1. What is goal programming? State clearly its assumptions.

2. Identify the major differences between linear programming and goal programming.

3. State some problem areas in management where goal programming might be applicable.

1. Company XYZ, produces two products. The maximum sales potential for product 1 and product 2 are 30 units and 40 units respectively. Write the goal constraints for achieving the sales goal by incorporating the deviational variables.

2. An office equipment manufacturer produces two kinds of products, chairs and lamps. Production of either a chair or lamp requires 1 hour of production capacity in the plant. The plant has a maximum production capacity of 10 hours per week. Because of the limited sales capacity, the maximum number of chairs and lamps that can be sold are 6 and 8 per week, respectively. The gross margin from the sale of a chair is Rs. 80 and Rs. 40 for a lamp.

The plant manager has set the following goals arranged in the order of importance:

- He wants to avoid any underutilization of production capacity.
- He wants to sell as many chairs and lamps as possible. Since the gross margin from the sale of chair is set a twice the amount of profit from a lamp, he has twice as much desire to achieve the sales goal for chairs as for lamps.
- He wants to minimize the overtime operation of the plant as much as possible.

Formulate this as a goal programming problem and then solve by both graphical and simplex method.

**3. Minimize z = P _{1}d_{1}^{−} + 2P_{2}d_{2}^{−} + P_{2}d_{3}^{−} + P_{3}d_{1}^{+}**

subject to

x_{1} + x_{2} + d_{1}^{−} - d_{1}^{+} = 45

x_{1} + d_{2}^{−} = 25

x_{2} + d_{3}^{−} = 35

x_{1}, x_{2}, d_{1}^{−},
d_{2}^{−}, d_{3}^{−},
d_{1}^{+} ≥ 0

**4. Minimize z = P _{1}d_{1}^{−} + 3P_{2}d_{2}^{−} + P_{2}d_{3}^{−} + P_{3}d_{1}^{+}**

subject to

x_{1} + x_{2} + d_{1}^{−} - d_{1}^{+} = 300

x_{1} + d_{2}^{−} = 140

x_{2} + d_{3}^{−} = 200

and x_{1}, x_{2}, d_{1}^{−},
d_{2}^{−}, d_{3}^{−},
d_{1}^{+} ≥ 0