A supply chain for refrigerator contains a number of plants, warehouses, collection centers, disposal center, and a set of customers as shown in Figure 1. The refrigerators are manufactured at plants located in three different sites in Saudi Arabia. Then they are transported to four (4) warehouses in different cities. After that, the refrigerators are shipped to seven customers to satisfy their needs. Some of these refrigerators are returned due to some faults. The returned items are collected and inspected at two collection centers to specify if they need recovery or not. The recoverable return products are sent to the plants and the unrecoverable return products are sent to the disposal center. From historical data, the rate of return refrigerators from the customers is 4 % of the demand and the rate of recoverable return refrigerators is (2% of the refrigerators inspected at collection centers).

The unit transportation cost between the other nodes in the network is estimated to be SR 7.

• Develop a mathematical model (in general form) to help the company determine which plants should be open, the number of refrigerators to produce in each factory, flow between each two nodes, number of refrigerators returned per day while minimizing the total cost including (fixed cost, production cost, rework cost, and transportation cost).
• Find the optimal solution using software.

  

Each plant has a production capacity, fixed cost, and production cost listed in the following table. Plant P1 P2 P3 Daily Production Capacity (Refrigerators) 950 800 750 Production cost(SR/ Refrigerators) 60 65 50 Rework cost(SR/ Refrigerators) 10 10 10 Fixed cost 7 million 6.5 million 6 million Each warehouses have a limited capacity given in table below: Warehouse W1 W2 W3 city 700 850 600