OD Transport Assignment LARGE 2010

7/24/2019 OD Transport Assignment LARGE 2010

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TRANSPORTATION ANDASSIGNMENT PROBLEMS


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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.

107


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P&T company problem

108


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Shipping data for P&T problem

109

Shipping cost per truckload in

Warehouse

1 2 3 4 Output

1 464 513 654 867 75

Cannery 2 352 416 690 791 125

3 995 682 388 685 100

Allocation 80 65 70 85 300


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Network representation

110


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Formulation of the problem

111

11 12 13 14 21 22

23 24 31 32 33 34

minimize 464 513 654 867 352 416690 791 995 682 388 685

Z x x x x x xx x x x x x

11 12 13 14

21 22 23 24

31 32 33 34

11 21 31

12 22 32

13 23 33

14 24 34

subject to 75

125

100

80

65

7085

and 0 ( 1, 2,3, 4; 1, 2,3, 4)ij

x x x x

x x x x

x x x x

x x x

x x x

x x xx x x

x i j


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Constraints coefficients for P&T Co.

112

1 1 1 1

1 1 1 1

1 1 1 11 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


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Transportation problem model

Transportation problem: distributesanycommodity

fromanygroup ofsources to any group ofdestinations, minimizing the total dist ribut ion cost.

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Prototype example General problem

Truckload of canned peas Units of a commodityThree canneries msources

Four warehouses ndestinations

Output from canneryi Supplysifrom source i

Allocation to warehousej Demanddjat destinationjShipping cost per truckload fromcannery ito 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 inat ioncan be added.

Similarly, if the demands represent maximum

amounts to be received, a dummy sourcecan beadded.

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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 directlyproportionalto the number of units distributed.

Thus, this cost is the unit costof distribution timesthe

number of units 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

117

Cost per unit distributed

Destination

1 2 n Supply

Source

1 c11 c12 c1n s12 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 aparameter t ableand 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

transportat ion can be formulated as a t ransportat ion

problem.

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119


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Formulation of the problem

Z: the total distribution cost

xij : number of units distributed from sourcei to destinationj

120

1 1

minimize ,m n

ij ij

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


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Coefficient constraints

121

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 ransportat ion simplex tableauhas onlym rows

andn columns!

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Transportation simplex tableau (TST)

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cij-ui-vj

Destination

1 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

Additionalinformation 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 areinitially under consideration for providing a basic

variable (allocation).

1. From the rows and columns still under consideration,

select the next basic variable (allocation) according toone 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 remainingdemand 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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l h d


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Optimalit y test :deriveui andvj by selecting the row

having the largest number of allocations, setting itsui=0, and then solving the set of equationscij= 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 aniteration.

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i i l h d


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Iteration:

1. Determine the entering basic variable: select thenonbasic 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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i i l h d


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Iteration:

3. Determine the new BF solution: add the value of theleaving 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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A i bl


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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.

131

A i f i bl


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

5. The objective is to determine how welln assignments

should be made to minimize the total cost.

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P t t l


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Prototype example

Job Shop Company has purchased three new

machines of different types, and there are fourdifferent 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 )

Location

1 2 3 4

1 13 16 12 11Machine 2 15 13 20

3 5 7 10 6

F l ti i t bl


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Formulation as an assignment problem

We need the dummy machine 4, and an extremely

large cost M:

Optimal solut ion: machine 1 to location 4, machine 2to location 3 and machine 3 to location 1 (total cost of

29 per hour).

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Location

1 2 3 4

1 13 16 12 11

Assignee

Machine)

2 15 13 20

3 5 7 10 6

4 D) 0 0 0 0

A i t bl d l


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Assignment problem model

Decision variables

135

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

i n

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 j

x

A i t T t ti b


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Assignment vs. Transportation prob.

Assignment problem is a special type of transportation

problem where sources= assigneesand destinations=tasksand:

#sourcesm = #destinationsn;

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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N t k t ti


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137

P t t bl i t t ti


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Parameter table as in transportation

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Cost per unit distributed

Destination

1 2 n Supply

Source

1 c11 c12 c1n 12 c21 c22 c2n 1

m = n c n1 cn2 cnn 1

Demand 1 1 1

C l di k


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Concluding remarks

A special algorithm for the assignment problem is the

Hungarian algorithm (more efficient). Therefore streamlined algorithmswere developed to

explore the special structureof 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.

OD Transport Assignment LARGE 2010

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