Question

Harry and Melissa Jacobson produce handcrafted furniture in a workshop on their farm. They have obtained

a load of 600 board feet of birch from a neighbor and are planning to produce round kitchen tables and ladder

back chairs during the next 3 months. Each table will require 30 hours of labor, each chair will require 18 hours,

and between them they have a total of 480 hours of labor available. A table requires 40 board feet of wood to

make, and a chair requires 15 board feet. A table earns the couple $575 in profit, and a chair earns $120 in profit.

Most people who buy a table also want four chairs to go with it, so for every table that is produced, at least four

chairs must also be made, although additional chairs can also be sold separately. Formulate and solve an integer

programming model to determine the number of tables and chairs the Jacobsons should make to maximize·profit.

Answer 1

ANSWER

First let us find out what is given to us :
Load acquired by Harry and Mellisa from neighbour	board feet	600
Amount of Labor required by Table	hours	30
Amount of Labor required by Chair	hours	18
Total amount of labor available	hours	480
Amount of board required by table	board feet	40
Amount of board required by chair	board feet	15
Profit on table	USD	575
Profit on chair	USD	120
Number of chairs produced for every table produced	numbers	4
Now let the number of tables to be produced is represented by	X1
& let the number of chairs to be produced are represented by		X2
& let profit on entire business is represented by P
Now here the object function will be :
max P = X1*575+X2*120
Where constraints are :
X1*30+X2*18=480
X1*40+X2*15=600
X2=4*X1
X1 & X2 >0
Now let us use excel solver to solve this problem
Here we will test all available options and since number of chairs to be produced has to four times that of Tables produced our total calculation will be based upon the number of
tables produced . Now let us go ahead and produce 1 table and see the results.
X1 to be produced	Numbers	1
X2 to be produced=4*X1	Numbers	4
			Board required	Board Consumed	Labor required	Labor Consumed	Profit	Total Profit
		X	A	B=A*X	C	D=C*X	p	P*X
			Board Feet/piece	Board Feet	Hours/piece	Hours	USD/Piece
X1 to be produced	Numbers	1	40	40	30	30	575	23000
X2 to be produced	Numbers	4	15	60	18	72	120	480
Total	100		102
Board & Labor hours available	600		480
Now in above we can see we have consumed only 100 board feet of board out of 680 and 102 hours of labor out of 480 , so let us try with 2 , 3 , 4 & 5 pieces of tables and stop where any of the constraints of board and labor hours is exceeded.
X1 to be produced	Numbers	2
X2 to be produced=4*X1	Numbers	8
			Board required	Board Consumed	Labor required	Labor Consumed	Profit	Total Profit
		X	A	B=A*X	C	D=C*X	p	P*X
			Board Feet/piece	Board Feet	Hours/piece	Hours	USD/Piece
X1 to be produced	Numbers	2	40	80	30	60	575	23000
X2 to be produced	Numbers	8	15	120	18	144	120	960
Total	200		204
Board & Labor hours available	600		480
X1 to be produced	Numbers	3
X2 to be produced=4*X1	Numbers	12
			Board required	Board Consumed	Labor required	Labor Consumed	Profit	Total Profit
		X	A	B=A*X	C	D=C*X	p	P*X
			Board Feet/piece	Board Feet	Hours/piece	Hours	USD/Piece
X1 to be produced	Numbers	3	40	120	30	90	575	23000
X2 to be produced	Numbers	12	15	180	18	216	120	1440
Total	300		306
Board & Labor hours available	600		480
X1 to be produced	Numbers	4
X2 to be produced=4*X1	Numbers	16
			Board required	Board Consumed	Labor required	Labor Consumed	Profit	Total Profit
		X	A	B=A*X	C	D=C*X	p	P*X
			Board Feet/piece	Board Feet	Hours/piece	Hours	USD/Piece
X1 to be produced	Numbers	4	40	160	30	120	575	23000
X2 to be produced	Numbers	16	15	240	18	288	120	1920
Total	400		408		24920
Board & Labor hours available	600		480
X1 to be produced	Numbers	5
X2 to be produced=4*X1	Numbers	20
			Board required	Board Consumed	Labor required	Labor Consumed	Profit	Total Profit
		X	A	B=A*X	C	D=C*X	p	P*X
			Board Feet/piece	Board Feet	Hours/piece	Hours	USD/Piece
X1 to be produced	Numbers	5	40	200	30	150	575	23000
X2 to be produced	Numbers	20	15	300	18	360	120	2400
Total	500		510		25400
Board & Labor hours available	600		480
Now as we can see we can produce a maximum of 4 tables and 16 chairs because if we are producing more than that then labor constraint is there.
Maximum profit we can earn is USD 24920

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