Question

2. The following model has been formulated to describe the manufacture of two products X and Y.

maximize 4X +5Y

subject to:

X +2Y ≤ 10 (labour in hours)

6X +6Y ≤ 36 (materials used in kg)

8X +4Y ≤ 40 (storage space in m^3 )

X,Y ≥ 0

Using Excel to solve gives the following output and sensitivity report:

	X	Y		OFV
revenue	$4.00	$5.00		$28.00
number	2	4
labour (hrs)	1	2	10	<=	10
material (kilo)	6	6	36	<=	36
space (m^2)	8	4	32	<=	40

Variable Cells

Cell	Name	Final Value	Reduced costs	Objective coefficient	Allowable Increase	Allowable Decrease
$C$3	number x	2	0	4	1	1.5
$D$3	number y	4	0	5	3	1

Constraints

Cell	Name	Final Value	Shadow Price	Constraint R.H Side	Allowable Increase	Allowable Decrease
$E$5	labour (hrs)	10	1	10	2	2
$E$6	materials (kilo)	36	0.5	36	4	6
$E$7	space (m^2)	32	0	40	1E+30	8

1) State the final recommendation as fully as you can.

2) Calculate and explain what happens to the optimal solution in each of the following situations:

(a) You acquire 2 additional kilos of material.

(b) You acquire an additional 1.5 hours of labour.

  (c) You give up 1 hour of labour and get 1.5 kilos of material.

   (d) The profits from X and Y are changed to $4.75 each.

(e) You introduce a new product that will sell for $2. Each unit of this item will use 1 hour of labour, 1 kilo of material and 2 m2 of space.

Answer 1

Decision Variable X,Y

Objective function , Total profit = maximize( 4X+ 5Y)

Subject to Constraint

Labours hour

X +2Y ≤ 10 ---------- 1

materials used in kg

6X +6Y ≤ 36 ---------2

storage space in m^3

8X +4Y ≤ 40----------3

Solver :

on Solving

Senstivity Analysis

a) Final recommendation is to produce X = 2 & Y = 4 number of products

2 a) on Acquiring 2 additional kilos of material

As per sensitivity analysis, Allowable increase of materials used in kg is 4 kg that means that increase of materials used till 4 kg will not change the optimal solutions. That means increase of 2 additional kilos of material will have no change in optiomal solution

X =2

Y =4

total profit = 28

You can also check by putting Material used limit from 36 to 38 . Solutions will be same

(b) You acquire an additional 1.5 hours of labour

As per sensitivity analysis, Allowable increase of labour hours  used in kg is 2hourthat means that increase of labour hour  used till 2 hour will not change the optimal solutions. That means increase of 1.5  additional labou r hour will have no change in optiomal solution

X =2

Y =4

total profit = 28

c)  You give up 1 hour of labour and get 1.5 kilos of material.

As per sensitivity analysis, allowable decrease of labour hour is 2 & allowable increase of material weights is 4 kg.

So decreasing 1 hour of labor and increasing 1.5 kilos of material will have no change in optiomal solution.

X =2

Y =4

total profit  = 28

d)  The profits from X and Y are changed to $4.75 each.

puttin profit from x 4.75 & Y 4.75 and keeping other constraint same

x = 4 , y=2

total profit  = 28.5

e ) let z be new product

On Solving the solver

Solutions coming as

x = 2, Y = 4 , Z= 0

Total profit = 28