Lost Maples Winery - Essay Example

It is very essential to ensure that these resources are carefully utilized by Lost Maples to attain the maximum possible profits.

 

Problem Definition:

 

From the given data, the contribution per case varies for each wine, $24 for Austin, $28 for St. Genevieve, and $20 for Los Almos. Hence the total contribution can be computed as

            Contribution C           = 24 Austin + 28 St. Genevieve + 20 Los Almos

Maximizing this Contribution C is the objective and hence termed as the objective function. This value can be maximized by producing the optimum amounts of the three wines.

 

It is to be noted that the amount of resources available is limited as indicated earlier. Hence these constraints are also to be taken into account. From the given data, 4 bushels are required for each case of Austin and 2 bushels for each case of Los Almos. Hence the total number of bushels (4 Austin + 2 Los Almos) cannot exceed 2,800 bushels. Similarly, the other constraints can also be specified as follows.

Grape A:         4 Austin + 2 Los Almos                                 <= 2800

Grape B:         4 St. Genevieve + 2 Los Almos                     <= 2040

Sugar:             1 Austin + 2 Los Almos                                 <= 800

Labour:           3 Austin + 1 St. Genevieve + 2 Los Almos   <= 1060

 

The objective function and the constraints form a linear programming problem aimed to arrive at the optimum output to maximize the contribution (de Heufville, Clark, and Field, 2008).

 

Maximum Contribution:

 

This linear programming problem has been solved using the Solver Tool in Microsoft Excel.

Target Cell (Max)

         

Cell

Name

Original Value

Final Value

     

$H$4

Contribution LHS

0

18680

                               

Adjustable Cells

         

Cell

Name

Original Value

Final Value

     

$B$12

Austin

0

183

     

$C$12

St. Genevieve

0

510

     

$D$12

Los Almos

0

0

                               

Constraints

         

Cell

Name

Cell Value

Formula

Status

Slack

 

$H$5

Variety A LHS

732

$H$5<=$F$5

Not Binding

2068

 

$H$6

Variety B LHS

2040

$H$6<=$F$6

Binding

0

 

$H$7

Sugar LHS

183

$H$7<=$F$7

Not Binding

617

 

$H$8

Labour LHS

1060

$H$8<=$F$8

Binding

0

 

It has been found that the optimum output comprises of 183 cases of Austin, 510 cases of St. Genevieve and no Los Almos wine to be produced. When this optimum amount of specific wine cases is produced by Lost Maples, the maximum contribution that will be attained is $ 18,680. It should also be noted that at this output, there is a surplus (slack) of 2068 bushels of Variety A grapes and 617 pounds of sugar.

 

Analysis of Usage of Sugar:

 

It is evident from the results of the linear programming that there are 617 pounds of sugar left out after the optimum output is reached. The dual price or the shadow price indicates the effect of reducing the sugar available by 1 unit. In this case, a reduction in sugar will not affect the maximum contribution, as there is surplus sugar available (Gal, 1986). Hence the sensitivity analysis indicates a shadow price of $0.

Adjustable Cells

           

 

 

Final

Reduced

Objective

Allowable

Allowable

 

Cell

Name

Value

Cost

Coefficient

Increase

Decrease

 

$B$12

Austin

183

0

24

60

12

 

$C$12

St. Genevieve

510

0

28

1E+30

12

 

$D$12

Los Almos

0

-6

20

6

1E+30

               

Constraints

           

 

 

Final

Shadow

Constraint

Allowable

Allowable

 

Cell

Name

Value

Price

R.H. Side

Increase

Decrease

 

$H$5

Variety A LHS

732

0

2800

1E+30

2068

 

$H$6

Variety B LHS

2040

5

2040

2200

2040

 

$H$7

Sugar LHS

183

0

800

1E+30

617

 

$H$8

Labour LHS

1060

8

1060

1550

549

 

 

The complimentary slackness condition indicates that the product of the slack/surplus and the dual price (shadow price) is equal to zero or in other words, either one of the slack or the dual price will be zero (Econ, 2008 and Alaouze, 1996). In this case, this condition has been proved as the shadow price value is zero and hence the product is also zero.

 

Lower Bound of Labour:

 

The allowable decrease in labor has been computed as 549 man-hours. Hence the lower bound to labor in the ranging (sensitivity) analysis is (1060 – 549) 511 man-hours. When the man-hours fall reach 510, the production of Austin has to be stopped and all the resources will be used only in the production of 510 cases of St. Genevieve wine (Higle and Wallace, 2003). This indicates that when the available labor falls to 510 man-hours or below, only St. Genevieve wine has to be produced to attain maximum contribution.

 

Upper Bound for Los Almos:

 

It is evident from the Limits Analysis that the upper bound for Los Almos cannot be more than zero. This indicates that the Los Almos wine should not be produced as it is the least profitable one with the available resources.

                     

 

Target

 

             

Cell

Name

Value

             

$H$4

Contribution LHS

18680

                                                     

 

Adjustable

 

 

Lower

Target

 

Upper

Target

 

Cell

Name

Value

 

Limit

Result

 

Limit

Result

 

$B$12

Austin

183

 

0

14280

 

183

18680

 

$C$12

St. Genevieve

510

 

0

4400

 

510

18680

 

$D$12

Los Almos

0

 

0

18680

 

0

18680

 This business report has thus indicated the optimum output to attain maximum contribution. 

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