a company sells sets of kitchen knives. A basic set of utility knives and 1 chef

Question

a company sells sets of kitchen knives. A basic set of utility knives and 1 chef's knife. A regular set consists of 2 utility knives, 1 chef's knife and 1 slicer. A Deluxe set consists of 3 utility knives, 1 chef's knife, and 1 slicer. The profit is $30 on a Basic Set, $40 on a Regular Set, and $50 on a Deluxe Set. The factory has on hand 900 utility knives, 300 chef's knives and 100 slicers. If all set will be sold, how many of each type should be made up in order to maximize the proft? What is the maximum profit?

How many Basic Sets should be made up

How many Regular sets should be made up

How many Deluxe Sets should be made up

What is the maximum profit

Explanation / Answer

One possibility is to make as many Delux sets as possible.
This is limited to 200 because there are only 200 slicers.
Knives remaining would be 200 utility and 200 Chef and no slicers.
From the remaining knives no (0) Regular sets can be made because there are no slicers left,
but there are enough knives to make 100 Basic sets.
100 chef's knives are left over.
This scenario gives a profit for
200 Delux plus 100 Basic = 200 x 60 + 100 x 30 = 1500.
I've put it in a chart. The column labeled P/t is profit or total depending on the row. The column labeled Rem is remainder.

Kniv Bas Reg Del P/t Rem
Sets 100 000 200 1500
Util 200 000 600 800 000
Chef 100 000 200 300 100
Slic 000 000 200 200 000

Another possibility is to make as many Regular sets as possible.
Again the slicers limit this to 200.
Remaining knives would be 400 utility and 200 Chef and no slicers.
From these remaining knives we can make 200 Basic sets.
The profit would be 200 x 40 + 200 x 30 = 1400.

Kniv Bas Reg Del P/t Rem
Sets 200 200 000 1400
Util 400 400 000 800 000
Chef 200 200 000 400 000
Slic 000 200 000 200 000

We could make as many Basic sets as possible. That would be 400.
This would use all Chef's and Utility knives. No other sets could be made.
This gave a profit of 400 x 30 = 1200.

Kniv Bas Reg Del P/t Rem
Sets 400 000 000 1200
Util 800 000 000 800 000
Chef 400 000 000 400 000
Slic 000 000 000 000 200

So far the first scenario gives the greatest profit.
Now we consider what would happen to the profit if we made one (or two) fewer Delux sets and used the extra knives to make the other sets.
Making one less Delux set frees 3 Utility knives, 1 chef's knife and 1 slicer.
With these we could make 1 Regular set (or one Basic set) and have 1 Utility knife left over. This reduces the profit.

If we make two fewer Delux sets, this frees up enough Utility knives to make three Regular (or Basic sets). However there won't be enough slicers to make the three regular sets.
Kniv Bas Reg Del P/t Rem
Sets 100 003 198 1500
Util 200 006 594 800 000
Chef 100 003 198 300 99
Slic 000 003 198 201 -1

The first scenario gives the maximum profit.

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