A big furniture company just purchased two new production plants and two warehouses. The sites are connected by the road network shown below. Symbols next to the arrows are the lengths of the roads in some unspecified units. P1 P2 C13 C24 C14 C25 3 4 5 C36 C46 C47 C57 C39 6 7 C69 C68 C78 W1 W2 (a) The company wants to know the distances between the warehouses and the production plants. Your responsibility as a consultant is to calculate the shortest distance between plant P2 and warehouse W2. Use this instance to Illustrate how and which of the optimisation techniques can be used to solve the shortest path problem. [10 marks] (b) Assume that the costs of delivering a piece of furniture from the plants to the warehouses are given. At the moment the company has N units of furniture available at plant P1 and N units available at plant P2. According to the existing contract with a customer, M units of the furniture are to be delivered to warehouse W1, and M units are to be delivered to warehouse W2. Unfortunately, the total supply (i.e. N+ N) is smaller than the total demand. For undelivered units, the company will pay penalties: p for each undelivered unit to W1, and q for each undelivered unit to W2. Formulate a linear programming problem to minimise the total costs of transporting the units and the penalties paid. [10 marks] (c) The company is investigating a possibility to balance the delivery along the four routes: it would be beneficial for the company to increase the delivery along the least busy route as much as possible. Enhance the model suggested in your answer to question (b) to incorporate the new objective.