Question

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 £10 per hour worked and craftsman time is costed at £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 £20 for X and £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.

Answer 1

Let 𝑥 be the number of items of 𝑋, 𝑦 be the number of items of 𝑌. Then the LP is maximise

20𝑥 + 30𝑦 − 10(𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑡𝑖𝑚𝑒 𝑤𝑜𝑟𝑘𝑒𝑑) − 2(𝑐𝑟𝑎𝑓𝑡𝑠𝑚𝑎𝑛 𝑡𝑖𝑚𝑒 𝑤𝑜𝑟𝑘𝑒𝑑)

subject to:

13𝑥 + 19𝑦≤ 40(60) 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑡𝑖𝑚𝑒

20𝑥 + 29𝑦≤ 35(60) 𝑐𝑟𝑎𝑓𝑡𝑠𝑚𝑎𝑛 𝑡𝑖𝑚𝑒

𝑥 ≥ 10 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡

𝑥,𝑦≥ 0

so that the objective function becomes maximise

20𝑥 + 30𝑦 −10(13𝑥 + 19𝑦)

60 −2(20𝑥 + 29𝑦)

60

i.e. maximise

17.1667𝑥 + 25.8667𝑦

subject to:

13𝑥 + 19𝑦≤ 2400

20𝑥 + 29𝑦≤ 2100

𝑥 ≥ 10

𝑥,𝑦≥ 0

It is plain from the diagram below that the maximum occurs at the intersection of 𝑥 =10 and

20𝑥 + 29𝑦≤ 2100.

Solving simultaneously, rather than by reading values off the graph, we have that 𝑥 =10 and 𝑦=65.52

with the value of the objective function being £1866.5.