An electronics company has factories in Cleveland and Toledo that manufacture three head and forehead VCRs. Each day the Cleveland factory produces 500 three head VCRs and 300 four head VCRs at a cost of $18000. Each day the Toledo factory produces 300 of each type of VCR at a cost of $15000. An order is received for 25,000 three head VCRs and 21,000 four head VCRs. For how many days should each factory operate to fill the order at the least cost?

The Toledo factory should work for 20 days

The Cleveland factory should work for 50 days

Explanation:

Let me use abbreviations to denote each of the VCR produces:

Three head  VCR = THV

Four head VCR = FHV

we were told that:

Cleveland in one day produces; 500 THV and 300 FHV at a price of $18000, while Toledo in one day produces; 300 THV and 300 FHV at a price of $15000.

Information on order received:

THV = 25,000

FHV = 21000

Next let us use the common factor between both company locations to divide the production days between them, and the common product produced equally by these two factories is FHV where each of them produce 300 in a day.

hence to fill an order of 21,000 FHV, each factory has to produce 21000 ÷ 2 = 10, 500 orders each.

Now let us find how many days it will take to produce 10,500 orders if they produce 300 orders each day:

300 FHV = 1 day

∴ 10,500 FHV = × = 35 days.

Therefore, if both factories were to be producing the same amount of both THV and FHV each it will take them 35 days each to fill the order, but because Cleveland factory produces 500 THV while Toledo produces 300 THV, this will not hold since at the end of 35 days:

the Cleveland factory will produce 35 × 500 = 17,500 THV

the Toledo factory will produce 35 × 300 = 10,500 THV, bringing the total number of THV to 28,000 which is 3000 more than the order of 25,000 THVs

Next, we have to work backwards.Since the Cleveland factory has an excess of 3000 THVs, let us see how many days it will take to produce the excess 3000 THVs and remove that number of days from the Cleveland factory, while adding that same number of days to the Toledo factory, to even things out.

So removing one day from Cleveland will reduce production of THVs by 500, while concurrently adding one day to FHV will increase production of THV by 300, creating a net production of 200 THVs being removed.

Remember that the excess THV produced was 3000, to get the total number of days to remove from Cleveland and to add to Toledo, we will divide 3000 by 200.

∴ 3000 ÷ 200 = 15.

hence we will subtract 15 days from the original 35 days of Cleveland while we add 15 days to the original 35 of Toledo giving us:

Cleveland: 35 - 15 = 20 days

Toledo: 35 + 15 = 50 days.

now let us test our answer.

for THV:

Cleveland working for 20 days will produce; 500 × 20 = 10000

Toledo working for 50 days will produce; 300 × 50 = 15000

giving a total of 10000 + 15000 = 25,000 three head VCRs.

for Four Head VCRs (FHV)

Cleveland working for 20 days will produce; 300 × 20 = 6,000

Toledo working for 50 days will produce; 300 × 50 = 15,000

therefore total Four head VCRs produced = 6,000 + 15,000 = 21,000 VCRs.

and the total cost of production:

Cleveland; 1 day = $18,000

∴ 20 days = 18,000 × 20 = $360,000

while Toledo in 50 days = 15000 × 50 = $750,000. Hence the total amount for production = $360,000 + $750,000 = $1,110,000