The Transportation and Assignment ProblemsChapter 8: Hillier and LiebermanDr. Hurley’s AGB 328 Course

Terms to KnowSources, Destinations, Supply,

Demand, The Requirements Assumption, The Feasible Solutions Property, The Cost Assumption, Dummy Destination, Dummy Source, Transportation Simplex Method, Northwest Corner Rule, Vogel’s Approximation Method, Russell’s Approximation Method, Recipient Cells, Donor Cells, Assignment Problems, Assignees, Tasks, Hungarian Algorithm

Case Study: P&T CompanyP&T is a small family-owned business

that processes and cans vegetables and then distributes them for eventual sale

One of its main products that it processes and ships is peas◦ These peas are processed in: Bellingham,

WA; Eugene, OR; and Albert Lea, MN◦ The peas are shipped to: Sacramento, CA;

Salt Lake City, UT; Rapid City, SD; and Albuquerque, NM

Case Study: P&T Company Shipping DataCannery Output Warehouse Allocation

Bellingham 75 Truckloads Sacramento 80 Truckloads

Eugene 125 Truckloads Salt Lake 65 Truckloads

Albert Lea 100 Truckloads Rapid City 70 Truckloads

Total 300 Truckloads Albuquerque 85 Truckloads

Total 300 Truckloads

Case Study: P&T Company Shipping Cost/Truckload

Warehouse

Cannery Sacramento Salt Lake Rapid City

Albuquerque Supply

Bellingham $464 $513 $654 $867 75

Eugene $352 $416 $690 $791 125

Albert Lea $995 $682 $388 $685 100

Demand 80 65 70 85

Network Presentation of P&T Co. Problem

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W4 -85

W2 -65

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Transportation ProblemsTransportation problems are

characterized by problems that are trying to distribute commodities from a any supply center, known as sources, to any group of receiving centers, known as destinations

Two major assumptions are needed in these types of problems:◦ The Requirements Assumption◦ The Cost Assumption

Transportation AssumptionsThe Requirement Assumption

◦Each source has a fixed supply which must be distributed to destinations, while each destination has a fixed demand that must be received from the sources

The Cost Assumption◦The cost of distributing commodities

from the source to the destination is directly proportional to the number of units distributed

Feasible Solution PropertyA transportation problem will

have a feasible solution if and only if the sum of the supplies is equal to the sum of the demands.◦Hence the constraints in the

transportation problem must be fixed requirement constraints met with equality.

The General Model of a Transportation ProblemAny problem that attempts to

minimize the total cost of distributing units of commodities while meeting the requirement assumption and the cost assumption and has information pertaining to sources, destinations, supplies, demands, and unit costs can be formulated into a transportation model

Visualizing the Transportation ModelWhen trying to model a

transportation model, it is usually useful to draw a network diagram of the problem you are examining◦A network diagram shows all the

sources, destinations, and unit cost for each source to each destination in a simple visual format like the example on the next slide

Network Diagram

Source 1

Source 2

Source 3

Source m

.

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.

Destination 1

Destination 2

Destination 3

Destination n

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Supply

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S2

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Demand

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General Mathematical Model of Transportation Problems

Minimize Z=Subject to: for I =1,2,…,m

Integer Solutions PropertyIf all the supplies and demands

have integer values, then the transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables◦This implies that there is no need to

add restrictions on the model to force integer solutions

Solving a Transportation ProblemWhen Excel solves a

transportation problem, it uses the regular simplex method

Due to the characteristics of the transportation problem, a faster solution can be found using the transportation simplex method◦Unfortunately, the transportation

simplex model is not programmed in Solver

Modeling Variants of Transportation ProblemsIn many transportation models,

you are not going to always see supply equals demand

With small problems, this is not an issue because the simplex method can solve the problem relatively efficiently

With large transportation problems it may be helpful to transform the model to fit the transportation simplex model

Issues That Arise with Transportation ModelsSome of the issues that may arise are:

◦ The sum of supply exceeds the sums of demand

◦ The sum of the supplies is less than the sum of demands

◦ A destination has both a minimum demand and maximum demand

◦ Certain sources may not be able to distribute commodities to certain destinations

◦ The objective is to maximize profits rather than minimize costs

Method for Handling Supply Not Equal to DemandWhen supply does not equal demand,

you can use the idea of a slack variable to handle the excess

A slack variable is a variable that can be incorporated into the model to allow inequality constraints to become equality constraints◦ If supply is greater than demand, then you

need a slack variable known as a dummy destination

◦ If demand is greater than supply, then you need a slack variable known as a dummy source

Handling Destinations that Cannot Be Delivered ToThere are two ways to handle the

issue when a source cannot supply a particular destination◦The first way is to put a constraint

that does not allow the value to be anything but zero

◦The second way of handling this issue is to put an extremely large number into the cost of shipping that will force the value to equal zero

Textbook Transportation Models ExaminedP&T

◦A typical transportation problem◦Could there be another formulation?

Northern Airplane◦An example when you need to use the Big M

Method and utilizing dummy destinations for excess supply to fit into the transportation model

Metro Water District◦An example when you need to use the Big M

Method and utilizing dummy sources for excess demand to fit into the transportation model

The Transportation Simplex MethodWhile the normal simplex method

can solve transportation type problems, it does not necessarily do it in the most efficient fashion, especially for large problems.

The transportation simplex is meant to solve the problems much more quickly.

Finding an Initial Solution for the Transportation SimplexNorthwest Corner Rule

◦Let xs,d stand for the amount allocated to supply row s and demand row d

◦For x1,1 select the minimum of the supply and demand for supply 1 and demand 1

◦ If any supply is remaining then increment over to xs,d+1, otherwise increment down to xs+1,d

For this next variable select the minimum of the leftover supply or leftover demand for the new row and column you are in

Continue until all supply and demand has been allocated

Finding an Initial Solution for the Transportation SimplexVogel’s Approximation Method

◦For each row and column that has not been deleted, calculate the difference between the smallest and second smallest in absolute value terms (ties mean that the difference is zero)

◦ In the row or column that has the highest difference, find the lowest cost variable in it

◦Set this variable to the minimum of the leftover supply or demand

◦Delete the supply or demand row/column that was the minimum and go back to the top step

Finding an Initial Solution for the Transportation SimplexRussell’s Approximation Method

◦For each remaining source row i, determine the largest unit cost cij and call it

◦For each remaining destination column j, determine the largest unit cost cij and call it

◦Calculate for all xij that have not previously been selected

◦Select the largest corresponding xij that has the largest negative ∆ij

Allocate to this variable as much as feasible based on the current supply and demand that are leftover

Algorithm for Transportation Simplex MethodConstruct initial basic feasible

solutionOptimality Test

◦Derive a set of ui and vj by setting the ui corresponding to the row that has the most amount of allocations to zero and solving the leftover set of equations for cij = ui + vj

If all cij – ui – vj ≥ 0 for every (i,j) such that xij is nonbasic, then stop. Otherwise do an iteration.

Algorithm for Transportation Simplex Method Cont.An Iteration

◦Determine the entering basic variable by selecting the nonbasic variable having the largest negative value for cij – ui – vj

◦Determine the leaving basic variable by identifying the chain of swaps required to maintain feasibility

◦Select the basic variable having the smallest variable from the donor cells

◦Determine the new basic feasible solution by adding the value of the leaving basic variable to the allocation for each recipient cell. Subtract this value from the allocation of each donor

cell

Assignment ProblemsAssignment problems are

problems that require tasks to be handed out to assignees in the cheapest method possible

The assignment problem is a special case of the transportation problem

Characteristics of Assignment ProblemsThe number of assignees and the

number of task are the sameEach assignee is to be assigned

exactly one taskEach task is to be assigned by exactly

one assigneeThere is a cost associated with each

combination of an assignee performing a task

The objective is to determine how all of the assignments should be made to minimize the total cost

General Mathematical Model of Assignment Problems

Minimize Z=Subject to: for I =1,2,…,m

Modeling Variants of the Assignment ProblemIssues that arise:

◦ Certain assignees are unable to perform certain tasks.

◦ There are more task than there are assignees, implying some tasks will not be completed.

◦ There are more assignees than there are tasks, implying some assignees will not be given a task.

◦ Each assignee can be given multiple tasks simultaneously.

◦ Each task can be performed jointly by more than one assignee.

Assignment Spreadsheet Models from Textbook Job Shop CompanyBetter Products Company

◦ We will examine these spreadsheets in class and derive mathematical models from the spreadsheets

Hungarian Algorithm for Solving Assignment ProblemsStep 1: Find the minimum from each row and

subtract from every number in the corresponding row making a new table

Step 2: Find the minimum from each column and subtract from every number in the corresponding column making a new table

Step 3: Test to see whether an optimal assignment can be made by examining the minimum number of lines needed to cover all the zeros◦ If the number of lines corresponds to the number of

rows, you have the optimal and you should go to step 6◦ If the number of lines does not correspond to the

number of rows, go to step 4

Hungarian Algorithm for Solving Assignment Problems Cont.Step 4: Modify the table by using

the following:◦Subtract the smallest uncovered

number from every uncovered number in the table

◦Add the smallest uncovered number to the numbers of intersected lines

◦All other numbers stay unchangedStep 5: Repeat steps 3 and four

until you have the optimal set

Hungarian Algorithm for Solving Assignment Problems Cont.Step 6: Make the assignment to the

optimal set one at a time focusing on the zero elements◦Start with the rows and columns that have

only one zero Once an optimal assignment has been given to

a variable, cross that row and column out Continue until all the rows and columns with

only one zero have been allocated Next do the columns/rows with two non crossed

out zeroes as above Continue until all assignments have been made

In Class Activity (Not Graded)Attempt to find an initial solution to

the P&T problem using the a) Northwest Corner Rule, b) Vogel’s Approximation Method, and c) Russell’s Approximation Method

8.1-3b, set up the problem as a regular linear programming problem and solve using solver, then set the problem up as a transportation problem and solve using solver

In Class Activity (Not Graded)Solve the following problem using

the Hungarian method.

Case Study: Sellmore Company Cont.The assignees for the task are:

◦Ann◦Ian◦Joan◦Sean

A summary of each assignees productivity and costs are given on the next slide.

Case Study: Sellmore Company Cont.

Required Time Per Task

Employee Word Processing

Graphics Packets Registration

Wage

Ann 35 41 27 40 $14

Ian 47 45 32 51 $12

Joan 39 56 36 43 $13

Sean 32 51 25 46 $15

Assignment of Variablesxij

◦i = 1 for Ann, 2 for Ian, 3 for Joan, 4 for Sean

◦j = 1 for Processing, 2 for Graphics, 3 for Packets, 4 for Registration

Mathematical Model for Sellmore Company

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