Question

The following constraints of a linear programming model have been graphed on the graph paper provided (same constraints found in problem #3) to form a feasible region:

2X    + 6Y     >=    120

10X + 2Y     > =   200

X      +     Y     <=    120

X                     <=    100

                 Y    <=      80

X,Y                  >=        0

Using the graphical method, determine the optional solution and the objective function value for the following objective functions. Graph the objective function as a dashed line on the feasible region described by the above constraints.

1. Maximize Profit: Z = 5X + 7Y, where the coefficients are the unit profit per quantity X and Y produced.

Optimal solution:                                                                                                                                                            

               

Objective Function Value at the optimal solution:                                                                                             

1. Maximize Profit:      Z = 3X + 2Y, where the coefficients are the unit profit per quantity X and Y produced.

Optimal solution:                                                                                                                                                            

               

Objective Function Value at the optimal solution:                                                                                             

Answer 1

(a)

Feasible region is the shaded in the figure.

We draw objective function for an arbitrary value (350 here). Objective function is drawn with dotted lines.

We move the objective function parallel to the dotted one and observe that it attains maximum at A (40, 80).

Thus optimal solution is x=40, y=80.

Objective function value at the optimal solution is 5*40+7*80 = 760.

(b)

Feasible region is the shaded in the figure.

We draw objective function for an arbitrary value (120 here). Objective function is drawn with dotted lines.

We move the objective function parallel to the dotted one and observe that it attains maximum at B (100, 20).

Thus optimal solution is x=100, y=20.

Objective function value at the optimal solution is 3*100+2*20 = 340.