaches of mathematical programming and simulation to solve the problem. Hill [24] proposed a mathematical model in which a manufacturer purchases raw materials, fabricates products, and distributes them on a regular basis to a single customer. The model’s purchasing and production schedules are output decisions with the goal of minimizing total costs, with each production run divided into multiple shipments. Bahroun et al. [25,26] considered the situation in a factory that produces multiple products and faces cyclic demand, i.e., the delivery schedule is repeated in the long run. They proposed a production model for creating production sequences. Li et al. [27] discussed a manufacturing environment in which machines operate in parallel and require periodic maintenance. They proposed mathematical programming models for creating production schedules. Sun et al. [28] proposed a novel mathematical programming model for scheduling production in the steel industry that takes energy consumption and carbon emissions into account in addition to traditional production performance indicators. As demonstrated in the preceding studies, the production schedule is inextricably linked to the delivery schedule. As a result, numerous researchers are also develop- ing simultaneous or integrated planning between production and delivery schedules. Li et al. [29] examined several integrated production, inventory, and delivery problems where customers specify delivery dates. They proposed an integrated model with the goal of ﬁnding an integrated schedule for processing orders, keeping ﬁnished goods, and delivering them to customers that minimizes inventory and delivery costs. Fu et al. [30] proposed an integrated production and delivery scheduling model, in which each job is occupied with release dates and deadlines. Nogueira et al. [31] discussed integrated production and delivery planning in which parallel batching machines are available and orders are generic in size and processing time. Masruroh et al. [32] proposed an integrated production and delivery scheduling model for manufacturers who produce many products and whose manufacturing setup is dependent on the production sequence. They aimed to reduce production costs to increase annual gross proﬁt. Mohammadi et al. [33] discussed an integrated production and delivery schedule, where the manufacturer is using a make- to-order strategy supported by multi-purpose machines. Their mathematical model has two objectives, which are minimizing total costs and minimizing precision on delivery. Despite the extensive research in the past reviewed in the previous paragraphs, a new situation has arisen that requires a solution, which is to develop a production schedule that adheres to the load-optimized delivery schedule while minimizing production variation and balancing production load. The purpose of this article is to propose a mathematical model for addressing the production scheduling problem. The model is proposed in the form of integer programming and can be solved using LINGO, a commercial mathematical programming software. This article also demonstrates the effectiveness of the proposed model through several examples. The mathematical model and its solution method for coordinating production and delivery schedules is the main contribution of this article. The remainder of this manuscript is organized as follows: In Section 2, Materials and Methods, the problem and mathematical model are deﬁned, and the solution method is explained. Section 3 demonstrates the effectiveness of the proposed mathematical model and solution method by applying them to solve several scheduling cases involving a variety of problem settings. The conclusion of this manuscript is presented in Section 4, along with some remarks for future development. 2. Materials and Methods As stated in the preceding section, the purpose of this paper is to propose a new mathematical model for optimizing the production schedule in a manufacturing industry Appl. Syst. Innov. 2022, 5, 59 3 of 14 where the delivery schedule has previously been optimized in terms of the container delivery schedule. The production scheduling issue involves a multitude of products and customers. It also planned for multiple time spans. In the mathematical model, the number of products, customers, containers, and time span can be set ﬂexibly, allowing the model to be applied to a variety of situations, i.e., problems with varying numbers of products, customers, containers, and time spans. After describing the problem and mathematical model in Sections 2.1 and 2.2 explains the solution method. 2.1. Proposed Mathematical Model This mathematical model is intended to solve the problem described below. The problem is to create a production plan for P periods, that is, from t = 1 to t = P. For these P periods, the shipping department has issued a tentative delivery schedule for K containers, each of which contains L products. It is called tentative because there are three possible delivery schedules for a container. Some containers must be shipped within a speciﬁc time frame, such as the 3rd period. There are also containers that can be shipped within a speciﬁc time frame, for example between the 5th and the 8th period. Finally, there are also containers that can be sent at any time during the span. The contents of each container have already been determined by the shipping depart- ment. In other words, the data input for this problem is the quantity of product i inside container j, denoted as Q . ij This problem has two sets of decision variables: the delivery schedule and the produc- tion schedule. The delivery schedule set the shipping delivery time for each container. The delivery schedule is speciﬁed in this mathematical model as binary variable x , which is jt deﬁned as 1, if container j is scheduled at period t x = (1) jt 0, otherwise The production schedule details which products are manufactured during each period, as well as the quantity produced for each product. In this problem, it is assumed that the completed product within a speciﬁed time is shipped immediately in the same time period. The production schedule is also speciﬁed as binary variable y , which is deﬁned as it 1, if product i is scheduled at period t y = (2) it 0, otherwise The company wants to minimize product variation in each period of the planning horizon. Therefore, the sum of decision variables y is being set to be minimized. it The quantity of product i to be shipped at period t can be calculated as Q x . Based on ij jt the assumption, this quantity is also equal to the quantity of product i to be manufactured at period t. The load factor is also deﬁned for product i or the total production time for one item product i as l . Therefore, the production load at period t can be calculated as L K l Q x (3) å iå ij jt i=1 j=1 The production load for each period needs to be balanced. If the total balance condition can be satisﬁed, then the production load in each period is equivalent to the average overall production load. The parameter a is introduced here to anticipate the variety of possible cases and to provide production planners with ﬂexibility, in which the production load at a certain period is set within fraction from the average. Mathematically, it is written as the following two equations. L K L K l Q x (1 + a) l Q (4) å iå ij jt å iå ij i=1 j=1 i=1 j=1 Appl. Syst. Innov. 2022, 5, 59 4 of 14 L K L K l Q x 1 a l Q (5) ( ) i ij jt i ij å å å å i=1 j=1 i=1 j=1 It is anticipated that by providing greater ﬂexibility to the production load balance, i.e., a greater value of , the problem is more relaxed, thereby increasing the likelihood of achieving a better objective function value, i.e., a smaller sum of decision variables y . In it the results section, these effects will be studied and discussed in detail. The complete mathematical programming formulation is presented below. Equation (6) is the objective function, while Equations (7)–(12) are the constraints. Equations (7) and (8) are the constraint for balancing the production load for each period. Equation (9) is deﬁned to ensure that each container is only being scheduled to be delivered in only a single period. Equation (10) is deﬁned to ensure that the delivery schedule can be supported by the production schedule, i.e., the product inside the container delivered at period t is scheduled to be manufactured in the same period. The parameter M on the right-hand side of this equation refers to a large positive number, like the parameter in the well-known Big-M method. Equations (11) and (12) are the deﬁnition of binary decision variables. L P Min Z = y (6) å å it t=1 i=1 L K L K l Q x (1 + a) l Q , 8t = 1 . . . P (7) å iå ij jt å iå ij i=1 j=1 i=1 j=1 L K L K l Q x (1 a) l Q , 8t = 1 . . . P (8) å iå ij jt å iå ij i=1 j=1 i=1 j=1 x = 1, 8j = 1 . . . K (9) jt t=1 Q x M y , 8i = 1 . . . L,8j = 1 . . . K, 8t = 1 . . . P (10) ij jt it x 2 f0, 1g, 8j = 1 . . . K, 8t = 1 . . . P (11) jt y 2 0, 1 , 8i = 1 . . . L, 8t = 1 . . . P (12) f g jt Additional constraints may be added to address the tentative delivery schedule. If a container must be shipped at a speciﬁc period, the respective delivery schedule decision variable for that container for that period is set equal to 1 and the variables for other periods are set equal to 0. For example, if container 3 must be shipped at period 2, the following constraints are set x = 1 (13) x = 0, 8t = 1, 3 . . . P (14) 3t Similarly, if container 4 can be shipped during periods 1 to 4, the following constraints are set x 1, 8t = 1 . . . 4 (15) 4t x = 0, 8t = 5 . . . P (16) 4t 2.2. Proposed Solution Method To assist the production planner in solving the proposed mathematical model pre- sented in the previous subsection, the following solution method is proposed. This method stores both the input and output data in Microsoft Excel spreadsheets. By using a Mi- crosoft Excel spreadsheet, the production planner gains the advantage of being able to conveniently enter data and read the output in a format that is commonly used in their company’s business processes. Hereinafter, the mathematical model described in Section 2.1 Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 5 of 15 x = 0 , ∀=tP 1, 3 3t (14) Similarly, if container 4 can be shipped during periods 1 to 4, the following con- straints are set x ≤ 1 , ∀= t 14 4t (15) x = 0 , ∀=tP 5 (16) 4t 2.2. Proposed Solution Method To assist the production planner in solving the proposed mathematical model pre- sented in the previous subsection, the following solution method is proposed. This method stores both the input and output data in Microsoft Excel spreadsheets. By using a Microsoft Excel spreadsheet, the production planner gains the advantage of being able Appl. Syst. Innov. 2022, 5, 59 to conveniently enter data and read the output in a format that is commonly used in 5 th of eir 14 company’s business processes. Hereinafter, the mathematical model described in Section 2.1 above is solved using a well-known mathematical programming solver, LINGO ver- sion 18. The designed solution procedure consists of the following steps: above is solved using a well-known mathematical programming solver, LINGO version 18. The designed solution procedure consists of the following steps: a. The data model is entered into the Excel spreadsheet file; b. The mathematical model is solved using LINGO; (a) The data model is entered into the Excel spreadsheet ﬁle; c. The production and delivery schedule output are read from the Excel spreadsheet (b) The mathematical model is solved using LINGO; file. (c) The production and delivery schedule output are read from the Excel spreadsheet ﬁle. In the first step, a special effort should be made to make the data input and output In the ﬁrst step, a special effort should be made to make the data input and output transferable between Microsoft Excel and LINGO, which uses the “Define Name” feature transferable between Microsoft Excel and LINGO, which uses the “Deﬁne Name” feature in in Microsoft Excel. All parameters and variables used in the mathematical model, which Microsoft Excel. All parameters and variables used in the mathematical model, which are P, are P, K, L, Q, l, α, x, and y, had to be identified by assigning a specific range in the work- K, L, Q, l, , x, and y, had to be identiﬁed by assigning a speciﬁc range in the worksheet to each sheet to name. each Figur name. e 1 demonstrates Figure 1 demon how strto ates ho deﬁne w to the d selected efine the selec range in ted the raworksheet nge in the as work- the parameter sheet as thQ e for parthe ame scheduling ter Q for th case e sche ofdu 18ling products case of and 18 pr 12 containers. oducts and 12 cont Before the ainers. screenshot Before in Figure 1 appears, two actions are required: selecting the cell range and clicking the the screenshot in Figure 1 appears, two actions are required: selecting the cell range and Formula tab’s “Deﬁne Name” button. clicking the Formula tab’s “Define Name” button. Figure 1. Defining parameter Q Range name in the worksheet. Figure 1. Deﬁning parameter Q Range name in the worksheet. In the second step, the mathematical model is solved using a well-known mathematical In the second step, the mathematical model is solved using a well-known mathemat- programming solver, LINGO version 18. Figure 2 illustrates a typical LINGO Model ical programming solver, LINGO version 18. Figure 2 illustrates a typical LINGO Model of the proposed mathematical model. The SETS...ENDSETS statement deﬁnes the sets, of the proposed mathematical model. The SETS...ENDSETS statement defines the sets, pa- parameters, and variables. The values of parameters and variables are deﬁned in the rameters, and variables. The values of parameters and variables are defined in the DATA...ENDDATA statement, which uses the @OLE function to retrieve data from a DATA...ENDDATA statement, which uses the @OLE function to retrieve data from a Mi- Microsoft Excel ® worksheet. Additionally, the @OLE function returns the values of decision crosoft Excel worksheet. Additionally, the @OLE function returns the values of decision variables x and y to the corresponding Excel worksheet. It is noted that the worksheet referenced in the illustration is “DATA.XLSX”, which is located on the computer ’s E drive. The remainder of the LINGO statements represents the mathematical programming expressed in Equations (5)–(10). For details on the syntax, please refer to the LINGO Manual book. It is also noted that this LINGO model is free from speciﬁc values for the number of products (L), containers (K), and periods (P). These input parameters are set already through the Microsoft Excel spreadsheet. Therefore, if there is a need to make a production and distribution plan with a different number of products, containers, or periods, the same LINGO model can be used directly, but the input parameters in the spreadsheet must be updated. In the third step, the production and delivery schedule as the solutions of the math- ematical model, which are deﬁned as decision variables x and y, can be obtained in the corresponding range name in the Excel spreadsheet ﬁle. The format of these decision variables in the Excel spreadsheet is also designed such that it is convenient to read by the production planner. Figure 3 displays an example format for the variable y, which represents the production schedule. It can be seen from the ﬁgure that the production schedule for each product can be read from its corresponding line. A value of zero in a Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 6 of 15 Appl. Syst. Innov. 2022, 5, 59 6 of 14 variables x and y to the corresponding Excel worksheet. It is noted that the worksheet column indicates that the product is not scheduled to be manufactured during that period. referenced in the illustration is “DATA.XLSX”, which is located on the computer’s E drive. Otherwise, a value of one indicates that the product is scheduled to be manufactured The remainder of the LINGO statements represents the mathematical programming ex- during that period. In Figure 3, for example, product P1 is scheduled to be manufactured pressed in Equations (5)–(10). For details on the syntax, please refer to the LINGO Manual in period 3 and product P9 is scheduled to be manufactured in period 1. book. Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 7 of 15 that period. In Figure 3, for example, product P1 is scheduled to be manufactured in pe- Figure 2. A typical LINGO Model of the proposed mathematical model. Figure riod 3 2. and product P A typical LINGO 9 is Model scheduled of theto be proposed manufacture mathematical d in pe model. riod 1. It is also noted that this LINGO model is free from specific values for the number of products (L), containers (K), and periods (P). These input parameters are set already through the Microsoft Excel spreadsheet. Therefore, if there is a need to make a produc- tion and distribution plan with a different number of products, containers, or periods, the same LINGO model can be used directly, but the input parameters in the spreadsheet must be updated. In the third step, the production and delivery schedule as the solutions of the math- ematical model, which are defined as decision variables x and y, can be obtained in the corresponding range name in the Excel spreadsheet file. The format of these decision var- iables in the Excel spreadsheet is also designed such that it is convenient to read by the production planner. Figure 3 displays an example format for the variable y, which repre- sents the production schedule. It can be seen from the figure that the production schedule for each product can be read from its corresponding line. A value of zero in a column indicates that the product is not scheduled to be manufactured during that period. Other- Figure 3. An example of production schedule (y) output format in Excel. Figure 3. An example of production schedule (y) output format in Excel. wise, a value of one indicates that the product is scheduled to be manufactured during 3. Results and Discussion In order to demonstrate the effectiveness of the proposed mathematical model and solution method, several scheduling cases are examined. Case 1 and Case 2 are the cases that originated from the manufacturing company that motivated this research. Case 3 is examined to demonstrate the generalizability of the model and solution method, as the load factors for all products in both Case 1 and Case 2 are similar. In this instance, each product has its own load factor. In Case 4, tentative delivery schedules are considered. The results for various values of α are presented at the end of this section. 3.1. Case 1 In this case, which is directly taken from the company, 43 types of products that are contained within 64 containers will be scheduled in 4 weeks. The quantity of each product inside each container is presented in Figure 4. It is also known that the load factor of all products is similar, i.e., l1 = l2 = … = l43 = 1. This case is solved using the proposed method described in Section 2 above using α equal to 0.005. Figure 4. Quantity of products inside containers, Case 1, in the form of an Excel spreadsheet. Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 7 of 15 that period. In Figure 3, for example, product P1 is scheduled to be manufactured in pe- riod 3 and product P9 is scheduled to be manufactured in period 1. Appl. Syst. Innov. 2022, 5, 59 7 of 14 Figure 3. An example of production schedule (y) output format in Excel. 3. Results and Discussion 3. Results and Discussion In order to demonstrate the effectiveness of the proposed mathematical model and In order to demonstrate the effectiveness of the proposed mathematical model and solution method, several scheduling cases are examined. Case 1 and Case 2 are the cases solution method, several scheduling cases are examined. Case 1 and Case 2 are the cases that originated from the manufacturing company that motivated this research. Case 3 is that originated from the manufacturing company that motivated this research. Case 3 is examined to demonstrate the generalizability of the model and solution method, as the examined to demonstrate the generalizability of the model and solution method, as the load factors for all products in both Case 1 and Case 2 are similar. In this instance, each load factors for all products in both Case 1 and Case 2 are similar. In this instance, each product has its own load factor. In Case 4, tentative delivery schedules are considered. The product has its own load factor. In Case 4, tentative delivery schedules are considered. results for various values of are presented at the end of this section. The results for various values of α are presented at the end of this section. 3.1. Case 1 3.1. Case 1 In this case, which is directly taken from the company, 43 types of products that are In this case, which is directly taken from the company, 43 types of products that are contained within 64 containers will be scheduled in 4 weeks. The quantity of each product contained within 64 containers will be scheduled in 4 weeks. The quantity of each product inside each container is presented in Figure 4. It is also known that the load factor of all inside each container is presented in Figure 4. It is also known that the load factor of all products is similar, i.e., l = l = . . . = l = 1. This case is solved using the proposed method 1 2 43 products is similar, i.e., l1 = l2 = … = l43 = 1. This case is solved using the proposed method described in Section 2 above using equal to 0.005. described in Section 2 above using α equal to 0.005. Figure 4. Quantity of products inside containers, Case 1, in the form of an Excel spreadsheet. Figure 4. Quantity of products inside containers, Case 1, in the form of an Excel spreadsheet. The results of the proposed method are presented in Figure 5, which are the value of decision variables x and y. The decision variable x represents the delivery schedule of containers. The interpretation of the results in Figure 5 are as follows: Containers C1, C2, . . . , C8 are scheduled to be delivered at week 4; Containers C9, C10, . . . , C13 are scheduled to be delivered at week 1; Container C14 is scheduled to be delivered at week 3; and so on. Meanwhile, the decision variable y represents the production schedule of products. Our interpretation of the results in Figure 5 is as follows: Products P1, P2, . . . , P14 are scheduled to be produced at week 4; Products P15, P16, . . . , P21 are scheduled to be produced at week 1; Products P22 and P23 are scheduled to be produced at week 1 and week 3; Products P24 and P25 are scheduled to be produced at week 4; and so on. A summary of the weekly decision variables and weekly production load are presented in Table 1. As seen in Table 1, all 64 containers have been scheduled completely within four weeks. Meanwhile, 43 products are scheduled to be manufactured within the next four weeks. It is also found that ﬁve products are scheduled for two distinct weeks, which are P22, P23, P28, P33, and P42. As a result, the total number of products, which is the objective function of this mathematical model, equals 48. Since the value is very small, the weekly production load is similarly comparable, in which the production load is in the range of Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 8 of 15 The results of the proposed method are presented in Figure 5, which are the value of decision variables x and y. The decision variable x represents the delivery schedule of containers. The interpretation of the results in Figure 5 are as follows: Containers C1, C2, …, C8 are scheduled to be delivered at week 4; Containers C9, C10, …, C13 are scheduled to be delivered at week 1; Container C14 is scheduled to be delivered at week 3; and so Appl. Syst. Innov. 2022, 5, 59 8 of 14 on. Meanwhile, the decision variable y represents the production schedule of products. Our interpretation of the results in Figure 5 is as follows: Products P1, P2, …, P14 are scheduled to be produced at week 4; Products P15, P16, …, P21 are scheduled to be pro- 1255 to 1265. The production load range is consistent with the deﬁnition of constraints (7) duced at week 1; Products P22 and P23 are scheduled to be produced at week 1 and week and (8). Following those constraints, the load must be within the range of 1252.7 to 1265.3. 3; Products P24 and P25 are scheduled to be produced at week 4; and so on. Figure 5. Output of decision variables x and y for Case 1 in the form of an Excel spreadsheet. Figure 5. Output of decision variables x and y for Case 1 in the form of an Excel spreadsheet. Table 1. A s Summary ummary of of the the decision weekly variables decision for v Case ariable 1. s and weekly production load are pre- sented in Table 1. As seen in Table 1, all 64 containers have been scheduled completely Week Number of Containers Number of Products Production Load within four weeks. Meanwhile, 43 products are scheduled to be manufactured within the Week 1 14 12 1258 next four weeks. It is also found that five products are scheduled for two distinct weeks, Week 2 16 10 1265 which are P22, P23, P28, P33, and P42. As a result, the total number of products, which is Week 3 19 8 1255 the objective function of this mathematical model, equals 48. Since the α value is very Week 4 15 18 1258 small, the weekly production load is similarly comparable, in which the production load Total 64 48 5036 is in the range of 1255 to 1265. The production load range is consistent with the definition of constraints (7) and (8). Following those constraints, the load must be within the range 3.2. Case 2 of 1252.7 to 1265.3. In this case, which is also directly taken from the company, 34 types of products that Table 1. Summary of the decision variables for Case 1. are contained within 47 containers will be scheduled in 5 weeks. The quantity of each product inside each container is presented in Figure 6. It is also known that the load factor Week Number of Containers Number of Products Production Load of all products is similar, i.e., l = l = . . . = l = 1. This case is also solved using the 1 2 34 Week 1 14 12 1258 proposed method described in Section 2 above using equal to 0.005. Week 2 16 10 1265 The result of this case is presented in Figure 7. It is implied from Figure 7 related to Week 3 19 8 1255 the delivery schedule: Containers C1, C2, C3, C4 are scheduled to be delivered at week 2; Week 4 15 18 1258 Container C5, C6, C7, C8 are scheduled to be delivered at week 5; Containers C9, C10, Total 64 48 5036 C11 are scheduled to be delivered at week 3; and so on. Furthermore, it is implied from the same ﬁgure related to production schedule: Products P1, P2, . . . , P8 are scheduled to be produced at week 2; Products P9, P10, P11, P12 are scheduled to be produced at week 5; Product 13 is scheduled to be produced at week 3 and week 5; Products P14, P15 are scheduled to be produced at week 3; and so on. A summary of the weekly decision variables and weekly production load are presented in Table 2. As seen in Table 2, all 47 containers have been scheduled completely within ﬁve weeks. Meanwhile, 34 products are scheduled to be manufactured within the next ﬁve weeks. It is also found that eight products are scheduled for two distinct weeks, which are P13, P16, P17, P18, P19, P21, P28, and P29. Furthermore, Product P22 is scheduled Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 9 of 15 3.2. Case 2 In this case, which is also directly taken from the company, 34 types of products that are contained within 47 containers will be scheduled in 5 weeks. The quantity of each product inside each container is presented in Figure 6. It is also known that the load factor of all products is similar, i.e., l1 = l2 = … = l34 = 1. This case is also solved using the proposed method described in Section 2 above using α equal to 0.005. Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 9 of 15 Appl. Syst. Innov. 2022, 5, 59 9 of 14 3.2. Case 2 for three distinct weeks. As a result, the total number of products, which is the objective In this case, which is also directly taken from the company, 34 types of products that function of this mathematical model, equals 44. Since the value is very small, the weekly are contained within 47 containers will be scheduled in 5 weeks. The quantity of each production load is similarly comparable, in which the production load is in the range of product inside each container is presented in Figure 6. It is also known that the load factor 1238 to 1249. The production load range is consistent with the deﬁnition of constraints (7) of all products is similar, i.e., l1 = l2 = … = l34 = 1. This case is also solved using the proposed and (8). Following those constraints, the load must be within the range of 1236.8 to 1249.2. method described in Section 2 above using α equal to 0.005. Figure 6. Quantity of products inside containers, Case 2, in the form of an Excel spreadsheet. The result of this case is presented in Figure 7. It is implied from Figure 7 related to the delivery schedule: Containers C1, C2, C3, C4 are scheduled to be delivered at week 2; Container C5, C6, C7, C8 are scheduled to be delivered at week 5; Containers C9, C10, C11 are scheduled to be delivered at week 3; and so on. Furthermore, it is implied from the same figure related to production schedule: Products P1, P2, …, P8 are scheduled to be produced at week 2; Products P9, P10, P11, P12 are scheduled to be produced at week 5; Product 13 is scheduled to be produced at week 3 and week 5; Products P14, P15 are Figure 6. scheduled Quan to tity be produced of products at week inside con 3; tai an nd so on ers, Case . 2, in the form of an Excel spreadsheet. Figure 6. Quantity of products inside containers, Case 2, in the form of an Excel spreadsheet. The result of this case is presented in Figure 7. It is implied from Figure 7 related to the delivery schedule: Containers C1, C2, C3, C4 are scheduled to be delivered at week 2; Container C5, C6, C7, C8 are scheduled to be delivered at week 5; Containers C9, C10, C11 are scheduled to be delivered at week 3; and so on. Furthermore, it is implied from the same figure related to production schedule: Products P1, P2, …, P8 are scheduled to be produced at week 2; Products P9, P10, P11, P12 are scheduled to be produced at week 5; Product 13 is scheduled to be produced at week 3 and week 5; Products P14, P15 are scheduled to be produced at week 3; and so on. Figure 7. Output of decision variables x and y for Case 2 in the form of Excel spreadsheet. Figure 7. Output of decision variables x and y for Case 2 in the form of Excel spreadsheet. Table 2. Summary of the decision variables for Case 2. Week Number of Containers Number of Products Production Load Week 1 8 7 1240 Week 2 13 18 1249 Week 3 7 6 1238 Week 4 8 6 1248 Week 5 11 7 1240 Figure 7. Output of decision variables x and y for Case 2 in the form of Excel spreadsheet. Total 47 44 6215 Appl. Syst. Innov. 2022, 5, 59 10 of 14 3.3. Case 3 As mentioned in the beginning of this section, this hypothetical case is introduced for testing the proposed method to deal with various load factors. This case is modiﬁed from Case 1. All data but the load factor are similar. The load factor for Case 3 is as follows: l = 1; l = 2; l = 3; . . . ; l = 43. The solution of this case with the parameter = 0.005 is 1 2 3 43 summarized in Table 3. It is shown that the objective value is equal to 49 and the weekly production load is distributed in the range of 33,667 to 33,929. Table 3. Summary of the result for Case 3. Week Number of Containers Number of Products Production Load Week 1 13 5 33,667 Week 2 10 6 33,858 Week 3 14 14 33,696 Week 4 27 24 33,929 Total 64 49 135,150 3.4. Case 4 As mentioned at the beginning of this section, this hypothetical case is introduced for testing the proposed method to deal with a tentative delivery schedule. This case is developed based on Case 2. Instead of a similar load factor for all products, the load factor for each product is presented in Table 4. In addition, there are additional constraints related to the delivery schedule, which are that Container C2 must be delivered in week 2 and Container C3 must be delivered in week 4 or earlier. The solution of this case with the parameter = 0.005 is displayed in Figure 8 and summarized in Table 5. It is shown that the objective value is equal to 44 and the weekly production load is distributed in the range 4098 to 4117. It is noted that both Container C2 and C3 are delivered in week 2 in the results. Table 4. Load Factor for Case 4. Product Load Factor Product Load Factor Product Load Factor P1 4 P13 5 P25 2 P2 1 P14 4 P26 3 P3 3 P15 5 P27 2 P4 5 P16 2 P28 4 P5 1 P17 1 P29 5 P6 2 P18 2 P30 2 P7 5 P19 3 P31 3 P8 1 P20 4 P32 5 P9 5 P21 5 P33 1 P10 4 P22 3 P34 2 P11 5 P23 4 P12 4 P24 1 Table 5. Summary of the result for Case 4. Week Number of Containers Number of Products Production Load Week 1 5 4 4106 Week 2 15 20 4101 Week 3 9 7 4117 Week 4 11 7 4098 Week 5 7 6 4108 Total 47 44 20,530 Appl. Syst. Innov. 2022, 5, x FOR PEER REVIEW 11 of 15 Table 4. Load Factor for Case 4. Product Load Factor Product Load Factor Product Load Factor P1 4 P13 5 P25 2 P2 1 P14 4 P26 3 P3 3 P15 5 P27 2 P4 5 P16 2 P28 4 P5 1 P17 1 P29 5 P6 2 P18 2 P30 2 P7 5 P19 3 P31 3 P8 1 P20 4 P32 5 P9 5 P21 5 P33 1 P10 4 P22 3 P34 2 Appl. Syst. Innov. 2022, 5, 59 11 of 14 P11 5 P23 4 P12 4 P24 1 Figure 8. Results of Case 4 in the form of Excel spreadsheet. Figure 8. Results of Case 4 in the form of Excel spreadsheet. 3.5. Results on Different Value of a Table 5. Summary of the result for Case 4. As mentioned before, changing the value of may have an impact on the results of Week Number of Containers Number of Products Production Load the proposed method. Setting a greater value of makes the constraints (7) and (8) more Week 1 5 4 4106 relaxed. Consequently, the likelihood of achieving a better objective function is increasing. Week 2 15 20 4101 Whenever Case 1 is solved with a bigger value of , i.e., = 0.1, the result is different Week 3 9 7 4117 from the result presented in Table 1. It results in a smaller objective function, which is 45. Week 4 11 7 4098 However, the weekly production load is widened to the range 1149 to 1371. The results Week 5 7 6 4108 using various values of are presented in Table 6. Total 47 44 20,530 Table 6. Results of Case 1 at various values of . Objective Function Value Weekly Production Load Range 0.0025 49 1257–1262 0.0050 48 1255–1265 0.0100 47 1249–1268 0.0500 46 1229–1299 0.1000 45 1149–1371 0.4000 44 925–1670 0.5000 43 744–1814 Similar results are obtained whenever Case 2, Case 3, and Case 4 are solved using various values of . A summary of the results is displayed in Tables 7–9, respectively. As predicted, there is a tendency that the larger the , the smaller the value of the optimum objective function. However, the larger the , the wider the range of weekly production loads obtained. Naturally, these results must be communicated to the solution’s users, for example, the company’s production manager. Finally, the manager can determine which value of is being used. For sure, not only the objective function value but also the acceptable range of weekly production load is becoming the criteria for determining the . Appl. Syst. Innov. 2022, 5, 59 12 of 14 Table 7. Results of Case 2 at various values of . Objective Function Value Weekly Production Load Range 0.0025 1257–1262 0.0050 44 1238–1249 0.0100 43 1232–1253 0.0250 43 1216–1273 0.0500 41 1190–1291 0.1000 41 1190–1299 0.4000 40 791–1685 0.5000 39 625–1834 Table 8. Results of Case 3 at various values of . Objective Function Value Weekly Production Load Range 0.0025 50 33,744–33,858 0.0050 49 33,667–33,929 0.0100 49 33,561–34,025 0.0250 49 33,310–34,316 0.0500 48 32,555–34,852 0.1000 47 30,970–36,787 0.4000 45 20,596–47,296 0.5000 45 18,189–46,064 Table 9. Results of Case 4 at various values of . Objective Function Value Weekly Production Load Range 0.0025 4098–4117 0.0050 44 4098–4117 0.0100 44 4094–4131 0.0250 43 4043–4207 0.0500 42 4020–4297 0.1000 42 3782–4461 0.4000 39 2950–5464 0.5000 39 2862–5473 4. Conclusions and Further Works Based on the results of the calculations for the four preceding cases, the mathematical model and proposed solution were successful in resolving the problem of delivery and production scheduling coordination. The four cases above illustrate a variety of possi- ble real-world scenarios, including variations in the number of products, the number of containers, the number of planning periods, and various tentative delivery schedules. In the proposed mathematical model, there is a parameter called , which is used to balance the production load across periods. The computational results over all cases considered concluded that the larger the , the smaller the value of the optimum objective function. However, that causes a wider range of production loads. It is essential to communicate with the company’s production manager in order to determine the value of used in the model so that the solution method can obtain the best objective function for the chosen value. This research can be expanded by considering some realistic aspects of production and delivery schedules that exist in the real world. These additional factors may impose con- straints on the mathematical model. Simultaneous scheduling of delivery and production is another area of future research. Despite the success of mathematical programming in coordinating delivery and pro- duction schedules, the size of the resulting mathematical models will grow exponentially as more containers, products, and planning periods are introduced. Case 1 involves 428 in- teger variables and 11,081 constraints, whereas Case 2 involves 405 integer variables and Appl. Syst. Innov. 2022, 5, 59 13 of 14 8048 constraints. The size of the model increases to 972 integer variables and 19,248 con- straints if the planning period for a case like Case 1 is increased from 4 to 12 periods. This larger problem size will eventually cause the mathematical modeling solver ’s computation time to increase. For this reason, it is still necessary to develop the solution of mathematical models using other techniques, such as metaheuristics or other computational techniques, so that mathematical models can be solved more quickly. Author Contributions: Conceptualization, T.J.A. and R.D.A.; methodology, T.J.A.; software, T.J.A.; validation, T.J.A. and R.D.A.; formal analysis, T.J.A. and R.D.A.; investigation, T.J.A. and R.D.A.; resources, R.D.A.; data curation, R.D.A.; writing—original draft preparation, T.J.A.; writing—review and editing, T.J.A. and R.D.A.; visualization, R.D.A.; supervision, T.J.A.; project administration, R.D.A. All authors have read and agreed to the published version of the manuscript. Funding: This research is partially supported by Universitas Atma Jaya Yogyakarta, Indonesia. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. 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Applied System Innovation – Multidisciplinary Digital Publishing Institute

**Published: ** Jun 22, 2022

**Keywords: **production schedule; integrated production and delivery; mathematical programming; integer programming

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