Lecture Introduction to Management Science with Spreadsheets: Chapter 6S - Stevenson, Ozgur

Chapter 6 - Supplement Transportation and assignment solution procedures, after completing this chapter, you should be able to: Use the transportation method to solve problems manually, deal with special cases in solving transportation problems, use the assignment (Hungarian) method to solve problems manually. deal with special cases in solving assignment problems.

Stevenson and Ozgur

First Edition

Introduction to Management Science

with Spreadsheets

McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved.

Chapter 6 Supplement

Transportation and  Assignment Solution  Procedures

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–3

Table 6S–1 Transportation Table for Harley’s Sand and Gravel

Table 6S–1 Transportation Table for Harley’s Sand and Gravel

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–4

Figure 6S–1 Overview of the Transportation Method

Figure 6S–1 Overview of the Transportation Method

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–5

Finding an Initial Feasible Solution:

The Northwest-Corner Method

Finding an Initial Feasible Solution:

The Northwest-Corner Method

• The Northwest-Corner Method

–is a systematic approach for developing an initial

feasible solution.

–is simple to use and easy to understand.

–does not take transportation costs into account.

–gets its name because the starting point for the

allocation process is the upper-left-hand (northwest) corner of the transportation table

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–6

Table 6S–2 Initial Feasible Solution for Harley Using Northwest-Corner

MethodTable 6S–2 Initial Feasible Solution for Harley Using Northwest-Corner

Method

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–7

Finding an Initial Feasible Solution:

The Intuitive Approach

Finding an Initial Feasible Solution:

The Intuitive Approach

1 Identify the cell that has the lowest unit cost.

2 Cross out the cells in the row or column that has been

exhausted (or both, if both have been exhausted), and adjust the remaining row or column total accordingly.

3 Identify the cell with the lowest cost from the

remaining cells.

4 Repeat steps 2 and 3 until all supply and demand

have been allocated.

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–8

Table 6S–3a Find the Cell That Has the Lowest Unit Cost

Table 6S–3a Find the Cell That Has the Lowest Unit Cost

Table 6S–3b Allocate 150 Units to Cell B–2

Table 6S–3b Allocate 150 Units to Cell B–2

Companies. All rights reserved.   McGraw­ Hill/Irwin  6S–9

Table 6S–4 200 Units Are Assigned to Cell C–3 and 50 Units Are

Assigned to cell A–1

Table 6S–4 200 Units Are Assigned to Cell C–3 and 50 Units Are

Assigned to cell A–1

Table 6S–5 Completion of the Initial Feasible Solution for the Harley

Problem Using the Intuitive ApproachTable 6S–5 Completion of the Initial Feasible Solution for the Harley

Problem Using the Intuitive Approach

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 10

Table 6S–6 Vogel’s Approximation Initial Allocation Tableau with Penalty

CostsTable 6S–6 Vogel’s Approximation Initial Allocation Tableau with Penalty

Costs

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 11

Table 6S–7 Initial Feasible Solution Obtained Using the Northwest-Corner

Method

Table 6S–7 Initial Feasible Solution Obtained Using the Northwest-Corner

Method

Table 6S–8 Evaluation Path for Cell B–1

Table 6S–8 Evaluation Path for Cell B–1

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 12

Table 6S–9 Evaluation Path for Cell C–1

Table 6S–9 Evaluation Path for Cell C–1

Table 6S–10 Evaluation Paths for Cells A–3 and C–2

Table 6S–10 Evaluation Paths for Cells A–3 and C–2

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 13

Table 6S–11 Initial Feasible Solution Obtained Using the Northwest-Corner

Method

Table 6S–11 Initial Feasible Solution Obtained Using the Northwest-Corner

Method

Evaluation Using the MODI Method

The MODI (MOdified DIstribution) method of evaluating a transportation

solution for optimality involves the use of index numbers that are

established for the rows and columns These are based on the unit costs of

the occupied cells The index numbers can be used to obtain the cell

evaluations for empty cells without the use of stepping-stone paths.

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 14

Table 6S–12 Index Numbers for Initial Northwest-Corner Solution to the

Harley Problem

Table 6S–12 Index Numbers for Initial Northwest-Corner Solution to the

Harley Problem

Rules for Tracing Stepping-Stone Paths

1 All unoccupied cells must be evaluated Evaluate cells one at a time.

2 Except for the cell being evaluated, only add or subtract in occupied cells (It is permissible to skip over occupied cells to find an occupied cell from which the path can continue.)

3 A path will consist of only horizontal and vertical moves, starting and

ending with the empty cell that is being evaluated.

4 Alternate + and - signs, beginning with a + sign in the cell being evaluated.

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 15

Table 6S–13 Cell Evaluations for Northwest-Corner Solution for the Harley

Problem

Table 6S–13 Cell Evaluations for Northwest-Corner Solution for the Harley

Problem

Table 6S–14 Stepping-Stone Path for Cell A–3

Table 6S–14 Stepping-Stone Path for Cell A–3

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 16

Table 6S–15 Distribution Plan after Reallocation of 50 Units

Table 6S–15 Distribution Plan after Reallocation of 50 Units

Table 6S–16 Index Numbers and Cell Evaluations

Table 6S–16 Index Numbers and Cell Evaluations

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 17

Summary of the Transportation Method

Summary of the Transportation Method

1 Obtain an initial feasible solution Use either the northwest-corner method, the intuitive method, or the Vogel’s approximation

method Generally, the intuitive method and Vogel’s approximation are the preferred approaches.

2 Evaluate the solution to determine if it is optimal Use either the stepping-stone method or MODI The solution is not optimal if any unoccupied cell has a negative cell evaluation.

3 If the solution is not optimal, select the cell that has the most

negative cell evaluation Obtain an improved solution using the

stepping-stone method.

4 Repeat steps 2 and 3 until no cell evaluations (reduced costs) are negative Once you have identified the optimal solution, compute its total cost.

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 18

Special Issues

Special Issues

1 Determining if there are alternate optimal solutions.

2 Recognizing and handling degeneracy (too few

occupied cells to permit evaluation of a solution).

3 Avoiding unacceptable or prohibited route

assignments.

4 Dealing with problems in which supply and demand

are not equal.

5 Solving maximization problems.

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 19

Table 6S–17a Index Numbers and Cell Evaluations

Table 6S–17a Index Numbers and Cell Evaluations

Table 6S–17b Alternate Optimal Solution

Table 6S–17b Alternate Optimal Solution

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 20

Table 6S–18 Harley Alternate Solution Modified for Degeneracy

Table 6S–18 Harley Alternate Solution Modified for Degeneracy

Table 6S–19 Solution to Harley Problem with a Prohibited Route

Table 6S–19 Solution to Harley Problem with a Prohibited Route

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 21

Table 6S–20 A Dummy Origin Is Added to Make Up 80 Units

Table 6S–20 A Dummy Origin Is Added to Make Up 80 Units

Table 6S–21 Solution Using the Dummy Origin

Table 6S–21 Solution Using the Dummy Origin

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 22

Table 6S–21 Solution Using the Dummy Origin

Table 6S–21 Solution Using the Dummy Origin

Table 6S–22 Solution Using the Dummy Origin

Table 6S–22 Solution Using the Dummy Origin

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 23

Table 6S–23 Row Reduction

Table 6S–23 Row Reduction

•The Hungarian Method

• provides a simple heuristic that can be used to find the optimal set

of assignments It is easy to use, even for fairly large problems It

is based on minimization of opportunity costs that would result from

potential pairings These are additional costs that would be

incurred if the lowest-cost assignment is not made, in terms of

either jobs (i.e., rows) or employees (i.e., columns).

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 24

The Hungarian Method

The Hungarian Method

• Provides a simple heuristic that can be used to find the optimal set of assignments

• Is easy to use, even for fairly large problems

• Is based on minimization of opportunity costs that would result from potential pairings

–These additional costs would be incurred if the cost assignment is not made, in terms of either jobs (i.e., rows) or employees (i.e., columns).

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 25

Requirements for Use of the Hungarian Method

Requirements for Use of the Hungarian Method

• Situations in which the Hungarian method can

be used are characterized by the following:

1 There needs to be a one-for-one matching of two

sets of items.

2 The goal is to minimize costs (or to maximize

profits) or a similar objective (e.g., time, distance, etc.).

3 The costs or profits (etc.) are known or can be

closely estimated.

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 26

Special Situations

Special Situations

• Special Situations

– Certain situations can arise in which the model

deviates slightly from that previously described.

• Among those situations are the following:

– The number of rows does not equal the number of

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 27

Table 6S–24 Column Reduction of Opportunity (Row Reduction) CostsTable 6S–24 Column Reduction of Opportunity (Row Reduction) Costs

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 28

Table 6S–25 Determine the Minimum Number of Lines Needed to Cover

the Zeros

Table 6S–25 Determine the Minimum Number of Lines Needed to Cover

the Zeros

Table 6S–26 Further Revision of the Cost Table

Table 6S–26 Further Revision of the Cost Table

Companies. All rights reserved.  

McGraw­

Hill/Irwin  6S– 29

Table 6S–27 Optimal Assignments

Table 6S–27 Optimal Assignments