Presented by:Dr. AadilKranteeMrudulRatish

Presented To:Dr. Col (Retd) Manjushree Kumar

2

Contents

• Introduction• History • Linear programming model formulation• Applications • Illustration (Diet problem)• Assignment problem• Assumptions of LP• Limitations of LP• Duality

3

Introduction

• Linear Programming is a mathematical modeling

technique used to determine a level of operational activity

in order to achieve an objective.

• Mathematical programming is used to find the best or

optimal solution to a problem that requires a decision or

set of decisions about how best to use a set of limited

resources to achieve a state goal of objectives.

4

History of linear programming

• It started in 1947 when G. B. Dantzig designed the “simplex

method” for solving linear programming formulations of

U.S. Air Force planning problems.

• It soon became clear that a surprisingly wide range of

apparently unrelated problems in production management

could be stated in linear programming terms and solved by

the simplex method.

5

LP Model Formulation

• Decision variables

– mathematical symbols representing levels of activity of an operation

• Objective function

– a linear relationship reflecting the objective of an operation

– most frequent objective of business firms is to maximize profit

– most frequent objective of individual operational units (such as a

production or packaging department) is to minimize cost

• Constraint

– a linear relationship representing a restriction on decision making

6

• Steps involved in mathematical programming

– Conversion of stated problem into a mathematical model that

abstracts all the essential elements of the problem.

– Exploration of different solutions of the problem.

– Find out the most suitable or optimum solution.

• Linear programming requires that all the mathematical

functions in the model be linear functions.

7

Applications

The Importance of Linear Programming

• Hospital management • Diet management• Manufacturing• Finance (investment)• Advertising• Agriculture• Military

8

Some applications in Hospital management :

1) Incinerators and Pollution Control. (What is the most economical way to make the necessary cutbacks? i.e. sulfur dioxide emissions must be limited to 400,000 units per day and particulate emissions to 50,000 units per day .)

2) Assignments to Hospitals .(draw up a disaster plan for assigning casualties to hospitals in the event of a disaster. How should the victims be assigned to minimize the total time lost in transporting them?)

3) The Diet Problem. (e.g.. The problem is to supply the required nutrients at minimum cost.)

4) The Transportation Problem (e.g..The problem is to meet the hospital or patient requirements at minimum transportation cost.)

9

5) The Activity Analysis Problem. ( e.g.. The problem is to choose the intensities which the various activities are to be operated to maximize the value of the output to the company subject to the given resources.)

6) The Optimal Assignment Problem. ( e.g.. The problem is to choose an assignment of persons to jobs to maximize the total value.)

7) The Product Mix Problem (The company would like to determine how many units of each product it should produce so as to maximize overall profit given its limited resources.)

8) Design of radiation therapy. (e.g.. Radiation therapy beams affects tissues. The goal of the design is to select the combination of beams to be used and the intensity of each one.)

10

Illustration 1. Diet problem

Question:

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of

calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A.

Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A.

The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol.

How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

11

Solution:

Let x and y be the number of packets of food P and Q respectively. Obviously x ≥ 0, y ≥ 0. Mathematical formulation of the given problem is as follows:

Minimize Z = 6x + 3y (vitamin A) subject to the constraints

12x + 3y ≥ 240 (constraint on calcium), i.e. 4x + y ≥ 80 ... (1)

4x + 20y ≥ 460 (constraint on iron.), i.e. x + 5y ≥ 115 ... (2)

6x + 4y ≤ 300 (constraint on cholestérol), i.e. 3x + 2y ≤ 150 ... (3)

x ≥ 0, y ≥ 0 ... (4)

12

• Let us graph the inequalities (1) to (4).The feasible region (shaded) determined by the constraints (1) to (4) is shown in the figure.

• The coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15) respectively. Let us evaluate Z at these points:

• From the table, we find that Z is minimum at the point (15, 20). Hence, the amount of vitamin A under the constraints given in the problem will be minimum, if 15 packets of food P and 20 packets of food Q are used in the special diet. The minimum amount of vitamin A will be 150 units.

13

Assignment problem

• The assignment problem refers to the class of linear programming problems that involve determining the most efficient assignment of assignees to perform tasks.

people to projectssalespeople to territoriescontracts to bidders jobs to machines, etc.• The objective is most often to minimize total costs or

total time of performing the tasks at hand.

14

• To fit the definition of an assignment problem, these kinds of applications need to be formulated in a way that satisfies the following assumptions.

1.The number of assignees and the number of tasks are the same. (This number is denoted by n.)

2.Each assignee is to be assigned to exactly one task.

3.Each task is to be performed by exactly one assignee.

4.There is a cost cij associated with assignee i (i1, 2,...,n) performing task j (j1, 2, . . . ,n).

5.The objective is to determine how all assignments should be made to minimize the total cost.

15

Assignment problem solving methods

1. Hungarian Method

2. Enumeration method

3. Simplex method

4. Transportation

16

Question:

We must determine how jobs should be assigned to machines to minimize setup times, which are given below:

AP (Hungarian method)

Job 1 Job 2 Job 3 Job 4

Machine 1 14 5 8 7

Machine 2 2 12 6 5

Machine 3 7 8 3 9

Machine 4 2 4 6 10

17

• Step 1: (a) Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each row. (b) Find the minimum cost in each column of the new matrix, and subtract this from each column. This is the reduced cost matrix.

18

Job 1 Job 2 Job 3 Job 4

Machine 1 14 5 8 7

Machine 2 2 12 6 5

Machine 3 7 8 3 9

Machine 4 2 4 6 10

Job 1 Job 2 Job 3 Job 4

Machine 1 9 0 3 2

Machine 2 0 10 4 3

Machine 3 4 5 0 6

Machine 4 0 2 4 8

Row Reduction

Step 1(a)

19

Job 1 Job 2 Job 3 Job 4

Machine 1 9 0 3 2

Machine 2 0 10 4 3

Machine 3 4 5 0 6

Machine 4 0 2 4 8

Job 1 Job 2 Job 3 Job 4

Machine 1 9 0 3 0

Machine 2 0 10 4 1

Machine 3 4 5 0 4

Machine 4 0 2 4 6

Step 1(b)

Column Reduction

20

Step 2

Draw the minimum number of lines that are needed to cover all the zeros in the reduced cost matrix. If m lines are required, then an optimal solution is available among the covered zeros in the matrix. Otherwise, continue to Step 3.

Job 1 Job 2 Job 3 Job 4

Machine 1 9 0 3 0

Machine 2 0 10 4 1

Machine 3 4 5 0 4

Machine 4 0 2 4 6

21

Step 3

Find the smallest nonzero element (say, k) in the reduced cost matrix that is uncovered by the lines. Subtract k from each uncovered element, and add k to each element that is covered by two lines. Return to Step 2.

22

Job 1 Job 2 Job 3 Job 4

Machine 1 9 0 3 0

Machine 2 0 10 4 1

Machine 3 4 5 0 4

Machine 4 0 2 4 6

Job 1 Job 2 Job 3 Job 4

Machine 1 10 0 3 0

Machine 2 0 9 3 0

Machine 3 5 5 0 4

Machine 4 0 1 3 5

Step 4

23

Job 1 Job 2 Job 3 Job 4

Machine 1 10 0 3 0

Machine 2 0 9 3 0

Machine 3 5 5 0 4

Machine 4 0 1 3 5

24

Assumptions of LP

1. Certainty: numbers in the objective and constraints are known with

certainty and do not change during the period being studied

2. Proportionality: exists in the objective and constraints constancy between production increases and resource

utilization3. Additivity:

the total of all activities equals the sum of the individual activities

25

Assumptions of LP cont…

4. Divisibility: solutions need not be in whole numbers (integers) solutions are divisible, and may take any fractional

value5. Non-negativity:

all answers or variables are greater than or equal to (≥) zero

negative values of physical quantities are impossible

26

Limitations of LP

•It treats all relationships among decision variables as linear.•There is no guarantee that we will get integer value solutions. e.g. 2.5 machines•LP does not take into consideration the effect of time & uncertainty.•In LP parameters are assumed to be constant ; but in real life situations majority of the times they are neither known nor constant.•LP deals with only single objective whereas in real life conflicting situations may have to be solved.

27

Duality in LP

•In context of LP, duality means each LP problem can be analyzed in two different ways.•LP problem can be stated in another equivalent problem based on the same data and new problem will be called as DUAL.•The main focus of a dual problem is to find best marginal value for each resource; also known as Shadow Prize.•The shadow prize is also defined as change in optimal objective function value with respect to unit change in availability of resource.

28

A linear programming model can

provide an insight and an intelligent

solution to the problem.

29

REFERENCES

• www.math.ucla.edu/~tom/LP.pdf• www.sce.carleton.ca/faculty/chinneck/po/Chapter2.• www.markschulze.net/LinearProgramming.pdf• web.ntpu.edu.tw/~juang/ms/Ch02.• cmp.felk.cvut.cz/~hlavac/Public/.../Linear

%20Programming-1.ppt• www.slideshare.net/

30

Questions/Queries?

31

Thank you