A local furniture shop makes chairs, tables, sofas and lamps The projected profi

Question

A local furniture shop makes chairs, tables, sofas and lamps The projected profits for the four products are, respectively, 20 per chair, 30 per table, 50 per sofa and 10 per lamp. The projected demand is 400 chairs, 100 tables, 50 sofas and 160 lamps. Each chair requires 2 cubic feet of type 1-wood, each sofa requires 4 cubic feet of type 1-wood, and each lamp requires 1 cubic feet of type 2- wood while, each table requires 5 cubic feet of type 2-wood. The shop has a total amount of 1,000 cubic feet of type-1 wood and 1500 cubic feet of type-2 wood in stock. How many chairs and tables should the shop make in order to maximize its profit? Use table form to solve the problem

Explanation / Answer

No of chairs = a

No of table = b

No of sofa = c

No of lamp = d

Let Profit be z = 20a + 30b + 50c + 10d

a 400 ; b 100 ; c 50 ; d 160

2a + 4c 1000

b + 5d 1500

Table #1
==========================================================
a b c d s1 s2 s3 s4 s5 s6 z   
==========================================================
1 0 0 0 1 0 0 0 0 0 0 400
0 1 0 0 0 1 0 0 0 0 0 100
0 0 1 0 0 0 1 0 0 0 0 50   
0 0 0 1 0 0 0 1 0 0 0 160
2 0 4 0 0 0 0 0 1 0 0 1000   
0 1 0 5 0 0 0 0 0 1 0 1500   
-20 -30 -50 -10 0 0 0 0 0 0 1 0

Table #2
==========================================================
a b c d s1 s2 s3 s4 s5 s6 z   
==========================================================
1 0 0 0 1 0 0 0 0 0 0 400
0 1 0 0 0 1 0 0 0 0 0 100
0 0 1 0 0 0 1 0 0 0 0 50   
0 0 0 1 0 0 0 1 0 0 0 160
2 0 0 0 0 0 -4 0 1 0 0 800
0 1 0 5 0 0 0 0 0 1 0 1500   
-20 -30 0 -10 0 0 50 0 0 0 1 2500   

Table #3
==========================================================
a b c d s1 s2 s3 s4 s5 s6 z   
==========================================================
1 0 0 0 1 0 0 0 0 0 0 400
0 1 0 0 0 1 0 0 0 0 0 100
0 0 1 0 0 0 1 0 0 0 0 50   
0 0 0 1 0 0 0 1 0 0 0 160
2 0 0 0 0 0 -4 0 1 0 0 800
0 0 0 5 0 -1 0 0 0 1 0 1400   
-20 0 0 -10 0 30 50 0 0 0 1 5500   

Table #4
==========================================================
a b c d s1 s2 s3 s4 s5 s6 z   
==========================================================
1 0 0 0 1 0 0 0 0 0 0 400
0 1 0 0 0 1 0 0 0 0 0 100
0 0 1 0 0 0 1 0 0 0 0 50   
0 0 0 1 0 0 0 1 0 0 0 160
0 0 0 0 -2 0 -4 0 1 0 0 0
0 0 0 5 0 -1 0 0 0 1 0 1400   
0 0 0 -10 20 30 50 0 0 0 1 13500

Table #5
==========================================================
a b c d s1 s2 s3 s4 s5 s6 z   
==========================================================
1 0 0 0 1 0 0 0 0 0 0 400
0 1 0 0 0 1 0 0 0 0 0 100
0 0 1 0 0 0 1 0 0 0 0 50   
0 0 0 1 0 0 0 1 0 0 0 160
0 0 0 0 -2 0 -4 0 1 0 0 0
0 0 0 0 0 -1 0 -5 0 1 0 600
0 0 0 0 20 30 50 10 0 0 1 15100

=> a = 400 , b = 100 , c = 50 and d = 160

=> Maximise Profit = 15100

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