transportation, assignment and transshipment...

Chapter 7Transportation, Assignment &

Transshipment ProblemsPart 2

Prof. Dr. Arslan M. ÖRNEK

7.5. Assignment Problems

Special type of LP, in fact a special type of Transportation problem.

Assignees (workers, processors, machines, vehicles, plants, time slots) are being assigned to tasks (jobs, classrooms, people).

Example: Machineco has four jobs to be completed. Each machine must be assigned to complete one job. The time required to setup each machine for completing each job is given. Machineco wants to minimize the total setup time needed to complete the four jobs.

(Also called the cost matrix)

The Model

Assignment problem: A balanced transportation problem where all supplies and demands are equal to 1.

All the supplies and demands for the Machineco problem (and for any assignment problem) are integers, so all variables in Machineco’s optimal solution must be integers. Solve with Transportation simplex.

Transportation simplex is often inefficient. For this reason The Hungarian Method is used for solving assignment problems.

7.5. Assignment Problems

The steps of The Hungarian Method:

Step1. Find a bfs. Find the minimum element in each row of the mxm cost matrix. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix (reduced cost matrix) by subtracting from each cost the minimum cost in its column.

Step2. Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are required, proceed to step 3.

Step3. Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step2.

Machineco Solution with Hungarian Method

Machineco Solution (cont.)


The smallest uncovered element equals 1, so we now subtract 1 from each uncovered element in the reduced cost matrix and add 1 to each twice-covered element.


To find an optimal assignment, observe that the only covered 0 incolumn 3 is x33, so we must have x33 = 1. Also, the only available covered zero in column 2 is x12, so we set x12 = 1.Now the only available covered zero in column 4 is x24. Thus, we choose x24 = 1. Finally, we choose x41 = 1.

Assignment Model ExampleProblem Definition and Data

Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game.

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xij = 1 if i is assigned to j, 0 otherwise. i= A, B, C, D, j=R, A, C, D

Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR +70xBA

+ 130xBD + 200xBC + 175xCR + 105xCA +140xCD

+ 170xCC + 80xDR + 65xDA + 105xDD + 120xDC

subject to: xAR + xAA + xAD + xAC = 1 xij 0xBR + xBA + xBD + xBC = 1xCR + xCA + xCD + xCC = 1xDR + xDA + xDD + xDC = 1xAR + xBR + xCR + xDR = 1xAA + xBA + xCA + xDA = 1xAD + xBD + xCD + xDD = 1xAC + xBC + xCC + xDC = 1

Assignment Model ExampleModel Formulation

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Assignment Model ExampleAssignment Network Solution

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Assignment ModelThe Hungarian Method

Develop the reduced cost matrix.

Subtract the minimum value in each row from every row element in the matrix.

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Next, subtract the minimum value in each column from every column element in the matrix.

Assignments can be made in the matrix where a 0 is present.


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Draw the minimum number of horizontal or vertical lines necessary to cross out all zeros through the rows and columns of the reduced cost matrix.


The three lines indicate that there are only three unique assignments, whereas four are required for an optimal solution.

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Next, subtract the minimum value that is not crossed out from all values not crossed out. Add this minimum value to those cells where two lines intersect.


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No matter how the lines are drawn now, at least four are required to cross out all the zeros. This indicates that four unique assignments can be made and that an optimal solution has been reached: Team A Atlanta, Team B Raleigh, Team C Durham, Team D Clemson with 450 miles total distance. ORTeam A Clemson, Team B Atlanta, Team C Durham, Team D Raleigh with 450 miles total distance.


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An assignment problem is unbalanced when supply exceeds demand or demand exceeds supply.

For example, assume that, instead of four teams of officials, there are five teams to be assigned to the four games. In this case a dummy column is added to the assignment tableau to balance the model.


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•In solving this model, one team of officials would be assigned to the dummy column. •If there were five games and only four teams of officials, a dummy row would be added instead of a dummy column. •The addition of a dummy row or column does not affect thesolution method.


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Prohibited assignments are possible in an assignment problem. In the assignment model, a Big M value is assigned as a large cost for the cell representing the prohibited assignment.

M


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7.6 Transshipment Problems

A transportation problem allows only shipments that go directly from supply points to demand points.

In many situations, shipments are allowed between supply points or between demand points. Sometimes there may also be points (called transshipment points) through which goods can be transshipped on their journey from a supply point to a demand point. The optimal solution to a transshipment problem can be found by solving a transportation problem.

S1

S2

D1

T1

T2


Define a supply point to be a point that can send goods to another point but cannot receive goods from any other point.

Similarly, a demand point is a point that can receive goods from other points but cannot send goods to any other point.

A transshipment point is a point that can both receive goods from other points and send goods to other points.


Example: Widgetco.

Two factories in Memphis and Denver, with supplies 150 per day, and 200 per day.

Customers in Los Angeles and Boston, with a demand 130 per day each.

Widgetco believes that it may be cheaper to first ship somewidgets to New York or Chicago and then ship them to their final destinations.

The costs of shipping a widget are given. Widgetco wants to minimize the total cost of shipping the required widgets to its customers.

Follow these steps in solving a transshipment problem:

Step1. If necessary, add a dummy demand point (demand equal to the problem’s excess supply) to balance the problem. Shipments to the dummy and from a point to itself will cost zero.

Step2. Construct a transportation tableau as follows: A row in the tableau will be needed for each supply point and transshipment point, and a column will be needed for each demand point and transshipment point.


Each supply point will have a supply equal to it’s original supply, and each demand point will have a demand to its original demand.

Let s= total available supply.

Then each transshipment point will have a supply equal to (point’s original supply)+s and a demand equal to (point’s original demand)+s.

Although we don’t know how much will be shipped through each transshipment point, we can be sure that the total amount will not exceed s.



Example: Widgetco.

Two factories in Memphis and Denver, with supplies 150 per day, and 200 per day.

Customers in Los Angeles and Boston, with a demand 130 per day each.

Widgetco believes that it may be cheaper to first ship somewidgets to New York or Chicago and then ship them to their final destinations.

The costs of shipping a widget are given. Widgetco wants to minimize the total cost of shipping the required widgets to its customers.

Include farms as second tier suppliers in the transportation example before. Now, the grain elevators are taking their grains from the two farms.

Grain Elevator

Farm 3. Kansas City 4. Omaha 5. Des Moines

$16

15 10 14

12 17

Transshipment Model ExampleProblem Definition and Data

1. Nebraska

2. Colorado

Shipping Costs

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Transshipment Model ExampleProblem Definition and Data

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xij = amount transported from i to j , i = 1, 2, 3Minimize Z = $16x13 + 10x14 + 12x15 + 15x23 + 14x24 + 17x25

+ 6x36 + 8x37 + 10x38 + 7x46 + 11x47 + 11x48

+ 4x56 + 5x57 + 12x58

subject to: x13 + x14 + x15 = 300 (supply point constraints)x23 + x24 + x25 = 300x36 + x46 + x56 = 200 (demand point constraints)x37 + x47 + x57 = 100x38 + x48 + x58 = 300x13 + x23 - x36 - x37 - x38 = 0 (transshipment point constraints)

x14 + x24 - x46 - x47 - x48 = 0x15 + x25 - x56 - x57 - x58 = 0xij 0

Transshipment Model ExampleLP Model Formulation

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Solution

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