1. A company can make two products P1 & P2. Profit: P1 – 15, P2 – 5 per unit.

 

Requirements per unit –

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  P1 P2 Maximum
Process 1 20 min. 10 min. 1200 min.
Process 2 40 min. 10 min. 1600 min.

Solve by simplex method.

 

 

  1. An electronics company wants to decide most profitable product mix. Profit per unit of Transistors – Rs. 120, Resistors – Rs. 60, CRT – Rs. 40.

 

Resource requirements per unit –

Product           Engg. Services (Hrs.)              Direct Labour (Hrs.)               Admin (Hrs.)

Transistor        1.0                                           10.0                                         2.0

Resistor           1.0                                           4.0                                           2.0

CRT                1.0                                           5.0                                           6.0

Total                100.0                                       600.0                                       300.0

Solve by simplex.

 

  1. ABC Ltd. Can make two products P1 & P2.

 

Product Casting Hrs. / unit Finishing Hrs. / unit Profit / unit Max. Sales (units)
P1 2 3 7
P2 1 3 4 200
  Max. 390 Max. 810    

 

Find optimal product mix to maximize profit by simplex method.

 

  1. Max. Z = 20 x1 + 6 x2 + 8 x3

 

Subject to constraints –

8 x1 + 2 x2 +3 x3 <= 250

4 x1 + 3 x2 <= 150

2×1 + x3 <= 50

x1, x2, x3 >= 0

 

  1. Max. Z = 10 x1 + 6 x2 + 6 x3

 

Subject to constraints –

3 x1 + 2 x2 +2 x3 <= 250

2 x1 + 3 x2 +3 x3 <= 270

x1 <= 60

x1, x2, x3 >= 0

 

  1. Max. Z = 30 x1 + 25 x2

 

Subject to constraints –

x1 + 2 x2 <= 80

3 x1 + 2 x2 <= 120

x1, x2 >= 0

 

  1. Minimum requirements of diet are 4000 units of vitamins, 50 units of minerals & 1400 units of calories per day. Two products A & B are available at a price of Rs. 4 & Rs. 2 per unit. 1 unit of product A contains 200 units of vitamins, 1 unit of minerals & 40 calories. 1 unit of product B contains 100 units of vitamins, 2 units of minerals & 40 calories. Find optimal product mix to minimize total cost by simplex method.

 

  1. Minimize Z = 27 x1 + 30 x2

 

Subject to constraints –

(2/3) x1 + (1/2) x2 >= 10

(1/3) x1 + (1/2) x2 >= 6

x1, x2 >= 0

 

  1. Product A offers a profit of Rs. 25/- per unit and product B yields a profit of Rs. 40/- per unit. To manufacture the products – leather, wood and glue are required in the amount shown below:

 

Product Resources for one unit
Leather

(in Kg.)

Wood

(in sq. Mts.)

Glue

(in lts.)

A 0.50 4 0.2
B 0.25 7 0.2

 

Available resources include 2100kgs. Of leathers, 28000 sq. metres of wood and 1,400 litres of glue;

  • State the objective function and constraints in mathematical form.
  • Find the optimum solution.
  • Which resources are fully consumed? How much of each resource remains unutilized?
  • What are the shadow prices of resources?

 

  1. Using simplex method solve the following LPP & explain the solution.

 

Max. Z = 6 x1 – 3 x2

 

Subject to constraints –

 

2 x1 – x2 <= 2

x1 <= 4

x1, x2 >= 0

 

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

 

Product Mix X1 X2 S1 S2 Quantity

(b1)

 

Ci Xi
5

 

X2 1 1 1 0 10
0

 

S2 1 0 -1 1 2
  Ci 4 5 0 0  
Zi 5 5 5 0
Ci – Zi -1 0 -5 0

 

Answer the following questions, giving reasons in brief:

  • Is there solution optimal?
  • Are there more than one optimal solution?
  • Is the solution degenerate?
  • Is the solution feasible?
  • If S1 is 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?
  • How many units of two products X1 and X2 are being produced and what is the total profit?
  • Machine A has to be shut down for repairs for 2 hours next week. What will be the effect on profits?
  • How much would you prepare to pay for another hour (per week) of capacity each on machine A and machine B?
  • 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 Rs. 3/- per unit.  Do you think it is advisable to introduce this product?

 

  1. Zigma Electronics produces two models of electronic products using Resistors, Capacitors and Chips. The following table gives the entire Technological and other details in this regard:
Resource Unit resource requirement Maximum Availability
Model 1 Model 2
Resistor

Capacitor

Chips

2

2

0

3

1

4

1200

1000

800

Unit profit (Rs). 3 4  

 

After formulating the above problem as a Linear Programming Problem the following optimal Simplex Solution table is obtained.

 

 

Profit

Coefficient

Basis

Variables

Solution

Values

C: 3 4 0 0 0
X: X1 X2 S1 S2 S3
C X B
3

0

4

X1

S3

X2

450

400

100

  1

0

0

0

0

1

-1/4

-2

1/3

¾

2

-1/2

0

1

0

Z=Rs.1750 Z   3 4 5/4 0.25 0  
= C-Z   0 0 -5/4 -0.25 0

 

  • Determine the value of each resource.
  • In terms of optimal profit, determine the worth of one Resistor, one Capacitor and one Chip.
  • Determine the range of the applicability of the shadow prices (duel prices) for each resource.
  • If the available number of chips is reduced to 350 units, will you be able to determine the new optimum solution directly from the given information? Explain.

 

  1. Standard Manufacturers produce three products P, Q, and R which generate profits of Rs. 20/-, Rs. 12/-, and Rs. 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 1300, 900 and 400 respectively. One of the Simplex Method Solution 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
= C-Z 0 0 -16/5 0 -4/5 -44/5

 

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

  • Which Machine is not fully utilized? If the balance working hrs. of this machine are shifted to M2, what will be the effect on the solution?
    • Retaining the optimality, find the range of working hrs. of the third Machine..
    • Within what range of profit of each product, the solution will remain optimal?
    • Keeping the Shadow Prices intact, find the range for the working hours of M2.
    • Without altering the optimality, is it possible to reduce the availability of the working hours of the M2 to 200 hours?
    • 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?

 

  1. An engineering company BMS Ltd. Produces three products A, B, and C using three machines M1, M2 and M3. The resource constraints on M1, M2, and M3 are 100, 40, and 60 hours respectively. The profits earned by the products. A, B, and C are Rs. 2, Rs.5 and Rs.8 per unit respectively.  A simplex optimal solution to maximize the profit is given below where x1, x2 and x3 are quantities of products A,B, and C produced by the company and s1, s2 and s3 represent the slack in the resources M1, M2, and M3.  Study the solution given below and answer the following questions.
C X

Variables in the basis

x1 x2 x3 s1 s2 s3 b

 

5 X2 1/3 1 0 1/6 -1/3 0 8/3
8 X3 5/6 0 1 2/3 0 56/3
0 S3 7/3 0 0 1/2 -1/3 1 44/3
  = C-Z -19/3 0 0 -1/6 -1/3 0  
  • Indicate the shadow price of each resource. Which of the resources are abundant and which are scare?
  • What profit margin for product A do you expect the marketing department to secure if it is to be produced, and justify your advice?
  • Within what range, the profit of product B can change for the above solution to remain optimal?
  • How would an increase of 10 hours in the resource M2 affect the optimality?
  • If the company BMS Ltd. Wishes to raise production which of the three resources should be given priority for enhancement?

 

  1. A business problem is formulated and expressed below as an LPP. (Profit is in Rs. and Resources are in units).

Objective Function: Maximize Z = 80 X1 + 100 X2

X1 + 2X2 < 750           ……… (Resource 1)

5X1 + 4X2 < 1800       ……… (Resource 2)

3X1 + X2 < 900           ……… (Resource 3)

X1 X2 > 0

Simplex algorithm of LPP, applied to the above problem yielded following solutions.

Basis           Bi
Cb Xb X1 X2 S1 S2 S3
100 X2 0 1 5/6 -1/6 0 300
80 X1 1 0 -2/3 1/3 0 120
0 S3 0 0 7/6 -5/6 1 240
  Cj 80 100 0 0 0  
  (= Cj – Zj ) 0 0 30 0 0  
  • Answer the following questions with justification.
    • Is the solution optimal and unique?
    • Is the above solution infeasible?
    • What is the maximum profit as per optimal solution?
    • Which resources are abundant and which are scare as per optimal solution?
  • Find out the range of coefficient of X1 in the objective function for which the above solution remains optimal.

Can you obtain the solution values of basic variables from the optimal solution when resource constraint (1) Changes to 800 units? If yes, find the new values of the basic variables.

 

Source:- Vipin Saboo Tutorials

For any further clarifications, please feel free to contact Prof Vipin Saboo on 9820779873

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Education Qualification: BMS- N M College (University Rank Holder) PGDBM- Sydenham College M Com- College topper Mr Vipin Saboo has been associated with the following institutes as a visiting faculty Lords college, Malad Patkar College, Goregoan Saraf college, Malad Dalmia college, Malad St Andrews College, Bandra Wilson College, Grant Road Thakur college, Kandivili L N College, Kandivili N K College, Malad Dhanukar College, Vile Parle St Xaviers College, Marine Lines Shroff College, Kandivili KES College, Khar Mr.Vipin Saboo also has more than 5 years of industry expertise with corporate like CRISIL, Motilal Oswal Investment Banking and Yes Bank. Mr. Saboo has also published a text book on Logistics and Supply Chain Management for TYBMS Students.
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