Please answer the following practice problem:

Maximize c'x

s.t. Ax

xi>= 0

Where A is an m by n matrix (m constraints and n variables), x is an n by 1 vector, and b is an m by 1 vector.

A' is the transpose matrix of A.

Its dual problem is

Minimize b'y

s.t. A'y >= c

yi >= 0

So now using this information, we need to fill out the dual table below and answer the following questions:

1. Which constraints are binding? Which resources are fully utilized?
2. How much increase of the objective value do you expect if the manufacturer is given an additional unit of resource 2? How about an additional one unit of resource 3? Solve the problem with one changed constraint (change the right side limit of constraint 2 from 12 to 13) in an LP solver, and compare the objective value with what you expect.
3. Solve the dual problem in an LP solver, and compare the optimal objectives in the primal and dual problems. Does your finding agree with the duality theorem 1?
4. Does duality theorem 2 hold in this example?
5. Use this example to show: In production problem, a positive opportunity cost (shadow price) of a resource is always to be associated with the full utilization of the resource in the optimal solution. If the total opportunity cost of producing a unit of product j is greater than the gross profit from that product, then product j should not be produced according to the optimal production strategy.

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