A medical equipment company produces a piece of diagnostic equipment at two plants, and must supply this product to three medical centers.
The problem was to determine how much of the product to ship from each plant to each customer, such that all demand is satisfied.
Let xij be the number of units shipped from Plant i to Customer j. This problem can then be formulated as:
minimize 600x11 + 800x12 + 700x13 + 400x21 + 900x22 + 600x23
subject to x11 + x12 + x13 ≤ 400
x21 + x22 + x23 ≤ 500
x11 + x21 = 300
x12 + x22 = 200
x13 + x23 = 400
x11,x12,x13,x21,x22,x23 ≥0
(Note: when using AMPL, entering the command ‘display varname, var;’ after solving the problem will display the names and values of all your variables at the optimal solution. Enter- ing the command ‘display conname, con;’ will display the names and shadow prices of your constraints.)
a) Formulate and solve this problem in AMPL. State the optimal solution and optimal objective function value.
b) Suppose Customer 1 decreases their order by 50 units. What impact will this have on the company’s total cost? (The optimal basis does not change).
c) Suppose the company has the option of purchasing a new machine that can add 100 units of production, at only one plant, at a cost of $12,000 (and suppose you know that making this purchase will not change the optimal basis). Which plant, if any, should this machine be purchased for? Provide a justification for your decision that does not involve resolving the problem. (Consider this separately from the change in part b), that is, assume Customer 1’s demand is still 300).
d) What would the cost of the machine in part c) have to be in order for your decision to change? Explain why briefly.
e) Suppose now that tweaking the production process at Plant 1 increases its capacity from 400 to 410. Without resolving the problem, can you determine what the slack variable for Plant 1’s production constraint will be in the new optimal solution? Explain why briefly. (Consider this change on the original formulation, separately from the changes considered in parts b, c and d).