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7/29/2019 Distn & Network Models
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Slide 2008 Thomson South-Western. All Rights Reserved
Slides by
JOHNLOUCKSSt. EdwardsUniversity
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Chapter 6, Part ADistribution and Network Models
Transportation ProblemNetwork RepresentationGeneral LP Formulation
Assignment Problem
Network Representation
General LP Formulation
Transshipment Problem
Network RepresentationGeneral LP Formulation
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Transportation, Assignment, andTransshipment Problems
A network model is one which can be represented
by a set of nodes, a set of arcs, and functions (e.g.costs, supplies, demands, etc.) associated with thearcs and/or nodes.
Transportation, assignment, transshipment
problems of this chapter as well as shortest-route,and maximal flow , minimal spanning tree andPERT/CPM problems (in others chapter) are allexamples of network problems.
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Transportation, Assignment, andTransshipment Problems
Each of the three models of this chapter can be
formulated as linear programs and solved bygeneral purpose linear programming codes.
For each of the three models, if the right-hand sideof the linear programming formulations are all
integers, the optimal solution will be in terms ofinteger values for the decision variables.
However, there are many computer packages(including The Management Scientist) that contain
separate computer codes for these models whichtake advantage of their network structure.
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Transportation Problem
The transportation problem seeks to minimize the
total shipping costs of transporting goods from morigins (each with a supply si) to n destinations(each with a demand dj), when the unit shippingcost from an origin, i, to a destination,j, is cij.
The network representation for a transportationproblem with two sources and three destinations isgiven on the next slide.
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Transportation Problem
Network Representation
2
c11
c12
c13
c21
c22
c23
d1
d2
d3
s1
s2
Sources Destinations
3
2
1
1
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Transportation Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from
origin i to destinationj
cij= cost per unit of shipping fromorigin i to destinationj
si = supply or capacity in units at origin i
dj = demand in units at destinationj
continued
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Transportation Problem
Linear Programming Formulation (continued)
1 1
Min
m n
ij iji j
c x
1
1,2, , Supplyn
ij ij
x s i m
1
1,2, , Demandm
ij ji
x d j n
xij > 0 for all i andj
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LP Formulation Special Cases
The objective is maximizing profit or revenue:
Minimum shipping guarantee from i toj:
xij > Lij
Maximum route capacity from i toj:
xij < LijUnacceptable route:
Remove the corresponding decision variable.
Transportation Problem
Solve as a maximization problem.
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Transportation Problem: Example #1
Acme Block Company has orders for 80 tons of
concrete blocks at three suburban locations
as follows: Northwood -- 25 tons,
Westwood -- 45 tons, and
Eastwood -- 10 tons. Acmehas two plants, each of which
can produce 50 tons per week.
Delivery cost per ton from each plant
to each suburban location is shown on the next slide.How should end of week shipments be made to fill
the above orders?
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Delivery Cost Per Ton
Northwood Westwood Eastwood
Plant 1 24 30 40
Plant 2 30 40 42
Transportation Problem: Example #1
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Partial Spreadsheet Showing Problem Data
Transportation Problem: Example #1
A B C D E F G H
1
2 Constraint X11 X12 X13 X21 X22 X23 RHS
3 #1 1 1 1 50
4 #2 1 1 1 505 #3 1 1 25
6 #4 1 1 45
7 #5 1 1 10
8 Obj.Coefficients 24 30 40 30 40 42 30
LHS Coefficients
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Partial Spreadsheet Showing Optimal Solution
Transportation Problem: Example #1
A B C D E F G
10 X11 X12 X13 X21 X22 X23
11 Dec.Var.Values 5 45 0 20 0 10
12 Minimized Total Shipping Cost 2490
1314 LHS RHS
15 50
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Optimal Solution
From To Amount Cost
Plant 1 Northwood 5 120
Plant 1 Westwood 45 1,350
Plant 2 Northwood 20 600
Plant 2 Eastwood 10 420
Total Cost = $2,490
Transportation Problem: Example #1
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Partial Sensitivity Report (first half)
Transportation Problem: Example #1
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30
$E$12 X13 0 4 40 1E+30 4
$F$12 X21 20 0 30 4 4
$G$12 X22 0 4 40 1E+30 4
$H$12 X23 10.000 0.000 42 4 1E+30
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30
$E$12 X13 0 4 40 1E+30 4
$F$12 X21 20 0 30 4 4
$G$12 X22 0 4 40 1E+30 4
$H$12 X23 10.000 0.000 42 4 1E+30
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Partial Sensitivity Report (second half)
Transportation Problem: Example #1
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$E$17 P2.Cap 30.0 0.0 50 1E+30 20
$E$18 N.Dem 25.0 30.0 25 20 20
$E$19 W.Dem 45.0 36.0 45 5 20
$E$20 E.Dem 10.0 42.0 10 20 10
$E$16 P1.Cap 50.0 -6.0 50 20 5
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$E$17 P2.Cap 30.0 0.0 50 1E+30 20
$E$18 N.Dem 25.0 30.0 25 20 20
$E$19 W.Dem 45.0 36.0 45 5 20
$E$20 E.Dem 10.0 42.0 10 20 10
$E$16 P1.Cap 50.0 -6.0 50 20 5
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Transportation Problem: Example #2
The Navy has 9,000 pounds of material in Albany,
Georgia that it wishes to ship to three installations:
San Diego, Norfolk, and Pensacola. They
require 4,000, 2,500, and 2,500 pounds,
respectively. Government regulations
require equal distribution of shipping
among the three carriers.
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The shipping costs per pound for truck, railroad,
and airplane transit are shown on the next slide.
Formulate and solve a linear program to
determine the shipping arrangements
(mode, destination, and quantity) thatwill minimize the total shipping cost.
Transportation Problem: Example #2
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Destination
Mode San Diego Norfolk Pensacola
Truck $12 $ 6 $ 5
Railroad 20 11 9
Airplane 30 26 28
Transportation Problem: Example #2
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Define the Decision Variables
We want to determine the pounds of material, xij ,to be shipped by mode i to destinationj. Thefollowing table summarizes the decision variables:
San Diego Norfolk PensacolaTruck x11 x12 x13
Railroad x21 x22 x23
Airplane x31 x32 x33
Transportation Problem: Example #2
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Define the Objective Function
Minimize the total shipping cost.
Min: (shipping cost per pound for each mode perdestination pairing) x (number of pounds shipped
by mode per destination pairing).Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23
+ 30x31 + 26x32 + 28x33
Transportation Problem: Example #2
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Define the Constraints
Equal use of transportation modes:(1) x11 + x12 + x13 = 3000
(2) x21 + x22 + x23 = 3000
(3) x31 + x32 + x33 = 3000
Destination material requirements:
(4) x11 + x21 + x31 = 4000
(5) x12 + x22 + x32 = 2500
(6) x13 + x23 + x33 = 2500Non-negativity of variables:
xij > 0, i = 1,2,3 and j = 1,2,3
Transportation Problem: Example #2
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The Management Scientist Output
OBJECTIVE FUNCTION VALUE = 142000.000
Variable Value Reduced Costx11 1000.000 0.000
x12 2000.000 0.000x13 0.000 1.000x21 0.000 3.000x22 500.000 0.000x23 2500.000 0.000
x31 3000.000 0.000x32 0.000 2.000x33 0.000 6.000
Transportation Problem: Example #2
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Solution Summary
San Diego will receive 1000 lbs. by truckand 3000 lbs. by airplane.
Norfolk will receive 2000 lbs. by truck
and 500 lbs. by railroad.
Pensacola will receive 2500 lbs. by railroad.
The total shipping cost will be $142,000.
Transportation Problem: Example #2
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Assignment Problem
An assignment problem seeks to minimize the total
cost assignment of m workers to m jobs, given thatthe cost of worker i performing jobj is cij.
It assumes all workers are assigned and each job isperformed.
An assignment problem is a special case of atransportation problem in which all supplies and alldemands are equal to 1; hence assignment problemsmay be solved as linear programs.
The network representation of an assignment
problem with three workers and three jobs is shownon the next slide.
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Assignment Problem
Network Representation
2
3
1
2
3
1c11
c12c13
c21c22
c23
c31 c32
c33
Agents Tasks
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Linear Programming Formulation
Using the notation:
xij = 1 if agent i is assigned to taskj
0 otherwise
cij= cost of assigning agent i to taskj
Assignment Problem
continued
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Linear Programming Formulation (continued)
Assignment Problem
1 1
Min
m n
ij iji j
c x
1
1 1,2, , Agentsn
ij
j
x i m
1
1 1,2, , Tasksm
iji
x j n
xij > 0 for all i andj
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LP Formulation Special Cases
Number of agents exceeds the number of tasks:
Number of tasks exceeds the number of agents:
Add enough dummy agents to equalize thenumber of agents and the number of tasks.The objective function coefficients for thesenew variable would be zero.
Assignment Problem
Extra agents simply remain unassigned.
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Assignment Problem
LP Formulation Special Cases (continued)
The assignment alternatives are evaluated in termsof revenue or profit:
Solve as a maximization problem.
An assignment is unacceptable:
Remove the corresponding decision variable.
An agent is permitted to work t tasks:
1
1,2, , Agentsn
ijj
x t i m
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An electrical contractor pays his subcontractors a
fixed fee plus mileage for work performed. On a givenday the contractor is faced with three electrical jobsassociated with various projects. Given below are thedistances between the subcontractors and the projects.
ProjectsSubcontractor A B CWestside 50 36 16Federated 28 30 18Goliath 35 32 20Universal 25 25 14
How should the contractors be assigned to minimizetotal mileage costs?
Assignment Problem: Example
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Network Representation
50
36
16
28
30
18
35 32
2025 25
14
West.
C
B
A
Univ.
Gol.
Fed.
ProjectsSubcontractors
Assignment Problem: Example
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Linear Programming Formulation
Min 50x11+36x12+16x13+28x21+30x22+18x23
+35x31+32x32+20x33+25x41+25x42+14x43s.t. x11+x12+x13 < 1
x21+x22+x23 < 1x31+x32+x33 < 1
x41+x42+x43 < 1
x11+x21+x31+x41 = 1
x12+x22+x32+x42 = 1x13+x23+x33+x43 = 1
xij = 0 or 1 for all i andj
Agents
Tasks
Assignment Problem: Example
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The optimal assignment is:
Subcontractor Project Distance
Westside C 16
Federated A 28
Goliath (unassigned)
Universal B 25
Total Distance = 69 miles
Assignment Problem: Example
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Transshipment Problem
Transshipment problems are transportation problems
in which a shipment may move through intermediatenodes (transshipment nodes)before reaching aparticular destination node.
Transshipment problems can be converted to largertransportation problems and solved by a specialtransportation program.
Transshipment problems can also be solved bygeneral purpose linear programming codes.
The network representation for a transshipment
problem with two sources, three intermediate nodes,and two destinations is shown on the next slide.
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Transshipment Problem
Network Representation
2
3
4
5
6
7
1
c13
c14
c23
c24c25
c15
s1
c36
c37
c46c47
c56
c57
d1
d2
Intermediate Nodes
Sources Destinationss2
DemandSupply
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Transshipment Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from node i to nodej
cij = cost per unit of shipping from node i to nodej
si= supply at origin node idj= demand at destination node j
continued
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Transshipment Problem
all arcs
Min ij ijc x
arcs out arcs in
s.t. Origin nodesij ij ix x s i
xij > 0 for all i andj
arcs out arcs in
0 Transhipment nodesij ijx x
arcs in arcs out
Destination nodesij ij jx x d j
Linear Programming Formulation (continued)
continued
all arcs
Min ij ijc x
h bl
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Transshipment Problem
LP Formulation Special Cases
Total supply not equal to total demandMaximization objective function
Route capacities or route minimums
Unacceptable routes
The LP model modifications required here are
identical to those required for the special cases in
the transportation problem.
h P bl l
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The Northside and Southside facilities
of Zeron Industries supply three firms(Zrox, Hewes, Rockrite) with customizedshelving for its offices. They both ordershelving from the same two manufacturers,
Arnold Manufacturers and Supershelf, Inc.Currently weekly demands by the users
are 50 for Zrox, 60 for Hewes, and 40 forRockrite. Both Arnold and Supershelf cansupply at most 75 units to its customers.
Additional data is shown on the nextslide.
Transshipment Problem: Example
T hi P bl E l
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Because of long standing contracts based on
past orders, unit costs from the manufacturers to thesuppliers are:
Zeron N Zeron SArnold 5 8
Supershelf 7 4
The costs to install the shelving at the variouslocations are:
Zrox Hewes RockriteZeron N 1 5 8Zeron S 3 4 4
Transshipment Problem: Example
T hi P bl E l
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Network Representation
ARNOLD
WASH
BURN
ZROX
HEWES
75
75
50
60
40
5
8
7
4
15
8
3
44
Arnold
SuperShelf
Hewes
Zrox
ZeronN
ZeronS
Rock-Rite
Transshipment Problem: Example
T hi P bl E l
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Linear Programming Formulation
Decision Variables Definedxij = amount shipped from manufacturer i to supplierj
xjk = amount shipped from supplierj to customer k
where i = 1 (Arnold), 2 (Supershelf)
j = 3 (Zeron N), 4 (Zeron S)
k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)
Objective Function Defined
Minimize Overall Shipping Costs:Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37
+ 3x45 + 4x46 + 4x47
Transshipment Problem: Example
T hi t P bl E l
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Constraints Defined
Amount Out of Arnold: x13 + x14 < 75Amount Out of Supershelf: x23 + x24 < 75
Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0
Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0
Amount Into Zrox: x35 + x45 = 50
Amount Into Hewes: x36 + x46 = 60
Amount Into Rockrite: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i andj.
Transshipment Problem: Example
T hi t P bl E l
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The Management Scientist Solution
Objective Function Value = 1150.000
Variable Value Reduced CostsX13 75.000 0.000
X14 0.000 2.000X23 0.000 4.000X24 75.000 0.000X35 50.000 0.000X36 25.000 0.000
X37 0.000 3.000X45 0.000 3.000X46 35.000 0.000X47 40.000 0.000
Transshipment Problem: Example
T hi t P bl E l
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Solution
ARNOLD
WASH
BURN
ZROX
HEWES
75
75
50
60
40
5
8
7
4
15
8
3 4
4
Arnold
SuperShelf
Hewes
Zrox
Zeron
N
ZeronS
Rock-Rite
75
Transshipment Problem: Example
T hi t P bl E l
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The Management Scientist Solution (continued)
Constraint Slack/Surplus Dual Prices
1 0.000 0.000
2 0.000 2.000
3 0.000 -5.0004 0.000 -6.000
5 0.000 -6.000
6 0.000 -10.000
7 0.000 -10.000
Transshipment Problem: Example
T hi t P bl E l
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The Management Scientist Solution (continued)
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper LimitX13 3.000 5.000 7.000
X14 6.000 8.000 No LimitX23 3.000 7.000 No LimitX24 No Limit 4.000 6.000X35 No Limit 1.000 4.000X36 3.000 5.000 7.000
X37 5.000 8.000 No LimitX45 0.000 3.000 No LimitX46 2.000 4.000 6.000X47 No Limit 4.000 7.000
Transshipment Problem: Example
T hi t P bl E l
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The Management Scientist Solution (continued)
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 75.000 75.000 No Limit
2 75.000 75.000 100.0003 -75.000 0.000 0.000
4 -25.000 0.000 0.000
5 0.000 50.000 50.000
6 35.000 60.000 60.0007 15.000 40.000 40.000
Transshipment Problem: Example
End of Chapter 6 Part A
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