Linear model Essay

THE CASE

Lost Maples Winery makes three varieties of contemporary Texas Hill County wines: Austin (a

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fine red), St. Genevieve (a table white) and Los Almos (a pink Zinfandel).

The raw materials plus labour requirements and unit contributions per case, for each of these

three wines are shown in the table below.

Per Case
Grapes: Variety bushels A
Grapes: Variety bushels

B
Pounds of sugar
Labour man-hours
Contribution per case $
Austin
4
0
1
3
24
St Genevieve

0
4
0
1
28
Los Almos
2
2
2
2
20

The winery has 2,800 bushels of Variety A grapes, 2040 bushels of Variety B grapes, 800

pounds of sugar and 1,060 man-hours of labour available during the next week. Its owners wish

to maximise their contribution and have asked you to advise them as to how to achieve this.

THE SOLUTION

a.        Based upon the above data, formulate the information provided as a Linear Programming problem.

The linear programming model is developed using following steps:

The first step towards solving the liner programming problem is to understand the question. The second step is to determine the objective function.

Definition of the decision variables:

X = Number of cases of Austin wine produced

Y = Number of cases of St. Genevieve wine produced

Z= Number of cases of Los Almos wine produced

Objective Function:

In the given scenario, the objective is to produce the Austin wine, St. Genevieve wine and Los Almos wine in such a combination, that the profit out of these three productions would be maximum. Mathematically,

24X + 28Y + 20Z

Constraints:

The next step is to determine the constraints in the problem. Besides the non-negativity constraints, there are four other major constraints. As defined in the question, the total quantity of Variety A grapes is 2800 Bushels provided that each Austin wine require 4 Bushel of variety A and Los Almos requires 2 Bushels of variety A. Moreover, the total amount of variety B available is 2040 Bushels, provided that each St. Genevieve wine require 4 Bushel of variety B and Los Almos requires 2 Bushels of variety B. The total quantity of Sugar is 800 pounds provided that each Austin wine require 1 pound of sugar and Los Almos requires 2 pounds of sugar.  For labor hours, the total availability is 1060 hours.  3 hours are required for each unit of Austin wine. 1 hour is required for each unit of St. Genevieve wine and 2 hours are required for Los Almos wine. Mathematically,

Variety A constraint:

4x + 2z ? 2800

Variety B constraint:

4y + 2z ? 2040

Sugar Constraint:

1x + 0y + 2z ? 800

Labor (men-hours):

3x + 1y + 2z ? 1060

X ? 0 (non-negativity constant)

Y ? 0 (non-negativity constant)

Z ? 0 (non-negativity constant)

b.      Solve the problem by computer.

Using computer programs, the following solution is obtained for the given problem:

Decision Variable
Optimal Value
Profit
Austin Wine
183
4392
St. Genevieve
510
14280
Los Almos
0
0
Total Profit
18672

ABC Company

Production Plan Proposal

For the period ___ to ___

The contemporary business scenario prevailing in this world has become very much competitive. In this context, it is important for any organization to ensure its survival and progress through achieving excellence at all levels of operations, besides thorough strategic planning.  One of the most important areas of operations, especially for manufacturing organizations is that of production. This report is intended to suggest a production plan for next period.

This plan is devised by using a number of quantitative analysis techniques so that the total profit for the firm can be maximized and doing so within the constraints the firm faces. In doing so, the first step was to determine the dependence of various factors on each other. In our production process, the profit is dependent on the quantity produced for each type of wine. Furthermore, the production of wine is dependent on the availability of various factors of production like raw material (grapes variety 1, grapes variety 2 and sugar) and labor hours. It is given that for the next period these are fixed and cannot be increased. Thus, we can deduce that our profit would be maximized with the certain production combination of these three wines when the given limitations are considered. Thus, our objective is to find out this combination.

For calculating this combination, following estimates were made:

Per Case
Grapes: Variety bushels A
Grapes: Variety bushels

B

Pounds of sugar
Labour man-hours
Contribution per case $
Austin
4
0
1
3
24
St Genevieve

0
4
0
1
28
Los Almos
2
2
2
2
20

Moreover, the constraints were:

·         All production would be zero or positive

·         The total quantity of Variety A grapes is 2800 Bushels provided that each Austin wine require 4 Bushel of variety A and Los Almos requires 2 Bushels of variety A.

·         The total amount of variety B available is 2040 Bushels, provided that each St. Genevieve wine require 4 Bushel of variety B and Los Almos requires 2 Bushels of variety B.

·         The total quantity of Sugar is 800 pounds provided that each Austin wine require 1 pound of sugar and Los Almos requires 2 pounds of sugar.

·         For labor hours, the total availability is 1060 hours.  3 hours are required for each unit of Austin wine. 1 hour is required for each unit of St. Genevieve wine and 2 hours are required for Los Almos wine.

After this quantitative analysis software was used and quantitative technique called ‘linear programming’ was applied.

The major assumptions of this model are:

·         Proportionality – a change in a variable results in a proportionate change in that variable’s contribution to the value of the function.

·         Additivity – the function value is the sum of the contributions of each term.

·         Divisibility – the decision variables can be divided into non-integer values, taking on fractional values. Integer programming techniques can be used if the divisibility assumption does not hold.

If done manually, various combinations of production would have been plotted on graph, had it had two variables, as graph can plot on two dimensions only. The software resulted in following answers:

Decision Variable
Optimal Value
Profit($)
Austin Wine
183
4392
St. Genevieve
510
14280
Los Almos
0
0
Total Profit
18672

Thus, 183 cases of Austin and 510 cases of St Genevieve wine is the optimal combination that maximizes the profit.  The net contribution from this combination is 18672$, which is highest for any combination, within the constraints considered.

800 pounds of sugar was available. However, only 183 pounds could be used. The slack is of 617 pounds. It is the slack, this means that even after optimal production, this much amount is left.  It is also deduced from the analysis that the lower bound for labor is 510 hours. This means that even if the amount of labor available falls to 510 from 1060, the optimal combination won’t change.  Likewise, the upper bound for Los Almos is 26. This means that till the point the profit from each unit of Los Almos is 26 or lesser, the optimal combination won’t change.

Thus, this way, the analysis not only tells the optimal combination, but also the flexibility available in terms of resources and production. However, there are certain limitations too. For example, in linear model we seek to determine the maximization of one objective at a time, however, in real world multiple objectives need to be considered simultaneously. Secondly, it is not always the case that the relationship between the variable is linear that is one changes in linear proportion with change in another variable. Therefore, after obtaining the analysis, one must use his managerial acumen and experience to make the decision.

Bibliography

Barnett, R., Byleen, K., ; Ziegler, M. (2007). College Mathematics for Business, Economics, Life Sciences ; Social Sciences (11th Edition). Alexandria, VA: Prentice Hall.
Etgen, G. (2007). Student’s Solutions Manual for College Mathematics for Business, Economics, Life Sciences & Social Sciences. Alexandria, VA: Prentice Hall.
Martinich, J. (1996). Production and Operations Management: An Applied Modern Approach. New York, NY: Wiley.