Finer Furniture produces three types of wooden furniture – benches, tables, and
ID: 3121588 • Letter: F
Question
Finer Furniture produces three types of wooden furniture – benches, tables, and chairs. The production of each of these involves both a carpentry process and a finishing process. The time required in each of these as well as lumber requirements for each product are given below. For example, it takes 3 carpentry hours, 2 finishing hours, and 10 feet of lumber to produce one bench. The profit per unit is also given in the table.
The company wishes to get the maximum profit possible, and a linear program was developed for this purpose. The variables were defined as:
B = number of benches produced each month
T = number of tables produced each month
C = number of chairs produced each month
The linear program is:
Maximize Profit = 30B + 60T + 50C
Subject to:
3B + 6T + 5C < 1800
2B + 5T + 4C < 1200
10B + 30T + 15C < 5400
B, T, C > 0
a. Suppose the company produced 60 benches, 60 tables, and 80 chairs. Find the total profit and the amount of each resource used. Put your answers on the answer sheet.
b. Find the optimal solution, and put the answers in the table on the answer sheet.
Benches Tables Chairs Amount available per month Carpentry hours 3 hr. 6 hr. 5 hr. 1800 Finishing hours 2 hr. 5 hr. 4 hr. 1200 Lumber (Feet) 10 ft. 30 ft. 15 ft. 5400 Profit $30 $60 $50Explanation / Answer
a. Given that Profit = 30B+60T+50C
Therefore, total profit from 60 benches, 60 tables and 80 chairs = 30*60+60*60+50*80
= 1800 + 3600+4000
= 9400
To produce one bench it requires 3hrs for carpentry work, 2 hrs for finishing work and 10 ft of Lumber.
Therefore to produce 60 benches, Carpentry hours required = 60* 3 = 180 hours
,, Finishing hours required = 60*2 = 120 hours
,, Lumber requirement = 60*10 = 600 ft.
To produce 60 tables, Carpentry hours required = 60* 6 = 360 hours
,, Finishing hours required = 60*5 = 300 hours
,, Lumber requirement = 60*30 = 1800 ft.
To produce 80 chairs, Carpentry hours required = 60* 5 = 300 hours
,, Finishing hours required = 60*4= 240 hours
,, Lumber requirement = 60*15 = 900 ft.
Therefore, Total Carpentry hours required = 180 + 360 + 300 = 840 hours
,, Finishing hours required = 120 +300 + 240 = 660 hours
,, Lumber requirement = 600 +1800 +900 = 3300 ft.
b.
The standard form of the LPP is
Max Z = 30B + 60 T+ 50C
s.t.
3B + 6T + 5C + S1 = 1800
2B + 5T + 4C + S2 = 1200
10B + 30T + 15 C + S3 = 5400
B, T, C, S1, S2, S3 >= 0.
Table 1
Cj
30
60
50
0
0
0
Basis
Cb
Xb
B
T
C
S1
S2
S3
MINIMUM RATIO
S1
0
1800
3
6
5
1
0
0
300
S2
0
1200
2
5
4
0
1
0
240
S3
0
5400
10
30
15
0
0
1
180
Zj - Cj
-30
-60
-50
0
0
0
Key column
Not all Zj – Cj values are >= 0. Hence the solution is not optimum. Hence go to next step
Table 2
Cj
30
60
50
0
0
0
Basis
Cb
Xb
B
T
C
S1
S2
S3
MINIMUM RATIO
S1
0
720
1
0
2
1
0
-5
360
S2
0
300
1/3
0
3/2
0
1
-1/6
200
T
60
180
1/3
1
½
0
0
1/30
360
Zj - Cj
-10
0
-20
0
0
2
Again solution is not optimum since some of the Zj-Cj values are negative. Hence the next table.
Table 3
Cj
30
60
50
0
0
0
Basis
Cb
Xb
B
T
C
S1
S2
S3
MINIMUM RATIO
S1
0
320
-3
0
0
1
-4/3
1/45
C
50
200
2
0
1
0
2/3
-1/9
T
60
80
-2/3
1
0
0
-1/3
4/45
Zj - Cj
60
60
50
0
40/3
14/9
In table 3, All Zj- Cj values are >=0.
Hence the solution is optimum.
The solution is
No. of Benches produced each month = 0
No. of Tables produced each month = 80
No. of chairs produced each month = 200
And the Maximum profit = 30 * 0 + 60 * 80 + 50 * 200 = 0+ 4800 + 10000 = 14800
Cj
30
60
50
0
0
0
Basis
Cb
Xb
B
T
C
S1
S2
S3
MINIMUM RATIO
S1
0
1800
3
6
5
1
0
0
300
S2
0
1200
2
5
4
0
1
0
240
S3
0
5400
10
30
015
0
0
1
180
Zj - Cj
-30
-60
-50
0
0
0
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