The company Mester uses linear programming in order to find the optimal proportions in four investment projects, K, L, M and N. Each of these projects may be accepted in full, in part or rejected. The objective of the company is to maximize the total net present value. There are capital constraints for the investment amounts in the first and in the second period, and avail-able working capital in the first period is also limited.

NOK 80 Mill. are available for investments in the first period, while NOK 20 Mill. are avail-able in the second period. The amount of working capital must not exceed NOK 4 Mill. Ac-cepted in full, the four projects, K, L, M and N, require an investment amount in the two pe-riods and a working capital of respectively NOK 40, 10 and 2 Mill. (Project K), NOK 40, 5 and 1 Mill. (Project L), NOK 60, 16 and 2 Mill. (Project M) and NOK 60, 10 and 3 Mill. (Project N). The net present value for the whole of Project K is NOK 3 Mill., while the cor-responding values for Project L and Project N are NOK 2 Mill. and NOK 4 Mill., respecti-vely.

From the solution to this problem, you have the following pieces of information: i) Both Project K and N are accepted in part, ii) There is slack in the working capital constraint, iii) The slack value of the constraint K 1 is 0.8, iv) The dual price of the constraint M 1 is 1.4.

a) Use information about Project K and Project N to find the dual prices for the two investment capital constraints. What are the optimal values for the four projects? What are the related reduced costs for the four projects?

b) What are the slack values for the three capital constraints and what are the slack values for the constraints L 1, M 1 and N 1? What is the optimal value of the objective function?

c) If Mester decides to transfer NOK 800,000 from the working capital budget to the investment budget in the first period, by how much will the objective function change? Check your answer by calculating the new optimal values for the projects and the related objective function value. (The solution implies that Project K and Project N will still be accepted in part and that the optimal values of Project L and Project M are unchanged.)

d) Why did the dual prices not change after the amount of NOK 800,000 was trans-ferred? Calculate the minimum net present value required for accepting a new Project P when you are informed that, accepted in full, the investment amount will be NOK 50 Mill. in the first period and NOK 6 Mill. in the second period, while working capital will amount to NOK 2 Mill.