P.N. Ram KumarE-mail: pnramkumar@gmail.com

Linear Programming problems – Formulation,

Graphical method - Sensitivity Analysis

Transportation, Transshipment and Assignment

problems

Network optimization models

Integer programming problems

Project management

Scarce resources – Labor, materials, money… Quest for best combination of inputs Minimizing costs or Maximizing profits Linear Programming (LP) problems

- Linear objective function

- Linear constraints Why only LP?

- Majority of real-life problems can be approximated

- Efficient solution procedures

- Easy Sensitivity Analysis

A small factory makes three products, soap,

shampoo and liquid soap. The processing is done in

two stages at two plants, I and II, each of which

works for 40 hours every week.

The time required for processing each product at

each plant and the profit of each product are shown

in the table

Formulate a linear program to maximize the profits

Plant I Plant II Profit / Kg

Soap 3 5 Rs.10

Shampoo 4 4 Rs. 13

Liquid Soap 4 2 Rs. 12

 Hours / week

40 hrs 40 hrs

1. Decision Variables???

2. Let X1, X2, and X3 be the quantities of soap,

shampoo and liquid soap manufactured per

week.

3. Constraints???

4. Time available per week (40 hours)

5. Last, the objective function (Max Profit or

Minimize Cost)

Maximize Z = 10X1 +13X2 + 12X3

Subject to

3X1 + 4X2 + 4X3 < 40 5X1 + 4X2 + 2X3 < 40

X1 > 0, X2 > 0, X3 > 0

X1, X2 , X3

Objective Function

Constraints

Non-Negativity Constraints

Decision Variables

A middle-aged woman, convalescing after a surgery, has been advised by her doctor to plan her diet in such a way as to ensure that she gets the prescribed quantities of vitamins.

The choice of foods available to her, the amounts of vitamins they contain, the cost of each unit and the daily requirement of vitamins are shown in the table

Cost Vitamin A Vitamin B Vitamin D

Eggs Rs 1.50/unit 15 mg/unit 40 mg/unit 10 mg/unit

Milk Rs. 14.00/lit 30 mg/lit 25 mg/lit 20 mg/lit

CerealsRs. 18/unit

(kg)10 mg/kg 55 mg/kg 30 mg/kg

Dailyrequirement

  100 mg 250mg 120mg

Formulate a linear programme to determine the

quantity of each food that the woman needs to buy

in order to minimize her total expenditure, ensuring

at the same time that she meets her daily

requirements of vitamins. Decision Variables? X1, X2, X3 be the quantities of eggs, milk and cereals

bought Constraints? Dietary requirements are to be satisfied Objective Function Minimization of expenditure

Let X1, X2, X3 be the quantities of eggs, milk and cereals bought.

The problem is of the form

(i) There is an objective function which describes what we desire

(ii) A set of constraints, usually in the form of inequalities

(iii) Non-negativity constraints for the decision variables

(iv) The objective function and the constraints are linear.

Max / Min Z = C1 x1 + C2 x2 +…..+ Cn xn

Subject to

a11 x1 + a12 x2 + ……+ a1n xn < or ≥ b1

a21 x1 + a22 x2 + ……+ a2n xn < or ≥ b2.

.

an1 x1 + an2 x2 + ……+ amn xn < or ≥ bm

x1 , x2 ,….., xn - Unrestricted in Sign b1 , b2 ,….., bm - Unrestricted in Sign

OR

1

/

1.....

1.....

x 1.....

j jj

ij j ij

i

j

Maximize Minimize Z C x

Subject to

a x or b i m

b Unrestricted in sign i m

Unrestricted in sign j n

A financial advisor who recently graduated from

IIMK received a call from a client who wanted to

invest a portion of a $ 150,000 inheritance

The client wanted to realize an annual income, but

also wanted to spend some of the money

After discussing the matter, the client and the

adviser agreed that a mutual fund, corporate

bonds, and a money market account would make

suitable investments.

The client was willing to leave allocation of the funds among these investment vehicles to the financial adviser, but with the following conditions:

At least 25 percent of the amount invested should be in the money

market account

A maximum of only 35% should be invested in corporate bonds

The investment must produce at least $ 12,000 annually (ROI)

The un-invested portion should be as large as possible The annual returns would be 11 percent for the mutual

fund, 8 percent for the bonds, and 7 percent for the money market

Formulate an LP model that will achieve the client’s requests. Ignore transaction costs, the adviser’s fee and so on.

Let X1, X2, X3 be the amounts to be invested in mutual fund,

corporate bonds and money market

Non-negativity condition: X1, X2, X3 ≥ 0

At least 25 percent of the amount invested should be in the

money market account

X3 / (X1+X2+X3) ≥ 0.25

A maximum of only 35% should be invested in corporate

bonds

X2 / (X1+X2+X3) ≤ 0.35

The investment must produce at least $12,000 annually (ROI)

0.11X1 + 0.08X2 + 0.07X3 ≥ 12000

Objective Function: Maximize Z = 150,000 – (X1 + X2 + X3)

A senior executive of a public sector company

recently quit his job under the VRS with a hefty

packet of Rs 1 Crore. Messrs. Dhana-chor Chit

Fund Company has offered the following

investment scheme for the benefit of such retired

people:

"Invest a certain sum (in lakhs of rupees) in any

month, invest half of that amount in the next

month and in the subsequent month, one would

get twice the amount invested originally in the

first month”

This scheme is available only for the next six

months (Encashment is possible on 181st day)

Returns received at the end of any month can be

used immediately for reinvesting either as a fresh

investment or as a follow-up investment

Let Xt = Fresh investment in the tth month

(t = 1,2,3,4,5) (in lakhs of Rs.)

Noting that, in any month, the cash outflow

must not exceed the cash on hand, the

following linear programme is formulated

On 181st day, the cash on hand must be

maximum

Hence the objective function is

A brewery blends three raw materials, A, B and C in

varying proportions to obtain three final products,

D, E and F. The salient data is given below.

Final Product

D E F

Net Profit/ Litre

$15 $9 $12

Ingredient

A B C

Cost/ Litre $10 $8 $7

Quantity Available

10,000 12,000 15,000

Formulate a linear programme to maximize the total revenue of the brewery.

Final Product

Specifications

D At least 25% of ingredient ‘A’Not more than 15% of ‘C’

E At least 50% of ingredient ‘B’Not more than 30% of ‘C’

F At least 65% of ingredient ‘C’Not more than 5% of ‘A’

Let Xij be the quantity of the ith raw material in jth the product

Solution to Example Problem No.5

A

B

C

A shipping company has to lift three types of

cargo whose details are given below.

Type ofCargo

Quantity(tons)

Volume perton

Profit per ton

A 300 1.2 m3 Rs. 10,000

B 500 0.8 m3 Rs. 7,000

C 400 1 m3 Rs. 8,000

Weight in aft, center, forward must be in

the same proportion as the holding

capacities by weight, i.e., 100 : 200 : 75

Formulate a linear programme to

maximize the profits of the shipping

company.

Solution to Example Problem No.6

Constraints on holding space

A final product is assembled with 4 units of component A and 3 units of component B.

The manufacturing shop runs three different processes, each of which requires varying amounts of raw materials and produce different amounts of A and B.

Two types of raw materials are used. 100 units of raw material I (RM I) and 200 units of raw material II (RM II) are available to the shop each day

Formulate a linear programme to maximize the

number of completed assemblies produced each

day.

Let X1, X2, X3 be the number of runs of each

process operated per day

Total quantity of A produced =

Total quantity of B produced =

No. of units of finished product??

A machine tool company conducts a job-training program for machinists

Trained machinists are used as teachers in the program at a ratio of one for every ten trainees. The training program lasts for one month (Assumption: No. of trainees – multiple of 10)

From past experience it has been found that out of ten trainees hired, only seven complete the program successfully (the unsuccessful trainees are released)

Trained machinists are also needed for machining and the company’s requirements for the next three months are as follows:

January 100 February 150 March 200

In addition, the company requires 250 trained machinists

by April. There are 130 trained machinists available at

the beginning of the year. Payroll costs per month are:

Each Trainee $ 400

Each trained machinist $ 700

(machining or teaching)

Each trained machinist idle $500

(Union contract forbids firing trained machinists)

Set up the linear programming problem that will produce

the minimum cost hiring and training schedule and meet

the company’s requirements.

Every month, a trained machinist can do one of the following:

(1) Work a machine, (2) Teach or (3) Stay idle Since the number of trained machinists for machining is

fixed, the only decision variables are the number teaching and the number idle for each month. Thus, the decision variables are:X1 – trained machinists teaching in January

X2 – trained machinists idle in January X3 - trained machinists teaching in February X4 – trained machinists idle in February X5 – trained machinists teaching in March X6 – trained machinists idle in March

Number machining + Number teaching + Number idle = Total trained machinists available at the beginning of the month

For January : 100 + X1 + X2 = 130For February : 150 + X3 + X4 = 130 + 7X1

For March : 200 + X5 + X6 = 130 + 7X1 + 7X3

Since the company requires 250 trained machinists by April,

130 + 7X1 + 7X3 + 7X5 = 250Objective Function: Minimize Z = 400 (10X1 + 10X3 +

10X5) +700 (X1 + X3 + X5) +500 (X2 + X4 + X6)

43

A company has two grades of inspectors, 1 and 2, who are to be assigned for a quality control inspection. It is required that at least 1800 pieces be inspected per 8-hour day. Grade 1 inspectors can check pieces at the rate of 25 per hour, with an accuracy of 98%. Grade 2 inspectors check at the rate of 15 pieces per hour, with an accuracy of 95%.

The wage rate of a Grade 1 inspector is $ 4.00 per hour, while that of a Grade 2 inspector is $ 3.00 per hour. Each time an error is made by an inspector, the cost to the company is $ 2.00. The company has available for the inspection job 10 Grade 1 inspectors, and 8 Grade 2 inspectors. The company wants to determine the optimal assignment of inspectors, which will minimize the total cost of the inspection.

A furniture shop manufactures tables and chairs.

The operations take place sequentially in two

work centers.

The associated profits and the man hours

required by each product at each work center are

shown in the table.

Formulate a linear program to determine the

optimal number of tables and chairs to be

manufactured so as to maximize the profit.

Furniture Shop Tables Chairs

Available

man hours

Profit / unit 8 6

Work Center-I 4 2 60

Work Center-II 2 4 48

1 2

1 2

1 2

1 2

8 6

4 2 60

2 4 48

, 0

Maximize Z x x

Subject to

x x

x x

x x

Max / Min Z = C1 x1 + C2 x2 +…..+ Cn xn

Subject to

a11 x1 + a12 x2 + ……+ a1n xn = b1

a21 x1 + a22 x2 + ……+ a2n xn = b2.

.

an1 x1 + an2 x2 + ……+ amn xn = bm

x1 , x2 ,….., xn - ≥ 0b1 , b2 ,….., bm - ≥ 0

Slack variable: Added to a ≤ constraint

2x1+ 3x2 ≤50

2x1+3x2+x3=50

Surplus variable: Added to a ≥ constraint

5x1+ 7x2 ≥120

5x1+ 7x2 -x3=120

Dealing with Unrestricted in Sign variables

x1 – URS

Let x2, x3 ≥ 0; x1 = x2 – x3

Infeasibility

◦ Occurs in problems where to satisfy one of the

constraints, another constraint must be violated.

Unbounded Problems

◦ Exists when the value of the objective function can be

increased without limit.

Multiple or Alternate Optimal Solutions

◦ Problems in which different combinations of values of the

decision variables yield the same optimal value.

Maximize Z = 10X1 +13X2 + 12X3

Subject to3X1 + 4X2 + 4X3 < 40

5X1 + 4X2 + 2X3 < 40

X1 > 0, X2 > 0, X3 > 0

◦ Enables the decision maker to determine how a change in

one of the values of a model will impact the optimal solution

and the optimal value of the objective function while

holding all other parameters constant.

◦ Provides the decision maker with greater insights about the

sensitivity of the optimal solution to changes in various

parameters of a problem.

◦ Change(s) in the value of objective function coefficient(s)

◦ Change(s) in the right-hand-side(RHS) value of constraint(s)

◦ Change in a coefficient of a constraint

The range of objective function values for which

the optimal values of the decision variables would

not change

A value of the objective function that falls within

the range of optimality will not change the

optimal mix of the variables, although the optimal

value of the objective function will change

The range of values over which the right-hand-side

(RHS) value can change without causing the shadow

price to change

Within the range of feasibility, the same decision

variables will remain optimal, although their values

and the optimal value of the objective function will

change

Analysis of RHS changes begins with determination

of a constraint’s shadow price in the optimal solution

Within the range of feasibility, the amount of

change in the optimal value of the objective

function per unit change of the RHS value of

a constraint

Maximization Problems

Minimization Problems

100% rule

A large automobile company owns 5 manufacturing plants in

India at the following locations:

- Jamshedpur

- Pune

- Lucknow

- Uttarakhand

- Sanand, Gujarat The automobiles (for simplicity, consider only one model)

produced are shipped to 4 regional warehouses located across

the country in:

- New Delhi (North)

- Kolkata (East)

- Bangalore (South)

- Mumbai (West) Weekly production in each plant is known Weekly demand at each warehouse is known

Possible distribution

routes

1

2

3

4

5

1

2

3

4

C11

C12

C14 C13

C21

C22

C23

C24

C51 C52

C53

C54

SUPPLY DEMAND

S1

S2

S3

S4

S5

D1

D2

D3

D4

Sources of Supply

Destinations

Notations:m - Number of sources

n - Number of destinations

Si - Supply at source i, i = 1, 2, 3, 4,5

Dj - Demand at destination j, j = 1, 2, 3, 4

Cij - Cost of transportation per unit from

source i to destination j

Xij - Number of units to be transported from source i

to destination j

The Transportation Problem seeks to find an allocation from the 5 sources to the 4 destinations such that the total cost of transportation is minimum

Destination

Source 1 2 3 4 Supply

1 C11 C12 C13 C14 S1

2 C21 C22 C23 C24 S2

3 C31 C32 C33 C34 S3

4 C41 C42 C43 C44 S4

5 C51 C52 C53 C54 S5

Demand D1 D2 D3 D4

11 11 12 12 21 21 22 22

31 31 32 32 41 41 42 42

51 51 52 52 53 53 54 54 55 55

11 12 13 14

..... .....

.... ......

Minimize Z C X C X C X C X

C X C X C X C X

C X C X C X C X C X

Subject to

X X X X

1 11 21 31 41 51 1

21 22 23 24 2 12 22 32 42 52 2

+ + D

+ + D

. .

.

S X X X X X

X X X X S X X X X X

51 52 53 54 5 15 25 35 45 55 5

.

+ + D

0 1,2,... 1, 2,...ij

X X X X S X X X X X

X i m and j n

1 1

1

1

, 1, 2,...

, 1, 2,...

0 1,2,... 1, 2,...

m n

ij iji j

n

ij ij

m

ij ji

ij

Minimize Z C X

Subject to

X S i m

X D j n

X i m and j n

1 1

1 1

1

1

, 1, 2,...

, 1, 2,...

0 1,2,... 1, 2,...

n n

i jj j

m n

ij iji j

n

ij ij

m

ij ji

ij

Suppose S D

Minimize Z C X

Subject to

X S i m

X D j n

X i m and j n

A company has seven manufacturing units situated in

different parts of the country.

Due to recession, it is proposed to close four of these and

to concentrate production in the remaining three units.

Production in these three units (E, F, and G) will actually

be increased from present levels and would require an

increase in the personnel employed in them.

Personnel at the closed units expressed their desire for

moving to any one of the remaining units and the

company is willing to provide them relocation expenses.

The retraining expenses would have to be incurred as

the technology in these units is different.

Not all existing personnel can be absorbed by transfer

and a few of them have to be retrenched.

As per labor laws, cost of retrenchment is given as a

general figure at each unit closed.

Formulate a LP model to determine the most economical

way to transfer/retrench personnel from units closed to

those units which will be expanded.

1

2

3

4

5

1

2

3

1

2

3

4

Destinations

Sources Transshipment nodes

Demand Constraints Supply Constraints

Transshipment nodes

11 21 31 1

12 22 32 2

13 23 33 3

14 24 34 4

D

D

D

D

Y Y Y

Y Y Y

Y Y Y

Y Y Y

11 12 13 1

21 22 23 2

31 32 33 3

41 42 43 4

51 52 53 5

X X X S

X X X S

X X X S

X X X S

X X X S

11 21 31 41 51 11 12 13 14

12 22 32 42 52 21 22 23 24

13 23 33 43 53 31 32 33 34

X X X X X Y Y Y Y

X X X X X Y Y Y Y

X X X X X Y Y Y Y

5 3 3 4'

1 1 1 1

4

1

3

1

5 4

1 1

1, 2,3,4 (Demand Constraints)

1, 2,3,4 (Supply Constraints)

1, 2,3 (Transshipment Constraints)

0 1,2,..

ij ij kl kli j k l

ij ji

ij ij

ij jki k

ij

Minimize C X C Y

Y D j

X S i

X Y j

X i

.5, 1,2,3

0 1,2,3, l 1, 2,3,4kl

j

Y k

A group of four boys and four girls are planning on a one day picnic. The extent of mutual happiness between boy i and girl j when they are together is given by the following matrix (data obtained from their previous dating experiences)

Girl

Boy

1 2 3 4

1 11 1 5 8

2 9 9 8 1

3 10 3 5 10

4 1 13 12 11

The problem is to decide the proper matching

between the boys and the girls during the picnic

that will maximize the sum of all the mutual

happiness of all the couples. Remember, a boy can

team-up with only one girl and vice-versa.

What are the decision variables?

Constraints?

Objective Function?

Decision Variable

1, if boy i teams-up with girl j

0, Otherwise

Constraints

Let Xij =

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

1

1

1

1

X X X X

X X X X

X X X X

X X X X

11 21 31 41

12 22 32 42

13 23 33 43

14 24 34 44

=1

=1

=1

=1

X X X X

X X X X

X X X X

X X X X

Objective Function

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

11 1 5 8

9 9 8 1

10 3 5 10

1 13 12 11

Maximize

X X X X

X X X X

X X X X

X X X X

1 1

1

1

/

1, 1,2,...

1, 1,2,...

1, if is assigned to

0, otherwise

n n

ij iji j

n

ijj

n

iji

ij

Maximize Minimize Z C X

Subject to

X i n

X j n

i jX

Minimum Spanning Tree Problem (MST)

Shortest Path Problem (SPP)

Maximal Flow Problem

In its simplest sense, a network is a set of

nodes/vertices with arcs/edges connecting them

Undirected/Directed

Connected Graph

Sub-graph

Tree: Connected sub-graph with no cycles

Spanning Tree

2

41

3

1. Dijkstra’s Algorithm – small problems

2. Mathematical Model - Complex networks

Constraints:1: X12+X13-X21-X31 = 12: X21+X23+X24-X12-X32-X42 = 03: X31+X32+X34+X35+X37-X13-X23-X43-X53-X73 = 04: X42+X43+X45+X46-X24-X34-X54-X64 = 05: X54+X53+X56+X57-X45-X35-X65-X75 = 06: X64+X65+X67-X46-X56-X76 = 07: X73+X75+X76-X37-X57-X67 = -1

1, if traversed from node to node

0, otherwise ij

i jX

1 1,

1; 1;

1, if is a source node

0, if is a transhipment node

1, if is a destination node

1, if traversed from to

0,

n n

ij iji j j i

n n

ij kij j i k k i

ij

Minimize Z C X

Subject to

i

X X i

i

i jX

otherwise

F

F

12 13 14

12 32 52 21 23 25

13 23 43 53 63 31 32 34 35 36

14 34 74 41 43 47

25 35 65 85 52 53 56 58

36 56 76 86 96 63 65 67 68 69

47 67 97 74 76 7

Maximize F

F f f f

f f f f f f

f f f f f f f f f f

f f f f f f

f f f f f f f f

f f f f f f f f f f

f f f f f f

9

58 68 98 10 8 85 86 89 8 10

69 79 89 10 9 96 97 98 9 10

8 10 9 10

f f f f f f f f

f f f f f f f f

f f F

12 21 13 31 14 41

25 52 23 32 34 43

35 53 36 63 47 74

20 0 15 0 22 0

18 0 20 10 0 5

20 0 13 0 18 0

f f f f f f

f f f f f f

f f f f f f

56 65 58 85 67 76

68 86 69 96 79 97

89 98 8 10 10 8 9 10 10

10 0 14 0 5 0

25 0 5 0 20 0

10 0 30 0 25

f f f f f f

f f f f f f

f f f f f f

9 0

Consider the following road network connecting six cities

1

2

3

4

5

6

(0,17,2)

(0,25,1)

(0,5,3)

(0,20,5)

(0,10,1)

(0,5,3)

(0,15,2)

(0,5,6)

(0,18,2)

(0,17,3)

The three numbers are the least amount you can ship along the arc, the maximum tonnage you can ship, and the cost in dollars per ton shipped along this arc, respectively. There are 30 tons of material at city 1 and it should be shipped to city 6 at minimum total cost. All materials originate at city 1 and end up in city 6.

Traveling Salesman Problem (TSP)

Facility Location Problem

Knapsack Problem

Set-Covering Problem

Set-Partitioning Problem

Integer programming problems do not

readily lend themselves to sensitivity

analysis as only a relatively few of the

infinite solution possibilities in a feasible

solution space will meet integer

requirements

Problem Description:

Given a list of cities and their pair-wise distances, the

problem is to find the shortest possible tour for a

salesman who starts from a city, visits each and every

city exactly once and comes back to the starting city

This problem is also known as Hamiltonian Circuit in

Graph Theory.

Number of feasible solutions for a ‘n’ city problem? For a 10 city problem:36,28,800 solutions For a 15 city problem:1,307,674,368,000!!!!! Optimal solution for 15,112 towns in Germany (2001)

- Network of 110 processors at Rice and Princeton

- Computational time equivalent: 22.6 years Optimal solution for 33,810 points on a PCB (2005) 85,900 points – Concorde TSP solver - current record

– April 2006 (136 CPU years)

Logistics: Vehicle routing

PCB manufacturing – Soldering using Robots

Gene mapping – DNA sequencing

Protein function prediction

1 1

1

1

1, 1,2,...

1, 1,2,...

1, if travelled from city to city

0, otherwise

Cost of traveling from city to city ( )

C Distance from cit

n n

ij iji j

n

ijj

n

iji

ij

ij

Minimize Z C X

Subject to

X i n

X j n

i jX

i j or

y to city (or)

Travel time between city to city

i j

i j

1 1

2 1

1 ( 1)(1 ) 1, 1

j

i j ij

u

u n j

u u n X i j

• Miller-Tucker-Zemlin (MTZ) Formulation (1960)

Decision 1:

Where to locate the

facilities?

Decision 2:

From each chosen facility,

how much should be

shipped to the destinations

for satisfying the demand?

Costs incurred:

1. Fixed cost of locating a

facility (Fi)

2. Transportation costs (Cij)

Mumbai

Chennai

Delhi

Bangalore

Kolkata

1

2

3

4

D1

D2

D3

D4

Destinations (n)

Set of Potential locations for facilities (m)

Hyderabad

Lucknow

Surat

Pune

Kochi

Vizag

Ahmedabad

1 1 1

1

1

Z=

1.......

1.......

1,if location is selected

0,otherwise

Quantity shipped from facility

m m n

i i ij iji i j

n

ij i ij

m

ij ji

i

ij

Minimize F X C y

Subject to

y S X i m

y D j n

iX

y

to destination

Supply at facility

Demand at destination i

j

i j

S i

D j

A thief breaks into a museum. Fabulous paintings,

sculptures, and jewels are everywhere. The thief has a good eye for the value of these objects,

and knows that each will fetch hundreds or thousands of

dollars on the clandestine art collector’s market. But, the thief has only brought a single knapsack to the

scene of the robbery, and can take away only what he

can carry. What items should the thief take to maximize the haul?

More formally:

The thief must choose among n items, where the i th

item is worth Vi dollars and weighs Wi pounds

Carrying at most ‘K’ pounds, maximize the value of

knapsack

Note: assume Vi, Wi, and K are all integers

An item must be taken or left in entirety

Applications: Resource utilization, Capital investments

and Financial portfolios, Cryptography etc

1

1

Z=

1,if object is selected

0,otherwise

n

j jj

n

j jj

j

Maximize V X

Subject to

W X K

jX

Node1 : 1, 2, 7, 8, 9

1

2 3

4

56

7

8

910

Node3 : 1, 2, 3, 4, 5, 10

Node5 : 3, 4, 5, 6, 9, 10

Node7 : 1, 5, 6, 7, 8, 9, 10

Node9 : 3, 6, 7, 8, 9,10

Cost of opening 5 nodes:Node 1: 125Node 3: 85Node 5: 70Node 7: 100Node 9: 110

Determine which nodes should be opened to

provide coverage to all neighborhoods at a

minimum cost?

1 3 5 7 9

Cost 125 85 70 100 110

1 1 1 0 1 0

2 1 1 0 0 0

3 0 1 1 0 1

4 0 1 1 0 0

5 0 1 1 1 0

6 0 0 1 1 1

7 1 0 0 1 1

8 1 0 0 1 1

9 1 0 1 1 1

10 0 1 1 1 1

Localities

Service Nodes

1 3 5 7 9

1 3 7

1 3

3 5 9

3 5

3 5 7

1,if node provides service (j = 1,3,5,7,9)

0,otherwise

Z=125 85 70 100 110

1

1

1

1

1

j

jX

Minimize X X X X X

Subject to

X X X

X X

X X X

X X

X X X

5 7 9

1 7 9

1 5 7 9

3 5 7 9

1

1

1

1

X X X

X X X

X X X X

X X X X

1

1

Z=

1 1.....

0,1

n

j jj

n

ij jj

j

Minimize C X

Subject to

a X i m

X

Applications:

1.Vehicle routing problems

2.Facility location

3.Airline crew scheduling

4.Circuit design

5.Resource allocation

6.Capital investment