Suppose you are shown the optimal solution to a linear program.

(a) A particular constraint has a negative dual price. Describe what happens (in complete sentences) to the program value if the objective is either maximization or minimization.

(b) How is it possible that the dual price of an inequality constraint in a primal LP has a different sign (sense) than the corresponding dual variable in the dual LP? Explain the possibilities clearly (in complete sentences) with respect to the program objective. The sensitivity report for the optimal solution youve been shown has two particular inequality constraints (material and labor) having respective shadow prices $3.75 (material) and -$4.68 (labor).

(c) What are the respective signs of the two inequalities? You may need to consider the LPs particular objective along with the sign of the material constraint and the labor constraint.

(d) Now, the allowable decrease and allowable increase of the first constraint (material) are 25 and 46, but are 18 and 34 for the second constraint (labor). Suppose 30 more units of material are gathered and brought into the problem, but labor availability also decreases by 6. (i) Explain if the range of feasibility for either constraint remains the same, and why. (ii) Determine the change in the value of the LP, if possible, given this data. Justify your work. If not possible, clearly explain why.