Jack, a ring maker, is interested in determining which items he should make for the upcoming month. His three most popular selections are a start sapphire cocktail ring, a royal ruby friendship band, and a single diamond placed in a Tiffany setting. These items require various combinations of diamonds, sapphires, rubies, and gold as indicated below.

	Unit Selling Price	Material Requirement
Item	Diamonds	Sapphires	Rubies	Gold (grains)
Star Sapphire Ring	1465	2	1	2	160
Royal Ruby Band	1500	2	0	3	180
Single Tiffany Diamond	400	1	0	0	150
Unit Cost		$250	$175	$200	$.375
Current Stock		60	45	50	10000

Note that the costs and on-hand availability of each raw material are also provided in the table above. Jack estimates that he can sell at most 70 star sapphire rings and at most 40 royal ruby friendship rings in the upcoming month. He is obligated to sell at least 25 Tiffany diamond rings in this time period.

The problem can be formulated to determine the product mix that maximize profit by defining S, R, and T to represent the number of sapphire rings, ruby bands, and Tiffany diamonds, respectively, to make for the upcoming month. The following linear programming results. (Note, we don't require the integer solution in this particular problem)

Max ?S+332.5R+93.75T

Subject to

2S+2R+T 60

S 45

2S+3R 50

160S+180R+150T 10000

S 70

R 40

T 25

S, R, T 0

When the problem is solved using Microsoft Excel, the following Answer Report and Sensitivity Report are obtained.

Target Cell (Max)
	Cell	Name	Original Value	Final Value
	$D$2	OBJ	0
Adjustable Cells
	Cell	Name	Original Value	Final Value
	$A$2	S	0	2.5
	$B$2	R	0	15
	$C$2	T	0	25
Constraints
	Cell	Name	Cell Value	Formula	Status	Slack
	$D$5	Diamond Availability	60	$D$5<=$E$5	Binding	0
	$D$6	Sapphire Availability	2.5	$D$6<=$E$6	Not Binding	42.5
	$D$7	Ruby Availability	50	$D$7<=$E$7	Binding	0
	$D$8	Gold Availability	6850	$D$8<=$E$8	Not Binding	3150
	$D$9	Upper Limit on S	2.5	$D$9<=$E$9	Not Binding	67.5
	$D$10	Upper Limit on R	15	$D$10<=$E$10	Not Binding	25
	$D$11	Lower Limit on T	25	$D$11>=$E$11	Binding	0

Adjustable Cells
			Final	Reduced	Objective	Allowable	Allowable
	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
	$A$2	S	2.5			2.5	45.83333333
	$B$2	R	15	0	332.5	68.75	2.5
	$C$2	T	25	0	93.75	68.75	1E+30
Constraints
			Final	Shadow	Constraint	Allowable	Allowable
	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
	$D$5	Diamond Availability	60	162.5	60	15	1.666666667
	$D$6	Sapphire Availability	2.5		45
	$D$7	Ruby Availability	50	2.5	50	2.5	15
	$D$8	Gold Availability	6850		10000	1E+30	3150
	$D$9	Upper Limit on S	2.5	0	70	1E+30	67.5
	$D$10	Upper Limit on R	15	0	40	1E+30	25
	$D$11	Lower Limit on T	25	-68.75	25	1.666666667	15

1. What is the reduced cost of S?

a.

The reduced cost is positive because S is at its upper bond.

b.

The reduced cost is zero .

c.

The reduced cost is positive because it is worth while to produce S.

d.

The reduced cost is negative.

2. What is the shadow price of Sapphire?

a.

130

b.

it is a positive value and we can use it to calculate the opportunity cost.

c.

it is a negative value.

d.

It is zero since Sapphire availability is a non biding constraint.

Question 3

What is the allowable increase of Sapphire availability?

a.

it is infinite since the Sapphire availability is a non-binding constraint.

b.

it is zero since Sapphire availability is a non-binding constraint.

c.

it is a positive number.

d.

We don't know unless we see the excel output.

Question 4

What is the allowable decrease of Sapphire availability?

a.

It is zero since the Sapphire availability is a slack constraint.

b.

It is a negative number.

c.

The allowable decrease is the amount of the slack

d.

It is infinite since the Sapphire availability is a slack constraint.

Question 5

Which statement is true?

a.

The Sensitivity Analysis is not available if there is an integer constraint.

b.

There is no alternate optimal solution in this problem because the allowable increase of T is 1E+30.

c.

The Sensitivity Analysis is available even with an integer constraint.

d.

There is an alternate optimal solution in this problem.

Question 6

If the unit price of the royal ruby friendship band increases to $1550

a.

The optimal solution will change and we have to resolve the problem to know the new optimal solution.

b.

The optimal solution will not change because the increase in profit is less than the allowable increase.

c.

The optimal solution will change because the increase in profit exceeds the allowable increase.

d.

The optimal solution will not change since we only change one parameter at a time.

Question 7

What happened if the minimum number of Tiffany Diamonds Ring obligated to sell in the upcoming month is reduced from 25 to 20?

a.

Unless we resolve the problem, we don't know what is the new optimal solution and new optimal objective value.

b.

The new optimal objective value will change, the new value is 7812.5, and we need to resolve the problem to know the new optimal solution.

c.

The new optimal objective value will change, the new value is 8500, and we need to resolve the problem to know the new optimal solution.

d.

The new optimal objective value will change, the new value is 8500, and the optimal solution will not change.

Question 8

By how much can the price of gold decrease without affecting the optimal solution?

a.

1.4535 cents

b.

1.5112 cents

c.

1.3533 cents

d.

1.7634 cents