# The Mixed Integer Linear Programming Model (Base Model)

The framework of the optimization model is as follows: The objective of the model is to maximize the total profit of the CLSC, defined as total revenue subtracted from the total cost to manufacture/refurbish and distribute products in the supply chain. Revenue of the supply chain comes from selling new and refurbished products. Total cost of the supply chain includes variable and fixed costs incurred in manufacturing and distributing the product. Variable costs include the cost of purchasing raw materials, cost of manufacturing, transportation cost between the facilities in the supply chain, and inspection cost at the recovery centers and hybrid facilities. Fixed cost includes the cost of opening the warehouses, hybrid facilities, and recovery centers.

The model is subject to the supply capacity restrictions at the suppliers, production capacity restriction at the manufacturing plants, flow conservation constraints for new products at the manufacturing plants, flow conservation constraints for refurbished products constraint at the manufacturing plants, capacity restriction and location selection constraint for the warehouses, flow conservation of new' products constraint at the warehouses, flowconservation of refurbished products constraint at the warehouses, capacity restriction and location selection constraint for the hybrid facilities, flowconservation of new products constraint at the hybrid facilities, and flowconservation of refurbished products constraint at the hybrid facilities.

The proposed mathematical model can be accessed from https://www. routledge.com/Design-and-Analysis-of-Closed-Loop-Supply-Chain-Networks/Pazhani/p/book/9780367537494 (see Base Model section). All the equation numbers are references from the web link.

Constraint (2.1) ensures that the raw material supply capacity at supplier i is not violated. The suppliers have a finite supply capacity. Constraint set (2.2) ensures that the sum of products manufactured at the plant m is less than or equal to its capacity. The products produced at plant m are defined by the sum of new' and refurbished products transported to warehouses and hybrid facilities from plant in. Constraint set (2.3) represents the flow' conservation constraints for new' products at manufacturing plant in. The raw material purchased from all suppliers in plant in should be equal to the quantity of new products flowing out of that plant to the warehouses and hybrid facilities. Similarly, constraint set (2.4) is the flow conservation constraints for recycled products at the manufacturer. The quantity of returns flowing into plant m from hybrid facilities and recovery centers should be equal to the quantity of refurbished products shipped out of that plant in the forward chain, to warehouses and hybrid facilities. Constraint set (2.5) represents the capacity and location selection of warehouses. Given the warehouse is selected for operation, this constraint set ensures that the quantity of new products and refurbished products shipped to a warehouse w does not exceed its storage capacity. The company can choose to build the warehouses at any of the capacity levels. Constraint set (2.6) ensures that only one of the capacity levels is selected if warehouse w is opened. Constraint set (2.7) represents the flow conservation constraint at the warehouses. The quantity of new products flowing into warehouse w should be equal to the new products flowing out of that warehouse. Similarly, constraint set (2.8) ensures flow conservation of refurbished products at warehouse w. Hybrid facilities also process product returns from the retailer. Constraint set (2.9) is the capacity constraints at the hybrid facilities. If a hybrid facility h is selected for operation, this constraint ensures that the flow of new and refurbished products and flow of return products into a hybrid facility h does not exceed its capacity. Constraint set (2.10) ensures that only one of the capacity levels is picked, if hybrid facility h is selected. Constraint set (2.11) and (2.12) is the flow conservation constraints for new and refurbished products at hybrid facility h in the forward channel. Constraint set (2.13) is the flow conservation constraint for returned products at the hybrid facility h in the return channel. Constraint set (2.14) represents the capacity and location selection constraints for recovery centers. Constraint set (2.15) is the flow conservation constraints for recovery center r. Constraint set (2.16) is the demand satisfaction constraints. Demand at the retailer c is satisfied with either new or refurbished products. Constraint set (2.17) ensures that total refurbishing activity should be less than or equal to the quantity of refurbished products accepted by customers. Constraint set (2.18) ensures that the returned products at the retailer are either sent to refurbishing, either via recovery centers or hybrid facilities, or are disposed-off at the retailer. Constraint set (2.19) describes non-negativity and binary conditions on the decision variables.

# Analysis of the Illustrative Example

This section will illustrate the model for the four-stage supply chain using a hypothetical example. The supply chain network has 20 potential suppliers for supplying raw materials to the manufacturing plants. The raw materials go into manufacturing new products. The company has five manufacturing plants for producing new products and refurbishing product returns. The supply chain consists of 16 potential warehousing facilities to distribute new and refurbished products to the retailers in the forward channel. Five potential recovery centers are considered to collect product returns from the retailers, inspect and distribute them to the manufacturing plants. Nine potential hybrid facilities are considered to distribute products in the forward channel and return channel. The network has 100 retailers, who face demand for new and refurbished products from the customers.

All the input parameters for the illustrative example are randomly generated from uniform distributions. Purchasing cost of raw materials from suppliers are uniformly generated from ~unif(\$600, \$800) per unit. Manufacturing and refurbishing costs at the manufacturers are ~unif(\$25, \$35) per unit and 'unif(\$5, \$10) per unit, respectively. Transportation costs between manufacturer and warehouses/hybrid facilities/recovery centers are 'unif(\$20, \$35) per unit. Transportation costs between warehouses/hybrid facilities/recovery centers and retailers are 'unif(\$45, \$55) per unit. Transportation cost increases as the products progress in the supply chain system closer to the retailers. This can be due factors like increase in product value, economies of scale in shipping, etc. Inspection cost at the hybrid facilities/recovery centers for the returned products are 'unif(\$2, \$5) per unit. The demand at the retailers is 'unif(500, 700) units. Both product return percentage and customer acceptance rates are 30% in this illustrative example. New' products are sold at the retailers for \$1,000 per unit. Refurbished products are sold at 25% discount from the new product price, that is, \$750 per unit.

Each potential warehouse and hybrid facility can be opened in one of the three different sizes (small, medium, large). Fixed cost for opening warehouses for the three different sizes are generated from 'unif(\$35O,OOO, \$450,000), ~unif(\$450,000, \$550,000), ~unif(\$55O,OOO, \$650,000). Fixed cost for opening hybrid facilities for the three different sizes are generated from ~unif(\$450,000, \$550,000), ~unif(\$550,000, \$650,000), ~unif(\$650,000, \$750,000). Fixed cost for opening recovery centers 'unif(\$450,000, \$550,000).

Production capacity at the manufacturers are assigned following uniform distribution using ~unif(15,000, 25,000) units as follows: Plant 1 has a capacity of 15,430 units. Plant 2 has a capacity of 23,295 units. Plant 3 has a capacity of 16,381 units. Plant 4 has a capacity of 18,922 units, and Plant 5 has a capacity of 23,583 units. \$upplier capacities are uniformly distributed between 'unif(5,000, 10,000) units. Table 2.1 show's the capacities of the suppliers.

Let td be the total demand and tr be the total expected returns across all the retailers. Capacities at the warehouses, for the three sizes, are generated as 'unif(10%, 20%) xtot_dem, ~unif(20%, 30%) x td. and 'unif(30%, 40%) x td. Capacities at the hybrid facilities, for the three sizes, are generated as ~unif(10%, 20%) x (td + tr), ~unif(20%, 30%) x (td + tr), and 'unif(30%, 40%) x (td + tr). Capacities at the recovery centers are generated

 TABLE 2.1 Capacity of the suppliers CAPACITY OF SUPPLIERS Supplier 1 7,109 Supplier 2 6,902 Supplier 3 6,648 Supplier 4 6,578 Supplier 5 7,928 Supplier 6 6,521 Supplier 7 8,631 Supplier 8 6,133 Supplier 9 6,382 Supplier 10 5,359 Supplier 11 7,481 Supplier 12 7,371 Supplier 13 6,857 Supplier 14 9,544 Supplier 15 9,269 Supplier 16 8,800 Supplier 17 7,921 Supplier 18 5,921 Supplier 19 5,079 Supplier 20 9,025

as ~unif(10%, 30%) x tr. Table 2.2 shows the capacity and fixed cost of warehouses. Table 2.3 shows the capacity and fixed cost of hybrid facilities. Table 2.4 shows the capacity and fixed cost of recovery centers. Production capacity data and data in Tables 2.1, 2.2, 2.3, and 2.4 will be used throughout the book in the illustrative examples.

The input parameters for the illustrative example are coded in Microsoft Visual C++ 6.0. The mathematical model is coded and solved using a commercial optimization software package. The mathematical model for this illustrative example has 467 constraints and 7,013 variables (with 6,933 continuous and 80 binary variables).

Optimal profit of the supply chain from the model for this example is \$21,125,300. We will discuss below the inferences from the mathematical model solution.

Suppliers 1. 5, 6, 8, 9, 10, and 15 are chosen to supply raw materials to the plants for producing new products. Supplier 15 is allocated 22.03% of the total volume, followed by supplier 1 (16.90%), supplier 6 (15.50%), supplier 9 (15.17%), supplier 8 (14.58%), supplier 10 (12.74%), and supplier 5 (3.08%). All the manufacturing plants are used to produce new products. Plants 1, 3,

 TABLE 2.2 i Capacity and fixed cost of warehouses SIZE 1 SIZE 2 SIZE 3 CAPACITY FIXED COST(S) CAPACITY FIXED COST (S') CAPACITY FIXED COST (S') Warehouse 1 9,927 415,379 14,108 485,024 21,159 602,485 Warehouse 2 10,189 419,745 15,071 501,072 19,101 568,214 Warehouse 3 9,111 401,781 16,999 533,186 20,208 586,648 Warehouse 4 8,259 387,590 16,147 519,002 21,062 600,878 Warehouse 5 7,338 372,253 16,650 527,372 18,516 558,459 Warehouse 6 6,551 359,148 17,748 545,677 18,610 560,024 Warehouse 7 11,429 440,403 16,342 522,251 18,049 550,683 Warehouse 8 11,905 448,339 17,123 535,253 19,397 573,142 Warehouse 9 8,528 392,081 17,437 540,491 20,912 598,379 Warehouse 10 9,414 406,834 12,146 452,351 18,984 566,264 Warehouse 11 10,872 431,128 16,382 522,916 22,186 619,599 Warehouse 12 6,434 357,184 15,361 505,906 23,119 635,151 Warehouse 13 7,176 369,558 13,283 471,290 23,461 640,838 Warehouse 14 11,765 445,996 14,413 490,108 22,130 618,672 Warehouse 15 7,880 381,276 12,422 456,948 19,374 572,760 Warehouse 16 11,280 437,919 16,099 518,195 23,635 643,737
 TABLE 2.3 Capacity and fixed cost of hybrid facilities SIZE 1 SIZE 2 SIZE 3 CAPACITY FIXED COST(S) CAPACITY FIXED COST(S) CAPACITY FIXED COST(\$) Hybrid facility 1 14,431 535,055 19,821 604,168 25,705 679,620 Hybrid facility 2 15,483 548,545 22,142 633,930 24,469 663,762 Hybrid facility 3 15,212 545,063 21,371 624,042 29,541 728,810 Hybrid facility 4 12,053 504,564 21,501 625,711 29,035 722,321 Hybrid facility 5 8,106 453,949 16,957 567,441 29,401 727,003 Hybrid facility 6 10,912 489,933 15,642 550,575 25,261 673,919 Hybrid facility 7 10,877 489,485 21,015 619,481 26,553 690,486 Hybrid facility 8 12,208 506,550 19,956 605,895 26,477 689,520 Hybrid facility 9 11,374 495,855 23,277 648,485 28,849 719,929

TABLE 2.4 Capacity and fixed cost of recovery centers

 CAPACITY FIXED COSTd) Recovery center 1 2,879 5,084 Recovery center 2 5,084 541,555 Recovery center 3 3,163 488,090 Recovery center 4 4,943 537,654 Recovery center 5 3,925 509,293

4, and 5 received returned products and are used in refurbishing these return products. There were no warehouses and recovery centers opened for product distribution in the optimal solution. Three hybrid facilities (at locations 6. 7, and 8), each of capacity level 3, are opened. These hybrid facilities distribute new and refurbished products to the retailers in the forward channel. They also collect, inspect, and distribute return products to the manufacturing plants in the return channel. The capacity utilization of these three selected hybrid facilities are greater than 99%. The demands at the retailers are satisfied from the selected hybrid facilities in the forward channel and the returns from the retailers are shipped to these hybrid facilities in the return channel. Hybrid facilities offer economic as well as practical advantages. Supply chain managers prefer having fewer numbers of facilities in the supply chain to reduce operations costs, risks related to monitoring and control of the facility, ease of tracking consignments, reduction in manpower and systems cost, and reduction in logistics and transportation risk in material handling. We can also observe that as the customer return and the acceptance rates are 30% each, there is potential to refurbish and sell all the returned items. As expected, in the solution, the returns were refurbished and sold to the customers.

However, with the objective of maximizing the supply chain profit, the model will recommend refurbishing if and only if it is profitable to refurbish, that is, the model will choose to dispose-off the returned products, if the cost of processing returns and refurbishing outweigh financial benefits from refurbishing. To illustrate this concept, we will now solve the example by setting customer return rate and the acceptance rate to zero. The solution from the models are compared in terms of transportation cost (cost of distributing products in both the forward and return channels), refurbishing cost (sum of inspection cost at the hybrid facilities and recovery centers, and the refurbishing cost at the plants), fixed cost (costs for opening warehouses, hybrid facilities, recovery centers), purchasing cost (raw material purchasing cost from the suppliers), and production cost at the manufacturers. The total profit of the supply chain without recycling was \$15,692,200 vs. profit of \$21,125,300 with recycling. Analyzing the solutions from the model, we observe that the transportation cost increases in the model with recycling due to inclusion of return and refurbished product flows. Fixed cost increases in the model with recycling, as facilities are opened to process and distribute returned and refurbished products. On the other hand, production cost is lower in the model with recycling as a portion of demand will be satisfied using refurbished products. There is also a huge reduction in raw material usage, that is, reduction in purchasing cost. The cost benefits from production and purchasing costs outweighs the cost of transportation, refurbishing, and additional fixed costs. Managers can use this model to show' the benefits of incorporating refurbishing activity in their supply chain and to categorically prove that it is possible to reduce the environmental impact/carbon footprint without compromising on the bottom line, that is, profits.

Note that the cost parameters can vary in practice based on the type of product and industry. In this example, purchasing costs account for 74.21% and transportation costs constitute 15.80% of the total supply chain cost. The quantity of returns and refurbishing activities in the supply chain are determined by customer return rate and acceptance rate parameters, which in turn affect these two major costs (purchasing and transportation) in the supply chain. Purchasing cost decreases with increase in customer return rate and acceptance rate. In our example, 30% of customer demand is satisfied using the refurbished products and raw materials are purchased to produce the remaining 70% of demand. Transportation costs in the supply chain increases with customer return and acceptance rates, as cost is incurred for handling the returns.