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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.20 No.2 pp.349-355

A New Mathematical Model for Joint Production and Distribution Optimization in a Multi-Echelon Supply Chain

A. Modenov, A. Boboshko*, A. Durandina, O. Kharchenko
Doctor of Economics, Saint Petersburg State University of Architecture and Civil Engineering, Russia
Candidate of Sciences in Economics, Saint Petersburg State University of Architecture and Civil, Engineering, Russia
*Corresponding Author, E-mail:
March 12, 2021 April 23, 2021 May 2, 2021


Nowadays, the field of production and services is faced with a change in competition pattern from independent firms with competition between supply chains. Accordingly, the importance of the flow of materials in the supply chain has attracted much attention. The factories that have many distributors at the city level or different cities and demand them are faced with the key question of how to implement the projects needed to meet their needs. The aim of this study is to present a correct linear programming model to determine the strategy. One of the distinguishing features of this model is that it tries to solve the production planning problem, inventory planning and transportation planning simultaneously. Considering the assumptions of the cost of production, the numerical results is provided to demonstrate the efficiency of the proposed mathematical model.



    The elevated global competition in an ever-changing environment doubles the requirement for proper reactions of organizations and manufacturing companies and emphasizes their flexibility against an uncertain external atmosphere. Today, national and global organizations need to employ appropriate models such as supply chain management to achieve competitive advantage and supply customer expectations. The efficient management of a supply chain is a leading factor in durability. Furthermore, the use of information technology in supply chain activities multiplies the importance of this topic. In summary, supply chain management focuses on increasing the flexibility of companies and can respond quickly and effectively to fluctuations in the market. In the 1990s, along with advances in manufacturing processes and the application of reengineering models, managers in many industries discerned that not only advanced internal processes and flexibility in the company’s skills are not sufficient to proceed in the market, but also suppliers of parts and materials should manufacture their commodities using the best quality materials and deliver them at the lowest price. Distributors of products must also be closely linked to the developmental policies of the manufacturer market (Vijayakumar, 2020;Nasiri and Ahmadi Daryake- nari, 2021;Sharma and Sharma, 2021). Thus, supply chain and management approaches were emerged according to this strategy. In supply chain management, a set of techniques are used for efficient integration and efficiency of suppliers, manufacturers, warehouses, and vendors to minimize system costs and supply service demands. In this system, goods are produced and distributed in the right number, at the right place, and at the right time. Reduction in costs and inventories, highlighting customer accountability, improved supply chain communication, diminishing production cycle time, and advanced coordination are the main tasks of supply chain management. In this manuscript, a linear planning (LP) model is proposed to integrate production and distribution planning. A multiproduct distribution center with multiple distributors is also considered in order to choose one of the products and reach a correct distribution of the product within a six-day period. The problem includes a central warehouse and six distributors. The volume of product that should be delivered to distributors and even the volume that should be produced within these six periods must be determined. The remainder of this paper is organized as follows. The literature review is presented in section II. In section III, the integrated production and distribution planning network model is discussed. The limitations and objective functions of the model are presented in section IV. Case studies are covered in section V. Ultimately in section VI, the manuscript is concluded and suggestions are offered for developing the model (Arabian et al., 2018).

    Decisions on production and transportation planning in supply chain environments are usually studied independently. The most accepted way is to first study production planning, determine the volume of production, and then focus on transportation to distribute products and goods to customers separately. In today's competitive supply chain and markets, firms must ensure the productivity of their resources, boost customer service, and diminish time and inventory constraints. Under this situation, the simultaneous examination of production planning activities and transportation planning in an integrated way may increase productivity and money savings (Díaz-Madroñero et al., 2015). Janan Researchers investigated different strategies for the inventory problem. They formulated this problem in the form of a complex integer-programming model. The purpose was to maximize profits from the delivery of goods to a wide range of customers using a single-product and multi-period model. They investigated the inventory problem by merging the problems of production planning, inventory planning, and distribution planning, in which the inventory flow variables of each period determine the volume of production in each period. They used C++ programming to solve their model (Bard and Nananukul, 2009). Armentano et al. (2011) studied the problem of integrating production planning and distribution planning in different periods with a definite time horizon. In their model, they considered the multi-product feature of the factory. The factories they modeled had limited capacity, the same equipment with limited capacity, and a limited number. They proposed variables of their model as the volume of production, the volume of inventory, and the volume of delivery to customers in each period. They solved their model using C programming (Armentano et al., 2011: Tarkhaneh et al., 2021).

    The problem of integrated production and transportation has been the subject of interest in recent years, with more than 80% of the works that have been carried out in the last decade. In a comprehensive review, these studies have been classified into five categories, including production, inventory, routing, and objective function modeling aspects, as well as solution techniques (Díaz- Madroñero et al., 2015). Production planning can be introduced as the design of activities to manufacture a product. These activities are required to convert raw materials into the final product according to customer demand and the most economical way.

    Production planning decisions are made on determining the volume of production for each product at a given time. The complexity of the production planning and modeling problem may be due to the number of production units in the manufacturing system, the number of production equipment, and production resources. However, analyzing problems with multiple products and capacity constraints makes the model more realistic. Most of the modeling in studies involves a single production site. The number of single-product models is slightly higher than the number of models with multiple products. Capacity constraints are apparent in all models, though there are differences in the formulation of these constraints. Some have formulated the maximum number of units for production over a period of time and some the maximum number of units for production time and some a combination of both. However, the maximum number of units is more general for production. The majority of articles have done inventory modeling both at the production site and in customer warehouses by means of an inventory balance constraint. Most studies have set their policy on the maximum inventory and considered the customer's storage capacity and the actual conditions of the industrial environment. Most studies have accepted having the same device group with a known capacity in the central warehouse as an assumption. In most articles, the number of trips per vehicle is limited to one at a given period, because it is difficult to complete multiple routes if the time is short. Most articles have utilized mixed-integer linear programming techniques, written their objective function based on cost minimization, and applied production, commissioning, inventory costs, and inter-node transportation costs more than any other cost (Díaz-Madroñero et al., 2015). Bredström et al. (2015) proposed a mixedinteger linear programming model for the transport of boats on the Rhine River, which is used as part of Omia’s decision-making system. This design problem covers a number of problems, including inventory, multiproduction, the existence of multiple vehicles for transportation, and the capacity of the boats. In their objective function, they minimized the sum of transportation costs and inventory maintenance costs (Bredström et al., 2015)

    Memari et al. proposed a green supply chain as a model to lessen environmental pollutants such as CO2 and transport in the supply chain. This model is split into three general parts, i.e., the display of products, the display of distribution, and the display of distributor. This model, which aims to optimize the green supply chain, has several general features. The model deals with bi-step production and distribution. The main part of the demand is at the start of the period, the capacity of factories and distributors is fixed, and transportation is done with containers with a fixed capacity (Memari et al., 2015). Chen (2015) proposed a model to study supply chain decentralization, which can include factories and distributors. Factories need to decide on the wholesale price of their products in which distributors must identify these prices. Factories distribute contract revenue among distributors according to the type of product offered. Formulation of the above factors in the supply chain forms a model for production and distribution with revenue sharing. This model defines the optimal plan to identify how many products shall be delivered to the distributor and identify the wholesale price received from the factories. In general, the model aims to maximize the profit of the supply chain. de Brito Junior et al. (2015) proposed a model in order to optimize logistics prices in the CIV port. The model included information on the number and size of containers, the supplier, the port, and the price of shipment by containers from the port. In this model, the first step is to show the shipping costs that the buyer accepts from the source port, The second step is to show the shipping costs from the source port to the CIV port, the third step is to show the shipping costs in the CIV port, the fourth step is to show the costs paid in LLINS, and the final step is costs paid by containers. Importantly, the products must reach the customer safely (de Brito Junior et al., 2015). Chen et al. (2009) introduced a nonlinear mathematical model for examining the production scheduling and routing of vehicles with time windows for perishable products. It was assumed that the demand of retailers is random and items start to deteriorate as soon as production is complete. Therefore, the supplier's profit was unknown and depends on the number of perishable products shipped to retailers. The model was aimed to maximize the total profit expected by the supplier. The optimal volume of production, production start time, and vehicle routes can be determined simultaneously from solving the model (Chen et al., 2009). Yan et al. (2010) developed an integrated production-distribution model for a perishable product in a two-echelon supply chain. The size of production stacks was considered an exact multiple of the delivery values. Precise cost functions were provided for the supplier, the buyer, and the entire supply chain. They proposed an approach to adopt the optimal decisions for the integrated supply chain, aiming to minimize the total cost of the system (Yan et al., 2010).

    Seyedhosseini and Ghoreyshi (2014) developed an innovative algorithm to solve a production and distribution- planning model for perishable products using inventory decision-making and routing (Seyedhosseini and Ghoreyshi, 2014).

    Ghasemkhani and Tavakkoli-Moghaddam (2019) presented an integrated production and routing problem with a mixed-integer programming model and included uncertainty in their model due to the limited service life of the products. Liang et al. (2020) proposed a model for the problem of vehicle routing, taking into account temperature and humidity for the distribution of perishable products, and proposed a neighborhood search method to solve the problem (Liang et al., 2020). Wang (2021) integrated production and delivery schedules into a supply chain using a nonlinear mathematical model of integer and particle swarm optimization (PSO) for its optimization. According to the results, the proposed mathematical model can remarkably diminish production and distribution costs. The aim of this study is to present a correct linear programming model to determine the strategy. One of the distinguishing features of this model is that it tries to solve the production planning problem, inventory planning and transportation planning simultaneously


    The model presented in this manuscript is proposed for the integration of production and distribution planning. In general, the problem of integrated production and distribution can be considered as a network G =(N, A) , where N identifies a set of nodes and represents the factory and customers. In this equation, A is a set of arcs connecting the nodes, so that A = { ( i , j ) : i , j N , i j } . Nodes are shown with i { 0 , 1 , , n } , where node zero belongs to the manufacturer and acts as a central warehouse, while customers are presented with j { 1 , , n } . In a finite planning horizon, consisting of equal scheduling periods denoted by t { 0 , 1 , , T } , the manufacturer factory generates products that can be stored in the factory warehouse or shipped to customers. Goods are transported by a set of the same vehicles t { 0 , 1 , , T } , which imposes the cost cij. This cost is for traveling from node i to node j. This model aims to minimize production, inventory, and transportation costs due to plant capacity constraints, inventory capacity constraints, and transportation constraints. The following items have been adapted from (Goli et al., 2021) for a better understanding of the nature of transportation in manufacturing systems.

    Variables in this study include the volume of production at the period t, the level of inventory in each node at the period t, and the rate of delivery to each node at the period t. It has been attempted to consider the transportation model in the objective function according to cost of transporting each shell. In general, limitations (1), (2), (3), (4), (5), (6), and (7) were applied in this study.

    In this section, a linear integer programming formulation is presented to solve this problem.

    Limitations (1) and (2) are respectively determining the inventory balance in the manufacturer factory and customer warehouses.

    I 0 t = I 0 , t 1 + p t i N c Q i t t T  

    I i t = I i , t 1 + Q i t d i t i N c , t T

    It is assumed that the initial inventory for the manufacturer factory and all customers is specified. Limitation (1) indicates that the level of product inventory in node zero at the end of the period t (I0t) is equal to the level of product inventory in node zero at the period t-1 ( I 0 , t 1 ), plus the quantity of product manufactured at the period t (Pt) minus the rate of delivery to the customer i at the period t (Qit). Limitation (2) indicates that the level of product inventory in node i at the end of the period t (Iit) is equal to the level of product inventory in node i at the period t-1 ( I i , t 1 ), plus the quantity of product delivered to the customer i at the period t (Qit) minus the demand for the product in node i at the period t (dit). Limitation (3) describes the volume of product delivered to all customers by the manufacturer factory at the period t that is limited by the level of inventory in the manufacturing factory at the period t-1. The given volume of product delivered to the customer i is also limited by the capacity of vehicle (Vcap) in limitation (10).

    i N c Q i t   I 0 , t 1     t T

    Limitation (4) limits production to the capacity of the manufacturer factory in each period. Variable “zero and one’’ has been added to the limitation for considering the implementation cost (one for implementation and zero for non-implementation).

    p t P c a p × γ t t T

    Limitation (5) ensures that the demand in period one can be supplied with production in the period zero.

    p 0 i N c ( d i 1 I i 0 ) I 00

    Limitations (6) and (7) ensure that if a customer receive a service in a period, there must be an alternative in his/her path, which may be the manufacturer factory. Limitation (7) is related to the flow of vehicles. If a vehicle reaches a customer i at the period t, it shall leave at the same period.

    j N , i j X i j t 1   i N c , t T

    j N , i j X i j t = j N , i j X j i t i   N c , t

    The number of vehicles that can leave the manufacturer factory in each period is k, according to limitation (8).

    j N c X 0 j t k t   T

    Limitations (9) and (10) are applied on the loading conditions in which Vcap is the vehicle capacity.

    w j t w i t Q i t + V c a p ( 1 X i j t ) i N c ,   j N , t T  

    w i t V c a p j N X i j t i N , t T

    Limitations (11) determines the inventory range for each node, and limitation (12) the capacity of loading by each vehicle.

    I i t I c a p i          i N ,     t T

    w i t V c a p t T

    Limitation (13) defines the lower bound and the correctness of the volume of production, inventory, and transportation. Limitation (14) defines zero-and-one variables, which relate the implementation, visiting customers, and zero arcs.

    p t , I i t , Q i t , w i t 0 , i n t e g e r i N , t T

    γ t , X i j t { 0 , 1 } i , j N , t T

    The objective function (15) is for minimizing total costs, including costs of manufacturing, implementation, maintenance of inventories in manufacturing factories, and transportation costs in the time horizon.

    min t T ( p c × p t + s c × γ t + i N i c i × I i t +   i N j N c i j X i j t )

    The annual production capacity is 1500 tons.

    Three distributors were considered. j =(0,1, 2, 3) Denotes nodes and j =(1, 2, 3) denotes distributors.

    The factory was considered as the manufacturing site or the central warehouse, where is i =(0,1, 2, 3) nodes and i = 0 is the manufacturer factory. The model was split in a five-day time horizon. Therefore, t =(0,1, 2, 3, 4, 5) is the set of time intervals.


    The cost of manufacturing each unit of the product was estimated at 1500 monetary units. Given that the problem parameters are based on the shell number, this value is multiplied by 12 to calculate the cost of manufacturing each unit per shell. The production capacity in the manufacturer factory was set on 500 shells.

    The model was implemented and solved in GAMS. Instead of using the MIP method, we worked with the RMIP method, as the RMIP method solves the model by LP and answers of the problem are integers, which is consistent to the limitation of being integer. Answers obtained by implementing this program are presented in tables below.


    In this study, we attempted to propose a model for the simultaneous investigation of production and distribution. The model presented includes problems of inventory, production, and transportation. As an advantage, this model is multi-period that enables including variations resulted from demands and expenses in the problem. This problem has been considered as a network, but all studies have assessed only the relationship between manufacturer factories and distributors. We can further assume a relationship between distributors and can supply their demands because costs of transportation between distributors might be lower than that of between factories and distributors. It is recommended to implement this model under uncertainty and fuzzy settings. Furthermore, a limitation can be applied to the cost of transportation, by which if the number of products delivered was less than a threshold, only the cost of that specified volume shall be paid.



    Parameters, sets, and variables of the model

    The maintenance cost per each shell in nodes (factory and distributors)

    The inventory (warehouse) per each node (factory and distributors)

    Demand in nodes of the distributor and at each period

    The cost of traveling between the factory and distributors (There is no relationship between distributors in this problem)

    The initial inventory in each node (the factory and distributors)

    The volume of production at each period (no production shall be done at periods 0, 4, and 5)

    The inventory level in each node (factory and manufacturers)

    The rate of delivery of goods to distributors at each period

    Costs of the target function


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