Steelcase Corporation manufactures 3 basic products: chairs, desks, and tables.
ID: 2881720 • Letter: S
Question
Steelcase Corporation manufactures 3 basic products: chairs, desks, and tables. Below is chart which summarizes the number of labor hours spent for each product in each division.
Chairs
Desks
Tables
Process
Carpentry
2
3
6
Finishing
1
1
1
Assembly
4
5
2
In a given week, Steelcase has 250 hours available for carpentry, 100 hours available for finishing, and 400 hours available for assembly.
Steelcase makes a profit of $66 on each chair, $75 on each desk, and $100 on each table that they sell.
Steelcase also needs to produce at least one chair for every desk they produce, and 4 chairs for every table they produce. The total number of chairs must be greater than or equal to the sum of the chairs needed for desks and tables. They can produce more chairs on their own too.
How many chairs, desks, and tables should Steelcase manufacture each week in order to maximize profit?
Chairs
Desks
Tables
Process
Carpentry
2
3
6
Finishing
1
1
1
Assembly
4
5
2
Explanation / Answer
let x be the number of chairs
y be the number of desks
z be the number of tables
the constraints are :
carpentary : 2x+3y+6z <= 250
Finishing : x+y+z <= 100
Assemble : 4x+5y+2z <= 400
atleast 1 chair for every desk : x>= y
atleast 4 chairs for every table : x >= 4z
total number of chairs : x >= y + 4z
x , y , z >= 0
the profit function maximize Z = 66x + 75y + 100z
we could use the simplex method to solve the linear programming model :
Tableau #1
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
1 -1 0 0 0 0 -1 0 0 0 0 0 0 0
1 0 -4 0 0 0 0 -1 0 0 0 0 0 0
1 -1 -4 0 0 0 0 0 -1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #2
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
1 0 -4 0 0 0 0 -1 0 0 0 0 0 0
1 -1 -4 0 0 0 0 0 -1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #3
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 4 0 0 0 0 1 0 0 0 0 0 0
1 -1 -4 0 0 0 0 0 -1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #4
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 4 0 0 0 0 1 0 0 0 0 0 0
-1 1 4 0 0 0 0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #5
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 4 0 0 0 0 1 0 0 0 0 0 0
-1 1 4 0 0 0 0 0 1 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #6
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 4 0 0 0 0 1 0 0 0 0 0 0
-1 1 4 0 0 0 0 0 1 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 -1 0 0 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 -1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #7
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
2 3 6 1 0 0 0 0 0 0 0 0 0 250
1 1 1 0 1 0 0 0 0 0 0 0 0 100
4 5 2 0 0 1 0 0 0 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 4 0 0 0 0 1 0 0 0 0 0 0
-1 1 4 0 0 0 0 0 1 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 -1 0 0 0 0 0 0 0 0 1 0 0 0
0 0 -1 0 0 0 0 0 0 0 0 1 0 0
-66 -75 -100 0 0 0 0 0 0 0 0 0 1 0
Tableau #8
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
3.5 1.5 0 1 0 0 0 0 -1.5 0 0 0 0 250
1.3 0.75 0 0 1 0 0 0 -0.25 0 0 0 0 100
4.5 4.5 0 0 0 1 0 0 -0.5 0 0 0 0 400
-1 1 0 0 0 0 1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 1 -1 0 0 0 0 0
-0.25 0.25 1 0 0 0 0 0 0.25 0 0 0 0 0
-1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 -1 0 0 0 0 0 0 0 0 1 0 0 0
-0.25 0.25 0 0 0 0 0 0 0.25 0 0 1 0 0
-91 -50 0 0 0 0 0 0 25 0 0 0 1 0
Tableau #9
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
1 0.43 0 0.29 0 0 0 0 -0.43 0 0 0 0 71
0 0.21 0 -0.36 1 0 0 0 0.29 0 0 0 0 11
0 2.6 0 -1.3 0 1 0 0 1.4 0 0 0 0 79
0 1.4 0 0.29 0 0 1 0 -0.43 0 0 0 0 71
0 -1 0 0 0 0 0 1 -1 0 0 0 0 0
0 0.36 1 0.071 0 0 0 0 0.14 0 0 0 0 18
0 0.43 0 0.29 0 0 0 0 -0.43 1 0 0 0 71
0 -1 0 0 0 0 0 0 0 0 1 0 0 0
0 0.36 0 0.071 0 0 0 0 0.14 0 0 1 0 18
0 -11 0 26 0 0 0 0 -14 0 0 0 1 6500
Tableau #10
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
1 0.75 0 -0.25 1.5 0 0 0 0 0 0 0 0 87
0 0.75 0 -1.2 3.5 0 0 0 1 0 0 0 0 37
0 1.5 0 0.5 -5 1 0 0 0 0 0 0 0 25
0 1.8 0 -0.25 1.5 0 1 0 0 0 0 0 0 87
0 -0.25 0 -1.3 3.5 0 0 1 0 0 0 0 0 37
0 0.25 1 0.25 -0.5 0 0 0 0 0 0 0 0 13
0 0.75 0 -0.25 1.5 0 0 0 0 1 0 0 0 87
0 -1 0 0 0 0 0 0 0 0 1 0 0 0
0 0.25 0 0.25 -0.5 0 0 0 0 0 0 1 0 13
0 -0.5 0 8.5 49 0 0 0 0 0 0 0 1 7000
Tableau #11
x y z s1 s2 s3 s4 s5 s6 s7 s8 s9 z
1 0 0 -0.5 4 -0.5 0 0 0 0 0 0 0 75
0 0 0 -1.5 6 -0.5 0 0 1 0 0 0 0 25
0 1 0 0.33 -3.3 0.67 0 0 0 0 0 0 0 17
0 0 0 -0.83 7.3 -1.2 1 0 0 0 0 0 0 58
0 0 0 -1.2 2.7 0.17 0 1 0 0 0 0 0 42
0 0 1 0.17 0.33 -0.17 0 0 0 0 0 0 0 8.3
0 0 0 -0.5 4 -0.5 0 0 0 1 0 0 0 75
0 0 0 0.33 -3.3 0.67 0 0 0 0 1 0 0 17
0 0 0 0.17 0.33 -0.17 0 0 0 0 0 1 0 8.3
0 0 0 8.7 47 0.33 0 0 0 0 0 0 1 7000
hence the maximum profit is Z =$ 7000
when x = 75 number of chairs
y = 17 number of desks
and z = 8.3 = 8 (nearest integer) number of tables are made
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