Question

In: Operations Management

Recall the Blue Ridge Hot Tubs problem we discussed in class. Let x1 = no. of...

Recall the Blue Ridge Hot Tubs problem we discussed in class. Let x1 = no. of Aqua-Spas produced, x2 = no. of Hydro-Luxes produced, and x3 = no. of Typhoon-Lagoons produced. Suppose we change some of the original conditions in the problem, and solve it using Solver. Use the sensitivity report given to answer the questions below:

Sensitivity Report
Name	Final Value	Reduced Cost	Objective Coefficient	Allowable Increase	Allowable Decrease
# Aqua-Spa	110	0	350	80	30
# Hydro-Lux	0	-20	300	20	Infinite
# Typhoon-Lagoon	75	0	320	50	10

Constraints
Name	Final Value	Shadow Price	Constant RHS	Allowable Increase	Allowable Decrease
# Pumps used	175	150	175	10	15
Hrs. Labor Used	1400	0	1500	Infinite	125
Ft. of Tubing Used	2550	7	2550	100	50

(a)What is the optimal decision? What is the value of the profit if this decision is implemented?

(b) What is the allowable set of values that the objective function coefficient of x3 can range over? How should this range be interpreted? Choose a value in that range, and calculate the value of the profit for that value.

(c) How much of each resource is being consumed to produce the optimal solution?

(d) What is the meaning of the shadow price for tubing?

(e) Why is the shadow price for labor equal to zero?

(f) What is the meaning of the reduced cost for Hydro-Luxes?

Expert Solution

(a) As seen from the table on the top for objective coefficients,

The optimal values can be seen under the column of FInal Values:

Aqua-Spa = 110

Hydro-Lux = 0

Typhoon-Lagoon = 75

This is the optimal decision.

The Objective coefficients of these items are:

Aqua-Spa = 350

Hydro-Lux = 300

Typhoon-Lagoon = 320

Hence, the value of the profit if this decision is implemented is 350 * 110 + 300 *0 + 320 * 75 = 62,500

(b) Allowable increase for x3 (Typhoon-Lagoon) = 50

Allowable decrease = 10

Hence, range of objective coefficients = 320 - 10 to 320 + 50 = 310 to 370 Any value out of these range will lead to change in th optimal fuction. Let the value considered be 350. Hence, total profit will be 350 * 110 + 300 *0 + 350 * 75 = 64,750. The changed value is shown in bold.

(c) The amount of resource consumed is seen under "Final Value" column in Constraints table:

Pumps used = 175

Hrs. Labor Used = 1400

Ft. of Tubing Used = 2550

(d) The shadow price for tubing is 7. This means that if 1 unit is increased for the availability of tubing, the objective value i.e. total profit value will increase by 7.

(e) The shadow price for labor hours is zero since it is not a binding constraint. The final value = 1400 which is less than the labor available = 1500. Hence, any increase in this 1500 value will not lead to an increase in total profit value. Hence, the shadow price is 0.

(f) Reduced cost of (-20) means that that objective coefficient has to increase by 20 i.e. 300 should be made 320 to make the final value as a non-zero number. This is also know as opportunity cost.

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