Example of Linear Programming - A Minimisation Problem

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A firm operates two types of aircraft:

• XJ100 - carry a maximum of 40 passengers and 30 tons of cargo / Cost £500 per Journey
• RZ45 - carry a maximum 60 passengers and 15 tons of cargo/ Cost £600 per Journey

The aircraft firm is contracted to carry at least 480 passengers and 180 tons of cargo each day.

What choice of aircraft will minimise overall cost?

Let:

Z = cost
x = XJ100
y = RZ45

Then:

(1) 500x + 600y = Z (objective function)
(2) 40x + 60y >= 480 (passenger constraint)
(3) 30x + 15y >= 180 (cargo constraint)

The procedure for a graphical solution is similar to the previous example (Example of Linear Programming: A Maximisation Problem).

The requirement, however, is to shift the ISO-Cost line as far to the left as possible and still have one point in the feasible region.

This occurs when the firm has 6 flights of the RZ45 and 3 flights of the XJ100 per day.

For example:

In equation (2) above:

40x + 60y = 480 - therefore x = 12 when y=0
40x + 60y = 480 - therefore y = 8 when x=0

∴ 40x + 60y ≤ 480 / 40x + 60y = 480 / 60y = 480 - 40x / y = 8 – 0.66x

In equation (3) above:

30x + 15y = 180 - therefore x = 6 when y=0
30x + 15y = 180 - therefore y = 12when x=0

∴ 30x + 15y ≤ 180 / 30x + 15y = 180 / 15y = 180 - 30x / y = 12 – 2x

The solution is shown in Figure 39:

The feasible region is the shaded area. Next is to identify the four vertices of the feasible solution region shown in Figure 39, which is: (0, 12), (3, 6), (30, 25) and (12, 0).

Then to use these vertices to evaluate the maximisation expression that was created for this problem: 500x + 600y = Z, which is shown in the mini-table in Figure 39. The largest number that appears in the third column, is the value that will be the minimum cost for this optimisation problem and the vertex.  The produced value will indicate the number of journeys are required to obtain the minimum cost.

The final solution to this minimisation problem is: x =3, y = 6 and Z = £5,100. Therefore, the aircraft firm should operate with 3 - XJ100 and 6 - RZ45 in order to produce the minimum cost possible, which in this case is £5,100.

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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 Minimisation Problem [Online]. dkode: United Kingdom, England, London. [Published on: 2013-02-07]. [Article ID: RSK666-0000098]. [Available on: dkode | Ravi - https://ravi.dkode.co/2013/02/example-of-linear-programming_7.html].

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