Define, constraints, set, decision variables and objective function.

4 9 An engineering factory makes seven products (PROD 1 to PROD 7) on the following machines: four grinders, two vertical drills, three horizontal drills, one borer and one planer. Each product yields a certain contribution to profit (defined as $/unit selling price minus cost of raw materials). These quantities (in $/unit) together with the unit production times (hours) required on each process are given below. A dash indicates that a product does not require a process. PROD1 PROD 2 PROD 3 PROD 4 PROD 5 PROD 6 PROD 7 Contribution to 10 6 8 11 3 Profit Grinding 0.5 0.7 0.3 0.2 0.5 Vertical drilling 0.1 0.2 0.3 - 0.6 Horizontal 0.2 0.8 - 0.6 drilling Boring 0.05 0.03 0.07 0.1 0.08 Planning 0.01 0.05 0.05 - In the present month (January) and the five subsequent months, certain machines will be down for maintenance. These machines will be as follows: January 1 Grinder February 2 Horizontal drills March 1 Borer April 1 Vertical drill May 1 Grinder and 1 Vertical drill June 1 Planer and 1 Horizontal drill There are marketing limitations on each product in each month. These are given in the following table: PROD PROD2 PROD 3 PROD 4 PROD 5 PROD 6 PROD January 500 1000 300 300 800 200 100 February 600 500 200 0 400 300 150 March 300 600 0 0 500 400 100 April 200 300 400 500 200 0 100 May 0 100 500 100 1000 300 0 June 500 500 100 300 1100 500 60 It is possible to store up to 100 of each product at a time at a cost of $0.5 per unit per month. There are no stocks at present, but it is desired to have a stock of 50 of each type of product at the end of June. The factory works a six days a week with two shifts of 8 hour each day. No sequencing problems need to be considered. a) When and what should the factory make in order to maximize the total product? Recommend any price increases and the value of acquiring any new machines. b) Instead of stipulating when each machine is down for maintenance, it is desired to find the best month for each machine to be down. Each machine must be down for maintenance in one month of the six apart from the grinding machines, only two of which need to be down in any six months. Extend the model to allow it to make these extra decisions. How much is the extra flexibility of allowing down times to be chosen worth