Consider a supplier order allocation problem under multiple sourcing, where it is required to buy 2000 units of a certain product from three different suppliers. The fixed set-up cost (independent of the order quantity), variable cost (unit price), and the maximum capacity of each supplier are given in Table 5.15 (two suppliers offer quantity discounts). The objective is to minimize the total cost of purchasing (fixed plus variable cost). Formulate this as a linear integer programming problem. You must define all your variables clearly, write out the constraints to be satisfied with a brief explanation of each and develop the objective function.

Reformulate the problem in Exercise 5.5 under the assumption that both suppliers 1 and 3 offer “all units” discount, as described in the following: • Supplier 1 charges $10/unit for orders up to 300 units and for orders more than 300 units, the entire order will be priced at $7/unit. • Supplier 3 charges $6/unit for orders up to 500 units and for orders more than 500 units, the entire order is priced at $4/unit. Exercise 5.5Consider a supplier order allocation problem under multiple sourcing, where it is required to buy 2000 units of a certain product from three different suppliers. The fixed set-up cost (independent of the order quantity), variable cost (unit price), and the maximum capacity of each supplier are given in Table 5.15 (two suppliers offer quantity discounts). The objective is to minimize the total cost of purchasing (fixed plus variable cost). Formulate this as a linear integer programming problem. You must define all your variables clearly, write out the constraints to be satisfied with a brief explanation of each and develop the objective function.

(Ravindran et al., 1987) A company manufacturers three products A, B, and C. Each unit of product A requires 1h of engineering service, 10h of direct labor, and 3 lb of material. To produce one unit of product B requires 2h of engineering, 4h of direct labor, and 2 lb of material. Each unit of product C requires 1h of engineering, 5h of direct labor, and 1lb of material. There are 100h of engineering, 700h of direct labor, and 400lb of materials available. The cost of production is a nonlinear function of the quantity produced as shown in Table 5.16. Given the unit selling prices of products A, B, and C as $12, 9, and 7 respectively, formulate a linear mixed integer program to determine the optimal production schedule that will maximize the total profit.

## Explain how the following conditions can be represented as linear constraints using binary variables

Explain how the following conditions can be represented as linear constraints using binary variables

(Ravindran et al., 1987) Convert the following nonlinear integer program to a linear integer program.

Explain the differences between set covering and set partitioning models. Give two applications of each.

(Adapted from Srinivasan, 2010) Consider a supply chain network with three potential sites for warehouses and eight retailer regions. The fixed costs of locating warehouses at the three sites are given as follows:Site 1: $100,000 Site 2: $80,000 Site 3: $110,000The capacities of the three sites are 100,000, 80,000, and 125,000 respectively. The retailer demands are 20,000 for the first four retailers and 25,000 for the remaining. The unit transportation costs ($) are given in Table 5.17: (a) Formulate a mixed integer linear program to determine the optimal location and distribution plan that will minimize the total cost. You must define your variables clearly, write out the constraints, explaining briefly the significance of each and write the objective function. Assume that the retailers can receive supply from multiple sites. Solve using any optimization software. Write down the optimal solution. (b) Reformulate the optimization problem as a linear integer program, assuming dedicated warehouses, that is, each retailer has to be supplied by exactly one warehouse. Solve the integer programming model. What is the new optimal solution? (c) Compare the two optimal solutions and comment on their distribution plans.

What is risk pooling? How does it affect safety stock and transportation costs? For what type of products risk pooling is beneficial?

Discuss the differences between the two types of quantity discounts. Is one better than the other? If so, under what conditions?

What are the pros and cons of discrete and continuous location models in facility location decisions? Discuss their practical applications.