transportation and assignment models

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* Property of STI

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Transportation and Assignment Models

TRANSPORTATION AND ASSIGNMENT

MODELS

Transportation Problem

Introduction

Concerned with selecting routes in a product-distribution network.

Involves assigning employees to tasks, salespersons to territories, contracts to bidders, or jobs to plants.

Assignment Problem

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Roxas Gravel Company

Problem

Schedule shipments from each plant to each

project to minimize the total transportation cost

within the constraints imposed by plant capacities

and project requirement.

Project Location Weekly

Requirement

Truckloads

A Manila 72

B Makati 102

C Parañaque 41

Total 215

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Roxas Gravel Company

Plant Location Weekly

Requirement

Truckloads

W Laguna 56

X Rizal 82

Y Batangas 77

Total 215

From To Project A To Project B To Project C

Plant W 4 8 8

Plant X 16 24 16

Plant Y 8 18 24

Cost per Truckload

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Linear Programming Formulation

Roxas

Gra

vel

Com

pany

Min

imiz

e:

4W

A +

8W

B +

8W

C +

16XA +

24XB +

16XC +

16YB +

24YC

Subje

ct

to:

W

A +

WB + W

C ≤

5

6

: P

lant

W

XA + X

B +

X

C ≤

8

2

: P

lant

X

YA

+ Y

A +

Y

C ≤

7

7

: P

lant

Y

W

A +

XA + Y

A ≥

7

2

: P

roje

ct

A

W

B +

X

B +

Y

B ≥

102

: Pro

ject

B

A

ll v

ari

able

s ≥

0

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Steps in Solving the Transportation

Problem

Step 1: Set-up the Transportation Tableau

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215

Project

Requirements

77

72 102 41

Plant X

Plant Y

Plant

Capacity

Project A Project B Project C

Plant W 56

82

X1

X4

X7

X2

X5

X8

X3

X6

X9

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Problem

Step 2: Develop an Initial Solution

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215

Plant

Capacity

56


Plant W

Requirements

82

Plant Y 77

72 102 41Project

Plant X

56

16 66

36 41

From

Plant

To

Project

Quantity

Truckload/Week

W A 56

X A 16

X B 66

Y B 36

Y C 41

Total 215

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Problem

Source - Destination

Combination

Quantity

Shipped

x Unit Cost = Total

Cost

WA 56 4 224

XA 16 16 256

XB 66 24 1,584

YB 36 16 576

YC 41 24 984

3,624 Total Transportation Cost

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Problem

Step 3: Test the Solution for Improvement

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215

Project A Project B Project C Plant

Capacity

Plant W 56

Plant X 82

Requirements

Plant Y 77

Project 72 102 41

56

16 66

36 41

+-

+ -

Computing the Improvement Index

Addition to cost: From plant W to project B 8

Front plant X to project A 16 24

Reduction in cost: From plant W to project A 4

From plant X to project B 24 28

-P4

OR

Improvement index for square WB = WB - WA + XA - XB

= P8 - 4 + 16 - 2

=-P4

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Problem

Steps in Evaluating any Unused Square

a. Choose the unused square to be evaluated.

b. Trace a closed path (moving horizontally and

vertically).

c. Assign plus (+) and minus (-) signs.

d. Determine the net change in costs.

e. Repeat the above steps until an improvement

index has been determined.

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215


Plant X

Requirements

Plant

Capacity

Plant W 56

82

Plant Y 77

Project 56 56 56

56

16 66

36 41

+-

+ -

+ -

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Problem

Improvement index for

WC =WC - WA + XA - XB + YB - YC

= P8 - 4 + 16 - 24 + 16 - 24

= -P12

Path for square XC: (+)XC (-)XB (+)YB(-)YC

Improvement index for XC = XC - XB + YB - YC

= P16 - 24 + 16 - 24

= -P16

Path for square YA: (+) YA (-)XA (+) XB (-) YB

Improvement index for YA = YA - XA + XB - YB

= P8 - 16 + 24 - 16

= P0

To

From

WA 4 WB 8 WC 8

-4 -12

XA 16 XB 24 XC 16

-16

YA 8 YB 16 YC 24

215

215


Capacity

Plant W 56

41

Plant X 82

Plant Y 77

Requirements

Project 72 102

56

16 66

36 41

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* Property of STI

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Problem

Step 4: Develop the Improved Solution

o Select the route with the most negative

improvement index.

o Reconstruct the closed path traced in

evaluating unused square.

XB 24 XC 16

YB 16 YC 24

66

36 41

+-

-+

XB - XC +

YB + YC -

66-41=

36+41=

0+41=25

77

41

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* Property of STI

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Problem

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215Requirements

Project 72 102 41

Plant X 82

Plant Y 77

Plant

Capacity

Plant W 56


56

16 25

77

41

Shipping

Assignments

Quantity

Shipped

x Unit Cost = Total

Cost

WA 56 4 224

XA 16 16 256

XB 25 24 600

XC 41 16 656

YB 77 16 1,232


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Problem

Unused

Square

Closed Path Computation of

Improvement Index

WB + WB - WA + XA - XB + 8 - 4 + 16 - 24 = -4

WC + WC - WA + XA - XC + 8 - 4 + 16 - 16 = +4

YA + YA - XA + XB - YB + 8 - 16 + 24 - 16 = 0

YC + YC - XC + XB - YB + 24 - 16 + 24 - 16 = +16

WA WB WC

XA XB XC

56

1625

41

+

-+

-

+WB - WA + XA - XB + - + -56 16 25

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Problem

WA WB

XA XB

25-25=

56-25= 0+25=

16+25=

31

77

25

0

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215


Capacity

Plant W 56

41

Plant X 82

Plant Y 77

Requirements

Project 72 102

31

41

25

77

41

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Problem

Shipping

Assignments

Quantity

Shipped

x Unit Cost = Total

Cost

WA 31 4 124

WB 25 8 200

XA 41 16 656

XC 41 16 656

YB 77 16 1,232


Unused

Square


Improvement Index

WC + WC - WA + XA - XC + 8 - 4 + 16 - 16 = +4

XB + XB - WB + WA - XA + 24 - 8 + 4 - 16 = +4

YA + YA - WA + WB - YB + 8 - 4 + 8 - 16 = -4

YC +YC-YB+WB-WA+XA-XC +24-16+8-4+16-16=+12

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* Property of STI

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Problem

To

From

WA 4 WB 8 WC 8

XA 16 XB 24 XC 16

YA 8 YB 16 YC 24

215

215


Capacity

Plant W 56

41

Plant X 82

Plant Y 77

Requirements

Project 72 102

31

41

56

46

41

Unused

Square


Improvement Index

WC + WC - WA + XA - XC + 8 - 4 + 16 - 16 = +4

XB + XB - WB + WA - XA + 24 - 8 + 4 - 16 = +4

YA + YA - WA + WB - YB + 8 - 4 + 8 - 16 = -4

YC +YC-YB+WB-WA+XA-XC +24-16+8-4+16-16=+12

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* Property of STI

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Problem

Shipping

Assignments

Quantity

Shipped

x Unit Cost = Total

Cost

WB 56 8 448

XA 41 16 656

XC 41 16 656

YA 31 8 248

YB 46 16 736


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Alternative Optimal Solutions

To

From

WA 4 WB 8 WC 8

+4 +8

XA 16 XB 24 XC 16

0

YA 8 YB 16 YC 24

-16

215

215


Capacity

Plant W 56

41

Plant X 82

Plant Y 77

Requirements

Project 72 102

72

41

56

5

41

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* Property of STI

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MODI Method

Ri Kj

To

From

R1 WA 4 WB 8 WC 8

R2 XA 16 XB 24 XC 16

R3 YA 8 YB 16 YC 24

215

215


82

Plant Y 77

Plant

Capacity

Plant W 56

Requirements

K1 K2 K3

Project 72 102 41

Plant X

4

1

16

56

3

6

6

6

Ri = value assigned to row i

Kj = value assigned to column j

Cij = cost in square ij (the square of row i and

column j)

Cij= Ri + Kj

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MODI Method

Ri Kj

K1=4

To

From

R1=0 WA 4 WB 8 WC 8

R2=12 XA 16 XB 24 XC 16

R1=4 YA 8 YB 16 YC 24

215

215


82

Plant Y 77

Plant

Capacity

Plant W 56

Requirements

K2=12 K3=20

Project 72 102 41

Plant X

4

1

16

56

3

6

6

6

Unused

Square

Cij - Rj - Kj Improvement Index

12 C12

- R1 - K

2

8 - 0 -12 -4

13 C13

- R1 - K

3

8 - 0 - 20 -12

23 C23

- R2 - K

3

16 - 12 - 20 -16

31 C31

- R3 - K

1

8 - 4 - 4 0

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Developing a New Solution

Procedures for Developing a New, Improved

Solution:

o Trace a closed path.

o Place plus and minus sign at alternate corners of

the path.

o The quantity in the smallest stone in a negative

position is added to all squares on the closed

path with plus signs and subtracted from those

with minus signs.

o Improvement indices are calculated.

Ri Kj

K1=4

To

From

R1=0 WA 4 WB 8 WC 8

R2=12 XA 16 XB 24 XC 16

R1=4 YA 8 YB 16 YC 24

215

215Requirements

K2=12 K3=4

Project 72 102 41

Plant X 82

Plant Y 77

Plant

Capacity

Plant W 56


16

56

77

25 41

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* Property of STI

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Developing a New Solution

Ri Kj

K1=4

To

From

R1=0 WA 4 WB 8 WC 8

+4

R2=12 XA 16 XB 24 XC 16

+4

R1=8 YA 8 YB 16 YC 24

-4 +12

215

215Requirements

K2=8 K3=4

Project 72 102 41

Plant X 82

Plant Y 77

Plant

Capacity

Plant W 56


41

31

77

25

41

Ri Kj

K1=4

To

From

R1=0 WA 4 WB 8 WC 8

+4 +8

R2=16 XA 16 XB 24 XC 16

0

R1=8 YA 8 YB 16 YC 24

+16

215

215Requirements

K2=8 K3=0

Project 72 102 41

Plant X 82

Plant Y 77

Plant

Capacity

Plant W 56


41

31 46

56

41

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Summary of MODI Method Steps

1. Compute R and K values

2. Calculate the improvement indices

3. Select the unused square

4. Trace the closed path

5. Develop an improved solution

6. Repeat steps 1 to 5

Data & Analytics

transportation and assignment models