PROBLEM FORMULATION Operation Research (2171901) Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1 Chapter 2. LINEAR PROGRAMMING 1. Production Allocation Problem A firm produces three products. These products are processed on three different machines. The time required to manufacture one unit of each of the three products and the daily capacity of the three machines are given in the table below: Machine Time per unit (minutes) Machine capacity (minutes/day) Product 1 Product 2 Product 3 M1 2 3 2 440 M2 4 - 3 470 M3 2 5 - 430 It is required to determine the daily number of units to be manufactured for each product. The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed that all the amounts produced are consumed in the market. Formulate the mathematical (L.P.) model that will maximize the daily profit. 2. Diet Problem A person wants to decide the constituents of a diet which will fulfil his daily requirements of proteins, fats and carbohydrates at the minimum cost. The choice is to be made from four different types of foods. The yields per unit of these foods are given in the table below: Food Type Yield per unit Cost per unit (Rs.) Proteins Fats Carbohydrates 1 3 2 6 45 2 4 2 4 40 3 8 7 7 85 4 6 5 4 65 Minimum requirement 800 200 700 Formulate linear programming model for the problem. 3. Blending Problem A firm produces an alloy having the following specifications: I. Specific gravity ≤ 0.98 II. Chromium ≥ 8 % III. Melting point ≥ 450°C Raw materials A, B and C having the properties shown in the table can be used to make the alloy. Property Properties of raw material A B C Specific gravity 0.92 0.97 1.04 Chromium 7 % 13 % 16 % Melting point 440°C 490°C 480°C
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PROBLEM FORMULATION
Operation Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 1
Chapter 2. LINEAR PROGRAMMING
1. Production Allocation Problem
A firm produces three products. These products are processed on three different machines.
The time required to manufacture one unit of each of the three products and the daily
capacity of the three machines are given in the table below:
Machine Time per unit (minutes) Machine capacity
(minutes/day) Product 1 Product 2 Product 3
M1 2 3 2 440
M2 4 - 3 470
M3 2 5 - 430
It is required to determine the daily number of units to be manufactured for each product.
The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed
that all the amounts produced are consumed in the market. Formulate the mathematical
(L.P.) model that will maximize the daily profit.
2. Diet Problem
A person wants to decide the constituents of a diet which will fulfil his daily requirements
of proteins, fats and carbohydrates at the minimum cost. The choice is to be made from
four different types of foods. The yields per unit of these foods are given in the table below:
Food Type Yield per unit
Cost per unit (Rs.) Proteins Fats Carbohydrates
1 3 2 6 45
2 4 2 4 40
3 8 7 7 85
4 6 5 4 65
Minimum requirement
800 200 700
Formulate linear programming model for the problem.
3. Blending Problem
A firm produces an alloy having the following specifications:
I. Specific gravity ≤ 0.98
II. Chromium ≥ 8 %
III. Melting point ≥ 450°C
Raw materials A, B and C having the properties shown in the table can be used to make
the alloy.
Property Properties of raw material
A B C
Specific gravity 0.92 0.97 1.04
Chromium 7 % 13 % 16 %
Melting point 440°C 490°C 480°C
PROBLEM FORMULATION
Operation Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 2
Costs of the various raw materials per ton are: Rs. 90 for A, Rs. 280 for B and Rs. 40 for
C. Formulate the L.P. model to find the proportions in which A, B and C be used to obtain
an alloy of desired properties while the cost of raw materials is minimum.
4. Advertising Media Selection Problem
An advertising company wishes to plan its advertising strategy in three different media –
television, radio and magazines. The purpose of advertising is to reach as large a number
of potential customers as possible. Following data have been obtained from market survey:
Television Radio Magazine I Magazine II
Cost of an advertising unit Rs. 30,000 Rs. 20,000 Rs.15,000 Rs.10,000
No. of potential customers
reached per unit 2,00,000 6,00,000 1,50,000 1,00,000
No. of female customers
reached per unit 1,50,000 4,00,000 70,000 50,000
The company wants to spend not more than Rs. 4,50,000 on advertising. Following are the
further requirements that must be met:
a. At least 1million exposures take place among female customers,
b. Advertising on magazines be limited to Rs.1,50,000,
c. At least 3 advertising units be bought on magazine I and 2 units on magazine II,
d. The number of advertising units on television and radio should each be between 5
and 10.
Formulate linear programming model for the problem.
5. Inspection Problem
A company has two grades of inspectors, I and II to undertake quality control inspection.
At least 1500 pieces must be inspected in an 8-hour day. Grade I inspector can check 20
pieces in an hour with an accuracy of 96%. Grade II I inspector checks 14 pieces in an hour
with an accuracy of 92%.
Wages of grade I inspector are Rs.5 per hour while those of grade II inspector are Rs. 4 per
hour. Any error made by an inspector costs Rs. 3 to the company. If there, are in all, 10
grade I inspectors and 15 grade II inspectors in the company, find the optimal assignment
of inspectors that minimizes the daily inspection cost.
6. Product Mix Problem
A chemical company produces two products, X and Y. Each unit of product X requires 3
hours on operation I and 4 hours on operation II, while each unit of product Y requires 4
hours on operation I and 5 hours on operation II. Total available time for operations I and
II is 20 hours and 26 hours respectively. The production of each unit of product Y also
results in two units of a by-product Z at no extra cost.
Product X sells at profit of Rs. 10/unit, while Y sells at profit of Rs. 20/unit. By-product Z
brings a unit profit of Rs. 6 if sold; in case it cannot be sold, the destruction cost is Rs.
4/unit. Forecasts indicate that not more than 5 units of Z can be sold. Formulate the L.P.
model to determine the quantities of X and Y to be produced, keeping Z in mind, so that
the profit earned is maximum.
PROBLEM FORMULATION
Operation Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 3
7. Trim Loss Problem
A paper mill produces rolls of paper used in making cash resisters. Each roll of paper is
100m in length and can be use in widths of 3, 4, 6 and 10cm. The company’s production
process results in rolls that are 24cm in width. Thus the company must cut its 24cm roll to
the desired widths. It has six basic cutting alternatives as follows:
Cutting alternatives Width of rolls (cm)
Waste (cm) 3 4 6 10
1 4 3 - - -
2 - 3 2 - -
3 1 1 1 1 1
4 - - 2 1 2
5 - 4 1 - 2
6 3 2 1 - 1
The minimum demand for the four rolls is as follows:
Roll width (cm) Demand
2 2000 4 3600
6 1600 10 500
The paper mill wishes to minimize the waste resulting from trimming to size. Formulate
the L.P. model.
CLASS TUTORIAL – GRAPHICAL METHOD
Operations Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 1
1. A firm manufactures two products A & B on which the profits earned per unit are Rs. 3 and
Rs. 4 respectively. Each product is processed on two machines M1 and M2. Product A
requires one minute of processing time on M1 and two minute on M2, while B requires one
minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs. 30
minutes, while machine M2 is available for 10 hrs. during any working day. Find the number
of units of products A and B to be manufactured to get maximum profit.
2. Mohan-Meakins Breveries Ltd. has two bottling plants, one located at Solan and the other at
Mohan Nagar. Each plant produces three drinks, whiskey, beer and fruit juices named A, B
and C respectively. The number of bottles produced per day are as follows:
Plant at
Solan (S) Mohan Nagar (M)
Whiskey, A 1500 1500
Beer, B 3000 1000
Fruit juices, C 2000 5000
A market survey indicates that during the month of April, there will be a demand of 20,000
bottles of whiskey, 40,000 bottles of beer and 44,000 bottles of fruit juices. The operating
costs per day for plants at Solan and Mohan Nagar are 600 and 400 monetary units. For how
many days each plant be run in April so as to minimize the production cost, while still
meeting the market demand?
3. Find the minimum value of
𝑍 = −𝑥1 + 2𝑥2,
Subject to,
−𝑥1 + 3𝑥2 ≤ 10,
𝑥1 + 𝑥2 ≤ 6,
𝑥1 − 𝑥2 ≤ 2,
𝑥1, 𝑥2 ≥ 0.
4. The standard weight of a special purpose brick is 5 kg and it contains two basic ingredients
B1 and B2. B1 costs Rs. 5/kg and B2 costs Rs. 8/kg. Strength considerations dictate that the
brick contains not more than 4 kg of B1 and a minimum of 2 kg of B2. Since the demand for
the product is likely to be related to the price of the brick, find graphically the minimum cost
of the brick satisfying the above conditions.
5. A firm uses lathes, milling machines and grinding machines to produce two machine parts.
Following table represents the machining times required for each part, the machining times
available on different machines and the profit on each machine part.
Type of machine
Machining time required for the
machine part (minutes)
Maximum time
available per week
(minutes) I II
Lathes 12 6 3000
Milling machines 4 10 2000
CLASS TUTORIAL – GRAPHICAL METHOD
Operations Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 2
Grinding machines 2 3 900
Profit per unit Rs. 40 Rs. 100
Find the no. of parts I and II to be manufactured per week to maximize the profit.
6. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 5𝑥1 + 4𝑥2,
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 𝑥1 − 2𝑥2 ≤ 1,
𝑥1 + 2𝑥2 ≥ 3,
𝑥1, 𝑥2 ≥ 0.
7. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 3𝑥 + 2𝑦,
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, −2𝑥 + 3𝑦 ≤ 9,
3𝑥 − 2𝑦 ≤ −20,
𝑥, 𝑦 ≥ 0.
8. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 3𝑥1 + 4𝑥2,
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 𝑥1 − 𝑥2 ≥ 0,
2.5𝑥1 − 𝑥2 ≤ −3,
𝑥1, 𝑥2 ≥ 0.
9. 𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑍 = 5𝑥1 + 8𝑥2,
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜, 3𝑥1 + 5𝑥2 = 18,
5𝑥1 + 3𝑥2 = 14,
𝑥1, 𝑥2 ≥ 0.
CLASS TUTORIAL – LINEAR PROGRAMMING
Operations Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 1
1. Solve the first example of graphical method by the Simplex method.
2. Use Simplex method to solve the following linear programming problem:
Maximize 𝑍 = 2𝑥1 + 5𝑥2, Subject to,
𝑥1 + 4𝑥2 ≤ 24, 3𝑥1 + 𝑥2 ≤ 21,
𝑥1 + 𝑥2 ≤ 9, 𝑥1, 𝑥2 ≥ 0.
3. A firm produces three products. These products are processed on three different machines.
The time required to manufacture one unit of each of the three products and the daily
capacity of the three machines are given in the table below:
Machine Time per unit (minutes) Machine capacity
(minutes/day) Product 1 Product 2 Product 3
M1 2 3 2 440
M2 4 - 3 470
M3 2 5 - 430
It is required to determine the daily number of units to be manufactured for each product.
The profit per unit for product 1, 2 and 3 is Rs. 4, Rs. 3 and Rs. 6 respectively. It is assumed
that all the amounts produced are consumed in the market. Formulate the mathematical (L.P.)
model that will maximize the daily profit and solve example by Simplex method.
4. Use Simplex method to solve the following linear programming problem:
Minimize 𝑍 = 𝑥1 − 3𝑥2 + 3𝑥3, Subject to,
3𝑥1 − 𝑥2 + 2𝑥3 ≤ 7, 2𝑥1 + 4𝑥2 ≥ −12,
−4𝑥1 + 3𝑥2 + 8𝑥3 ≤ 10, 𝑥1, 𝑥2, 𝑥3 ≥ 0.
5. A food processing company produces three canned fruit products mixed fruit fruit
cocktail and fruit delight The main ingredients in each product are pears and peaches Each
product is produced in lots and must go through three processes, mixing, canning
and packaging The resource requirement for each product and each process are shown in the
following L.P. formulation
Maximize Z= 10x1 + 6x2 + 8x3, (profit., Rs)
subject to 20x1+10x2+16x3 ≤ 320 (pears, kg)
10x1+ 20x2+ 16x3 ≤ 400, (peaches, kg)
x1 + 2x2 +2x3 ≤ 43, (mixing, hr)
x1 + x2 + x3 ≤ 60, (canning., hr)
CLASS TUTORIAL – LINEAR PROGRAMMING
Operations Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 2
2x1 + x2 + x3 ≤ 40, (packaging, hr)
x1, x2, x3 ≥ 0
Cj 10 6 8 0 0 0 0 0
Basic
variables
Solution
values x1 x2 x3 s1 s2 s3 s4 s5
Cb B b (=Xb)
10 x1 8 1 0 8/15 1/15 1/30 0 0 0
6 x2 16 0 1 8/15 -1/30 1/15 0 0 0
0 s3 3 0 0 2/15 0 -1/10 1 0 0
0 s4 36 0 0 -1/15 1/30 1/30 0 1 0
0 s5 8 0 0 -8/15 1/10 0 0 0 1
Zj 10 6 128/15 7/15 1/15 0 0 0
Cj - Zj 0 0 -8/15 -7/15 -1/15 0 0 0
On the basis of above information answer the following questions
(i) Is the above solution feasible?
(ii) Is the above solution optimal? If yes, what is it?
(iii) Is the above solution unbounded?
(iv) Is the above solution degenerate?
(v) Doesn’t have multiple solutions?
(vi) Determine the amount of used and unused resources.
6. Food X contains 6 units of vitamin A per gram & 7 units vitamin B per gram and costs 12
paise per gram. Food Y contains 8 units of vitamin A per gram & 12 units of vitamin B per
gram and costs 20 paise per gram. The daily minimum requirement of vitamin A and vitamin
B is 100 units and 120 units respectively. Find the minimum cost of product mix by the
simplex method.
7. Maximize Z = 3x1 - x2,
subject to 2x1 + x2 ≤ 2,
x1 + 3x2 ≥ 3,
x2 ≤ 4,
x1, x2 ≥ 0.
8. Use the two-phase simplex method to
Maximize Z = 5x1 + 3x2,
Subject to the constraints 2x1 + x2 ≤ 1,
x1 + 4x2 ≥ 6,
x1, x2, ≥ 0.
CLASS TUTORIAL – LINEAR PROGRAMMING
Operations Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 3
9. Use the two-phase simplex method to
Maximize Z = 5x1 - 4x2 + 3x3,
Subject to the constraints 2x1 + x2 – 6x3 = 20,
6x1 + 5x2 + 10x3 ≤ 76,
8x1 – 3x2 + 6x3 ≤ 50,
x1, x2, x3 ≥ 0.
10. Construct the dual of the problem
Minimize Z = 3x1 + 5x2
Subject to the constraints 2x1 + 6x2 ≤ 50,
3x1 + 2x2 ≤ 35,
5x1 - 3x2 ≤ 10
X2 ≤ 20
x1, x2 ≥ 0.
11. Construct the dual of the problem
Minimize Z = 3x1 + 17x2 + 9x3
Subject to the constraints x1 - x2 + x3 ≥ 3,
-3x1 + 2x3 ≤ 1,
x1, x2, x3 ≥ 0.
12. Construct the dual of the problem
Maximize Z = 3x1 – 2x2 + 4x3,
Subject to the constraints 3x1 + 5x2 + 4x3 ≥ 7,
6x1 + x2 + 3x3 ≥ 4,
7x1 – 2x2 – x3 ≤ 10,
x1 – 2x2 + 5x3 ≥ 3,
4x1 + 7x2 – 2x3 ≥ 2,
x1, x2, x3 ≥ 0.
CLASS TUTORIAL – TRANSPORTATION PROBLEM
Operation Research (2171901)
Department of Mechanical Engineering
Darshan Institute of Engineering and Technology, Rajkot 1
Unit 3(a). TRANSPORTATION MODEL
1. A firm owns facilities at seven places. It has manufacturing plants at places A, B and C with
daily output of 500, 300 and 200 units of an item respectively. It has warehouses at places P,
Q, R and S with daily requirements of 180, 150, 350 and 320 units respectively. Per unit
shipping charges on different routes are given below:
To: P Q R S
From A: 12 10 12 13
From B: 7 11 8 14
From C: 6 16 11 7
The firm wants to send the output from various plants to warehouses involving minimum