OPERATION RESEARCH

Second Year BIO

Operations research is a quantitative approach to decision making based
on the scientific method of problem solving.
 Operations Research applies analytical (Mathematical) methods to
help organizations make better decisions when solving complex
problems.
 Managers use Operations Research to make their organizations more
productive and efficient.
 Operations Research is the application of scientific methods,
techniques and tools to complex problems to management of large
number of men, machines, materials and money, in industry,
business, government and defense.
 Mathematical models represent real world problems through a system
of mathematical formulas and expressions.
 Generally, experimenting with models (compared to experimenting
with the real situation):
 Requires less time
 Is less expensive
 Involves less risk
 The linear programming is useful not only in industry and business
but also in non-profit sectors such as education, government, hospital,
and libraries
 Linear programming determines the way to achieve the best outcome
(such as maximum profit or lowest cost) in a given mathematical
model and given some list of requirements represented as linear
equations.
 The linear programming method is applicable in problems
characterized by the presence of decision variables.

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 In mathematics, linear programming (LP) is a technique for
optimization of a linear objective function, subject to linear equality
and linear inequality constraints.
Formulation of Linear Programming Model
The LP model has three basic components:
1. Decision variables that we seek to determine.
2. Objective (goal) that we need to optimize (maximize or minimize)
3. Constraints that the solution must satisfy.
4. Add non-negativity restrictions or constraints
i.e. The objective function, the set of constraints and the non-negativity
restrictions together form an LP model.

𝑎𝑖𝑗 Technical coefficients

𝑏𝑗 Available amounts
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  Example 1
Suppose an industry is manufacturing two types of products P1 and P2.
The profits per kg of the two products are rs.30 and rs.40 respectively.
These two products require processing in three types of machines. The
following table shows the available machine hours per day and the time
required on each machine to produce one kg of P1 and P2. Formulate the
problem in the form of linear programming model which will maximize
the profit.
Total available
Profit Product 1 Product 2
Machine hours/day
Machine 1 3 2 6
Machine 2 3 5 8
Machine 3 5 6 11
Solution
• Variables
Let 𝑋1 = amount of P1
𝑋2 = amount of P2
Z = profit
• Objective
Maximize Z = 30 X1 + 40 X 2
• 3. Constraints
3 x1 + 2 x2 < 6 (1)
3x1 + 5 x2 < 8 (2)
5 x1 +6 x2 < 11 (3)
X1 ≥ 0, x2 > 0 (4)

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  Example 2
Two types of products A and B. Profit is 4 on type A profit is 5 on type
B. Both A and B are produced by X and Y machines
Products Machine X Machine Y
A 2 minutes 3 minutes
B 2 minutes 2 minutes
Total available Machine hours/day 5 hours 8 hours

Formulate the problem as a LP problem to maximize the profit.

Solution
• Variables
Let 𝑋1 = product A
𝑋2 = product B
Z = profit
• Objective
Maximize Z = 4 X1 + 5 X 2
• 3. Constraints
2 x1 + 2 x2 < 5*60 (1)
3x1 + 2 x2 < 8*60 (2)
X1 ≥ 0, x2 > 0 (3)

 Example 3
Reddy mikks produces both interior and exterior paints from two raw
materials M1 and M2.The following table provides the basic data of the
problem:

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 Tons of raw material per ton Maximum
Exterior paints Interior paints daily
availability
(tons)
Raw materials M1 6 4 24
Raw materials M2 1 2 6
Profit per ton 5 4 ??
($1000)

A market survey indicates that the daily demand for interior paint cannot
exceed that of exterior paint by more than 1 ton. Also, the maximum
daily demand for interior paint is 2 tons.
Reddy Mikks wants to determine the optimum (best) product mix of
interior and exterior paints that maximizes the total daily profit?
Solution
1. Variables
𝑋1 = tons produced daily of exterior paint
𝑋2 = tons produced daily of interior paint
Z represent the total daily profit ($1000)
2. Objective:
maximize Z = 5 𝑋1 + 4 𝑋2
3. Constraints
6 𝑋1 + 4 𝑋2 < 24 (1)
𝑋1 + 2 𝑋2 < 6 (2)
− 𝑋1 + 𝑋2 < 1 (3)
𝑋2 < 2 (4)
𝑋1 , 𝑋2 > 0 (5)
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  Example 4
Ozark farms uses at least 800 Ib of special feed daily. The special feed is
a mixture of corn and soybean meal with the following compositions:
Ib per Ib of feedstuff

Feedstuff Protein Cost($/Ib)

Fiber
corn 0.09 0.02 0.3

soybean meal 0.6 0.06 0.9

The dietary requirements of the special feed are at least 30% protein and
at most 5% fiber.
Ozark farms wishes to determine the daily minimum cost feed mix?
Solution
1. Variables
𝑋1 = Ib of corn in the daily mix
𝑋2 = Ib of soybean meal in the daily mix
Z represent the total daily cost ($) of the feed mix
2. Objective:
minimize Z = 0. 3𝑋1 + 0. 9𝑋2
3. Constraints
𝑋1 + 𝑋2 ≥ 800 (1)
0. 09𝑋1 + 0. 6𝑋2 ≥ 0.3 ( 𝑋1 + 𝑋2 )
Or 0. 21𝑋1 -0.3𝑋2 ≤ 0 (2)
0.02 𝑋1 + 0. 06𝑋2 < 0.05( 𝑋1 + 𝑋2 )

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Or 0. 03𝑋1 -0.01𝑋2 ≥ 0 (3)
𝑋1 , 𝑋2 > 0 (4)
 Example 5
Company samyoung produces knives & forks. The company wants to
determine the production levels of the two products which maximize the
total profit. One case of knives provides profit of $8 and one case of
forks provides profit of $6. They are produced after two processes called
press process and polishing process. During the next week, the company
can use 70 hours of press process and 100 hours of polishing process.
Production of one case of knives requires 12 minutes of press process
and 30 minutes of polishing process. And production of one case of
forks requires 24 minutes of press process and 15 minutes of polishing
process. The company wants to determine the amounts of knives and
forks production which maximize the total profit of the next week?
Solution
1.Variables
𝑋1 : amount of knives produced during the next week.
𝑋2 : amount of knives produced during the next week.
Z : total profit obtained during the next week.
2. Objective
Maximize total profit Z =8 𝑋1 +6 𝑋2
3.Constraints
0.2 𝑋1 +0.4 𝑋2 < 70 (1)
0.5 𝑋1 +0.25 𝑋2 < 100 (2)
𝑋1 > 0 (3)
𝑋2 > 0 (4)

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  Example 6
This problem is looking for a daily diet for a person such that it satisfies
the minimum daily requirement of each nutrient and requires the
smallest cost. In this example, they would like to construct a diet of 4
basic foods which are milk, tuna, bread, and spinach. The diet should
satisfy the minimum requirements of 4 nutrients which are vitamin A, C,
D and iron. The table shows nutrient contents of the 4 basic foods and
their prices and the minimum daily requirements of the four nutrients.
Since this diet is for human being, the taste of the diet is also very
important. Hence, they decided to include at least 0.1kg of tuna and half
roll of bread
Nutriment Milk Tuna Bread Spinach Requirements
(L) (kg) (roll) (kg) per day

Vitamin A (IU) 1600 500 0 70000 5000

Vitamin C (mg) 10 0 0 140 30
Vitamin D (IU) 120 0 0 0 100
Iron (mg) 7 14 13 16 12
Price ($) 1 3 0.65 0.6 ??
Solution
1.Variables
𝑋1 : consumption of milk
X 2 : consumption of tuna
X 3 : consumption of bread
X 4 : consumption of spinach
2. Objective
Minimize Z = X1 +3X 2 +0.65X 3 +0.6X 4
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3.Constraints
1600X1 +500X 2 +70000X 4 ≥5000 (1)
10X1 +140X 2 ≥30 (2)
120X1 ≥100 (3)
7𝑋1 +14𝑋2 +13𝑋3 +16𝑋4 ≥12 (4)
𝑋2 ≥0.1 (5)
𝑋3 ≥0.5 (6)
𝑋1 , 𝑋2 , 𝑋3 , 𝑋4 ≥0 (7)
summary
• In operations research linear programming is a versatile technique
with wide applications in various management problems.
• Linear programming problem has a number of characteristics. That
is first we have to identify the decision variable. The problem must
have a well defined objective function, which are expressed in
terms of the decision variables.
• The objective function may have to be maximized when it
indicates the profit or production or contribution. If the objective
function represents cost, in this case the objective function has to
be minimized.
• The management problem is expressed in terms of the decision
variables with the objective function and constraints.

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  Examples
1-A financial consulting company manages funds for various companies
and private investors. The company develops an investment strategy for
each client based on his request. Today, an investor asked the company
to manage $1,200,000 in stock fund or income fund. The stock fund is
operated in the unit of $50 and its expected rate of return is 10%. The
basic unit of income fund is $100 and it is expected to have 4% annual
return. This client wants to invest at least $300,000 to income fund and
minimize risk as far as total return is expected to be at least $60,000.
The company evaluated the risk indices for stock fund and income fund
as 8 and 3 per $10 of investment, respectively.
Develop a linear programming model which can minimize the total risk /
generate an optimal investment strategy for this client
Solution:
1. Variables
𝑋1 : Stock funds
𝑋2 : Income fund
Z represents the total risk
2. Objective:
Min. Z = 0.8 𝑋1 + 0.3 𝑋2
3. Constraints
𝑋1 + 𝑋2 < 1 200 000 (1)
𝑋2 >300 000 (2)
0.1𝑋1 + 0.04𝑋2 > 600 000 (3)
𝑋1 > 0 (4)
𝑋2 > 0 (5)

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2-Jack is an aspiring freshman at Uiern University. He realizes that all
work and no play make jack a dull boy. As a result, Jack wants to
apportion his available time of about 10 hours a day between work and
play. He estimates that play is twice as much fun as work. He also,
wants to study at least as much as he plays.
However, Jack realizes that if he is going to get all his homework
assignments done, he cannot play more than 4 hours a day.
How should Jack allocate his time to maximize his pleasure from both
work and play?
Solution:
1. Variables
𝑋1 : play hours per day
𝑋2 : work hours per day
Z represents the total time
2. Objective:
Max. Z = 2𝑋1 +𝑋2
3. Constraints
X1 + X 2 ≤ 10 (1)
X1 ≤ 4 (2)
X 2 ≥ X1 or X 2 − X1 ≥ 0 (3)
X1 ≥ 0 (4)
X2 ≥ 0 (5)

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3-A company produces two products A and B. The sales volume for A is
at least 80% of the total sales of both A and B. However, the company
cannot sell more than 100 units of A per day. Both products use one raw
material, of which the maximum daily availability is 240 Ib. The usage
rates of the raw material are 2 Ib per unit of A and 4 Ib per unit of B.
The profit unit for A and B are $20 and $50, respectively. Determine the
product mix of A and B that maximizes the total daily profit?
Solution:
1. Variables
𝑋1 : Numbers of unit of A
𝑋2 : Numbers of unit of B
Z represents the total daily profit
2. Objective:
Max. Z = 20𝑋1 + 50 𝑋2
3. Constraints
𝑋1 > 0.8(𝑋1 + 𝑋2 )
Or -0.2𝑋1 +0.8 𝑋2 ≤ 0(1)
2𝑋1 + 4𝑋2 ≤ 240(2)
𝑋1 ≤ 100 (3)
𝑋1 > 0 (4)
𝑋2 > 0 (5)

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4-A nutritionist advises an individual who is suffering from iron and
vitamin B deficiency to take at least 2400 milligrams (mg) of iron, 2100
mg of vitamin B1, and 1500 mg of vitamin B2 over a period of time. Two
vitamin pills are suitable, brand-A and brand-B. Each brand-A pill costs
6 cents and contains 40 mg of iron, 10 mg of vitamin B1, and 5 mg of
vitamin B2. Each brand-B pill costs 8 cents and contains 10 mg of iron
and 15 mg each of vitamins B1 and B2. What combination of pills should
the individual purchase in order to meet the minimum iron and vitamin
requirements at the lowest cost?

Solution:
1. Variables
𝑋1 : Brand A 𝑋2 : Brand B
C represents the total cost
2. Objective:
Min. C = 6𝑋1 + 8 𝑋2
3. Constraints
40𝑋1 + 10𝑋2 ≥ 2400 (1)
10𝑋1 + 15𝑋2 ≥ 2100 (2)
5𝑋1 +15𝑋2 ≥ 1500 (3)
𝑋1 , 𝑋2 > 0 (4)

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5- Giapetto's wooden soldiers and trains. Each soldier sells for $27, uses
$10 of raw materials and takes $14 of labor & overhead costs. Each train
sells for $21, uses $9 of raw materials, and takes $10 of overhead costs.
Each soldier needs 2 hours finishing and 1 hour carpentry; each train
needs 1 hour finishing and 1 hour carpentry. Raw materials are
unlimited, but only 100 hours of finishing and 80 hours of carpentry are
available each week. Demand for trains is unlimited; but at most 40
soldiers can be sold each week. How many of each toy should be made
each week to maximize profits?
Solution:
1. Variables
𝑋1 : number of soldiers produced per week
𝑋2 : number of trains produced per week
Z represents the total profit
2. Objective:
Max. Z = 3𝑋1 +2𝑋2
3. Constraints
2X1 + X 2 ≤ 100 (1)
X1 + X 2 ≤ 80 (2)
X1 ≤ 40 (3)
X1 ≥ 0 (4)
X2 ≥ 0 (5)

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