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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.2 pp.270-278

Multi-Product Shipment and Production Scheduling Mathematical Model under Different Distribution Policies

Tung-Cheng Huang*, Tzu-Chia Chen
BA Program, International College, Krirk University, Bangkok , Thailand
Soi Ramindra 1, Khwaeng Anusawari, Khet Bang Khen, Bangkok 10220, Thailand
*Corresponding Author, E-mail:

February 27, 2021 April 1, 2021 April 19, 2021


Nowadays, integration of production and distribution has been important in the supply chain. Besides, due to their limited lifespan, integration in the supply chain of perishable items has led to more studies. This study aims to present integrated production and shipment models under full truckload capacity (TL) and less than truckload (LTL) policies combining perishable and non-perishable products in a supply chain, simultaneously. The optimization objective is to minimize the total relevant cost of manufacturer such as the production setup, inventory holding, and transportation costs per time unit. After presenting the models, different numerical example has been used to evaluate based on the change of truck capacity, lifespan as well as costs, and increasing the number of perishable products via Gams software. The results obtained illustrate that the priority of policies TL and LTL depends on the capacity of the truck and shipping costs.



    In the past, different parts of the supply chain had separate responsibilities, meaning that people in charge of each of these parts carried out their executive and planning duties independently from other sections of the supply chain. In other words, each layer of the supply chain only focused on the profit and loss of its operating units alone in the past. Therefore, this leads to poor performance and low return in the supply chain most of the time due to the conflict in the goals of various sections. The need for integration in the supply chain increased with the passing of time and a higher competition among various companies to meet customers’ needs promptly and appropriately (Christopher, 2016). As such, managerial methods of the supply chain that lacked integration were gradually eliminated. Today, supply chain management includes a set of methods that aim to coordinate and integrate various parts and layers of the supply chain, including suppliers, producers, inventories, and retailers. Integration leads to the production of the required number of products in a specific time and location and their provision to consumers, which decreases the total costs of the supply chain and meets customers’ needs favorably (Chandra and Fisher, 1994). Shelf life and perishability of products have long been neglected in scientific research and articles related to the supply chain. Meanwhile, raw materials, intermediate goods, and products are of a perishable kind in numerous industries from discrete production to the processing industry. This fact has caused many limitations in the series of various processes of the supply chain, including procurement, production planning, inventory management, and distribution (Baghernejad and Fiuzat, 2021;Abdollahbeigi and Asgari Bajgirani, 2020;Barmasi, 2020).

    Today, inventory items are classified into three categories; the first category includes items with indefinite shelf life, which can be stored for unlimited time and will not be destroyed over time. The second category includes obsolete items that go out of style after a while. Finally, the third category includes perishable products, the features, and amounts of which change and decrease over time (Goyal and Giri. 2001). Lack of integration in the supply chain of perishable items leads to inefficient production and distribution, the results of which will be overstock, waste production, or a shortage in the supply chain. In the present article, perishable products are those items that have a fixed shelf life after which they are considered unsuitable for utilization. Some of the most important studies performed on the integrated supply chain of normal and perishable items are presented below.

    In recent studies, proper attention has been paid to coordination in production, inventory, and delivery, and there are various models for integrated decision-making in areas of production, inventory, and delivery. For instance, Jonrinaldi and Zhang (2017) proposed a model and solution method for coordinating integrated production and inventory cycles in a completely manufacturing supply chain, which included several raw materials, suppliers, distributors, retailers, and consumers (Zhang, 2013). In another research, Liu and Papageorgiou (2013) developed a multi-objective mixed-integer linear programming (MILP) approach with the total cost, total flow time and total lost sales as key objectives in order to make optimal decisions about production, distribution and expansion of capacity (Liu and Papageorgiou, 2013). Farahani et al. (2014) provided a review of supply chain network design (SCND) literature, presented solution methods, and applicable programs (Farahani et al., 2014). Recently, Jonrinaldi and Zhang (2017) developed a mathematical model to integrate production, inventory and transportation decision in a completely green manufacturing supply chain (Jonrinaldi and Zhang, 2017). In another research, Wei et al. (2017) formed a heuristics algorithm based on the MILP, which dealt with integrated production, inventory and transportation (Wei et al., 2017). Similarly, Sağlam and Banerjee (2018) presented a MILP model for production and various delivery policies (Chung and Kwon, 2016). Chung and Kwon (2016) proposed an integrated supply chain management framework to explicitly consider the impact of product perishability on a broad scale that includes manufacturers, distribution centers, wholesalers, and demand markets. The framework proposed enabled the mentioned researchers to consider the oligopolistic competition across wholesalers that drives price and demand fluctuations (Chung and Kwon, 2016). A study was performed by Ekşioğlu and Jin (2006) to address a production and distribution-planning problem in a dynamic, two-stage supply chain. The factors considered in the model included product perishability, limited shelf life and unlimited capacity. In the end, the problem was formulated as a network flow problem with a fixed charge cost function (Ekşioğlu and Jin, 2006).

    Today, one of the most challenging responsibilities of the food industry is controlling product quality throughout the food supply chain. Rong et al. (2011) integrated food quality in decision-making on production and distribution in a food supply chain. These scholars presented a method that evaluated reduced food quality along with the production and distribution of products in a MILP. The obtained model was used in a case study and could be employed to design and establish food distribution systems and evaluate cost and quality of products (Rong et al., 2011). Chen et al. proposed a nonlinear mathematical model to consider production scheduling and vehicle routing with time windows for perishable food products in the same framework. The demands at retailers were assumed stochastic and perishable goods were going to deteriorate once they were produced. Therefore, the revenue of the supplier was uncertain and depended on the value and the transaction quantity of perishable products when carried to retailers. The model was primarily developed to maximize the expected total profit of the supplier, as well as the optimal production quantities, the time to start producing and the vehicle routes simultaneously (Chen et al., 2009: Arabian et al., 2020).

    In another research, Yan et al. (2011) developed an integrated production-distribution model for a perishable item in a two-echelon supply chain. The supplier’s production batch size was restricted to an integer multiple of the discrete delivery lot quantity to the buyer. In addition, they established exact cost functions for the supplier, the buyer and the entire supply chain. In the end, they outlined a procedure for determining the optimal supply chain decisions with the objective of minimizing the total system cost (Kharisma and Perdana, 2019). Moreover, Seyedhosseini and Ghoreyshi (2014) formulated a heuristic algorithm to tackle the problem of production and distribution planning models for perishable products using inventory routing decision-making (Seyedhosseini and Ghoreyshi, 2014). In another study, Ghasemkhani et al. (2019) present an integrated production inventory routing problem with a MILP model, using the condition of uncertainty due to perishable products (Ghasemkhani et al., 2019). In another research, Liang et al. introduced a model for the vehicle routing problem with time windows and configurations of temperature and humidity for perishable product distribution. They proposed an adaptive large neighborhood search heuristic to solve the problem (Liang et al., 2020). In a recent study, Wang (2021) considered an integrated scheduling problem in an ecommerce supply chain. To this end, a MILP model was proposed and a hybrid particle swarm algorithm was employed to optimize the model. According to the results, the proposed model could lead to a substantial reduction of both the total and penalty costs (Wang, 2021).

    The present study will propose mathematical models to integrate problems of production scheduling, stack size determining, and external loading (product delivery) in a multi-product state (with perishable products), the result of which is to minimize production, storage and delivery costs in various delivery scenarios and their evaluation in the presence of various products. A literature review revealed that perishable and non-perishable products have been separately evaluated under various conditions in previous studies. However, the integrated state will lead to the development of the existing model, and, thereby, is considered an innovation. The use of this model in the pharmaceutical industry, where products last from a few days to a few years, cannot be ignored. In other words, the most important achievement of the current research, which distinguishes it from other studies in the perishable product supply chain area, include:

    • 1. Adding a shelf-life constraint to integrated production and shipment models by TL and LTL

    • 2. Simultaneous consideration of perishable and non-perishable products in integrated production and shipment models

    The present study focuses on a specific supply chain model, where a manufacturer produces numerous products to satisfy customers’ needs in several retail stores. Transportation and shipment of products occurred based on full truckload (TL) and less than truckload (LTL) policies. To state the current problem, we first describe the conditions governing the issue:

    • •Products are produced by one center.

    • •The number of products of the manufacturer can be more than one.

    • •Some of the products are perishable.

    • •There are definite market demands for various products.

    • •The set of products is produced in a production unit with separate capacity and production rates vary for different items.

    • •In the case of perishable products, they can be produced in the cycle several times.

    • •The last product that is produced and loaded should be of non-perishable type.

    • •According to the TL policy, a combination of all products that form a full load of a truck is delivered to all retail situations based on the sales order. On the other hand, the LTL state considers the direct delivery of any product to each retailer.

    • •These two distribution methods impose different transportation costs. The TL model includes a truck with a specific capacity, where only a fixed cost is paid for all of the products in each shipment. Meanwhile, each shipment cost in the LTL method is directly based on the variable load-depended cost. Moreover, for simplicity without losing generality, we assumed that the variable speed of sending a given product was the same for all retail locations.

    Considering these conditions, the present study aimed to develop a model through mathematical formulation, which could simultaneously reduce costs of establishment and production, as well as storage and distribution (shipment).

    In the TL policy, modeling occurred in two different states:

    • 1. All items are produced once throughout a cycle.

    • 2. Perishable products can be produced more than once in case of need.

    In the LTL policy, the product is produced once throughout the production cycle due to the direct shipment of the product from the manufacturer to the retailer.

    Proposed Model’s Indexes:

    • i : index of the type of products (perishable and non-perishable)

    • j : index of non-perishable products

    • p : index of perishable products

    Proposed Model’s Parameters:

    • Di : demand rate for the i-th product

    • Pi : product rate for the i-th product

    • δi : Cost of setting up construction in each production group for the i-th product

    • λi : inventory storage cost for the i-th product

    • C : FTL capacity, the maximum storable capacity in each truckload

    • wi : weight or volume of each unit of the i-th product

    • γ : fixed cost of a truck shipment in the TL policy

    • vi : shipment cost of each unit of the i-th product in the LTL policy

    • Lp : shelf life of each product P

    • nt : counter of sending periods

    Decision Variables of the Proposed Model:

    • Ii : average inventory level of the i-th product

    • Qi : the amount of the i-th product in each shipment

    • Β : a positive integer showing the number of shipments in each cycle

    • T : shipment interval per unit time

    • KQi : production lot size of the i-th product

    • KT : duration of cycle per time unit

    • Rp : a positive integer, showing the number of productions of the p product in each cycle

    • Xp : an integer and T multiple, showing the distance between two consecutive productions

    Objective Function:

    • TRC: total costs


    This part of the article presents a multi-product shipment and production scheduling mathematical model for perishable products under the distribution policy:

    In this model, all items (either perishable or nonperishable) were produced once in the production cycle.

    M i n i m i z e T R C ( Q , β ) = β i = 1 n 1 D i λ i [ Q i 2 P i + j = i + 1 n 1 Q i P j ] + Q n P n i = 1 n 1 D i λ i + i = 1 n D i δ i β Q i + ( β 1 ) Q n P n i = 1 n 1 Q i λ i 2 + Q n λ n 2 [ D n P n ( 2 β ) + ( β 1 ) ] + γ 1 T


    Q 1 D 1 + Q 2 D 2 = = Q n D n = T

    i = 1 n D i P i 1

    i = 1 n 1 Q i P i + Q n P n β T

    i = 1 n w i Q i = C

    K T M i n { L P }

    K 1 , K Z +

    Equation (1) shows the objective function of the model, the first part of which was related to the total setting up cost of the production unit. The second part includes total inventory costs, which were calculated by multiplying the storage cost of each unit of products into the average inventory. The last part shows the total transportation cost, which was obtained by multiplying the fixed cost of a truckload into the total number of transportation periods per time unit.

    Equation (2) demonstrates that the time of the shipment cycle is equal for all products. On the other hand, Equation (3) ensures sufficient production capacity for the production of all products. Meanwhile, constraints (4) were related to the feasibility of the production plan, meaning that the total production time of all products should be less than the period of the production cycle (the production setting up cost was assumed to be insignificant). In addition, constraints (5) demonstrates that the truckload capacity is limited by the total volume or weight of products. Moreover, Equation (6) shows that the period of the cycle should be less than the shelf life of all products so that no product perishes during the cycle. Equation (7) ensures that at least one shipment should occur during a production period.

    Due to constraints (6) in the previous state, the number of shipments and duration of the cycle (KT) depended on the shelf life of perishable items. However, the model described in this section is useful when the limited shelf life of items does not affect the number of shipments, the transportation time, and the length of the cycle. To this end, the model presented in the previous section was solved without the limited shelf life of items (constraints [6]) and values of Qi, T, and K were obtained. Afterwards, the X variable (the distance between two consecutive production processes) was obtained considering that the shelf life of perishable items was in which part of the KT range.

    I F           L P β T x =

    IF L P i = 1 n 1 K Q i P i + Q n P n x = 1

    IF i = 1 n 1 β Q i P i + Q n P n + ( n T 1 ) T L P i = 1 n 1 β Q i P i + Q n P n + n T T

    According to Equation (10), if the shelf life of perishable products was in the middle range of the cycle (KT), the following two states occur depending on the total duration of the products’ shelf life and length of the production process. In fact, we had to determine which of the following conditions was established, which was followed by estimating X that fits the state using Equation (11).

    ( n T 1 ) Q P P P + L P i = 1 n 1 β Q i P i + Q n P n + n T T X = n T ( n T 1 ) Q P P P + L P i = 1 n 1 β Q i P i + Q n P n + n T T X = n T + 1

    In the second state, the objective function will be, as follows:

    M i n i m i z e T R C ( Q , β ) = β j = 1 n 1 D j λ j [ Q j 2 P j + e = j + 1 m 1 Q e P e ] Q m P m j = 1 m 1 D j λ j + j = 1 n D J δ j β Q j + ( β 1 ) Q m P m j = 1 m 1 Q j λ j 2 + Q m λ m 2 [ D m P m ( 2 β ) + ( β 1 ) ] + P = 1 C R P . D P δ P K Q P + P = 1 C λ P R P ( β   D P ( Q P 2 P P ) + Q P 2 ( X P 1 ) ) + γ 1 T

    In this state, the cost of several production times and storage of perishable items was added to the objective function. The value of R (number of production times) was obtained by dividing the number of shipments by the interval between the two shipments and the upward rounding.

    M i n i m i z e T R C ( T , β ) = 1 T i = 1 n δ i + T i = 1 n D i λ i 2 × { D i P i ( 2 β 1 ) + 1 1 β } + i = 1 n w i D i v i

    i = 1 n D i P i 1

    i = 1 n β   Q i P i T

    KT Min { L P }

    K 1 , K Z +

    Equation (13) shows the model’s objective function. The first part of the objective function was related to the total cost of setting up a production unit. The second part involved the total inventory costs, and the final part showed the total transportation costs. Equation (14) ensures that the production capacity was sufficient for producing all products. Constraints (15) are related to the feasibility of the production plan, which means that the total production cost of all products should be less than the production cycle. Equation (16) shows that the cycle’s time should be shorter than the shelf life of all products so that no product perishes during the cycle. In addition, equation (17) ensures a minimum of one shipment during each production process.

    3. RESULTS

    In this section, an experimental problem is proposed and solved with different modes of TL and LTL policies in order to validate the proposed models. In addition, attempts were made to observe the matching of different parameters to achieve logical and acceptable answers. To this end, we used numerical examples of articles. The problem under study included three products, and Table 1 shows the data related to the example. In the TL policy, the model mentioned in the previous section was a mixed and non-linear one. In this section, we used the GAMS optimization software to achieve an optimal solution. The output of the model was the number of products per TL shipment (Qi) and the number of shipments per cycle (β) and delivery interval (T). Similarly, the model mentioned in the previous part was a mixed and non-linear one in the LTL policy, for which we also used the GAMS. The output of the model was the optimal T and βi values, estimation of which simplified the calculation of the Qi amount. In the TL policy, the truck capacity varied from 4535.9237 Kg to 31751.4659 Kg nd with a 2267.962 Kg interval. Meanwhile, the cost of each unit of the product changed from 0.14$ to 0.24$ per Kg of load and increased with a 0.01-dollar interval. Tables 2 and 3 summarize the computational results of various states of TL policy per different truckloads. In addition, Table 4 presents the computational results related to the LTL policy per different transportation costs.

    According to Tables 2 and 3, the TL shipment period varied from 0.00735 years (2.68 days) to 0.05147 years (18.79 days). However, the T duration increased with an increase in truck capacity, and the number of shipments (K) decreased during the cycle (KT). As expected, TL was normal in the first state, and the fixed cost of each truck increased with the increase of truck capacity. As mentioned, the length of the T-period increased as the truck capacity increased, and the length of storage time of the product increased, which led to a higher inventory cost. On the other hand, an increase of capacity led to a decrease in the number of shipments, which reduced the delivery cost. According to Diagram (1), the cost function in the first case TL behaved convexly as capacity increased. As observed in Tables 2-4, the total cost first decreased with an increase in the capacity, meaning that reducing the number of shipments led to a decrease in the total cost. However, after a certain point (e.g., after C=50000 in this problem), the savings from reducing the number of shipments cannot compensate for the fixed cost increase, inventory cost, and the total cost by increasing capacity. In the second TL policy state, as observed in Table 3 and Diagram (1), the number of productions of perishable products was less than the number of shipments due to a shorter delivery period. In addition, there was a need for re-production at shorter intervals. Therefore, there was a higher slope of cost reduction in low capacities. However, after a certain point (e.g., C = 25000 onward in this example), an increase in T led to equal numbers of productions and shipments, meaning that one product was required for each time of delivery. This led to an insignificant cost for perishable products. Moreover, increased capacity led to a lower number of shipments, which decreased the number of productions and the endproduct costs. Contrary to the first TL state, a decrease in cost still occurred with a mild slope in high capacities. In the LTL state, as observed in Table 4, a minimum objective function value was obtained when the number of shipments per product was equal to 1, and the delivery cost of each unit of the product had no impact on the values of Qi and T and only affected the objective function. The equal values of Qis were due to the use of the fixed cycle approach.

    As observed in diagrams (2) and (3), the increased shelf life of perishable products led to a decrease in the total costs and vice versa in all three models. Table (5) shows problem solving per every two perishable products with shelf lives of 0.1 and 0.06 at an equal truck capacity of 10000 lbs. As expected, the total cost was lower when there was only one perishable product with a shelf life of 0.1, compared to the state where there were two perishable products with shelf lives of 0.1 and 0.06.


    Inventory costs are considered the most important costs of a supply chain. Up to several years ago, the research did not distinguish different types of inventory and considered all of them to have an unlimited shelf life. Today, however, the necessity of integration of the perishable product supply chain is known to all. This issue has drawn much attention, especially in the past few years due to people’s concerns about consuming healthy and fresh products. In addition, the pharmaceutical products supply chain has received special attention due to its relevance to people’s lives. The present study focused on a specific supply chain scenario, where a manufacturer produces several products to satisfy customers’ demands in numerous retail stores. The factory can produce only one product at a time, but depending on the shelf life of perishable products under certain conditions, only perishable items are allowed to be produced more than once. Products were delivered through TL and LTL shipping policies. The present study aimed to minimize the total related costs, such as setting up, inventory, storage, and distribution costs of products by TL and LTL policies. The fact that the model presented in this article considered several types of product simultaneously, which could have different shelf lives or some of them could be of normal type (unlimited shelf life), distinguished it from other studies performed in this area. Evidently, adding perishable items to the problem, and proposed solutions of the models with a limited shelf life of some items, led to a change in the number of shipments and the delivery cycle or a change in the number of productions, all of which resulted in an increase in total cost, compared to states that included no perishable items. However, the rate of increase in total cost varies in different situations, such as changes in truck capacity, changes in fixed truck costs, and variable costs that depend on the load volume. Moreover, this difference in costs was observed in various states of TL and LTL policies. One of the features of the present research, which distinguished it from previous studies, was presenting a model that can estimate changes in the total cost in the presence of perishable and nonperishable products simultaneously and choose the best product delivery based on the current situations (e.g., truck capacity and fixed and variable costs).

    In the present study, an experimental problem was first solved with a perishable product among the products with three different shelf lives to compare these costs in different shipment policies. In each of these states, the problem was solved per 13 different capacities and fixed costs in TL shipping policy and 11 different variable costs in the LTL shipping policy. The results of solving these models with different conditions were presented in full in the previous section.


    It is concluded that a shortened shelf life led to an increase in total costs. In order to evaluate the increased number of perishable products in the supply chain, the problem was solved at a fixed truck capacity while assuming that the two products were of perishable type. In the end, we realized that an increased number of perishable products led to an increase in the total cost. In each problem, we were able to determine the more optimal shipping policy based on the total cost diagram in terms of truck capacity. The LTL policy only depended on the load-dependent variable cost and could be in its optimum form at each capacity. The superiority of TL policies over each other was also optimal in the low capacities of the first TL state and after the intersection of the second TL state. Notably, the intersection depended on the shelf life and the number of perishable products. It is recommended that research be conducted in the future and the results be expanded in more complicated and realistic supply chain environments. Other recommendations include considering the market demand randomly, integrating the routing problem with TL and LTL shipping policies, and use of unequal cycle approach, which might be more efficient even though it increases complexity.


    The author received financial support from Ming Zhong International Education (Thailand) Co., Ltd. for the research, authorship, and/or publication of this article.



    Comparison of different TL and LTL states with a perishable product and shelf life of 0.8.


    Comparison of the effect of shelf life on cost in the first TL policy state.


    Experimental problem parameters

    First TL policy state with a perishable product and shelf life of 0.8

    Second TL policy state with a perishable product and shelf life of 0.8

    LTL policy with a perishable product and shelf life of 0.6

    Comparison of the total cost with an increased number of perishable products


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