MODULE 5: MATHEMATICS FOR
EFFICIENCY
Linear Programming,
Graph and Shortest
Paths and Assignment
Problems
GEMATMW
MR. MARK ANTHONY GARCIA
Department of Mathematics and Statistics
MAG
What is Linear Programming (LP)?
Linear programming is a mathematical
technique for optimizing linear functions of
several variables such as revenue, profit and
cost.
• It involves solving systems of linear
inequalities using graphical methods and the
simplex method.
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Brief History of LP
In 1939, Soviet mathematician and
economist Leonid Kantorovich
formulated a problem that is
similar to the general linear
programming problem.
He later, during the World War II, used it to plan
expenditures and returns in order to reduce costs of the
army and to increase losses imposed on the enemy.
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Brief History of LP
In
1947,
George
Dantzig
developed
general
linear
programming formulation to use
for planning problems in the US
Air Force.
He provided a formal proof of “A Theorem in Linear
Inequalities” and it was published in 1951. In the post war
years, many industries applied it in their daily planning.
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Some Applications of LP
Diet Problems – finding the cheapest
combination of foods to satisfy nutritional
requirements.
Portfolio Problems – maximizing returns and
minimizing the risk of losing in an investment
portfolio.
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Some Applications of LP
Scheduling Problems – an airline has to assign
crews to its flights with goal of minimizing
logistical costs.
Manufacturing and Transportation Problems –
maximizing the production costs of a product
while minimizing the costs of product transfers
from one location to another.
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Elements of Linear Programming
Decision Variables
• The decision variables represent the unknown
decisions to be made.
• Quantity of products to be produced and sold
• Quantity of products to be consumed
• Allocation of money in investment problems
• We normally use the notations 𝒙𝟏 , 𝒙𝟐 , … , 𝒙𝒏 to
represent decision variables. For LP problems
involving only two decision variables, we can
simply use 𝒙 and 𝒚.
Elements of Linear Programming
Objective Function
This is the goal of the problem which is
represented by an equation. This function is
either minimized or maximized.
•
•
•
•
•
•
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Maximize profit or revenue
Minimize cost of production/transportation
Minimize overtime payment of employees
Maximize use of resources/ingredients/materials
Minimize labor turnover changes
Minimize machine downtime
Elements of Linear Programming
Constraints
• The constraints represent limitation in
resources and decision variables. It is
represented by a system of inequalities.
• There are two types of constraints: resource
constraints and non-negativity constraints.
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Elements of Linear Programming
The resource constraints are limitations in
elements that affect production,
consumption, money allocation.
•
•
•
•
•
•
Limited raw material/ingredients
Limited money or budget
Limited employee work time or machine time
Limited personnel/employees
Limited ability or skills of employees
Limited available capacity in shipments
Elements of Linear Programming
The non-negativity constraints are constraints
that assures nonnegative values for the
decision variables.
• Most of the time, decision variables represent
quantities of products produced and consumed and
money allocation, these values should be
nonnegative.
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Example 1: Formulation and
Solution of LP
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A company produces two types of tables, T1 and
T2.
It takes 2 hours to produce the parts of one unit of
T1, 1 hour to assemble and 2 hours to polish.
It takes 4 hours to produce the parts of one unit of
T2, 2.5 hours to assemble and 1.5 hours to polish.
Per month, 7000 hours are available for producing
the parts, 4000 hours for assembling the parts and
5500 hours for polishing the tables.
The profit per unit of T1 is $90 and per unit of T2 is
$110.
How many of each type of tables should be
produced in order to maximize the total monthly
profit?
Example 1: Formulation and
Solution of LP
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A company produces two types of tables, T1 and T2.
𝒙
Quantity for table T1
𝒚
Quantity for table T2
Example 1: Formulation and
Solution of LP
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The profit per unit of T1 is $90 and per unit of T2 is
$110.
How many of each type of tables should be
produced in order to maximize the total monthly
profit?
Example 1: Formulation and
Solution of LP
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It takes 2 hours to produce the parts of one unit of
T1, 1 hour to assemble and 2 hours to polish.
It takes 4 hours to produce the parts of one unit of
T2, 2.5 hours to assemble and 1.5 hours to polish.
Per month, 7000 hours are available for producing
the parts, 4000 hours for assembling the parts and
5500 hours for polishing the tables.
Table T1
Production Time
Assembly Time
Polishing Time
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Table T2
Limitation
Example 1: Formulation and
Solution of LP
𝒙
𝒚
Quantity for table T1
Quantity for table T2
Maximize total monthly profit
𝒛 = 𝟗𝟎𝒙 + 𝟏𝟏𝟎𝒚
𝟐𝒙 + 𝟒𝒚 ≤ 𝟕𝟎𝟎𝟎
𝒙 + 𝟐. 𝟓𝒚 ≤ 𝟒𝟎𝟎𝟎
𝟐𝒙 + 𝟏. 𝟓𝒚 ≤ 𝟓𝟓𝟎𝟎
𝒙 ≥ 𝟎, 𝒚 ≥ 𝟎
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Example 1: Formulation and
Solution of LP
(𝟎, 𝟏𝟔𝟎𝟎)
(𝟏𝟓𝟎𝟎, 𝟏𝟎𝟎𝟎)
(𝟐𝟑𝟎𝟎, 𝟔𝟎𝟎)
(𝟎, 𝟎)
(𝟐𝟕𝟓𝟎, 𝟎)
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Example 1: Formulation and
Solution of LP
𝒛 = 𝟗𝟎𝒙 + 𝟏𝟏𝟎𝒚
(𝟎, 𝟏𝟔𝟎𝟎)
𝒛 = 𝟗𝟎 𝟎 + 𝟏𝟏𝟎 𝟏𝟔𝟎𝟎 = 𝟏𝟕𝟔, 𝟎𝟎𝟎
(𝟏𝟓𝟎𝟎, 𝟏𝟎𝟎𝟎)
𝒛 = 𝟗𝟎 𝟏𝟓𝟎𝟎 + 𝟏𝟏𝟎 𝟏𝟎𝟎𝟎 = 𝟐𝟒𝟓, 𝟎𝟎𝟎
(𝟐𝟑𝟎𝟎, 𝟔𝟎𝟎)
𝒛 = 𝟗𝟎 𝟐𝟑𝟎𝟎 + 𝟏𝟏𝟎 𝟔𝟎𝟎 = 𝟐𝟕𝟑, 𝟎𝟎𝟎
(𝟐𝟕𝟓𝟎, 𝟎)
𝒛 = 𝟗𝟎 𝟐𝟕𝟓𝟎 + 𝟏𝟏𝟎 𝟎 = 𝟐𝟒𝟕, 𝟓𝟎𝟎
(𝟎, 𝟎)
𝒛 = 𝟗𝟎 𝟎 + 𝟏𝟏𝟎 𝟎 = 𝟎
• The company should produce 2300 units of Table T1 and 600 units of Table T2.
• This implies that the maximum monthly profit is $273,000.
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Example 2: Formulation and
Solution of LP
• The liquid portion of a diet is to provide at least 300
calories, at least 36 units of vitamin A, and at least
90 units of vitamin C daily.
• A cup of dietary drink X provides 60 calories, 12
units of vitamin A, and 10 units of vitamin C.
• A cup of dietary drink Y provides 60 calories, 6 units
of vitamin A, and 30 units of vitamin C.
• Now, suppose that dietary drink X costs Php12 per
cup and drink Y costs Php15 per cup.
• How many cups of each drink should be consumed
each day to minimize the cost and still meet the
stated daily requirements?
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Example 2: Formulation and
Solution of LP
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Now, suppose that dietary drink X costs Php12 per
cup and drink Y costs Php15 per cup.
How many cups of each drink should be consumed
each day to minimize the cost and still meet the
stated daily requirements?
𝒙
𝒚
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No. of cups for Drink X
No. of cups for Drink Y
Example 2: Formulation and
Solution of LP
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The liquid portion of a diet is to provide at least
300 calories, at least 36 units of vitamin A, and at
least 90 units of vitamin C daily.
A cup of dietary drink X provides 60 calories, 12
units of vitamin A, and 10 units of vitamin C.
A cup of dietary drink Y provides 60 calories, 6
units of vitamin A, and 30 units of vitamin C.
Drink X
Calories
Vitamin A
Vitamin C
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Drink Y
Min. Req.
Example 2: Formulation and
Solution of LP
𝒙
𝒚
No. of cups for Drink X
No. of cups for Drink Y
Minimize total cost
𝒛 = 𝟏𝟐𝒙 + 𝟏𝟓𝒚
𝟔𝟎𝒙 + 𝟔𝟎𝒚 ≥ 𝟑𝟎𝟎
𝟏𝟐𝒙 + 𝟔𝒚 ≥ 𝟑𝟔
𝟏𝟎𝒙 + 𝟑𝟎𝒚 ≥ 𝟗𝟎
𝒙 ≥ 𝟎, 𝒚 ≥ 𝟎
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Example 2: Formulation and
Solution of LP
(𝟎, 𝟔)
(𝟏, 𝟒)
(𝟑, 𝟐)
(𝟗, 𝟎)
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