The solution to LPP is based on a number of deterministic assumptions like the prices are known exactlyand are fixed, resources are known with certainty and time needed to manufacture/assemble/produce a product is fixed. In real life situations, which are dynamic and changing, the effect of variation of these variables must be studied and understood. This process of knowing the impact of variables on the outcome of optimal result is known as sensitivity analysis of linear programming problems. Let us say, for example, that if originally we had assumed the cost per unit to be Rs. 10 but it turns out be Rs. 11, how will the final profit and solution mix vary. Also, if we start with the assumption of certain fixed resources like man hours or machine hours and as we proceed we realise the availability can be improved, how will this change our optimal solution.
Sensitivity analysis can be used to study the impact of changes in
a) Addition or deletion of variables initially selected.
b) Change in the cost or price of the product under consideration.
c) Increase or decrease in the resources.
Sensitivity Analysis uses the following two approaches:
a) It involves solving the entire problem by trial and error approach and involves very cumbersome calculations.
Every time data of a variable is changed, it becomes another set of the problem and has to be solved independently.
b) The last simplex table may be investigated. This reduces completion and computations considerably.
Limitations of Sensitivity Analysis
Sensitivity analysis does take into account the uncertainty element, yet, it suffers from the following limitations.
 Only one variable can be taken into account at one time. Hence, the impact of many variables changing cannot be considered simultaneously.
 It suffers from the linearity limitations as only linear relationship between the variable is considered.
 The extent of uncertainty cannot be studied.
 As the result can be judged by individual analysts depending upon their skills and experience, it is to that extent subjective in nature.
REVIEW
 1. Explain the following terms.
 Basic feasible solution.
 Optimal solution.
 2. Explain step by step the method used in solving LPP using simplex method.
 3. Explain the use of slack, surplus and artificial variables when are these used and why?
 4. How are the key column, key row and key element (number) selected?
 5. Explain the use of simplex method in solving the maximisation and minimisation problems. What are the differences in the approach?
 6. What do you understand by a redundant constraint? Do these constraints influence analysis and final solution of a LPP?
 7. What are the limitations of LPP. Give examples to support your argument.
 8. What do the coefficient in a simplex table represent? Why is it necessary to compute a new set of coefficients for each table in the analysis.
 9. Explain the terms decision variables, basic variables, entering and departing variables.
 10. Write a detailed note on the sensitivity analysis.
 11. Maximise x_{1} + 2x_{2} + 3x_{3} x_{4}
Subject to x_{1} + 2x_{2} + 3x_{3} = 15
2x_{1} + x_{2} + 5x_{3} = 20
x_{1} +2x_{2}+x_{3}+x_{4} = 10
using simplex method.
 1. Maximise Z=x_{1}+2x_{2}+3x_{3}– x_{4}
Subject to x_{1} + 2x_{2} + 3x_{3} = 15
2x_{1} + x_{2} + 5x_{3} = 20
x_{1}+2x_{2}+x_{3}+X_{4} =10
x_{1}, x_{2}, x_{3} , x_{4} ≥ 0
using simplex method.
 1. The ABC company makes two products P_{1} and P_{2} with contribution per unit of Rs, 15 and Rs. 11 respectively. Each of the products is made from two raw materials A and B. P_{1} and P_{2} require the raw material in the following amounts.
Kgs. 



A 
B 
Product 
P_{1} 
4 
3 
P_{2} 
2 
1 

Availability in Kgs 
400 
500 
Find the optimum product mix for maximum profit.
 2. ABC manufacturing company makes three products x_{1}, x_{2} and x_{3} with contribution per unit to profit Rs. 2, Rs. 4 and Rs. 3 respectively. Each of three products passes through three centres as part of production process. Time required in each centre to procedure one unit of each product is as given below.
Hours per unit 

Product 
Centre 1 
Centre 2 
Centre 3 
X_{1} 
3 
2 
1 
X_{2} 
4 
1 
3 
X_{3} 
2 
2 
2 
Time available (Hours) 
60 
40 
80 
Determine the optimal mix for next week production.
 1. Explain the simplex method by carrying out the iteration in the followllng problem.
Maximise Z = 5x_{1} + 2x_{2} + 3x_{3} – x_{4} + x_{5}.
Subject to x_{1} + 2x_{2} + 3x_{3} + x_{4} = 8
3x_{1} +4x_{2}+ x_{3}+ x_{5} =7
x_{1} to x_{5} ≥ 0
 1. A firm manufactures three products A, B and C. The profits are Rs. 3, Rs. 2 and Rs. 4 respectively.
The firm has two machines and below is the required processing time in minutes for each machine on each product.
Product 

Machine 
A 
B 
C 
G 
4 
3 
5 
H 
3 
2 
4 
Machines G and H have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A’s, 200B’s and 50C’s but no more than 150A’s. Set up an LP problem to maximise profit. Find solution.
1. For the following production given in the table, formulate the problem as linear programming and solve.
Products 
Machine Time (hours) 
Profit per product (Rs.) 

A 
B 
C 


P 
8 
4 
2 
20 
Q 
2 
3 
0 
6 
R 
3 
0 
1 
8 
Available M/C hours per week 
250 
150 
50 
1. A plant makes two products A and B which are routed through four process centres shown by the solid lines in figure shown below.
Given the information below, how should production be scheduled so as to maximise profits. Centres 1 and 4 run up to 16 hours a day, centres 2 and 3 run up to 12 hours a day. The shipping facility limit the daily output of A and B to a total of 2500 litres.
Product 
Centre 
Input (Litres/hour) 
% Recovery 
Running cost per year 
A 
1 
300 
90 
150 
2 (First pass) 
450 
95 
200 

4 
250 
85 
180 

2 (Second pass) 
400 
80 
220 

3 
350 
75 
250 

B 
1 
500 
90 
300 
3 
480 
85 
250 

4 
400 
80 
240 

Product 
Rate material cost/year 
Sale price per finished litre 
Maximum daily sales litres of finished product 

A 
5 
20 
1700 

B 
6 
18 
1500 
Formulate the problem as an LP problem. Do not solve.
 1. A firm makes two types of furniture chairs and tables. The contributions for each product as calculated by accounting department are Rs. 20 per chair unit and Rs. 30 per table. Both products are processed on three machines M_{1}, M_{2} and M_{3}. The time required by each product and total time available per week on each machine is as follows.
Machine 
Chair 
Table 
Available Time (hours) 
M_{1} 
3 
3 
36 
M_{2} 
5 
2 
50 
M_{3} 
2 
6 
60 
How should the manufacture schedule his production in order to maximise contribution?
 2. The products A, B and C are produced on three machines centres X, Y and Z. Each product involves operations on each of the machine centres. The time required for each operation for unit amount of each product is given below.
Machine centres 

X 
Y 
Z 

Products 
A 
10 
7 
2 
B 
2 
3 
4 

C 
1 
2 
1 

(Time in hours) 
There are 100, 77 and 80 hours available at machine centres X, Y, Z respectively.
The profit per unit of A, B and C is Rs. 12, Rs 3 and Rs. 1 respectively. Formulate the problem as LPP (Linear Programming Problem) and find the profit maximisation product mix.
 3. A company is engaged in producing three products viz A, B and C. The following data are available.
Product 
A 
B 
C 
Sale price (per unit) 
10 
12 
15 
Cost (per unit) 
6 
9 
10 
The wholesaler who is responsible for selling to the customer is to be paid Rs. 150 per day irrespective of the quantities sold of each of the products.
The products are processed in three different operations. The time (hours) required to produce one product in each of the operations and the daily capacity (hours) available for each operation centre are given given below.
Operation 
Product 
Daily capacity 

A 
B 
C 

1 
2 
3 
2 
400 
2 
3 
2 
2 
350 
3 
1 
4 
2 
300 
What product would yield maximum profit and how much?
 1. A pharmaceutical company has 100 kgs of A, 180 kgs of B, and 120 kgs of C available per month. They can use these materials to make three basic pharmaceutical products, namely 5 – 10 – 5, 5 – 5 – 10 and 2 5 – 10 where the numbers in each case represent the percentage by weight of A, Band C respectively in each of the products. The costs of these raw materials are given below
Ingredients 
Cost per kg (Rs.) 
A 
80 
B 
20 
C 
50 
Inert ingredients 
20 
Selling prices of these products are Rs. 40·50, Rs. 43 and Rs. 45 per kg respectively. There is a, capacity restriction of the company for product 5 – 10 5, so as they cannot produce more than 30 kg per month. Determine how much of each of the product they should produce in order to maximise their monthly profit.
 2. Noap’s Boats makes three different kinds of boats, all can be made profitably in this company. But, the company monthly production is constrained by the limited amount of labour, wood and screws available; each month. The director will choose the combination of boats that maximises his revenue in view of the information given in the following table.
Input  Row boat  Comoe  Kayak  Monthly availability 
Labour (Hours)  12  7  9  1260 hours 
Wood (Board feet)  22  18  16  19008 board feet 
Selling price (Rs.)  4000  2000  5000 
(a) Formulate the above as a LPP.
(b) Solve it by the simplex method from the optimal table of the solved LPP, answer the following questions.
 i. How many boats of each type will be produced and what will be the resulting revenue?
 ii. Which, if any of the resources are not fully utilised ? If so how much of spore capacity is left?
 iii. How much of wood will be used to make all of the boats given in the optimal solution?
 1. An agriculturist has a farm with 125 acres. He produces Radish; Mutter and Potato. Whatever he raises is fully sold in the market. He gets Rs. 5 for radish per kg, Rs. 4 for mutter per kg, and Rs. 5 for potatoes per kg. The average yield is 1500 kg of radish per acre, 1800 kg of mutter per acre and 1200 kg of potato per acre. To produce each 100 kg of radish and mutter and to produce each 80 kg of potatoes, a sum of Rs. 12.50 has to be used for manure. Labour required for each acre to raise the crop is 6 man days for radish and potatoes each and 5 man days for mutter. A total of 500 man days of labour at rate of Rs. 40 per man day are available.
Formulate items as a LP model to maximise the agriculturist’s total profit.
 2. The cost of production per unit of products A, B and Care Rs. 10, Rs. 15 and Rs. 18 respectively and their selling prices per unit are Rs. 16, Rs. 24 and Rs. 28 respectively but the agency commission of 25% on selling price should be borne by the company. These products are produced in 3 operations and their installed capacity with effective utilisation and working hours are given below.
Operations 
No. of machines installed 
No. of working hours per day 
Effective utilisation of machines 
I 
3 
15 
80% 
II 
4 
16 
75% 
III 
2 
24 
75% 
The effective time required to produce one product in each of these operations are given below.
Operations 
Product A 
Product B 
Product C 
I II III 
4 1 2 
3 5 4 
2 3 5 
Demand is no constraint for all the three products. Find the optimum product mix per day that gives the maximum profit.
 3. A factory works 8 hours a day, producing three products, viz A, Band C. Each of these products is processed in three different operations viz 1, 2 and 3. The processing time in minutes for each of these, products in each of the operations are given below along with utilisation of the process and the cost and price rupees for each of these three products which have unlimited demand.
Product 
Processing time (minutes) 
Cost per unit (Rs.) 
Price per unit (Rs.) 

1 
2 
3 

A B C 
4 2 3 
3 1 4 
1 4 5 
10 8 5 
16 12 10 
Utilisation 
80% 
70% 
90% 
 Determine the optimal product mix using simplex method.
 Give interpretation for the value obtained in the final simplex table.
 4. A resourceful home decorator manufactures two types of lamps say A and B. Both the lamps go through two technicals, first a cutter and second, a finisher.
Lamp A requires 2 hours of cutter’s time and I hour of finisher’s time. Lamp B requires 1 hour of cutter’s and 2 hours of finisher’s time. The cutter has 104 hours and finisher 76 hours of available time each % month. Profit on one lamp A is Rs 6.00 and on one lamp B is Rs. 11·00. Assuming that he can sell all that he produces, how many of each type lamp should be manufactured to obtain the best return?
 1. A product is manufactured by blending three different raw materials. The finished product should meet certain requirements. Given the following data, what is your recommendation with regard to quantity, for raw materials to be blended, which will meet the quality requirement with minimum cost?
Quantity characteristics 
Contribution of quality by each unit of raw materials 
Minimum Quality Requirement 

A 
B 
C 

1 2 3 
3 5 1 
0 1 2 
1 2 0 
10 15 8 
Cost of raw materials per unit in Rs. 
2 
5 
3 
 2. A TV company operates two assembly lines. Line I and Line II. Each line is used to assemble the components of three types of TV’s, colour, standard, economy. The expected daily production on each line is as follows.
TV model 
LineI 
LineII 
Colour Standard Economy 
3 1 2 
1 1 6 
The daily running cost for two lines average Rs. 6000 for line – [and Rs. 4000 for line [I. It is given that the company must produce at least 24 colour, 16 standard and 48 economy TV sets for which an order is pending. You are required to formulate the above problem as LPP model taking the objective function a minimisation of total cost. Also, determine the number of days that the two lines should be seen to meet the requirement.
 1. The owner of fancy goods shop is interested to determine how many advertisements to release in magazines A, B and C. His main purpose is to advertise in such a way that the total exposure to principal buyers of his goods is maximised. Percentage of readers for each magazine are known Exposure in any particular magazine is the number of advertisements released multiplied by the number of principal buyers. The following data are available.
Particulars 
Magazines 

A 
B 
C 

ReadersPrincipal buyersCost per advt. 
1.0 lakh 20% Rs. 8,000 
0.6 lakhs 15% Rs. 6,000 
0.4 lakhs 8% Rs.5,000 
The budgeted amount is at the most Rs. 1.0 lakh for the advertisement. The owner has already decided that magazine A should have no more than 15 advertisement and B and C each gets at least 8 advertisements, Formulate the Linear Programming Problem model and solve it.
 1. The HandyDandy company wishes to schedule the production of kitchen appliances which requires two resources, labour and material. The company is considering three different models and its production engineering department has furnished the following data

Model 


A 
B 
C 
Labour (Hours/unit) 
7 
3 
6 
Material (kg/unit) 
4 
4 
5 
Profit (Rs./unit) 
4 
2 
3 
The supply of raw material is restricted to 200 kg per day. The daily availability of man power is 150 hours. Formulate the Linear Programming model to determine the daily production rate of the various models in order to maximise the total profit.
 1. A firm manufactures two qualities of tweed. A yard of quality A tweed requires 8 ozs of grey wool 2·5 oz of red wool and 2 oz of green wool, one yard of tweed B is made up of 10 oz of grey wool, 1 oz of red wool and 4 oz of green wool. The availability of wool in given periods are 5,000 lbs of grey, 250 lbs red and 1875 lbs of green. Both tweeds can be produced on the same machines and both can be woven the rate of 12 yards per hour. A total of 750 machine hours are available in given period. The contribution towards profit is Rs. 2 per yard of tweed. A and Rs.’ 4 for tweed B. Given that the company has firm order for and is obliged to produce, at least 300 yards of tweed ; what is the optimal production policy for the firm?
 2. Degeneracy in L.P. problem
 3. Slack and surplus variables.
 4. Convex set.
 5. Convex and Concave sets.
 6. Slack variables.
 7. Explain Degeneracy in L.P.P.
 8. Define Degeneracy
 9. Degenerate solution of L.P.P.
 10. (i) Unbounded solution (ii) Key element.
 11. Explain briefly simplex method of solving a Linear Programming problem. Why is simplex method considered superior to graphic method?
 12. Define slack, surplus and artificial variables in a LPP and explain how they help in, finding a basic feasible solution to LPP using Simplex Method.
 13. Give sequence of steps in Simplex Method for solving a linear problem.
 14. Why is the simplex method a better technique than a graphical approach for most real cases? Discuss the advantages and limitations of Linear Programming.
 15. Briefly describe the graphic and simplex methods of solving a linear programming problem. Why is simplex method considered superior to graphic method.
 16. A farm is engaged in breeding pigs. This pigs are fed in various products grown on the farm. Because of the need to ensure certain nutrient constituents, it is necessary to buy additional one or two products A and B. The nutrient constituents (vitamin a proteins) in each unit of the products are given.
Nutrient 
Nutrient Constituents in the products 
Minimum Account on Nutrient 

A 
B 

1 
36 
6 
108 
2 
3 
12 
36 
3 
20 
10 
100 
Product A costs Rs. 20 per unit and ‘product B costs Rs. 40 per unit. How much of products A and B should be purchased at lowest possible cost as to provide the pigs nutrients not less than given in the table?
 1. The HandyDandy company wishes to schedule the production of kitchen appliances which requires two resources labour and material. The company is considering three different models and its production engineering department has furnished the following data :
A 
B 
C 

Labour (Hours/unit) 
7 
3 
6 
Material (Kg/unit) 
4 
4 
5 
Profit (Rs./unit) 
4 
2 
3 
The supply of the raw material is restricted to 200 kg. per day. The daily availability of the manpower is 150 hours. Formulate the Linear Programming model to determine the daily production rate of the various models in order to maximize the total profit.
1. A firm manufactures two qualities of tweed. One yard of a tweed A requires 8 ozs. of gray wool, 2.5 ozs of red wool, and 2 ozs. of green wool, yard of tweed B is made up 10 ozs. of gray wool, I oz. of red wool and 4 ozs. of green wool. The availability of wool in a given period are 5,000 lbs. of gray, 1,250 lbs of red 1,875 obs. of green. Both tweeds can be produced on the same machines and both can be woven at the rate of 12 yards per hour. A total of 750 machine hours are available in the given period. The contribution towards profit is Rs. 2 per yard of tweed A and Rs. 4 of tweed B. Given that the company has firm order for, and is obliged to produce, at least 3,000 yards of tweed A, what is the optimal production policy for firm? (Note 1 lbs. = 16 ozs.)
1. A company produces the products P, Q and R from three raw materials A, B and C One unit of product P requires two units of A and three units of B. One unit or, requires two units of B and five units of C and one unit of product R needs three unit, of A, 2 units of Band 4 units of C. The company has 8 units of A, 10 units of B and 15 units of C available. Profit per unit of products P, Q and R are Rs. 3, Rs. 5 and Rs. 4 respectively. How many units of each product should be produced to maximize profits?
1. ABC Manufacturing company makes three products X_{1}, X_{2} and X_{3} with contribution per unit to profit Rs. 2, Rs 4. and Rs. 3 respectively. Each of these products passes through three centres as part of production process. Time required in each centre to produce one unit of each product is as given below:
Hours per unit 

Product 
Centre I 
Centre 2 
Centre 
X_{1} 
3 
2 
1 
X_{2} 
4 
1 
3 
X_{3} 
2 
2 
2 
Time Available (Hours) 
60 
40 
80 
Determine the optimal product mix for next week product ion.
 1. The ABC company makes two products P_{1} and P_{2} with contribution per unit of Rs. 15 and Rs. 11 respectively. Each of the product is made from two raw materials A and B P_{1} and P_{2} require the raw materials in the following amounts.
Kgs 

A 
B 

Product 
P_{1} 
4 
3 
P_{2} 
2 
1 

Availability in Kgs. 
400 
500 
Find the optimum product mix for maximum profit.
 2. ABC manufacturing company makes three products X_{1}, X_{2} and X3 with contribution per unit to profit Rs. 2 Rs. 4 and Rs. 3 respectively. Each of these products passes through three centers as part of production process. Time required in each centre to produce one unit of each product is as given below:
Hours per unit 

Product 
Centre 1 
Centre 2 
Centre 3 
X_{1} 
3 
2 
1 
X_{2} 
4 
1 
3 
X_{3} 
2 
2 
2 
Time Available (Hours) 
60 
40 
80 
Find the optimum product mix.
 1. A firm has 240, 370 and 280 kg. of wood, plastic and steel respectively. The firm produces two products A and B. Each unit of A requires 1, 3 and 2 kg. of wood, plastic and steel respectively. The corresponding requirement for each unit of B is 3, 4 and 1 kg. respectively. If A sells for Rs. 4 and B sells for Rs. 6 per unit, then what product mix should the firm produce in order to have max. gross income, ? Formulate this as a LPP and solve.
 2. Discuss the assumption of proportionality, additively, continuity, certainty and finite choices in the context of LPPs.
 1. Degeneracy in L.P. problem
 2. Degeneracy in LPP.
 3. Slack and surplus variables
 4. Convex set
 5. (i) Convex and Concave sets (ii) Degeneracy in LPP
 6. Why is the simplex method a better technique than the graphical approach for most real cases? Discuss the advantage and limitations of Linear Programming.
 7. Explain briefly simplex method of solving a Linear Programming problem, Why is simplex method considered superior to graphic method?
 8. Explain briefly two phase simplex method.
 9. Explain briefly simplex method of solving a Linear Programming problem. How it is better than graphic method?
 10. Explain briefly two phase method of solving a Linear Programming program.
 11. Illustrate the following and their importance in dealing with linear programming problem.
 Slack I Surplus variables
 Artificial Variables
 Basic Variables and
 Non basic variables
 12. A company produces the products P, Q and R from the three raw material A, B and C One unit of product P requires two units of A and three units of B. One unit of Q requires two units of B and five units of C and one unit. of product R needs three units of A, 2 units of Band 4 units of C. The company has 8 units of A, 10 units of B and 15 units of C available. Profit per unit of products P, Q and R are Rs. 3, Rs. 5 and Rs. 4 respectively. How many units of each product should be produced to maximize profit?
 1. The ABC company makes two products P_{1} and P_{2} with contribution per unit of Rs. 15 and Rs. 11 respectively. Each of the products is made from two raw materials A and B. P_{1} and P_{2} require the raw material in the following amounts Kgs.


A 
B 
P_{1} 
4 
3 

P_{2} 
2 
1 

Availability in Kgs 

400 
500 
Find the optimum product mix for maximum profit.
 2. ABC manufacturing company makes three products X_{1}, X_{2} and X_{3} with contribution per unit to profit Rs. 2 Rs. 3 and Rs. 3 respectively. Each of these products passes through three centres as part of production process. Time required in each centre to produce one unit of each product is as given below:
Hours per unit 

Product 
Centre 1 
Centre 2 
Centre 
X_{1} 
3 
2 
1 
X_{2} 
4 
1 
3 
X_{3} 
2 
2 
2 
Time available (Hours) 
60 
40 
80 
Determine the optimal product mix for next week production.
 1. A firm manufactures thee products, A, B and C. The profits are Rs. 3, Rs. 2 and Rs. 4 respectively. The firm has two machines and below is the required processing time is minutes for each machine on each product.
Product 

Machine 
A 
B 
C 
G 
4 
3 
5 
H 
3 
2 
4 
Machine G and H have 2,000 and 2,500 machine minutes respectively. The firm must manufacture 100 A’s, 200 B’s and 50 C’s, but no more than ISO A’s Set up an L.p. problem to maximize profit. Find solution.
 1. For the following information given in the table, formulate the problem as a linear programming problem and solve.
Products 
Machining time Under M/cs (in hours) 
Profit per Product (Rs.) 

A 
B 
C 

P Q R Available M/c hours per week 
8 2 3 250 
4 3 0 150 
2 0 1 50 
20 6 8 
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