ABC has been present with the following LP model: Minimize Z = 30A + 45 B (cost)
ID: 2958766 • Letter: A
Question
ABC has been present with the following LP model:Minimize Z = 30A + 45 B (cost)
Subject to
5a + 2b = 100
4a + 8b = 240
b = 20
a and b = 0
Questions:
What are the optimal values for A and B? what is the minimum cost?
If B could be reduced to $42 per unit how many units of B would be optimal? What would the minimum cost be?
What is the shadow price for RHS of the first constraint? Over what range is it valid?
By what amount would cost change, and in what direction, if the first constraint changed to 110?
Explanation / Answer
Well to start, I’m going to assume your formulation of the model is incorrect. The formulation you have posted doesn’t have a solution. However, if, like most minimization LPs, if I assume the constraints are all “>=,” then the solution would be a = 10 and b = 25, with an optimal objective value of $1425. If the coefficient of b in the objective function dropped to $42, then the new optimal solution would be the same as the Simplex method would put the value of a as large as possible before assigning values to b since it has a smaller penalty. The minimum cost with the new coefficient would be $1350. The shadow price for the 1st constraint would be 2.25. The values this shadow price would be valid for are from 100 up to infinity. The cost would go up to $1372.50 if the first constraint changed to 110. This can easily be done in Excel or in modeling programs like AMPL. AMPL is free and can be found at ampl.com. Excel’s Solver add-on can also do this. It comes with full sensitivity analysis as well, so you can answer these questions with a single click!