Graphical Method: Case Studies

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In this article we are going study some special types of problems having some exceptional behaviour.

Case 01: Unbounded Solution Space

Consider the problem,
 

Minimize Z= 20X + 40Y

Subject to,36X +  6Y >= 108
 3X + 12Y >= 36
20X + 10Y >= 100
X, Y >=0

Considering the in-equation as equation we get following equations,
36X +  6Y = 108....(i)
 3X + 12Y = 36.....(ii)
20X + 10Y = 100....(iii)

See the graph (figure 01), having unbounded solution space.

The optimal solution can be found by calculating the values of objective function at Y1, P, Q and X2. Also we can apply Iso-Cost/ Iso-profit line to find the optimal solution. 

 

Case 02: No Solution Space, Unique Solution

Consider the problem,
 

Minimize Z= 3X + 5Y
 

Subject to,
X + Y = 200
X <= 80
Y >= 80
X >= 0

Considering the in-equation as equation we get following equations,
X + Y = 200....(i)
X = 80.........(ii)
Y = 80.........(iii)

See the graph (figure 02), having no feasible solution space, but has only one feasible solution at P.

 

Case 03: Multiple Solutions

Consider the problem,
 

Minimize Z= 4X + 3Y
 

Subject to,
8X + 6Y <=48
X <= 6
X, Y >= 0


Considering the in-equation as equation we get following equations,
8X + 6Y =48....(i)
X = 6..........(ii)

See the graph (figure 03) for feasible solution space, having multiple optimal solution as the line 8X+6Y=48 is parallel to the Iso-Cost/ Iso-profit line 4X+3Y=12, joining the points (3, 0) and  (0, 4).

At point (0, 8), the solution is 4*0 + 3*8= 24
At point (6, 0), the solution is 4*6 + 3*0= 24
At point (3, 4), the solution is 4*3 + 3*4= 24

So, there are multiple solutions.

Case 04: Infeasible Solution

Consider the problem,
 

Maximize Z= 2X + 3Y

Subject to, 
X + Y <= 13X + Y >= 3X, Y >= 0 
See the graph (figure 04) , there are no feasible solution region, thus, no feasible solution.