OD Transport Assignment 2012

8/13/2019 OD Transport Assignment 2012

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TRANSPORTATION ANDASSIGNMENT PROBLEMS

Transportation problem

Example

P&T Company produces canned peas.

Peas are prepared at three canneries (Bellingham,

Eugene and Albert Lea).

Shipped by truck to four distributing warehouses(Sacramento, Salt Lake City, Rapid City and

Albuquerque). 300 truckloads to be shipped.

Problem: minimize the total shipping cost.

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P&T company problem

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Shipping data for P&T problem

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Shipping cost per truckload in Warehouse

1 2 3 4 Output1 464 513 654 867 75

Cannery 2 352 416 690 791 125

3 995 682 388 685 100

Allocation 80 65 70 85 300


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Constraints coefficients for P&T Co.

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1 1 1 1

1 1 1 1

1 1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

A

Coefficient of:

11 12 13 14 21 22 23 24 31 32 33 34x x x x x x x x x x x x

Cannery

constraints

Warehouse

constraints

Transportation problem model

Transportation problem: distributesanycommodity

fromanygroup ofsources to any group of

destinations, minimizing the total dist ribution cost.

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Prototype example General problemTruckload of canned peas Units of a commodity

Three canneries msources

Four warehouses ndestinations

Output from canneryi Supplysifrom source i

Allocation to warehousej Demanddjat destinationj

Shipping cost per truckload fromcanneryito warehousej

Cost cijper unit distributed fromsource ito destinationj


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Each source has a certain supply of units to distribute

to the destinations.

Each destination has a certain demand for units to be

received from the source .

Requirements assumption: Each source has a fixed

supplyof units, which must be entirely distributed to

the destinations. Similarly, each destination has a fixed

demandfor units, which must be entirely received

from the sources.

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The feasible solutions property: a transportation

problem has feasible solution if and only if

If the supplies represent maximumamounts to be

distributed, a dummy dest inationcan be added.

Similarly, if the demands represent maximum

amounts to be received, a dummy sourcecan be

added.

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1 1

m n

i j

i j

s d


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The cost assumption: the cost of distributing units

from any source to any destination is directly

proportionalto the number of units distributed.

Thus, this cost is the unit costof distribution timesthe

number of unit s dist ributed.

Integer solution property: for transportation problems

where everysianddjhave an integer value, all basic

variables in everyBF solution also have integervalues.

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Parameter table for transp. problem

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Cost per unit distributedDestination

1 2 n Supply

Source

1 c11 c12 c1n s1

2 c21 c22 c2n s2

m cm1 cm2 cmn sm

Demand d 1 d2 dn


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The model: any problem fits the model for a

transportation problem if it can be described by a

parameter tableand if it satisfies both the

requirements assumption and thecost assumption.

The objective is to minimize the total cost of

distributing the units.

Some problems that have nothing to do with

t ransportation can be formulated as a t ransportation

problem.

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Network representation

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Formulation of the problem

Z: the total distribution cost

xij : number of units distributed from sourcei to destinationj

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1 1

minimize ,m n

ij i j

i j

Z c x

1

1

subject to

, for 1,2, , ,

, for 1,2, , ,

n

ij i

j

m

ij j

i

x s i m

x d j n

and 0, for all and .ij

x i j

Coefficient constraints

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1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

A

Coefficient of:

11 12 1 21 22 2 1 2... ... ... ...n n m m mnx x x x x x x x x

Supply

constraints

Demand

constraints


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Example: solving with Excel

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Example: solving with Excel

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Transportation simplex method

Version of the simplex called transportation simplex

method.

Problems solved by hand can use a transportation

simplex tableau.

Dimensions:

For a transportation problem withm sources andn

destinations, simplex tableauhasm +n + 1 rows and

(m + 1)(n + 1) columns.

The t ransportation simplex tableauhas onlym rows

andn columns!

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Transportation simplex tableau (TST)

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cij-ui-vj

Destination1 2 n Supply ui

Source

1 c11 c12 c1n s1

2 c21 c22 c2n s2

m cm1 cm2 cmn sn

Demand d1 d2 dn Z=

vj

Additional

information to be

added to each cell:

cij

xij

cij

Ifxijis a basic

variable:

Ifxijis a nonbasic

variable:


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Initialization: construct an initial BF solution. To begin,

all source rows and destination columns of the TST are

initially under consideration for providing a basic

variable (allocation).

1. From the rows and columns still under consideration,

select the next basic variable (allocation) according to

one of the criteria:

Northwest corner rule

Vogels approximation method

Russells approximation method

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2. Make that allocation large enough to exactly use up

the remaining supply in its row or the remaining

demand in its column (whatever is smaller).

3. Eliminate that row or column (whichever had the

smaller remaining supply or demand) from further

consideration.

4. If only one row or one column remains under

consideration, then the procedure is completed.

Otherwise return to step 1.

Go to the optimality test.

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Opt imalit y test :deriveui andvj by selecting the row

having the largest number of allocations, setting its

ui=0, and then solving the set of equations cij= ui+ vjfor each (i,j) such thatxij is basic. Ifcij - ui - vj 0 for

every (i,j) such thatxij isnonbasic, then the current

solution is optimal and stop. Otherwise, go to an

iteration.

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Iteration:

1. Determine the entering basic variable: select the

nonbasic variablexij having thelargest(in absolute

terms)negative value ofcij - ui - vj.

2. Determine the leaving basic variable: identify the

chain reaction required to retain feasibility when the

entering basic variable is increased. From the donor

cells, select the basic variable having thesmallest

value.

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Iteration:

3. Determine the new BF solution: add the value of the

leaving basic variable to the allocation for each

recipient cell. Subtract this value from the allocation

for each donor cell.

4. Apply the optimality test.

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Assignment problem

Special type of linear programming where assignees

are being assigned to perform tasks.

Example: employees to be given work assignments

Assignees can be machines, vehicles, plants or even

time slots to be assigned tasks.

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Assumptions of assignment problems

1. The number of assignees and the number of tasks are

the same, and is denoted byn.

2. Each assignee is to be assigned to exactly onetask.

3. Each task is to be performed by exactly oneassignee.

4. There is a costcij associated with assigneei

performing taskj (i,j = 1, 2, ,n).

5. The objective is to determine how well n assignments

should be made to minimize the total cost.

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Prototype example

Job Shop Company has purchased three new

machines of different types, and there are four

different locations in the shop where a machine can be

installed.

Objective: assign machines to locations.

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Cost per hour of material handling (in )

Location1 2 3 4

1 13 16 12 11

Machine 2 15 13 203 5 7 10 6


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Formulation as an assignment problem

We need the dummy machine 4, and an extremely

large cost M:

Optimal solution: machine 1 to location 4, machine 2

to location 3 and machine 3 to location 1 (total cost of

29 per hour).

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Location1 2 3 4

1 13 16 12 11

AssigneeMachine) 2 15 13 203 5 7 10 64 D) 0 0 0 0

Assignment problem model

Decision variables

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1 1

minimize ,n n

ij ij

i j

Z c x

1

1

subject to

1, for 1,2, , ,

1, for 1,2, , ,

n

ij

j

n

ij

i

x i n

x j n

and 0, for all and

( binary, for all and )

ij

ij

x i j

x i j

1 if assignee performs task ,

0 if not.ij

i jx


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Assignment vs. Transportation prob.

Assignment problem is a special type of transportation

problem where sources= assigneesand destinations=

tasksand:

#sourcesm = #destinations n;

every supplysi = 1;

every demanddj = 1.

Due to theinteger solution property, sincesi anddj are

integers,every BF solution is an integer solution for an

assignment problem. We may delete the binary

restriction and obtain alinear programming problem!

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Network representation

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Parameter table as in transportation

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Cost per unit distributedDestination

1 2 n Supply

Source1 c11 c12 c1n 1

2 c21 c22 c2n 1

m = n c n1 cn2 cnn 1Demand 1 1 1

Concluding remarks

A special algorithm for the assignment problem is the

Hungarian algorithm (more efficient).

Therefore st reamlined algorithmswere developed to

explore the special st ructureof some linear

programming problems: t ransportat ion or assignment

problems.

Transportation and assignment problems are special

cases ofminimum cost flow problems.

Network simplex methodsolves this type of problems.

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OD Transport Assignment 2012

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