A company manufactures two types of slurry pumps: type A and type B. Profits of $60 and $80 are obtained for producing a unit of type A and type B pumps respectively. The production process requires 3 and 2 hours of labour to produce one unit of types A and B pumps respectively. Production of a unit of pump type A requires 5 kg of steel and a unit of type B pump requires 6 kg of steel. There are 480 hours of labour and 1,000 kg of steel available for production daily. The manufacturing company has a contract to meet a daily demand of 80 type A and 90 type B pumps. The company incurs penalty costs if it does not produce enough to meet daily demand, as per the contract agreement.
Management of the company wants to know how many of each pump type to be produced and has set the following goals in order of priority.
1. Achieve a profit target of $9,000 per day.
2. Avoid overtime as much as possible.
3. Avoid incurring penalty cost from inability to meet contract agreement. Use differential weights. The penalty costs for type A and type B pumps are related to their profitability.
4. Avoid the purchase of extra steel because of storage problems.
Formulate this as a goal programming model.