1. Question:-
A company assembles four products (1, 2, 3, 4) from delivered components. The profit per unit for each product (1, 2, 3, 4) is £10, £15, £22 and £17 respectively. The maximum demand in the next week for each product (1, 2, 3, 4) is 50, 60, 85 and 70 units respectively.
There are three stages (A, B, C) in the manual assembly of each product and the man-hours needed for each stage per unit of product are shown below
Stages Products A B C
The nominal time available in the next week for assembly at each stage (A, B, C) is 160, 200 and 80 man-hours respectively.
It is possible to vary the man-hours spent on assembly at each stage such that workers previously employed on stage B assembly could spend up to 20% of their time on stage A assembly and workers previously employed on stage C assembly could spend up to 30% of their time on stage A assembly.
Production constraints also require that the ratio (product 1 units assembled)/(product 4 units assembled) must lie between 0.9 and 1.15.
Formulate the problem of deciding how much to produce next week as a linear program.

2. Decision Variables:- Objective:- Maximize profit for the week
x1=amount of product 1 produced
x2= amount of product 2 produced
x3= amount of product 3 produced
x4= amount of product 4 produced
Tba = amount of time transferred from B to A
Tca = amount of time transferred from C to A
Objective:-
Maximize profit for the week
Z= 10 x1+ 15x2+ 22x3+ 17x4

3. Constraints:- Maximum demand for each product x1<=50 x2<=60
Ratio of amount of x1 assembled to the amount of x4 assembled
0.9<=(x1/x4)<=1.15
i.e. 0.9x4<=x1 & x1<= 1.15x4
Limit on transfer of man hours
Tba<=0.2(200)
Tca<=0.3(80)

4. Contd…… Work time available during the week for each stage
2x1+2x2+x3+x4 <= 160+Tba+Tca Stage A
2x1+4x2+x3+2x4<= 200-Tba Stage B
3x1+6x2+x3+5x4<= 80-Tca Stage C
All variables used x1,x2,x3,x4, Tba,Tca >= 0