Developing a multi-item, multi-product, and multi-period supply chain planning optimization model

Objectives : To develop a multi-item, multi-product, and multi-period mathematical model that optimizes supply, production, distribution, and inventory planning for a supply chain network. Methods: Mixed Integer Linear Program-ming has been used to formulate the proposed model, while Excel and Evolver solvers have been used to solve problems regarding supply chain network (con-sisting of two suppliers and two retailers) for proﬁt maximization. Findings: The analysis showed that the developed model has the ability to eﬃciently and eﬀectively solve real-life problems of the multi-item, multi-product, and multi-period supply chain. Novelty: The model considered multi-item, multi-product, and multi-period. In addition, it considers initial, ﬁnal, and intermittent inventory in multi-periods. The model is solved using both Excel and Evolver solvers. The model assists organizations involved in the supply chain to design and plan their network eﬃciently.


Introduction
Research on Supply chain management (SCM) has gained remarkable attention among academics and practitioners (1) . SCM is responsible for coordinating different business functions through a supply chain and within a company to optimize the supply chain and individual companies' productivity (2) . The supply chain is a network that connects material sources, manufacturing facilities with suppliers and retailers (3) .
To reduce the environmental and economic impacts of supply chains in (4) , a singleitem comprehensive green supply chain planning optimization model is created. In (5) , a multi-objective, single-item, single-period optimization model in the strategic supply network planning process that includes the environmental investment decision is suggested. In (6) , a multi-period multi-echelon forward-reverse logistics network design under risk for only single item products is proposed. In (7) , a mathematical model for a single item, single product, multi-period, multi-echelon supply network considering perishability has been developed. In (8) , only one product has been considered in a formulated MINLP model for a two-echelon vehicle routing problem. A single item https://www.indjst.org/ and a single objective are considered in (9) since a mathematical model has been proposed to minimize the silo establishment costs. In (10) , a four objective MILP model considering only a single item was developed. In (11) , the cost and benefit optimization of a multi-product single-item SCN is studied. No prior research has designed a model for multi-item, and multi-product supply chain design and planning. So, it has been decided that a multi-item, multi-product, and multi-period mathematical model be developed to optimize supply, production, distribution, and inventory planning for a supply chain that consists of two suppliers and two retailers to maximize the profit. The proposed model has been formulated using MILP and solved using Evolver solver.
This work is an extension of the work done in (4,12) . It entails the supply chain network's configuration and influence of performance considering the stated objectives (13) . So, it was decided that the profit be maximized directly in this work.
The proposed supply chain consists of three stages of potential suppliers, and retailers in a factory as shown in Figure 1. The following assumptions were considered: • Each product is composed of multi-items.
• The model aims to maximize profit.
• All products may have Initial/final inventory. • All facilities have limited capacity for each period.
• The customers' demands are deterministic and known.

Sets:
P: Set of products, I: Set of items, S: Set of potential suppliers, C: Set of potential retailers, T: Set of periods,

Parameters:
FFCt: facility's fixed cost at period t DEM cpt : demand of retailer c from product p in period t (unit/period) REQip: Required amount of item i for product p(unit) IIf p : the initial inventory of product p (unit) FIf p : the final inventory of product p (unit) P pct : the unit price of product p at retailer c in period t ($) W p : the weight of product p (kg) MH p : manufacturing hours for product p (hour) D s f : the linear distance between supplier s and the facility (km) D f c : the linear distance between the facility and retailer c (km) CAP sit : the capacity of supplier s for item i in period t (kg) CAPHf t : manufacturing capacity of the facility in period t (hour) CAPMf t : raw material storing capacity of the facility in period t (kg) CAPFSf t : the finished good storing capacity of the facility in period t (kg) MATCsit: material cost per unit of item i supplied by supplier s in period t ($/kg) https://www.indjst.org/ MC f t : manufacturing cost per hour for the facility in period t($/hour) MH p : manufacturing hours for product p (hour) NUCCf: non-utilized manufacturing capacity cost per hour of the facility ($/hour) SCPU p : back-ordering cost per unit per period ($/unit/period) HC: holding cost per unit weight per period at the facility store ($/kg/period) B si : the batch size of item i transported from supplier s to the factory(unit) Bf p : batch size transported from the facility for product p to retailer (unit) TC mt : transportation cost for the transportation mode per kilometer in period t ($/km) Decision Variables: QSF sit : number of batches of item i transported from suppliers to the facility in period t QFC cpt : number of batches of product p transported from the facility to retailer c in period t IFF pt : number of batches transported from the facility to its store for product p in period t IFC cpt : number of batches transported from store of the facility to retailer c for product p in period t Rf pt : facility store a residual inventory of product p in period t The profit is calculated by subtracting the total cost from the total revenue. The total revenue is calculated using Equation 1.
The total cost is the summation of the following costs.
• Fixed Cost • Material costs After subtracting the cost of the final inventory material, the material costs include both summaries of the material supplied to the factory and the cost of the initial inventory.
• Manufacturing costs The manufacturing costs include both the amounts of the factory's manufacturing costs distributed to all suppliers and the manufacturing costs of the initial inventory after deducting the manufacturing costs of the final inventory.
• Non-Utilized capacity cost (for the facility) The cost of the facility's non-utilized capacity is calculated by multiplying the depreciation per hour of machines during non-utilized time by the non-utilized capacity hours.
• Back-ordering cost (for retailers) The back-ordering cost is determined by multiplying the shortage quantities and commodity overall times for all retailers by the shortage cost per unit per day.
• Transportation costs For all shipments in all transportation types at all times for transporting both raw materials from manufacturers and finished products to retailers, transportation costs are determined by multiplying the distance traveled by the transportation cost per unit of distance.
• Inventory holding costs Except for the previous year, inventory costs are determined using the weights of the remaining commodity inventory at the end of each year in addition to the retention of the initial inventory.

Constraints
To ensure flow balance and capacity limits, this model considered two types of constraints.

Balancing Constraints
RF pT B f p = FI f p , ∀p ∈ P (12) Constraint (9-13) ensures that materials and goods flow in a balanced manner.

Capacity Constraints:
(  18 as the ratio between the total weights of products sent to all retailers during the planning horizon and the weight of products requested during the same planning horizon.

Model verification
The efficacy of the model has been verified through the following comprehensive example.

Verification example inputs
To verify the model, the following random example is solved, and the results are analyzed. The assumed demands are shown in Table 1, while Table 2 represents other parameters.

Verification example Results and discussion
The model is solved using Evolver software on an Intel ® Core TM i7-7700 CPU @3.60 GHz 4 Core(s) 8 logical processors (8 GB of RAM). Figure 2 shows that the demands of the first product have been satisfied, while the demand of the second product has not been fully satisfied as shown in Figure 3, where the unit contribution of the first product is larger than that of the second product.
For more discussion, the demand of each retailer from each product has been studied individually. From Figures 4 and 6, it can be observed that the demands for the first product of both retailers have been satisfied, while the demands of the second product of both retailers have not been fully satisfied as shown in Figures 5 and 7, respectively. It is also clear to observe that the shortage of the first retailer is less than that of the second retailer because of its closeness to the factory which minimizes the transportation cost contrary to the second retailer's demands which are partially satisfied as shown in Figures 5 and 7. https://www.indjst.org/  Balancing of the supplied material and the received products during all periods have been presented in Table 3 verifies the developed model. Table 3 represents the relationships between facilities' capacities, demand, received quantities, and the residual inventory for all periods.

Computational Results and analysis
In this section, the effect parameter change has on system behavior has been studied. The following parameters are shown in Table 4 has been assumed to simplify the discussion.
The effect of demand on the total revenue, cost, profit, and OSL have been studied by assuming equal demands for all products and customers in all periods. The results of this study have been presented in Figure 8 from which it was observed that the https://www.indjst.org/   Dsf, Dfc 50 Km revenue increases as the demand increase till the demand of almost 1500 item/period, where the capacity of the factory has been consumed as observed from the OSL graph. Thereafter, the increasing rate of revenue has been decreased due to the shortage in satisfying the demand till a demand of 5000 units per period. However, the total revenue remains constant because of the stability in production. Furthermore, the total cost increases as long as the demand increases even after reaching the maximum capacity, the rate increased significantly due to the shortage cost. Accordingly, the profit increases as a result of a rise in profit and decreases on reaching the limiting capacity.

Conclusion
The developed multi-item, multi-product, and multi-period mathematical model has successfully optimized the supply, production, distribution, and inventory planning for a multi-echelon supply chain of two suppliers, one factory, and two retailers to maximize the profit.
The efficiency of this model has been verified by solving and analyzing the results of a comprehensive example. Also, the effect of demand on the total revenue, cost, profit, and OSL have been studied and analyzed. Limitations: