Supply Chain Network Design Optimization Model for Multi-period Multi-product Under Uncertainty

This research is a development of a stochastic mixed integer linear programming (SMILP) model considering stochastic customer demand, to tackle the multi-product SCND problems. It also considers multi-period, multi-echelons, products inventories, considering locations capacities and associated cost elements. The model represents both location and allocation decisions of the supply chain which maximize the total expected profit. The effect of demand mean on the total expected profit and the effect of the number of scenarios on the CPU time are studied. The results have shown the effect of customers’ demands for each product in each period on the quantities of material delivered from each supplier to each factory, the quantities of products delivered from each factory and factory store to each distributor, the inventory of each product in each factory and distributor, the quantities of each type of product delivered from each distributor to each customer in each period. The model has been verified through a detailed example.

SCND with objectives of minimizing total cost and minimization of the total environmental impact [7].
Badri, H. et al. developed a new multi-commodity SCND model with different time resolutions for strategic and tactical decisions. In addition, a mathematical technique based on the Lagrangian relaxation method was developed to solve the problem [2].
Wu, J., & Li, J. studied dynamic factory location and supply chain planning through minimizing the costs of factory location, path selection and transportation of coal under demand uncertainty [11].
Xia R. & Matsukawa H. investigated a supplier-retailer supply chain that experiences disruptions in supplier during the planning horizon. While determining what suppliers, parts, processes, and transportation modes to select at each stage in the supply chain, options disruption must be considered [13].
Adabi, F., & Omrani, H. formulated a mixed integer programming model considering two objective functions where the first one maximizes the efficiency of the supply chain and the second one minimizes the cost of facility layout as well as the production of different products [1].
Serdar E. T. & Al-Ashhab M. S. mathematically modeled an SCN in a mixed integer linear programming (MILP) form considering deterministic demand maximizing the total profit [9].
This research is a development of a stochastic mixed integer linear programming (SMILP) model considering stochastic customer demand, to tackle the multi-product SCND problems. It also considers multi-period, multiechelons, products inventories, considering locations capacities and associated cost elements. The model represents both location and allocation decisions of the supply chain which maximize the total expected profit. The nature of the logistic decisions encompasses procurement of raw materials from suppliers, production of finished product at factories, distribution of finished product to customers via distributors, and the storage of end product at factories and distributors. The proposed scheme of supply chain consists of three echelons (three suppliers, three factories, and three distributors) to serve four customers as shown in Figure 1.

Model Assumptions and Limitations
The problem is formulated using multi-stage stochastic mixed integer linear programming (SMILP) and it is solved using Xpress-SP software which uses Mosel language [12].
The following assumptions are considered: a. Customers' demands are stochastic and known for all product in all periods. b. The model is multi-product, where actions and flow of materials take place for multi-product. c. Product weight affected material, transportation, holding d. Costs parameters (fixed costs, material costs, manufacturing costs, non-utilized capacity costs, shortage costs, transportation costs, and inventory holding costs) are known for each location, each product at each period. e. The shortage cost depends on the shortage quantity for each product and time. f. The manufacturing cost depends on the manufacturing hours for each product and manufacturing cost per hours The stochastic demand of the customers is normally distributed with mean µ and standard deviation σ, it is discretized into N points.

Model Formulation
The model involves the following sets, parameters, and variables: Sets: S, F, D, and D: potential number of suppliers, factories, distributors, and first customers, T: number of periods, indexed by t. P: number of products, indexed by p. Parameters: F i : fixed cost of opening location i, DEMAND cpt : demand of customer c from product p in period t, µ ct : demand mean of customer c in period t, σ ct : demand standard deviation of customer c in period t, P pct : unit price of product p at customer c in period t, W p: weight of product p, MH p : manufacturing hours for product p, D sf : distance between supplier s and factory f, D fd : distance between factory f and distributor d, Q ijpt : flow of batches from location i to location j of product p in period t, I fpt : flow of batches from factory f to its store of product p in period t, I fdpt : flow of batches from the store of factory f to distributor d of product p in period, R fpt : residual inventory of the period t at the store of factory f for product p, R dpt : residual inventory of the period t at distributor d for product p.

Objective Function
The objective of the model is to maximize the total expected profit of the supply chain network.
Total expected profit = Total expected income -Total expected cost

Total Expected Cost
Total expected cost = fixed costs + material costs + manufacturing costs + non-utilized capacity costs + shortage costs + transportation costs + inventory holding costs.

i. Fixed Costs
vi. Transportation Costs vii. Inventory Holding Costs

Effect of Demand Mean and Number of Scenarios
The relationship between demand mean and total expected profit has been studied at different values of scenarios of 1, 8, 27 and 64. Figures 2-6 show that the general shape of the relation between demand means and total expected profit is almost the same for a different number of scenarios. In general, the increase in demand mean increases the total expected profit as shown in Figure 6. The total expected profit is linearly proportional to the total demand. At      The effect of changing number of scenarios on the total expected profit has been studied for different values of demand means. The changing percentage is calculated by dividing the resulted total expected profit of the given number of scenarios by the resulted total expected profit of the 1-scenario case. Figure 7 depicts the effect of the number of scenarios on the total expected profit. As it is noticed in Figure 8 the most changes happened for demand means of 200, 400 and 500 respectively which are located in the transient ranges since the increase of the total expected profit in the high demand scenarios is not equal to the decrease of it in the low demand scenarios.

Computational Results
In this section, we describe numerical experiment using the proposed model for solving a supply chain network design problem.

Model Inputs
The model has been verified through the following example where the input parameters are considered as shown in Table 1. In this case, demand means and standard deviations for all customers in all periods for all products are assumed to be the same to simplify discussion, demand mean is assumed to be 200 units and demand standard deviation is assumed to be 10 units

Model Outputs
In this section, the model outputs are presented. One of the outputs is the probabilities of scenarios generated by the model and they are as shown in Table 2. The model also generates the demand for each scenario according to the given distribution which is assumed to be normal in this research. Figure 9 depicts the generated scenario tree. The resulted optimal supply chain network obtained from the model is shown in Figure 10 where it is decided to open the second supplier, factory, and distributor to serve the four customers. The resulted total expected profit is 691870. Table 3 presents the number of material batches transferred from the supplier to the factory in all scenarios. Table 4 presents the number of batches transferred from the factory to the distributor from product 1 in all scenarios while the numbers of batches transferred from product 2 in all scenarios are presented Table 5, and Table 6 presents the number of batches transferred from product 3 in all scenarios.   There are a huge amount of data representing the number of batches transferred from the distributor to customers so, the data of only first 25 scenarios are represented by the paper and the full data are represented in appendix A or through this link https://drive.google.com/open?id=0B448W9rNzRcPRk9iWHlPN0dQa1U. Table 7 represents the number of batches (units) transferred from the distributor to the first and second customers for the three products for only 25 scenarios and Table 8 represents the number of batches (units) transferred from the distributor to the third and fourth customers for the three products for only 25 scenarios. distributor (I fdpt ), and the residual batches in both factory (R fpt ) and distributor (R dpt ) stores for the three products for the first 25 scenarios are shown in Table 9.   Table 10 represents the mean demand and its required material weight in kilograms and the required manufacturing hours. It can be noticed that the total expected required material equals 1200 kg which is smaller than the supplying capacity of one supplier so it is reasonable to open only one supplier. The total expected required manufacturing hours equals 1200 hour which is smaller than the manufacturing capacity of one factory so also, it is reasonable to open only one factory and one distributor to transfer them to customers. Considering transportation cost, the model optimally decided to open the second raw of facilities as shown in Figure 10 to reduce the total transportation cost to the four customers.

Model Results Analysis
Verification of the network flow balancing is done for only the middle scenario which has the most probability. Demands of all products for all customers are 200, 200 and 200 in the three periods. Table 11 depicts the quantities of batches and weights of raw material transferred from supplier 2 to factory 2, and the quantities of batches and weights of products transferred from factory 2 to all customers through distributor 2. Balancing is noticed in Table 11 for the transferred weights since the transferred amount from any echelon to other are the same of 14,400 kilograms.  The effect of changing the number of scenarios on the processing time is studied and it was as shown in Figures 11 and 12 from which it is noticed that increasing the number of scenarios dramatically increases CPU time. Figure 12 shows that the processing time of 125-scenarios is too big comparing to other values. The CPU time of this example is 485,925 seconds, 135 hours or 5.6 days.

Conclusion
The proposed model is successful in designing supply chain networks while considering multi-product, multi-period stochastic demand with three echelons (suppliers, facilities, and distributors). It can only be used for single item problems. The model is flexible to solve larger problems; however, it requires more powerful hardware since the CPU time increases exponentially with the increase of the number of scenarios which increases by increasing the number of periods.
The application of the proposed model showed that the total expected profit is directly affected by demand mean for a given capacity of the network. The proposed design model is capable of supply chain networks while considering inventory at the factory and distribution centers.
The proposed design model takes into account different types of costs like the non-utilized capacity cost for factories, transportation cost between all nodes, and the holding cost of inventory in both factories and distributors and shortage cost to enhance customers' satisfaction.