Question

In: Operations Management

The Valley Wine Company produces two kinds of wine—Valley Nectar and Valley Red. The wines are...

The Valley Wine Company produces two kinds of wine—Valley Nectar and Valley Red. The wines are produced from 64 tons of grapes the company has acquired this season. A 1000-gallon batch of Nectar requires 4 tons of grapes and a batch of Red requires 8 tons. However, production is limited by the availability of only 50 cubic yards of storage space for aging and 120 hours of processing time. A batch of each type of wine requires 5 cubic yards of storage space. The processing time for a batch of Nectar is 15 hours and the processing time for a batch of Red is 8 hours. Demand for the two types of wine is limited to 7 batches, each. The profit for a batch of Nectar is $9,000 and the profit for a batch of red is $12,000. The company wants to determine the number of 1,000--gallon batches of Nectar and Red to produce in order to maximize profit.

a. Formulate a linear programming model for this problem.

b. Use POM for windows to obtain the solutions list and results printouts for this problem.

c. Answer the questions below from the printouts you obtained in b above and submit both the printout and answers to the questions to Blackboard by the time and date specified.

1. How much is produced of each type of wine to maximize profits and what is the maximum profit?

2. How much of each type of resource is left over?

3. What is the value for an additional unit of each resource?

4. Over what ranges are the values in 3) above valid?

5. What is the profit-per-unit at which the production plan will change for each type of wine?

Solutions

Expert Solution

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Formulate a linear programming model for this problem.

X1 = number of batches for valley nectar

X2 = number of batches for valley red

MAX: 9,000X1 + 12,000X2

ST: 4X1 + 8X2 ≤ 64 (tons of grapes)

5X1 + 5X2 ≤ 50 (cubic yards of storage)

15X1 + 8X2 ≤ 120 (hours of processing)

X1 ≤ 7 (demand, valley nectar batches)

X2 ≤ 7 (demand, valley red batches)

X1, X2 ≥ 0

Use POM for windows to obtain the solutions list and results printouts for this problem.

Variable              Status                   Value

X1                           Basic                      4

X2                           Basic                      6

slack 1                   NONBasic            0

slack 2                   NONBasic            0

slack 3                   Basic                      12

slack 4                   Basic                      3

slack 5                   Basic                      1

Optimal Value (Z)                             108000

Variable               Value    Reduced Cost    Original Val         Lower Bound     Upper Bound

Val. Nectar          4              0                              9000                       6000                       12000

Val. Red               6              0                              12000                    9000                       18000

Constraint           Dual Value          Slack/Surplus     Original Val         Lower Bound     Upper Bound

Tons of Grapes 750                         0                              64                           57.1429 68

Storage                                1200                       0                              50                           45                           52.7273

Proc. Time           0                             12                           120                         108                        Infinity

Demand VN       0                              3                              7                              4                              Infinity

Demand VR        0                              1                              7                              6                              Infinity

Answer the questions below from the printouts you obtained in b above and submit both the printout and answers to the questions to Blackboard by the time and date specified.

How much is produced of each type of wine to maximize profits and what is the maximum profit?

4 batches of Valley Nectar and 6 batches of Valley Red. The maximum profit is $108,000.

How much of each type of resource is left over?

0 tons of grapes

0 cubic yards of storage

12 hours of processing time

3 batches for demand of Valley Nectar

1 batch for demand of Valley Red

What is the value for an additional unit of each resource?

Tons of Grapes 750                        

Storage                                1200                      

Proc. Time          0                            

Demand VN       0                             

Demand VR        0

Over what ranges are the values in 3) above valid?

Tons of Grapes 57.1429 to 68

Storage                                45 to 52.7273

Proc. Time          108 to Infinity

Demand VN       4 to Infinity

Demand VR        6 to Infinity

What is the profit-per-unit at which the production plan will change for each type of wine?

The production plan for Valley Nectar would change if the profit per unit is less than 6,000 or greater than 12,000. The production plan for Valley Red would change if the profit per unit is less than 9,000 or greater than 12,000.

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keosha answered 3 months ago
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