Numericals on Sensitivity Method of Linear Programming


0

(Sensitivity)

 

1.         The   Simplex   Tableau   for   a   maximization   problem   of   linear programming is given below:

Product Mix

X1

X2

S1

S2

Quantity (bi)

Cj

Xj

5

X2

1

1

1

0

10

0

S2

1

0

-1

1

3

Cj

4

5

0

0

Zj

5

5

5

0

Cj – Zj

-1

0

-5

0

Answer the following questions, giving reasons in brief:

(i)         Is the solution optimal?

(ii)        Are there more than one optimal solutions?

(iii)       Is the solution degenerate?

(iv)       Is the solution feasible?

(v)        If S1 is the slack in machine A (in hours/week) and S2 is slack in machine B (in hours/week), which of these machines is being used to the full capacity when producing according to this solution?

(vi)       How many units of two products X1 and X2 are being produced and what is the total profit?

(vii)      Machine A has to be shut down for repairs for 2 hours next week. What will be the effect on profits?

(viii)    How much would you prepare to pay for another hour (per week) of capacity each on machine A and machine B?

(ix)      A new product is proposed to be introduced which would require processing time of 45 minutes on machine A and 30 minutes on machine B. It would yield a profit of ` 3/- per unit. Do you think it is advisable to introduce this product?

 

2.         Hardware Fabricators, a manufacturer of boilers, produces and sells three models of cookers. While market demands pose no restrictions, the capacity to produce is currently constrained by the limited supplies of special grade steel to 1500 kg per week and machine processing time limited to 1200 hours per week.Model 1 requires 6 kg. of steel and three hours of machine processing. Similarly, these figures for model 2 are 3 kg. and 4 hours where for model 3 these figures are 5 kg and 5 hours each. The profit contributions per boiler for these 3 models are ` 60,   ` 40 and ` 80 respectively. In order to determine the optimal product-mix to maximize weekly contribution, a linear programming model is formulated from which by using the simplex method, the following table was obtained.

60

40

80

0

0

CB

XB

b

x1

x2

x3

S1

S2

*

x1

100

1

-1/3

0

1/3

-1/3

*

x3

180

0

1

1

-1/5

2/5

Zj

*

*

*

*

*

a)           Fill in all the numerical values in starred positions.

b)           Is the current solution optimal?

c)           Analyze the sensitivity of the optimal solution to the following changes, giving the new solution.

i)           Due to a machine breakdown, the machine hours available gets reduced to 1050 hours.

ii)          An additional quantity of 150 kg of aluminium can be obtained.

iii)        The second model does not feature in the current optimal solution. What should be the minimum increase in unit contribution on this model for this to feature in the optimal solution?

iv)        What will be optimal solution if the constraints of steel and machine hours are changed to 2000 kg and 1500 hrs respectively?

3.         Standard Manufacturers produce three products P, Q and R which generate profits of ` 20/-, Rs 12/- and ` 8/- per unit. Three operations are needed for each product on three machines M1, M2 and M3. The maximum working hours available for each of these three machines are 1200, 900 and 400 respectively. One of the Simplex Method solutions is given in the following table:

c

x

b

20

12

8

0

0

0

 

 

 

X1

X2

X3

S1

S2

S3

0

S1

160

0

0

 4/5

1

-4/5

4/5

12

X2

120

0

1

 3/5

0

 2/5

-3/5

20

X1

140

1

0

 1/5

0

 -1/5

4/5

Z

20

12

 56/5

0

 4/5

44/5

D = C – Z

0

0

-16/5

0

-4/5

-44/5

On the basis of above table, answer the following questions:

(a)       Which Machine is not fully utilized? If the balance working hrs of this machine is shifted to M2, what will be the effect on the solution?

(i)        Retaining the optimality, find the range of working hrs of the third Machine.

(ii)       Within what range of profit of each product, the solution will remain optimal?

(iii)      Keeping the Shadow Prices intact, find the range for the working hours of M2.

(iv)      Without altering the optimality, is it possible to reduce the availability of the working hours of the M2 to 200 hours?

(v)        If it is decided to increase the capacities of all three machines by 25% of their respective present capacities, what will be the new product mix?

 


Like it? Share with your friends!

0
MT UVA BMS

MT UVA- University, Vocational and Affiliated Education for BMS

One Comment


Warning: Undefined array key "html5" in /home/bmsnewco/public_html/wp-content/plugins/facebook-comments-plugin/class-frontend.php on line 140

Facebook comments:

This Website Is For Sale. Email us an offer we cannot refuse on [email protected] :)

X
Choose A Format
Personality quiz
Series of questions that intends to reveal something about the personality
Trivia quiz
Series of questions with right and wrong answers that intends to check knowledge
Poll
Voting to make decisions or determine opinions
Story
Formatted Text with Embeds and Visuals
List
The Classic Internet Listicles
Countdown
The Classic Internet Countdowns
Open List
Submit your own item and vote up for the best submission
Ranked List
Upvote or downvote to decide the best list item
Meme
Upload your own images to make custom memes
Video
Youtube and Vimeo Embeds
Audio
Soundcloud or Mixcloud Embeds
Image
Photo or GIF
Gif
GIF format