# companies around the world. The last stage in the production process is to produce the jet engines and then to install them (a very fast operation) in the completed airplane frame. The company has been working under some contracts to deliver a considerable number of airplanes in the near future, and the production of the jet engines for these planes must now be scheduled for the next 4 months.

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The NORTHERN AIRPLANE COMPANY builds commercial airplanes for various airline

companies around the world. The last stage in the production process is to produce the jet engines and then to install them (a very fast operation) in the completed airplane frame. The company has been working under some contracts to deliver a considerable number of airplanes in the near future, and the production of the jet engines for these planes must now be scheduled for the next 4 months.

Production scheduling data for Northern Airplane Co.

Cost is expressed in millions of dollars

To meet the contracted dates for delivery, the company must supply engines for installation

in the quantities indicated in the second column of the Table. Thus, the cumulative number of engines produced by the end of months 1, 2, 3, and 4 must be at least 10, 25, 50, and 70, respectively.

The facilities that will be available for producing the engines vary according to other production, maintenance, and renovation work scheduled during this period. The resulting monthly differences in the maximum number that can be produced and the cost (in millions of dollars) of producing each one are given in the third and fourth columns of the Table.

Because of the variations in production costs, it may well be worthwhile to produce some of the engines a month or more before they are scheduled for installation, and this possibility is being considered. The drawback is that such engines must be stored until the scheduled installation (the airplane frames will not be ready early) at a storage cost of \$15,000 per month (including interest on expended capital) for each engine,1 as shown in the rightmost column of the Table.

The production manager wants a schedule developed for the number of engines to be produced in each of the 4 months so that the total of the production and storage costs will be minimized.

Because the units being distributed are jet engines, each of which is to be scheduled for production in a particular month and then installed in a particular (perhaps different) month,

Let us denote by:

Source i=production of jet engines in month i (i =1, 2, 3, 4).

Destination j=installation of jet engines in month j ( j =1, 2, 3, 4).

a-Define the Decision Variables.

b-Calculate the cost of producing a jet engine in month iand install it in month j. (Build a table).

c-Build the Mathematical Model.

Given that,

​​​​$\Rightarrow$ The demand for 4 months are 10, 15, 25 and 20, put that in the sheet of demand row.

$\Rightarrow$ The production capacity in these 4 months are 25, 35, 30, and 10 put these in total capacity.

$\Rightarrow$ The Production cost in first month is 1.08 , Now if an item produced in period 1 and it is help for period 2 .

The cost for the production = Period 1 cost + Unit Cost of the production. ￼

Similarly if item is produced in period 1, but it is held in till period 2 is

Similarly for item production in period 1, but sold in period 2.

Put these values in small boxes.

$\rightarrow$ Now if an item is produced in period 2 , it can’t be sold in period 1. Hence no value in the box for period 2 produced and install in period 1. Grayed out the areas.

$\rightarrow$ Now for an item produced in period 2, and utilised in period 2 , Cost=1.08 Utilized in period 3,

Utilized in period 4,

$\rightarrow$ For an item produced in period 3 and utilized in period 3, Cost = 1.08 utilized in period 4

$\rightarrow$ For an item produced in period 4 and utilized in period 4, Cost = 1.08.

$\rightarrow$ Now let us optimise the production , so that less has to be put in storage.

$\rightarrow$ For period 4 Demand is 20, but only 10 can be produced So 10 have to be carried over from previous months for period 3 , demand is 25 and 30 can be produced. So, 30 will have be produced , so that 5 can be produced in period 4.

$\rightarrow$ For period 2, demand is 15 but 35 can be produced. We still have 5 pending for period 4. Hence 20 will have be produced in period 2, pending for period 4, Requirement is 10 and 25 can be produced, so we produced 10 for requirement as all the latters period have been taken as correct.

$\rightarrow$ In the table cells highleted in orange are the inventory form previous period. This will involves a storage cost all the calculations shown in the below table.

$\Rightarrow$ The sum of total capacity and unused demand should be equal.

$\Rightarrow$ The total production cost = 75.6 and total cost = 0.225.