A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840,respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work fromDept. II, and 2 hours of work from Dept. III.)Product AProduct BProduct CDept. IDept. IIDept. IIIProfit212121$96$64$80How many units of each product should the company produce in order to maximize its profit?unitsProduct AProduct BunitsProduct CunitsWhat is the largest profit the company can realize?Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)labor in Dept. Ilabor-hourslabor-hourslabor in Dept. IIlabor in Dept. IIIlabor-hours

Question
Asked Oct 3, 2019
A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840,
respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from
Dept. II, and 2 hours of work from Dept. III.)
Product A
Product B
Product C
Dept. I
Dept. II
Dept. III
Profit
2
1
2
1
2
1
$96
$64
$80
How many units of each product should the company produce in order to maximize its profit?
units
Product A
Product B
units
Product C
units
What is the largest profit the company can realize?
Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)
labor in Dept. I
labor-hours
labor-hours
labor in Dept. II
labor in Dept. III
labor-hours
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Image Transcriptionclose

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 900, 1080, and 840, respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows. (For example, to make 1 unit of product A requires 2 hours of work from Dept. I, 3 hours of work from Dept. II, and 2 hours of work from Dept. III.) Product A Product B Product C Dept. I Dept. II Dept. III Profit 2 1 2 1 2 1 $96 $64 $80 How many units of each product should the company produce in order to maximize its profit? units Product A Product B units Product C units What is the largest profit the company can realize? Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.) labor in Dept. I labor-hours labor-hours labor in Dept. II labor in Dept. III labor-hours

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Expert Answer

Step 1

Let the company produce x1 , x2  and  x3 units of product A, B and C in order to maximize the profit.

According to the question the equation is written as,

 

MaxZ 96x,64x, +80x3
Subjected_to
2.r, +x, +2.r, <900
3r1 +x 2r3 <1080
211 2x2x840
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MaxZ 96x,64x, +80x3 Subjected_to 2.r, +x, +2.r, <900 3r1 +x 2r3 <1080 211 2x2x840

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Step 2

Problem converted into the canonical form by adding slack, surplus and artificial variables.

After introducing the slack variables.

96x64.x, +80x3 +0S + 0S2 +0S
MaxZ
- OS, +OS, +0S
Subjected_to
2x, x, 2x S 900
3x1 +x2 +2x3
+S2 = 1080
2x12x2x
+S3 840
help_outline

Image Transcriptionclose

96x64.x, +80x3 +0S + 0S2 +0S MaxZ - OS, +OS, +0S Subjected_to 2x, x, 2x S 900 3x1 +x2 +2x3 +S2 = 1080 2x12x2x +S3 840

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Step 3

Now, make the iterati...

Iteration
1
В
96
64
80
0
0
Min Ratio
с,
Сs х,
SX
xx3SS
12
0
900
2
0
0
900
=450
2
(3)
| 2
0 1
0
1080
1
1080
=360
3
00
| 1
0
840
2
2
1
840
= 420
2
Z =0
00
Z
0
0
0
0
0
Z,-с,
-961 -6480 0
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Image Transcriptionclose

Iteration 1 В 96 64 80 0 0 Min Ratio с, Сs х, SX xx3SS 12 0 900 2 0 0 900 =450 2 (3) | 2 0 1 0 1080 1 1080 =360 3 00 | 1 0 840 2 2 1 840 = 420 2 Z =0 00 Z 0 0 0 0 0 Z,-с, -961 -6480 0

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