2 Linear Programming Part 1

7/29/2019 2 Linear Programming Part 1

http://slidepdf.com/reader/full/2-linear-programming-part-1 1/21

ECS716/QMT710: Operations Research

Pn Pa’ezah Hamzah

1



Part 1:

What is Linear Programming (LP)?

General form of LP model

Assumptions

of

LP

model

Formulation of LP model

Graphical Solution ‐ only for LP problems in 2 variables

Part 2:

Simplex Solution

Computer Solution

2



At the end of this class, students should be able

to: describe the basic approach of LP

explain the

types

of

problems

that

can

be

solved using LP formulate LP problems solve LP problems in 2 varibles by the

graphical method

3



Explain the meaning of the terms:▪ Linear programming

▪ Objective function

▪

Constraints▪ Feasible solutions

▪ Optimal solution

▪ Equality and inequality ▪ Graphical method

▪ Isoprofit/isocost function

▪ Corner point (extreme point)

4



Linear programming is a mathematical programming

and

an

optimization

technique

that has been widely used by business and

industry for

many

years.

It involves allocating scarce resources on the

basis of a given criterion of optimality. Most often, this criterion is either maximum profit

or minimum

cost.

5



Examples of problems solved by LP: allocation

of

production

facilities,

production

mix

determination, blending, manpower allocation and

assignment, personnel allocation, transportation

planning, scheduling, advertising budget allocation.

LP uses mathematical model to describe the

problem of concern. The

word

linear implies

that

all

mathematical

relationships in the model are linear. The word programming does not refer to

computer programming;

it

refers

to

planning.

6



Every LP problem consists of:

decisions that must be made

an objective to be achieved

a set

of

restrictions (or constraints) to

consider

7



The general LP model in variables X i for allocating mresources with n constraints can be formulated as follows:

MAXIMIZE (or MINIMIZE) Z = c1 X 1 + c2 X 2 + … + cn X n

subject to: a11 X 1 + a12 X 2 + … + a1n X n ≤ b1

a21 X 1 + a22 X 2 + … + a2n X n ≤ b2ak1 X 1 + ak2 X 2 + … + akn X n ≤ bk …am1 X 1 + am2 X 2 + … + amn X n ≤ bm

X i ≥ 0 where i =1,…,n

constraints

non‐negativity constraints

the objective function

Note:

All mathematical relationships are linear. “ >” and “ < “ relationships are not allowed (since LP model is deterministic.)

8



If

all

the

functions in

the

model

are

linear

⇒ Linear Programming (LP) problem.

If all

the

variables in

the

LP

model

are

integer

⇒ Integer Linear Programming (ILP) problem.

If some

of

the

variables are

integer

⇒Mixed Integer Linear Programming (MILP) problem.

9



1. Certainty ‐ all parameters of the model are known with certainty.

2. Additivity ‐the

total

consumption

of

each

resource,

as

well

as

the

overall

objective value are the aggregates of the resource consumptions and the contributions to the problem objective, resulted from

carrying out each activity independently.

3. Proportionality ‐ the contribution of each variable in the objective function or its

usage of the resources are directly proportional to the value of the variable.

4. Divisibility ‐ the decision variables can take on any fractional values.

5. Nonnegativity ‐the

decision

variables

can

take

on

any

value

greater

than

or

equal

to zero.

10



Step 1: Identify and define the decision variables

(unknowns) of

the

problem.

Step 2: Write the objective function to be optimized

(maximize/minimize) as a linear function of

the decision

variables.

Step 3: Translate the requirements, restrictions, or

wishes,

that

are

in

narrative

form

to

linear

equations or inequalities in terms of the

decision variables.

Step 4:

Add

the

non

‐negativity

constraints.

11



ABC Inc. produces two types of chandelier: Design 1 and Design 2. The critical resources are available labor hours, crystals and

switches for the next production cycle. The following table outlines

usage factors

and

unit

profit:

There are 200 switches, 1566 hours of labor, and 2880 pieces of crystals available.

Formulate an

LP

model

to

determine

the

quantity

of

each

type

of

chandelier to produce in order to maximize profit.

12

Design 1 Design 2Switch 1 1

Labor 9 hours 6 hours

Crystal 12 pieces 16 pieces

Unit Profit RM350 RM300



Maxi Design Sdn. Bhd. has been awarded a contract to design a label for a new

computer produced by KC Computer Company. Maxi Design estimates that at least 160 hours will be required to complete the project. Three of the firm’s

graphic designers

are

available

for

assignment

to

this

project.

Aida,

a senior

team

leader; Bakri, a senior designer; and Carol, a junior designer.

Since Aida has worked on several projects for KC, the management has specified

that Aida must be assigned with at least 45% of the total number of hours

assigned to

the

two

senior

designers.

To

provide

label

‐designing

experience

for

Carol, Carol must be assigned at least 20% of the total project time. However, the

number of hours assigned to Carol must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Aida has a maximum of 50 hours available to work on this project.

Hourly salary rates are RM40 for Aida, RM30 for Bakri, and RM20 for Carol.

Formulate an LP model to determine the number of hours each graphic designer should be assigned to the project in order to minimize the total salary.

13



There are TWO methods to solve by graphing:

A. Isoprofit /Isocost Line Method

▪isoprofit line

method

for

maximization

problems.

▪ isocost line method for minimization problems.

B. Corner (Extreme)Point Method▪ for both maximization and minimization problems.

14



Step 1: Formulate the problem.

Step 2: For each

constraint,

draw

a graph

of

the

feasible

solutions.

The solution set of the LP problem is that region (or set of ordered pairs), which satisfies ALL the constraints

simultaneously. This region is called the area of feasible solution (or feasible

region.)

Step 3: Draw an

objective

function

line.

Step 4: Determine the optimum point.

15



Step 4: Determine the optimum point. For maximization problems:

Move parallel objective function lines toward

larger objective function values without entirely

leaving the

feasible

region.

For

minimization

problems:Move parallel objective function lines toward

smaller objective function values without entirely

leaving

the

feasible

region.16



17

Objective Function

x

y

Feasible

Region

Optimum point



Step 1: Graph the constraints.

Step 2:

Locate all

the

corner

points

of

the

graph.

The coordinates of the corners will be determined algebraically.

It is important to note that the optimal point is obtained at the

boundary

of

the

feasible

region

and

furthermore

at

the

corner

points.

For linear programs, it can be shown that the optimal point will always be obtained at corner points.

Step 3: Determine the optimal value.Test all the corner points to see which one yields the optimum

value for the objective function.

18



Solve the ABC Co. problem by graphing.a. If the company can sell all the chandeliers it can

produce, how many of each design should it make in

order to maximize the profit?

b. Determine the quantity of each resource used at the

optimal production

level.

Identify

all

resources

that

are

fully utilized.

19



A dietician is considering a simple breakfast menu (excluding beverages)

consisting of

scrambled

egg

and

curry

puffs,

for

participants

of

an

orientation program. An adequate amount of the following nutrients in

the breakfast: vitamin A, vitamin B, and iron must be included. Each

small scoop of scrambled egg contains 3 milligrams of vitamin A, 4

milligrams of

vitamin

B,

and

1 milligram

of

iron.

Each

curry

puff

contains

3 milligrams of vitamin A, 2 milligrams of vitamin B, and 2 milligram of

iron. The minimum requirements for vitamin A, vitamin B and iron are 30

milligrams, 24 milligrams and 12 milligrams respectively. Each scoop of

scrambled egg

costs

30

sen

and

each

curry

puff

costs

35

sen.

The

dietician would like to determine how much of each type of food to serve

in order to meet the nutrient requirements and also to minimise the total

cost. The following table summarises the relevant information.

20



a. Formulate a linear

programming

model

for

this

problem.

b. Determine the optimal solution. What is the lowest cost that meets the nutrient requirements?

c. Determine the quantity of each nutrient in the optimal solution.

21

Nutrient Content (mg.) Nutrient Requirement (mg.)

Nutrient Scrambled Egg Curry Puff

Vitamin A 3 3 30

Vitamin B 4 2 24

Iron 1 2 12

Unit Cost 30 sen 35 sen

Documents

2 Linear Programming Part 1