Excel Solver - Linearize a Non linear equation due to multiplication of variable

Hello Everyone,

I am looking help in solving part C. How do I linearize the non linear equation which is product of two variable. The base model is in attachment

A company has two factories, one each at Bristol and Leeds. The factories produce paints which are sold to five wholesalers. The wholesalers are either supplied directly from the factories or through one of the company warehouses, the transportation costs being paid by the company. The company has three warehouses, one each in London, Birmingham and Glasgow. Table 1 shows the transportation costs per ton for deliveries from the factories to the warehouses or wholesalers and also from the warehouses to the wholesalers, omitting entries when delivery from a certain supplier or warehouse is impossible for some destination. Warehouse Wholesaler Supplier London Birmingham Glasgow 1 2 3 4 5 Bristol 25 23 - 80 - 90 100 86 Leeds 30 27 30 - 70 54 - 100 London - - - 37 31 - 40 44 Birmingham - - - 36 40 43 40 46 Glasgow - - - 45 42 30 - 36 The two factories at Bristol and Leeds can produce up to 40,000 and 50,000 tons per week respectively. No more than 20,000, 15,000 and 12,000 tons can be moved each week through the warehouse in London, Birmingham and Glasgow, respectively. Wholesalers 1, 2, 3, 4 and 5 require at least 15,000, 20,000, 13,000, 14,000 and 16,000 tons per week respectively.

a) Formulate a linear programming model to determine the minimum cost transportation schedule. Explain clearly the variables you use and the constraints you construct. What is the minimum cost transportation schedule and what are the corresponding costs? b)

c) The management of the company is considering the possibility of closing down one of the warehouses as this is expected to result in substantial labour and maintenance savings. Further, the manager of the Birmingham warehouse is considering sub-letting some of the capacity of this warehouse. Such sub-lets would have to be in exact multiples of 1000 tons. It is estimated that each 1000 tons of capacity could be let for £21,000 per week. Formulate a mixed integer linear programming model - o r, if necessary, different model variants - to examine and evaluate the alternative courses of action. What would you recommend the company to do, and why? Discuss the alternatives, also taking into account the solution from part (a) and explain which additional information you might need (if any) to give the company more specific advice.