transportation and assignment models
TRANSCRIPT
D0010
* 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
D0010
* Property of STI
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Transportation and Assignment Models
Transportation Problem
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
D0010
* Property of STI
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Transportation and Assignment Models
Transportation Problem
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
D0010
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Transportation and Assignment Models
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
D0010
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Transportation and Assignment Models
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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Transportation and Assignment Models
Steps in Solving the Transportation
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
Project A Project B Project C
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
D0010
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Transportation and Assignment Models
Steps in Solving the Transportation
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
D0010
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Transportation and Assignment Models
Steps in Solving the Transportation
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
D0010
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Transportation and Assignment Models
Steps in Solving the Transportation
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
Project A Project B Project C
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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Transportation and Assignment Models
Steps in Solving the Transportation
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
Project A Project B Project C Plant
Capacity
Plant W 56
41
Plant X 82
Plant Y 77
Requirements
Project 72 102
56
16 66
36 41
D0010
* Property of STI
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Transportation and Assignment Models
Steps in Solving the Transportation
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
D0010
* Property of STI
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Transportation and Assignment Models
Steps in Solving the Transportation
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
Project A Project B Project C
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
2,968 Total Transportation Cost
D0010
* Property of STI
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Transportation and Assignment Models
Steps in Solving the Transportation
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
D0010
* Property of STI
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Transportation and Assignment Models
Steps in Solving the Transportation
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
Project A Project B Project C Plant
Capacity
Plant W 56
41
Plant X 82
Plant Y 77
Requirements
Project 72 102
31
41
25
77
41
D0010
* Property of STI
Page 15 of 23
Transportation and Assignment Models
Steps in Solving the Transportation
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
2,868 Total Transportation Cost
Unused
Square
Closed Path Computation of
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
D0010
* Property of STI
Page 16 of 23
Transportation and Assignment Models
Steps in Solving the Transportation
Problem
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
41
Plant X 82
Plant Y 77
Requirements
Project 72 102
31
41
56
46
41
Unused
Square
Closed Path Computation of
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
D0010
* Property of STI
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Transportation and Assignment Models
Steps in Solving the Transportation
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
2,744 Total Transportation Cost
D0010
* Property of STI
Page 18 of 23
Transportation and Assignment Models
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
Project A Project B Project C Plant
Capacity
Plant W 56
41
Plant X 82
Plant Y 77
Requirements
Project 72 102
72
41
56
5
41
D0010
* Property of STI
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Transportation and Assignment Models
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
Project A Project B Project C
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
D0010
* Property of STI
Page 20 of 23
Transportation and Assignment Models
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
Project A Project B Project C
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
D0010
* Property of STI
Page 21 of 23
Transportation and Assignment Models
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
Project A Project B Project C
16
56
77
25 41
D0010
* Property of STI
Page 22 of 23
Transportation and Assignment Models
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
Project A Project B Project C
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
Project A Project B Project C
41
31 46
56
41