The following is a linear programming model for analyzing the product mix of Maxine's Hat Company, which produces three hat types:

Maximize: Z =

$7X1

+

$8X2

+

$6X3

Subject to:

2X1

+

4X2

+

2X3

120

(machine A time)

5X1

+

3X2

+

2X3

400

(machine B time)

2X1

+

2X2

+

4X3

110

(machine C time)

X1

0,

X2

0, and

X3

0

The POM for Windows printout below shows the optimal solution to the problem.

(a) Results Screen

	X1	X2	X3		RHS	Shadow Price
Maximize	7	8	6
Machine A	2	4	2	<=	120	0.5
Machine B	0.20	0.05	0.10	<=	400	0
Machine C	0.15	0.10	0.10	<=	110	3
Solution	50	5	0		390

(b) Ranging screen

Variable	Value	Reduced Cost	Original Val	Lower Bound	Upper Bound
X1	50	0	7	4.6667	8
X2	5	0	8	7	14
X3	0	7	6	-Infinity	13
Constraint	Shadow Price	Slack/Surplus	Original Val	Lower Bound	Upper Bound
Machine A	.5	0	120	110	220
Machine B	0	135	400	265	Infinity
Machine C	3	0	110	60	120

Consider each of the following statements independently, and state whether it is true or false. Explain each answer.

a. If the price of hat 3 were increased to

$12.25,

it would be part of the optimal product mix.

(Hint:

Hat 3 is represented by

X3

and its optimal value currently is 0, which means that it is not to be produced (and not part of the optimal product mix).)

A.

True, because the price of Hat 3 increased more than 50% of its current price.

B.

True, because the reduced cost of Hat 3 is $7.00.

C.

Unknown, because the optimal value of Hat 3 should be recalculated with the new selling price.

D.

False, because the optimal value of Hat 3 would remain at its current value until its price increases above $13.00.

b. The capacity of machine B can be

reduced

to

300

hours without affecting profits.

A.

True, because the new RHS value is within the range in which the shadow price of 0 would remain valid.

B.True, because the new limit is

lower

than its previous value.

C.False, because the new limit is

lower

than its previous value and the constraint would become

more

restrictive.

D.False, because the new limit is

lower

than its previous value and its shadow price would change.c. If machine

C

has a capacity of

115

hours, the production output would remain unchanged.

A.False, because constraint

C

is a binding constraint.

B.True, because the new RHS value is within the range in which the shadow price of

3

would remain valid.

C.

True, because the new limit is higher than its previous value and the constraint would become less restrictive.

D.

False, because the new limit is higher than its previous value and its shadow price would change.

Click to select your answer.