A company is involved in the production of two items (X and Y). The resources need to produce X and Y are twofold, namely machine time for automatic processing and craftsman time for hand finishing. The table below gives the number of minutes required for each item:
Machine time Craftsman time Item
X 13 20
Y 19 29
The company has 40 hours of machine time available in the next working week but only 35 hours of craftsman time. Machine time is costed at Β£10 per hour worked and craftsman time is costed at Β£2 per hour worked. Both machine and craftsman idle times incur no costs. The revenue received for each item produced (all production is sold) is Β£20 for X and Β£30 for Y. The company has a specific contract to produce 10 items of X per week for a particular customer. Formulate the problem of deciding how much to produce per week as a linear program. Solve this linear program graphically.
Let π₯ be the number of items of π, π¦ be the number of items of π. Then the LP is maximise
20π₯ + 30π¦ β 10(πππβπππ π‘πππ π€πππππ) β 2(πππππ‘π πππ π‘πππ π€πππππ)
subject to:
13π₯ + 19π¦β€ 40(60) πππβπππ π‘πππ
20π₯ + 29π¦β€ 35(60) πππππ‘π πππ π‘πππ
π₯ β₯ 10 ππππ‘ππππ‘
π₯,π¦β₯ 0
so that the objective function becomes maximise
20π₯ + 30π¦ β10(13π₯ + 19π¦)
60 β2(20π₯ + 29π¦)
60
i.e. maximise
17.1667π₯ + 25.8667π¦
subject to:
13π₯ + 19π¦β€ 2400
20π₯ + 29π¦β€ 2100
π₯ β₯ 10
π₯,π¦β₯ 0
It is plain from the diagram below that the maximum occurs at the intersection of π₯ =10 and
20π₯ + 29π¦β€ 2100.
Solving simultaneously, rather than by reading values off the graph, we have that π₯ =10 and π¦=65.52
with the value of the objective function being Β£1866.5.
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