1

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 andAlbuquerque). 300 truckloads to be shipped.

Problem: minimize the total shipping cost.

107

2

P&T company problem

108

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

3

Network representation

110

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 7512510080657085

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

x x x xx x x xx x x x

x x xx x xx x xx x x

x i j

4

Constraints coefficients for P&T Co.

112

1 1 1 11 1 1 1

1 1 1 11 1 1

1 1 11 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

Canneryconstraints

Warehouseconstraints

Transportation problem model

Transportation problem: distributes any commodityfrom any group of sources to any group ofdestinations, minimizing the total distribution cost.

113

Prototype example General problemTruckload of canned peas Units of a commodityThree canneries m sourcesFour warehouses n destinationsOutput from cannery i Supply si from source iAllocation to warehouse j Demand dj at destination jShipping cost per truckload fromcannery i to warehouse j

Cost cij per unit distributed fromsource i to destination j

5

Transportation problem model

Each source has a certain supply of units to distributeto the destinations.

Each destination has a certain demand for units to bereceived from the source .

Requirements assumption: Each source has a fixedsupply of units, which must be entirely distributed tothe destinations. Similarly, each destination has a fixeddemand for units, which must be entirely receivedfrom the sources.

114

Transportation problem model

The feasible solutions property: a transportationproblem has feasible solution if and only if

If the supplies represent maximum amounts to bedistributed, a dummy destination can be added.

Similarly, if the demands represent maximumamounts to be received, a dummy source can beadded.

115

1 1

m n

i ji j

s d

6

Transportation problem model

The cost assumption: the cost of distributing unitsfrom any source to any destination is directlyproportional to the number of units distributed.

Thus, this cost is the unit cost of distribution times thenumber of units distributed.

Integer solution property: for transportation problemswhere every si and dj have an integer value, all basicvariables in every BF solution also have integer values.

116

Parameter table for transp. problem

117

Cost per unit distributed

Destination1 2 … n Supply

Source

1 c11 c12 … c1n s1

2 c21 c22 … c2n s2

… … … … …m cm1 cm2 … cmn sm

Demand d1 d2 … dn

7

Transportation problem model

The model: any problem fits the model for atransportation problem if it can be described by aparameter table and if it satisfies both therequirements assumption and the cost assumption.

The objective is to minimize the total cost ofdistributing the units.

Some problems that have nothing to do withtransportation can be formulated as a transportationproblem.

118

Network representation

119

8

Formulation of the problem

Z: the total distribution cost

xij : number of units distributed from source i to destination j

120

1 1minimize ,

m n

ij iji j

Z c x

1

1

subject to

, for 1,2, , ,

, for 1,2, , ,

n

ij ij

m

ij ji

x s i m

x d j n

and 0, for all and .ijx i j

Coefficient constraints

121

1 1 11 1 1

1 1 11 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

Supplyconstraints

Demandconstraints

9

Example: solving with Excel

122

Example: solving with Excel

123

10

Transportation simplex method

Version of the simplex called transportation simplexmethod.

Problems solved by hand can use a transportationsimplex tableau.

Dimensions:For a transportation problem with m sources and ndestinations, simplex tableau has m + n + 1 rows and(m + 1)(n + 1) columns.

The transportation simplex tableau has only m rowsand n columns!

124

Transportation simplex tableau (TST)

125

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

Additionalinformation to beadded to each cell:

cij

xij

cij

If xij is a basicvariable:

If xij is a nonbasicvariable:

11

Transportation simplex method

Initialization: construct an initial BF solution. To begin,all source rows and destination columns of the TST areinitially under consideration for providing a basicvariable (allocation).

1. From the rows and columns still under consideration,select the next basic variable (allocation) according toone of the criteria:

Northwest corner rule

Vogel’s approximation method

Russell’s approximation method

126

Transportation simplex method

2. Make that allocation large enough to exactly use upthe remaining supply in its row or the remainingdemand in its column (whatever is smaller).

3. Eliminate that row or column (whichever had thesmaller remaining supply or demand) from furtherconsideration.

4. If only one row or one column remains underconsideration, then the procedure is completed.Otherwise return to step 1.

Go to the optimality test.

127

12

Transportation simplex method

Optimality test: derive ui and vj by selecting the rowhaving the largest number of allocations, setting itsui=0, and then solving the set of equations cij = ui+ vjfor each (i,j) such that xij is basic. If cij - ui - vj ³ 0 forevery (i,j) such that xij is nonbasic, then the currentsolution is optimal and stop. Otherwise, go to aniteration.

128

Transportation simplex method

Iteration:

1. Determine the entering basic variable: select thenonbasic variable xij having the largest (in absoluteterms) negative value of cij - ui - vj.

2. Determine the leaving basic variable: identify thechain reaction required to retain feasibility when theentering basic variable is increased. From the donorcells, select the basic variable having the smallestvalue.

129

13

Transportation simplex method

Iteration:

3. Determine the new BF solution: add the value of theleaving basic variable to the allocation for eachrecipient cell. Subtract this value from the allocationfor each donor cell.

4. Apply the optimality test.

130

Assignment problem

Special type of linear programming where assigneesare being assigned to perform tasks.

Example: employees to be given work assignments

Assignees can be machines, vehicles, plants or eventime slots to be assigned tasks.

131

14

Assumptions of assignment problems

1. The number of assignees and the number of tasks arethe same, and is denoted by n.

2. Each assignee is to be assigned to exactly one task.

3. Each task is to be performed by exactly one assignee.

4. There is a cost cij associated with assignee iperforming task j (i, j = 1, 2, …, n).

5. The objective is to determine how well n assignmentsshould be made to minimize the total cost.

132

Prototype example

Job Shop Company has purchased three newmachines of different types, and there are fourdifferent locations in the shop where a machine can beinstalled.

Objective: assign machines to locations.

133

Cost per hour of material handling (in €)Location

1 2 3 41 13 16 12 11

Machine 2 15 – 13 203 5 7 10 6

15

Formulation as an assignment problem

We need the dummy machine 4, and an extremelylarge cost M:

Optimal solution: machine 1 to location 4, machine 2to location 3 and machine 3 to location 1 (total cost of29€ per hour).

134

Location1 2 3 4

1 13 16 12 11Assignee(Machine)

2 15 M 13 203 5 7 10 6

4(D) 0 0 0 0

Assignment problem model

Decision variables

135

1 1minimize ,

n n

ij iji j

Z c x

1

1

subject to

1, for 1,2, , ,

1, for 1,2, , ,

n

ijj

n

iji

x i n

x j n

and 0, for all and

( binary, for all and )ij

ij

x i jx i j

1 if assignee performs task ,0 if not.ij

i jx

16

Assignment vs. Transportation prob.

Assignment problem is a special type of transportationproblem where sources = assignees and destinations =tasks and:

#sources m = #destinations n;

every supply si = 1;

every demand dj = 1.

Due to the integer solution property, since si and dj areintegers, every BF solution is an integer solution for anassignment problem. We may delete the binaryrestriction and obtain a linear programming problem!

136

Network representation

137

17

Parameter table as in transportation

138

Cost per unit distributed

Destination1 2 … n Supply

Source

1 c11 c12 … c1n 12 c21 c22 … c2n 1… … … … … …

m = n cn1 cn2 … cnn 1Demand 1 1 … 1

Concluding remarks

A special algorithm for the assignment problem is theHungarian algorithm (more efficient).

Therefore streamlined algorithms were developed toexplore the special structure of some linearprogramming problems: transportation or assignmentproblems.

Transportation and assignment problems are specialcases of minimum cost flow problems.

Network simplex method solves this type of problems.

139