Transcript Chapter 11 Supplement Operational Decision-Making Tools: Transportation and Transshipment Models Operations Management - 6hh Edition Roberta Russell & Bernard W.

Chapter 11 Supplement
Operational Decision-Making Tools:
Transportation and Transshipment Models
Operations Management - 6hh Edition
Roberta Russell & Bernard W. Taylor, III
Copyright 2006 John Wiley & Sons, Inc.
Beni Asllani
University of Tennessee at Chattanooga
Just how do you make
decisions?
 Emotional direction
 Intuition
 Analytic thinking
 Are you an intuit, an analytic, what???
 How many of you use models to make
decisions??
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-2
Problems
 Arise whenever there is a perceived
difference between what is desired and
what is in actuality.
 Problems serve as motivators for doing
something
 Problems lead to decisions
42
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-4
Model Classification Criteria
 Purpose
 Perspective





Use the perspective of the targeted decision-maker
Degree of Abstraction
Content and Form
Decision Environment
{This is what you should start any modeling
facilitation meeting with}
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-5
Purpose
 Planning
 Forecasting
 Training
 Behavioral research
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-6
Perspective
 Descriptive


“Telling it like it is”
Most simulation models are of this type
 Prescriptive


“Telling it like it should be”
Most optimization models are of this type
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-7
Degree of Abstraction
 Isomorphic

One-to-one
 Homomorphic

One-to-many
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Supplement 10-8
Content and Form
 verbal descriptions
 mathematical constructs
 simulations
 mental models
 physical prototypes
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Supplement 10-9
Decision Environment
 Decision Making Under Certainty

TOOL: all of mathematical programming
 Decision Making under Risk and
Uncertainty

TOOL: Decision analysis--tables, trees,
Bayesian revision
 Decision Making Under Change and
Complexity

TOOL: Structural models, simulation models
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-10
Mathematical Programming
 Linear programming
 Integer linear programming

some or all of the variables are integer variables
 Network programming (produces all
integer solutions)




Nonlinear programming
Dynamic programming
Goal programming
The list goes on and on

Geometric Programming
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-11
A Model of this class
 What would we include in it?
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-12
Management Science Models
 A QUANTITATIVE REPRESENTATION
OF A PROCESS THAT CONSISTS OF
THOSE COMPONENTS THAT ARE
SIGNIFICANT FOR THE PURPOSE
BEING CONSIDERED
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-13
Mathematical programming models
covered in Ch 11, Supplement
 Transportation Model
 Transshipment Model
Not included are:
Shortest Route
Minimal Spanning Tree
Maximal flow
Assignment problem
many others
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-14
Transportation Model
 A transportation model is formulated for a class of
problems with the following characteristics



a product is transported from a number of sources to a
number of destinations at the minimum possible cost
each source is able to supply a fixed number of units of
product
each destination has a fixed demand for the product
 Solution (optimization) Algorithms

stepping-stone
modified distribution

Excel’s Solver

Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-15
Transportation Method: Example
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-16
Transportation Method: Example
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-17
Problem
Formulation
Using Excel
Total Cost
Formula
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Supplement 10-18
Using Solver
from Tools
Menu
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Supplement 10-19
Solution
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Supplement 10-20
Modified
Problem
Solution
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Supplement 10-21
The Underlying Network
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-22
For problems in which there is an
underlying network:
 There are easy (fast) solutions

An exception is the traveling salesman
problem
 The solutions are always integer ones
 {How about solving a 50,000 node
problem in less than a minute on a
laptop??}
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-23
CARLTON PHARMACEUTICALS
 Carlton Pharmaceuticals supplies drugs and other
medical supplies.
 It has three plants in: Cleveland, Detroit,
Greensboro.
 It has four distribution centers in:
Boston, Richmond, Atlanta, St. Louis.
 Management at Carlton would like to ship cases
of a certain vaccine as economically as possible.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-24
 Data

Unit shipping cost, supply, and demand
From
From
Cleveland
Cleveland
Detroit
Detroit
Greensboro
Greensboro
Demand
Demand
Boston
Boston
$35
$35
37
37
40
40
1100
1100
Richmond
Richmond
30
30
40
40
15
15
400
400
To
To
Atlanta
Atlanta
40
40
42
42
20
20
750
750
St.
St.Louis
Louis
32
32
25
25
28
28
750
750
Supply
Supply
1200
1200
1000
1000
800
800
 Assumptions




Unit shipping cost is constant.
All the shipping occurs simultaneously.
The only transportation considered is between
sources and destinations.
Total supply equals total demand.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-25
Sources
NETWORK
REPRESENTATI
ON
Destinations
D1=1100
Boston
Cleveland
S1=1200
Richmond
D2=400
Detroit
S2=1000
Atlanta
D3=750
Greensboro
S3= 800
Copyright 2006 John Wiley & Sons, Inc.
St.Louis
D4=750
Supplement 10-26
• The Associated Linear Programming Model

The structure of the model is:
Minimize <Total Shipping Cost>
ST
[Amount shipped from a source] = [Supply at that source]
[Amount received at a destination] = [Demand at that
destination]

Decision variables
Xij = amount shipped from source i to destination j.
where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro)
Copyright 2006 John Wiley & Sons, Inc.
j=1 (Boston), 2 (Richmond), 3 (Atlanta),Supplement 10-27
Supply from Cleveland X11+X12+X13+X14 = 1200
Supply from Detroit X21+X22+X23+X24
= 1000
Supply from Greensboro X31+X32+X33+X34 = 800
The supply constraints
Boston
D1=1100
X11
Cleveland
S1=1200
X12
X13
X21
X31
Richmond
X14
X22
Detroit
S2=1000
D2=400
X32
X23
X24
Atlanta
X33
St.Louis
Greensboro
S3= 800
Copyright 2006 John Wiley & Sons, Inc.
D3=750
X34
D4=750
Supplement 10-28
• The complete mathematical programming model
Minimize 35X11+30X12+40X13+ 32X14 +37X21+40X22+42X23+25X24+
40X31+15X32+20X33+38X34
ST
Supply constrraints:
X11+ X12+ X13+ X14
X21+ X22+ X23+ X24
X31+ X32+ X33+ X34
Demand constraints:
X11+
X12+
X13+
X21+
X31
X22+
X32
X23+
X14+
X33
X24+
X34
= 1200
= 1000
= 800
= 1000
= 400
= 750
= 750
All Xij are nonnegative
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-29
Excel Optimal Solution
CARLTON PHARMACEUTICALS
UNIT COSTS
BOSTON RICHMOND ATLANTA ST.LOUIS
CLEVELAND
$
35.00 $
30.00 $
40.00 $
32.00
DETROIT
$
37.00 $
40.00 $
42.00 $
25.00
GREENSBORO $
40.00 $
15.00 $
20.00 $
28.00
DEMANDS
1100
400
750
750
SHIPMENTS (CASES)
BOSTON RICHMOND ATLANTA ST.LOUIS
CLEVELAND
850
350
0
0
DETROIT
250
0
0
750
GREENSBORO
0
50
750
0
TOTAL
1100
400
SUPPLIES
1200
1000
800
750
TOTAL
1200
1000
800
750
TOTAL COST =
Copyright 2006 John Wiley & Sons, Inc.
84000
Supplement 10-30
WINQSB Sensitivity Analysis
If this path is used, the total cost
will increase by $5 per unit
shipped along it
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Supplement 10-31
Shadow prices for warehouses - the cost resulting from 1 extra case of vaccine
demanded at the warehouse
Shadow prices for plants - the savings incurred for each extra case of vaccine available at
the plant
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-32
Transshipment
Model
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-33
Transshipment Model: Solution
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-34
Copyright 2006 John Wiley & Sons, Inc.
All rights reserved. Reproduction or translation of this work beyond that
permitted in section 117 of the 1976 United States Copyright Act without
express permission of the copyright owner is unlawful. Request for further
information should be addressed to the Permission Department, John Wiley &
Sons, Inc. The purchaser may make back-up copies for his/her own use only and
not for distribution or resale. The Publisher assumes no responsibility for
errors, omissions, or damages caused by the use of these programs or from the
use of the information herein.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-35
DEPOT MAX
A General Network Problem
 Depot Max has six stores.



Stores 5 and 6 are running low on the
model
65A Arcadia workstation, and need a total
of 25 additional units.
Stores 1 and 2 are ordered to ship a total
of 25 units to stores 5 and 6.
Stores 3 and 4 are transshipment nodes
with no demand or supply of their own.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-36
 Other restrictions


There is a maximum limit for quantities
shipped on various routes.
There are different unit transportation costs
for different routes.
 Depot Max wishes to transport the
available workstations at minimum total
cost.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-37
• DATA:
20
10
1
7
3
5
Arcs: Upper bound and lower bound constraints:
6
5
12
0  X ij  U ij
2
15
4
11
7
15
–Supply nodes:
Network
presentation
6
Net flow out
of the node] nodes:
= [Supply at the node]
–Intermediate
transshipment
Transportation
X12
+ X13
X15node]
- X21= =[Total
10 flow into the
(Node
1)
[Total
flow
out of+ the
node]
–Demand
nodes:
unit cost
X21
- X12
= 15
(Node
[Net flow
into +the
node]
= [Demand
for the node]
X34+X35
=X24
X13
(Node
3) 2)
X15 +X46
X35= +X65
X56 = 12
(Node 5)
X24 +- X34
(Node 4)
X46 +X56 - X65 = 13
(Node 6) Supplement 10-38
Copyright 2006 John Wiley & Sons, Inc.
 The Complete mathematical model
Minimize 5X12  10X13  20X15  6X21 15X24  12X34  7X35  15X46  11X56  7X65
ST
X12 + X13 + X15 - X21
- X12
= 10
+ X21 + X24
- X13
= 15
+ X34 + X35
- X24
- X15
- X34
= 0
+ X46
- X35
= 0
+ X56 - X65 = -12
- X46
- X56 + X65 = -13
0  X12  3; 0  X13  12; 0  X15  6; 0  X21  7; 0  X24  10; 0  X34  8; 0  X35  8;
0  X46  17; 0  X56  7; 0  X65  5
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-39
WINQSB Input Data
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-40
WINQSB Optimal Solution
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-41
MONTPELIER SKI COMPANY
Using a Transportation model for production
scheduling




Montpelier is planning its production of skis for the
months of
July, August, and September.
Production capacity and unit production cost will
change from
month to month.
The company can use both regular time and
overtime to produce skis.
Production levels should meet both demand
forecasts
and end-of-quarter inventory Supplement 10-42
Copyright 2006 John Wiley & Sons, Inc.
 Data:



Initial inventory = 200 pairs
Ending inventory required =1200 pairs
Production capacity for the next quarter = 400 pairs in
regular time.
= 200 pairs in
overtime.

Holding cost rate is 3% per month per ski.

Production
Production
Costs
ProductionForecasted
capacity, and
forecasted
demand
Forecasted
Production
Production
Costs for this
Month
Demand
Capacity
Month
Demand
Capacity Regular
RegularTime
Time Overtime
Overtime
quarter
July
400
1000
25
30
Julypairs of skis),
400 and production
1000
25 per unit
30 (by
(in
cost
August
600
800
26
32
August
600
800
26
32
months)
September
1000
400
29
37
September
1000
400
29
37
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-43
 Analysis of demand:

Net demand to satisfy in July = 400 - 200 = 200
pairs
Initial inventory
• Analysis of Unit costs
Unit cost = [Unit production cost] +
 Net demand in August = 600
[Unit holding cost per month][the number of months stays in
 Net demand in September
= 1000In+house
1200
= 2200
Forecasted demand
inventory
inventory]
pairs
Example: A unit produced in July in Regular time and sold in
September costs 25+ (3%)(25)(2 months) = $26.50
 Analysis of Supplies:


Production capacities are thought of as supplies.
There are two sets of “supplies”:

Set 1- Regular time supply (production capacity)
Supplement 10-44
Copyright 2006 John Wiley & Sons, Inc.
Production
Month/period
1000
800
July
O/T
Aug.
R/T
25
25.75
26.50
0
30
30.90
31.80 +M
0
26
26.78
400
Aug.
O/T
Month
sold
July
+M
+M
32.96
200
Sept.
R/T
0
0
Aug.
600
Sept.
2200
Dummy
300
+M
0
29
400
+M
+M
32
200
Demand
Production Capacity
500
July
July
R/T
R/T
Network representation
37
0
Sept.
O/T
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-45
Source: July production in R/T
Destination: July‘s demand.
Unit cost= $25 (production)
Source: Aug. production in O/T
Destination: Sept.’s demand
32+(.03)(32)=$32.96
Unit cost =Production+one month holding cost
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-46
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-47
 Summary of the optimal solution

In July produce at capacity (1000 pairs in R/T, and 500
pairs in O/T). Store 1500-200 = 1300 at the end of July.

In August, produce 800 pairs in R/T, and 300 in O/T.
Store additional 800 + 300 - 600 = 500 pairs.

In September, produce 400 pairs (clearly in R/T). With
1000 pairs
retail demand, there will be
(1300 + 500) + 400 - 1000 = 1200 pairs available for
shipment to
Ski Chalet
Inventory
+ .Production Copyright 2006 John Wiley & Sons, Inc.
Demand
Supplement 10-48
Problem 4-25
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-49
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-50
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-51
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-52
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-53
6.3 The Assignment Problem
 Problem definition

m workers are to be assigned to m jobs

A unit cost (or profit) Cij is associated with worker i
performing job j.

Minimize the total cost (or maximize the total
profit) of assigning workers to job so that each
worker is assigned a job, and each job is
performed.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-54
BALLSTON ELECTRONICS
 Five different electrical devices produced on five
production lines, are needed to be inspected.
 The travel time of finished goods to inspection
areas depends on both the production line and the
inspection area.
 Management wishes to designate a separate
inspection area to inspect the products such that
the total travel time is minimized.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-55
 Data: Travel time in minutes from
assembly
lines to
inspection areas.
Assembly
Assembly
Lines
Lines
11
22
33
44
55
AA
10
10
11
11
13
13
14
14
19
19
Copyright 2006 John Wiley & Sons, Inc.
BB
44
77
88
16
16
17
17
Inspection
Inspection Area
Area
CC
66
77
12
12
13
13
11
11
DD
10
10
99
14
14
17
17
20
20
EE
12
12
14
14
15
15
17
17
19
19
Supplement 10-56
NETWORK REPRESENTATION
Assembly Line
S1=1
1
Inspection Areas
A D1=1
S2=1
2
B
S3=1
3
C D3=1
S4=1
4
D
D4=1
S5=1
5
E
D5=1
Copyright 2006 John Wiley & Sons, Inc.
D2=1
Supplement 10-57
 Assumptions and restrictions

The number of workers equals the number of
jobs.

Given a balanced problem, each worker is
assigned exactly once, and each job is performed
by exactly one worker.
For an unbalanced problem “dummy” workers (in
case there are more jobs than workers), or
“dummy” jobs (in case there are more workers
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-58

 Computer solutions


A complete enumeration is not efficient
even for moderately large problems (with
m=8, m! > 40,000 is the number of
assignments to enumerate).
The Hungarian method provides an
efficient solution procedure.
 Special cases
A worker is unable to perform a particular
job.
 A worker can be assigned to more than
one job.
Copyright
John Wiley & Sons, Inc. assignment problem. Supplement 10-59
 A2006
maximization

6.5 The Shortest Path Problem
 For a given network find the path of
minimum
distance, time, or cost from a starting
point,
the start node, to a destination, the
terminal node.
 Problem definition
There are n nodes, beginning with start node
1 and
ending with terminal node n.
 Bi-directional
arcs connect connectedSupplement
nodes
Copyright
2006 John Wiley & Sons, Inc.
10-60i

Fairway Van Lines
Determine the shortest route from Seattle to El
Paso over the following network highways.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-61
1
180
Seattle
497
3
432
Portland
5
Sac.
599
691
Boise
420
4
138
Reno
6
345
Bakersfield
114
13
Los Angeles
621
Denver 9
Las Vegas
11
108
155
Barstow
14
452
Kingman
469
15
207
Albuque.
Phoenix
386
425
Copyright 2006 John Wiley & Sons, Inc.
12
403
16
17
8
102
432
118
San Diego
440
7
526
280
Cheyenne
Salt Lake City
291
10
Butte
2
Tucson
18
314
19
Supplement
10-62
El Paso
 Solution - a linear programming approach
Decision variables
1 if a truck travels on the highway from city i to city j
X ij  
0 otherwise
Objective = Minimize S dijXij
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-63
Subject to the following constraints:
1
180
Seattle
497
3
432
Portland
Butte
599
2
Boise
4
345
Salt Lake City
7
[The number of highways traveled out of Seattle (the start node)] = 1
X12 + X13 + X14 = 1
In a similar manner:
[The number of highways traveled into El Paso (terminal node)] = 1
X12,19 + X16,19 + X18,19 = 1
[The number of highways used to travel into a city] =
[The number of highways traveled leaving the city].
For example, in Boise (City 4):
X14 + X34 +X74 = X41 + X43 + X47. Supplement 10-64
Nonnegativity
constraints
Copyright
2006 John Wiley & Sons, Inc.
WINQSB Optimal Solution
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-65
 Solution - a network approach
The Dijkstra’s algorithm:
 Find the shortest distance from the “START” node
to every other node in the network, in the order of
the closet nodes to the “START”.
 Once the shortest route to the m closest node is
determined, the shortest route to the (m+1) closest
node can be easily determined.

This algorithm finds the shortest route from the start
to all the nodes in the network.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-66
An illustration of the Dijkstra’s algorithm
+ 420
SLC.=
SLC
599
BUT.
BUT
691
+
CHY.
=
345 =
+ SLC
SLC.
SLC
497
SEA.
BOI
BOI
BOI.
1
Seattle
497
180
138
POR.
POR
180
180
+ 432 =
BOIBOI
2
Butte
691
Boise
420
3
432
Portland
5
Sac.
599
4
Reno
345
Salt Lake City
440
7
526
6
102
621
291
10
Bakersfield
Denver 9
… and so on
until the
Kingman
Barstow
whole network
15
12
14
Albuque.
isPheonix
covered.
Las Vegas
11
280
108
452
155
469
207
Copyright 2006 John Wiley & Sons, Inc.
8
432
114
+ 602 =
SACSAC.
Cheyene
13
Los Angeles
386
San Diego
403
16
118
17
425
Tucson
18
19
314
Supplement
10-67
El Paso
6.6 The Minimal Spanning Tree
 This problem arises when all the nodes
of a given network must be connected to
one another, without any loop.
 The minimal spanning tree approach is
appropriate for problems for which
redundancy is expensive, or the flow
along the arcs is considered
Copyright
2006 John Wiley & Sons, Inc.
Supplement 10-68
instantaneous.
THE METROPOLITAN TRANSIT DISTRICT
 The City of Vancouver is planning the development
of a
new light rail transportation system.
 The system should link 8 residential and
commercial
centers.
 The Metropolitan transit district needs to select the
set
of lines that will connect all the centers at a
minimum total cost.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-69
SPANNING TREE
NETWORK
North Side
PRESENTATION
3
34
University
50
5
Business
District
39
4
West Side
45
1
8
35
2
City
Center
41
7
Copyright 2006 John Wiley & Sons, Inc.
6
Shopping
Center
East Side
South Side
Supplement 10-70
 Solution - a network approach



The algorithm that solves this problem is a very
easy
(“trivial”) procedure.
It belongs to a class of “greedy” algorithms.
The algorithm:
Start by selecting the arc with the smallest arc
length.
 At each iteration, add the next smallest arc length to
the set
of arcs already selected (provided no loop is
constructed).
 Finish when all nodes are connected.

Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-71
WINQSB Optimal Solution
Copyright 2006 John Wiley & Sons, Inc.
Supplement 10-72
OPTIMAL SOLUTION
NETWORK
REPRESENTATION
3
North Side
34
West Side
University
50
5
Business
District
39
4
45
Loop
1
8
35
2
City
Center
41
6
Shopping
Center
East Side
Total Cost = $236 million
7
Copyright 2006 John Wiley & Sons, Inc.
South Side
Supplement 10-73