A company sells sets of kitchen knives. A Basic Set consists of 2 utility knives

Question

A company sells sets of kitchen knives. A Basic Set consists of 2 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 $20 on a Basic Set, $60 on a Regular Set, and $80 on a Deluxe Set. The factory has on hand 800 utility knives, 400 chefs knives, and 200 slicers. If all sets will be sold, how many of each type should be made up in order to maximize profit? 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

Solution:

Let x be the total number of basic set of knives

Let y be the total number of regular set of knives

Let z be the total number of Deluxe set of knives

As per given condtion

set of all utility kinves

2x +2y +3z <=800

for chef's knives

1x + y +z <= 400

for slicer

0x + y + z <=200

Now we have obejctive function : p = 20x + 60y + 80z which has to be maximized

using simplex method

Maximize p = 20x +60y+80z subject to
2x +2y +3z <=800
x + y +z <= 400
y+z <= 200

Tableau #1
x y z s1 s2 s3 p   
2 2 3 1 0 0 0 800
1 1 1 0 1 0 0 400
0 1 1 0 0 1 0 200
-20 -60 -80 0 0 0 1 0

Tableau #2
x y z s1 s2 s3 p   
2 -1 0 1 0 -3 0 200
1 0 0 0 1 -1 0 200
0 1 1 0 0 1 0 200
-20 20 0 0 0 80 1 16000

Tableau #3
x y z s1 s2 s3 p   
1 -0.5 0 0.5 0 -1.5 0 100
0 0.5 0 -0.5 1 0.5 0 100
0 1 1 0 0 1 0 200
0 10 0 10 0 50 1 18000

therefore from above we conclude that

Basic set x= 100,

Regular set y = 0,

Deluxe set z = 200

optimal solution:the maximum profit p = $ 18000

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