Example of Linear Programming - A Maximisation Problem

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A farmer produces White and Wild Mushrooms.

Labour:

• 5 employee who work a 40 hour week (200 hours a week in total)
• The production of:
  • White Mushrooms requires 3 metres of land and 2 hours of labour time
  • Wild Mushrooms requires 5 metres of land and 5 hours of labour time

Land:

• The farmer owns 225 metres of land

Profit:

• White Mushrooms is £1
• Wild Mushrooms is £2

What is the maximum profit that the farmer can make and what is the minimum profit?

Let:

Z = profit
x = number of White Mushrooms produced
y = number of Wild Mushrooms produced

Then:

(1) 2x + y = Z (objective function)
(2) 5x + 2y <= 200 (labour constraint)
(3) 5x + 3y <= 225 (land constraint)

To solve this problem, graphically, requires drawing the four constraints on suitable graph in order to define the feasible region. This is the region encompassing the combination of White and Wild Mushrooms that it is possible to produce per week, given the resources available.

To draw each constraint on the graph you must determine two points between which to draw the line. The easiest points to find are the one where the line crosses an axis (when: x=0 or when y=0).

For example:

In equation (2) above:

5x + 2y = 200 - therefore x = 40 when y=0
5x + 2y = 200 - therefore y = 100 when x=0

∴ 5x + 2y ≤ 200 / 5x + 2y = 200 / 2y = 200 - 5x / y = 100 - 2.5x

In equation (3) above:

5x + 3y = 225 - therefore x = 45 when y=0
5x + 3y = 225 - therefore y = 75 when x=0

∴ 5x + 3y ≤ 225 / 5x + 3y = 225 / 3y = 225 - 5x / y = 75 – 1.67x

The resulting graph is shown in Figure 36:

The feasible region is the shaded area. Next is to identify the four vertices of the feasible solution region shown in Figure 36, which is: (0, 0), (40, 0), (30, 25) and (0, 75).

Then to use these vertices to evaluate the maximisation expression that was created for this problem: 2x + y = Z, which is shown in the mini-table in Figure 36. The largest number that appears in the third column, is the value that will be the maximum profit for this optimisation problem and the vertex.  The produced value will indicate the number of mushrooms are required to obtain the maximum profit.

The final solution to this maximisation problem is: x = 30, y = 25 and Z = £80. Therefore, the farmer should produce and 25 White Mushrooms and 30 Wild Mushrooms in order to produce the maximum profit possible, which in this case is £80.

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If a contract was agreed with the farmer to deliver per week:

• 10 White Mushrooms
• 20 Wild Mushrooms

Then

(4) x >= 10 (contract constraint)
(5) y >= 20 (contract constraint)

The resulting graph is shown in Figure 37:

In order to determine the points of maximum and minimum profit it is necessary to establish an ISO-Profit line on the graph, for example: a line along which the profit does not change. (The word “iso” means “equal” as in isobar, isomer, etc.)

The slope of this line can be derived from the objective function, as follows:

• 2x + y = Z
• 2x + y = 0 for Z=0 (zero profit)
• y = -2x
• y/x = -2/1 so the slope of the line is –2

This gradient can be plotted as a line from y=2, x=0 to y=0, x=1.

However, this is not inside the feasible region in Figure 37 (and very hard to see). Therefore, we multiply the gradient by 30/30 (and subsequently the points on each axis) to give:

• y/x= -60/30 (plotted as a line between y=60,x=0 and y=0, x=30).

The ISO-Profit line is shown as a dashed line in Figure 37.

To obtain the maximum profit point, the ISO-Profit line is moved out to the right as far as it can go, with at least one point on the line still being in the feasible region. This is shown in Figure 38:

This shows that profits are maximised, if 25 White Mushrooms and 30 Wild Mushrooms are produced each week. The value of the profit is, 2(30) + 1(25) = £85.

This assumes that all production over that needed to fulfil the contract can be sold. In order to find the minimum point, the ISO-Profit line needs to he moved as far to the left as possible with at least one point still in the feasible region. Therefore 20 White Mushrooms and 10 Wild Mushrooms will be produced, giving a minimum profit of 2(10)+1(20)=£40

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Reference(s)
	Book		Campbell, D. J. & Craig, T. (2005) Organisations and the Business Environment. 2nd Edition. Elsevier: Netherlands, North Holland, Amsterdam. [ISBN: 9780750658294]. [Available on: Amazon: https://amzn.to/3VHJupz].
	Book		Pfaffenberger, B. (2002) Computers in Your Future 2003. 5th Edition. Prentice Hall: United States of America (USA), New Jersey (NJ), Bergen, Upper Saddle River. [ISBN: 9780139227820]. [Available on: Amazon: https://amzn.to/3gv8n7D].
	Web		Wacha, D. M. (2007) Using Microsoft Excel to Graph Optimization Problems [Online]. Monmouth University: United States of America (USA), New Jersey (NJ), Monmouth, West Long Branch. [Accessed on: 2013-02-07]. [Available on: Monmouth: http://zorak2.monmouth.edu/~dwacha/E07h-LinearProgramming.pdf].

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Reference (or cite) Article
	Kahlon, R. S. (2013) Example of Linear Programming: A Maximisation Problem [Online]. dkode: United Kingdom, England, London. [Published on: 2013-02-07]. [Article ID: RSK666-0000097]. [Available on: dkode | Ravi - https://ravi.dkode.co/2013/02/example-of-linear-programming.html].

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