Question: Consider the following parametric linear programming problem. Maximize Z() (10 4)x1 (4 )x2 (7 )x3, subject to 3x1 x2

Consider the following parametric linear programming problem.

Maximize Z() (10 4)x1 (4 )x2 (7 )x3, subject to 3x1 x2 2x3 7 (resource 1), 2x1 x2 3x3 5 (resource 2), and x1 0, x2 0, x3 0, where can be assigned any positive or negative values. Let x4 and x5 be the slack variables for the respective constraints. After we apply the simplex method with 0, the final simplex tableau is

(a) Determine the range of values of over which the above BF solution will remain optimal. Then find the best choice of

within this range.

(b) Given that is within the range of values found in part (a), find the allowable range to stay feasible for b1 (the available amount of resource 1). Then do the same for b2 (the available amount of resource 2).

(c) Given that is within the range of values found in part (a), identify the shadow prices (as a function of ) for the two resources. Use this information to determine how the optimal value of the objective function would change (as a function of

) if the available amount of resource 1 were decreased by 1 and the available amount of resource 2 simultaneously were increased by 1.

(d) Construct the dual of this parametric linear programming problem. Set 0 and solve this dual problem graphically to find the corresponding shadow prices for the two resources of the primal problem. Then find these shadow prices as a function of [within the range of values found in part (a)] by algebraically solving for this same optimal CPF solution for the dual problem as a function of .

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