1 A company makes products 1 and 2 from two resources. The linear programming model for determining the amounts of product 1 and 2 to produce (i.e. X1 and X2 ) is

Maximize Z = 8x1 + 12x2 (profit, $)

subject to

4x1 + 5x2 ≤ 20 (resource1, lb)

2x1 + 6x2 ≤ 18 (resource2, 2h)

3x1 + 4x2 ≤ 24 (resource3, ft)

X1, X2 ≥ 0

Using the graphical solution method described in class,

a. Plot the constraints to identify the feasible region;

b. Plot two Iso-profit lines to identify the optimal corner point

c. Solve for the primal solution using elimination and back substitution

d. Construct the matrix of detached coefficients

e. Solve for the shadow prices for the binding constraints, including units of measure.

2 Consider the following three linear programming problems in two variables that illustrate various special cases. By means of the graphical solution method , indicate which is infeasible, which has unbounded solutions, and which has multiple optimal solutions?.

a. Max X1+X2 subjet to : X1+X2 ≥ 400; -X1+2X2 ≤400; X1, X2≥0

b. Max 45X1 + 300X2 subject to: X1 + X2 ≤ 200

9X1+6X2 ≤1,566; 12X1+16X2 ≤ 2880; x1, x2 ≥ 0

c. Max 7X1+5X2 subject to: 4X1 + 5X2 ≤20; 5X1+4X2≥28; X1+X2≥ 0

Maximize Z = 8x1 + 12x2 (profit, $)

subject to

4x1 + 5x2 ≤ 20 (resource1, lb)

2x1 + 6x2 ≤ 18 (resource2, 2h)

3x1 + 4x2 ≤ 24 (resource3, ft)

X1, X2 ≥ 0

Using the graphical solution method described in class,

a. Plot the constraints to identify the feasible region;

b. Plot two Iso-profit lines to identify the optimal corner point

c. Solve for the primal solution using elimination and back substitution

d. Construct the matrix of detached coefficients

e. Solve for the shadow prices for the binding constraints, including units of measure.

2 Consider the following three linear programming problems in two variables that illustrate various special cases. By means of the graphical solution method , indicate which is infeasible, which has unbounded solutions, and which has multiple optimal solutions?.

a. Max X1+X2 subjet to : X1+X2 ≥ 400; -X1+2X2 ≤400; X1, X2≥0

b. Max 45X1 + 300X2 subject to: X1 + X2 ≤ 200

9X1+6X2 ≤1,566; 12X1+16X2 ≤ 2880; x1, x2 ≥ 0

c. Max 7X1+5X2 subject to: 4X1 + 5X2 ≤20; 5X1+4X2≥28; X1+X2≥ 0

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