Operational decision making tools transportation and transshipment models
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Chapter 11 Supplement. Operational Decision-Making Tools: Transportation and Transshipment Models. Beni Asllani University of Tennessee at Chattanooga. Operations Management - 6 hh Edition. Roberta Russell & Bernard W. Taylor, III. Just how do you make decisions?. Emotional direction

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Operational decision making tools transportation and transshipment models

Chapter 11 Supplement

Operational Decision-Making Tools: Transportation and Transshipment Models

Beni AsllaniUniversity of Tennessee at Chattanooga

Operations Management - 6hh Edition

Roberta Russell & Bernard W. Taylor, III

Copyright 2006 John Wiley & Sons, Inc.


Just how do you make decisions

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.


Problems

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


Operational decision making tools transportation and transshipment models

Problem

Problem

MS Model

Mental Model

Mental Model

Decision

Decision

Action

Action

Copyright 2006 John Wiley & Sons, Inc.


Model classification criteria

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.


Purpose

Purpose

  • Planning

  • Forecasting

  • Training

  • Behavioral research

Copyright 2006 John Wiley & Sons, Inc.


Perspective

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.


Degree of abstraction

Degree of Abstraction

  • Isomorphic

    • One-to-one

  • Homomorphic

    • One-to-many

Copyright 2006 John Wiley & Sons, Inc.


Content and form

Content and Form

  • verbal descriptions

  • mathematical constructs

  • simulations

  • mental models

  • physical prototypes

Copyright 2006 John Wiley & Sons, Inc.


Decision environment

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.


Mathematical programming

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.


A model of this class

A Model of this class

  • What would we include in it?

Copyright 2006 John Wiley & Sons, Inc.


Management science models

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.


Mathematical programming models covered in ch 11 supplement

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.


Transportation model

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.


Transportation method example

Transportation Method: Example

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Transportation method example1

Transportation Method: Example

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Problem Formulation Using Excel

Total Cost Formula

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Operational decision making tools transportation and transshipment models

Using Solver from Tools Menu

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Operational decision making tools transportation and transshipment models

Solution

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Modified Problem Solution

Copyright 2006 John Wiley & Sons, Inc.


The underlying network

The Underlying Network

Copyright 2006 John Wiley & Sons, Inc.


For problems in which there is an underlying network

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.


Carlton pharmaceuticals

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.


Operational decision making tools transportation and transshipment models

  • Data

    • Unit shipping cost, supply, and demand

  • 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.


Operational decision making tools transportation and transshipment models

Destinations

Boston

Sources

35

Cleveland

30

Richmond

40

S1=1200

32

37

40

Detroit

42

25

S2=1000

Atlanta

35

15

20

St.Louis

Greensboro

28

S3= 800

D1=1100

NETWORK

REPRESENTATION

D2=400

D3=750

D4=750

Copyright 2006 John Wiley & Sons, Inc.


The associated linear programming model

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) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Supply from Cleveland X11+X12+X13+X14 = 1200

Supply from Detroit X21+X22+X23+X24 = 1000

Supply from Greensboro X31+X32+X33+X34 = 800

X11

Cleveland

X12

X31

S1=1200

X21

X13

X14

X22

X32

Detroit

X23

S2=1000

X24

X33

Greensboro

S3= 800

X34

The supply constraints

Boston

D1=1100

Richmond

D2=400

Atlanta

D3=750

St.Louis

D4=750

Copyright 2006 John Wiley & Sons, Inc.


The complete mathematical programming model

The complete mathematical programming model

=

=

=

=

=

=

=

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Excel Optimal Solution

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Range of optimality

WINQSB Sensitivity Analysis

If this path is used, the total cost will increase by $5 per unit

shipped along it

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Range of feasibility

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.


Transshipment model

Transshipment Model

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Transshipment Model: Solution

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

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.


Operational decision making tools transportation and transshipment models

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.


Operational decision making tools transportation and transshipment models

  • 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.


Operational decision making tools transportation and transshipment models

  • DATA:

20

10

7

1

3

5

Arcs: Upper bound and lower bound constraints:

6

5

12

11

7

2

4

6

15

15

  • Supply nodes: Net flow out of the node] = [Supply at the node]

  • X12 + X13 + X15 - X21 = 10 (Node 1)X21 + X24 - X12 = 15(Node 2)

Network presentation

  • Intermediate transshipment nodes: [Total flow out of the node] = [Total flow into the node]

  • X34+X35 = X13 (Node 3)X46 = X24 + X34 (Node 4)

Transportation

unit cost

  • Demand nodes:[Net flow into the node] = [Demand for the node]

  • X15 + X35 +X65 - X56 = 12 (Node 5)X46 +X56 - X65 = 13 (Node 6)

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

  • The Complete mathematical model

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

WINQSB Input Data

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

WINQSB Optimal Solution

Copyright 2006 John Wiley & Sons, Inc.


Montpelier ski company using a transportation model for production scheduling

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 requirement.

  • Management would like to schedule production to minimize its costs for the quarter.

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

  • 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 capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)

Copyright 2006 John Wiley & Sons, Inc.


Operational decision making tools transportation and transshipment models

Initial inventory

  • Analysis of Unit costs

    • Unit cost = [Unit production cost] +

  • [Unit holding cost per month][the number of months stays in inventory]

  • Example: A unit produced in July in Regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50

    • Analysis of demand:

      • Net demand to satisfy in July = 400 - 200 = 200 pairs

      • Net demand in August = 600

      • Net demand in September = 1000 + 1200 = 2200 pairs

    • Analysis of Supplies:

      • Production capacities are thought of as supplies.

      • There are two sets of “supplies”:

        • Set 1- Regular time supply (production capacity)

        • Set 2 - Overtime supply

    Forecasted demand

    In house inventory

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Network representation

    Production

    Month/period

    Month

    sold

    July

    R/T

    July

    R/T

    25

    25.75

    26.50

    0

    1000

    200

    July

    July

    O/T

    30

    30.90

    31.80

    0

    500

    +M

    26

    26.78

    0

    +M

    +M

    37

    0

    +M

    +M

    29

    0

    Aug.

    R/T

    600

    +M

    32

    32.96

    0

    Aug.

    800

    Demand

    Production Capacity

    Aug.

    O/T

    400

    2200

    Sept.

    Sept.

    R/T

    400

    Dummy

    300

    Sept.

    O/T

    200

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Source: July production in R/T

    Destination: July‘s demand.

    Source: Aug. production in O/T

    Destination: Sept.’s demand

    Unit cost= $25 (production)

    32+(.03)(32)=$32.96

    Unit cost =Production+one month holding cost

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • 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 -

    Demand

    Copyright 2006 John Wiley & Sons, Inc.


    Problem 4 25

    Problem 4-25

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    Operational decision making tools transportation and transshipment models

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Copyright 2006 John Wiley & Sons, Inc.


    6 3 the assignment problem

    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.


    Ballston electronics

    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 thatthe total travel time is minimized.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • Data: Travel time in minutes from assembly lines to inspection areas.

    Copyright 2006 John Wiley & Sons, Inc.


    Network representation

    S1=1

    S2=1

    S3=1

    S4=1

    S5=1

    NETWORK REPRESENTATION

    Assembly Line

    Inspection Areas

    D1=1

    1

    A

    2

    B

    D2=1

    3

    C

    D3=1

    D4=1

    4

    D

    D5=1

    5

    E

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • 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 than jobs) are added to balance the problem.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • 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.

      • A maximization assignment problem.

    Copyright 2006 John Wiley & Sons, Inc.


    6 5 the shortest path problem

    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 connected nodes i and jwith nonnegative distances, d i j.

      • Find the path of minimum total distance that connects node 1 to node n.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Fairway Van Lines

    Determine the shortest route from Seattle to El Paso over the following network highways.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Seattle

    Butte

    599

    1

    2

    497

    691

    Boise

    180

    420

    3

    4

    Cheyenne

    345

    Salt Lake City

    432

    Portland

    440

    Reno

    7

    8

    526

    6

    138

    102

    432

    5

    621

    Sac.

    291

    Denver

    9

    Las Vegas

    11

    280

    10

    108

    452

    Bakersfield

    Kingman

    155

    Barstow

    114

    469

    15

    207

    12

    14

    13

    Albuque.

    Phoenix

    Los Angeles

    386

    403

    16

    118

    19

    17

    18

    San Diego

    425

    314

    Copyright 2006 John Wiley & Sons, Inc.

    Tucson

    El Paso


    Operational decision making tools transportation and transshipment models

    • Solution - a linear programming approach

      Decision variables

    • Objective = Minimize S dijXij

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    Butte

    Seattle

    2

    599

    1

    497

    Boise

    180

    3

    4

    345

    Salt Lake City

    432

    Portland

    7

    Subject to the following constraints:

    • [The number of highways traveled out of Seattle (the start node)] = 1X12 + X13 + X14 = 1

    In a similar manner:

    [The number of highways traveled into El Paso (terminal node)] = 1X12,19 + 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.

    Copyright 2006 John Wiley & Sons, Inc.

    Nonnegativity constraints


    Operational decision making tools transportation and transshipment models

    WINQSB Optimal Solution

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • 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.


    Operational decision making tools transportation and transshipment models

    420

    +

    =

    SLC.

    1119

    SLC

    599

    BUT.

    599

    BUT

    +

    =

    691

    1290

    CHY.

    345

    +

    SLC

    SLC.

    =

    SLC

    497

    497

    BOI.

    BOI

    BOI

    +

    =

    432

    612

    BOI

    BOI

    180

    POR.

    180

    POR

    +

    =

    602

    SAC.

    782

    SAC

    An illustration of the Dijkstra’s algorithm

    842

    SEA.

    … and so on until the

    whole network

    is covered.

    Copyright 2006 John Wiley & Sons, Inc.


    6 6 the minimal spanning tree

    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 instantaneous.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    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 commercialcenters.

    • The Metropolitan transit district needs to select the set of lines that will connect all the centers at a minimum total cost.

    • The network describes:

      • feasible lines that have been drafted,

      • minimum possible cost for taxpayers per line.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    SPANNING TREE

    NETWORK

    PRESENTATION

    55

    North Side

    University

    50

    3

    5

    30

    Business

    District

    39

    38

    4

    33

    34

    West Side

    45

    32

    1

    8

    28

    43

    35

    2

    6

    East Side

    City

    Center

    Shopping

    Center

    41

    40

    37

    44

    36

    7

    South Side

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    • 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.

    • Computer solution

      • Input consists of the number of nodes, the arc length, and the network description.

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    WINQSB Optimal Solution

    Copyright 2006 John Wiley & Sons, Inc.


    Operational decision making tools transportation and transshipment models

    OPTIMAL SOLUTION

    NETWORK

    REPRESENTATION

    55

    University

    50

    3

    5

    30

    North Side

    Business

    District

    39

    38

    4

    33

    34

    West Side

    45

    Loop

    32

    1

    8

    28

    43

    35

    2

    6

    East Side

    City

    Center

    Shopping

    Center

    41

    40

    37

    44

    36

    Total Cost = $236 million

    7

    South Side

    Copyright 2006 John Wiley & Sons, Inc.


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