OD3 PL Transport Assignment

OPTIMIZAO E DECISO 10/11PL #3 Transportation and Assignment Problems

Alexandra Moutinho

(from Hillier & Lieberman Introduction to Operations Research, 8th edition)

1

METRO WATER DISTRICT is an agency that administers water distribution in a large geographicregion. The region is fairly arid, so the district must purchase and bring in water from outside theregion. The sources of this imported water are the Colombo, Sacron, and Calorie rivers. The districtthen resells the water to users in the region. Its main customers are the water departments of thecities of Berdoo, Los Devils, San Go, and Hollyglass.

It is possible to supply any of these cities with water brought in from any of the three rivers, with theexception that no provision has been made to supply Hollyglass with Calorie River water. However,because of the geographic layouts of the aqueducts and the cities in the region, the cost to thedistrict of supplying water depends upon both the source of the water and the city being supplied.The variable cost per acre foot of water (in tens of dollars) for each combination of river and city isgiven in Table 1.

Cost (tens of $) per Acre FootBerdoo Los Devils San Go Hollyglass Supply

Colombo River 16 13 22 17 50Sacron River 14 13 19 15 60Calorie River 19 20 23 - 50

Minimum needed 30 70 0 10 (in units of 1 millionacre feet)Requested 50 70 30

Table 1 Water resources data for Metro Water District.

Despite these variations, the price per acre foot charged by the district is independent of the sourceof the water and is the same for all cities.

The management of the district is now faced with the problem of how to allocate the available waterduring the upcoming summer season. In units of 1 million acre feet, the amounts available from thethree rivers are given in the rightmost column of Table 1. The district is committed to providing acertain minimum amount to meet the essential needs of each city (with the exception of San Go,which has an independent source of water), as shown in the minimum needed row of the table. Therequested row indicates that Los Devils desires no more than the minimum amount, but that Berdoowould like to buy as much as 20 more, San Go would buy up to 30 more, and Hollyglass will take asmuch as it can get.

Management wishes to allocate all the available water from the three rivers to the four cities in sucha way as to at least meet the essential needs of each city while minimizing the total cost to thedistrict.

1. Formulate this problem as a transportation problem by constructing the appropriate parametertable.

2. Using Russells approximation method, apply the transportation simplex method to obtain theoptimal solution for this problem.

3. Use Excel Solver to solve this problem and confirm the solution obtained in 2.

Optimizao e Deciso 09/10 - PL #3 Transportation and Assignment Problems - Alexandra Moutinho 2

Resolution:

1. Table 1 already is close to the proper form for a parameter table, with the rivers being the sourcesand the cities being the destinations. However, the one basic difficulty is that it is not clear whatthe demands at the destinations should be. Hollyglass has an upper bound given by the differencebetween the available water supply and the water requested by the other districts:(50 + 60 + 50)? (30 + 70 + 0) = 60Since the demand quantities must be constants, let us temporarily suppose that it is notnecessary to satisfy the minimum needs, and assume the upper bounds as constraints. However,considering the Hollyglass new request of 60, we now have excess demand capacity, so we needto adjust the problem introducing a dummy source. The imaginary supply quantity is thedifference between the sum of the demands and the sum of the real supplies:(50 + 70 + 30 + 60) ? (50 + 60 + 50) = 50The resulting parameter table is given in Table 2. The cost entries in the dummy row are zerobecause there is no cost incurred by the fictional allocations from this dummy source. On theother hand, a huge unit cost M is assigned to the Calorie River Hollyglass spot, preventing thisallocation.

Cost (tens of $) per Acre FootDestination

Berdoo Los Devils San Go Hollyglass Supply

Source

Colombo River 16 13 22 17 50Sacron River 14 13 19 15 60Calorie River 19 20 23 M 50Dummy 0 0 0 0 50

Demand 50 70 30 60

Table 2 Parameter table without minimum needs.

Let us now go back and check each citys minimum needs:

? San Go has no minimum needs;? Hollyglass, with a demand of 60 exceeds the dummys source supply of 50 by 10, which

corresponds to the minimum request provided by the real sources in any feasible solution;? Los Devils entire demand of 70 must be supplied from real sources, so we must assign a huge

unit cost M to the Los Devils Dummy spot;? Berdoo: the dummy source could supply some minimum need in addition to extra requested

amount. In order to prevent the dummy source from supplying more than 20 of Berdoo totaldemand of 50, we split Berdoo into 2 destinations, one with the minimum request of 30 and adummy cost of M, and one extra with the remaining request of 20 and a dummy cost of zero.

The final parameter table is given in Table 3.


Berdoo(min)

Berdoo(extra)

Los Devils San Go Hollyglass Supply

Source

Colombo River 16 16 13 22 17 50Sacron River 14 14 13 19 15 60Calorie River 19 19 20 23 M 50Dummy M 0 M 0 0 50

Demand 30 20 70 30 60

Table 3 Final parameter table.


2. Summary of the Transportation Simplex Method

Initialization: Construct an initial BF solution by the general procedure for constructing an initial BFsolution described below. Go to the optimality test.

Optimality test: Derive ?? and ?? by selecting the row having the largest number of allocations,setting its ?? = 0, and then solving the set of equations ??? ? ?? ? ?? for each(?? ?) such that ??? is basic. If ??? ? ?? ? ?? ? 0 for every (?? ?) such that ??? isnonbasic, then the current solution is optimal, so stop. Otherwise, go to aniteration.

Iteration:

1) Determine the entering basic variable: Select the nonbasic variable ??? having the largest (inabsolute terms) negative value of ??? ? ?? ? ??.

2) Determine the leaving basic variable: Identify the chain reaction required to retain feasibilitywhen the entering basic variable is increased. From the donor cells (cells with minuscontribution), select the basic variable having the smallest value.

3) Determine the new BF solution: Add the value of the leaving basic variable to the allocationfor each recipient cell. Subtract this value from the allocation for each donor cell.?

General procedure for constructing an initial BF solution

To begin, all source rows and destination columns of the transportation simplex tableau 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 to some criterion (in this case, Russells approximation method described below).

2) Make that allocation large enough to exactly use up the remaining supply in its row or theremaining demand in its column (whichever is smaller).

3) Eliminate that row or column (whichever had the smaller remaining supply or demand) fromfurther consideration. (If the row and column have the same remaining supply and demand,then arbitrarily select the row as the one to be eliminated. The column will be used later toprovide a degenerate basic variable, i.e., a circled allocation of zero.)

4) If only one row or only one column remains under consideration, then the procedure iscompleted by selecting every remaining variable (i.e., those variables that were neitherpreviously selected to be basic nor eliminated from consideration by eliminating their row orcolumn) associated with that row or column to be basic with the only feasible allocation.Otherwise, return to step 1.?

Russells approximation method: For each source row ? remaining under consideration, determineits ???, which is the largest unit cost ??? still remaining in that row. For each destination column? remaining under consideration, determine its ???, which is the largest unit cost ??? still remainingin that column. For each variable ??? not previously selected in these rows and columns, calculate???? ??? ? ??? ? ???. Select the variable having the largest (in absolute terms) negative value of ???.(Ties may be broken arbitrarily.)?Using Russells approximation method in step 1 of the General procedure for constructing an initialBF solution, at iteration 1, the largest unit cost in row 1 is ??? = 22, the largest in column 1 is??? ? ?, and so forth. Thus,

???? ??? ? ??? ? ??? = 16 ? 22?? ? ????Calculating all the ??? values for ? = 1, 2, 3, 4 and ? = 1, 2, 3, 4, 5 shows that ???= 0? ?? has thelargest negative value, so ??? = 50 is selected as the first basic variable (allocation). Thisallocation exactly uses up the supply in row 4, so this row is eliminated from further consideration.


Note that eliminating this row changes ??? and ??? for the next iteration. Therefore, the seconditeration requires recalculating the ??? with ? = 1, 3 as well as eliminating ? = 4. The largestnegative value now is

???? ??? ? ??? ? ??? = 17? 22 ?? ? ????so ??? = 10 becomes the second basic variable (allocation), eliminating column 5 from furtherconsideration.

Proceed similarly for the subsequent iterations. The initial BF solution obtained by Russellsapproximation method is shown in Table 4.

Iteration ??? ??? ??? ??? ??? ??? ??? ??? ???

LargestNegative ??? Allocation

1 22 19 M M M 19 M 23 M ???= ?2? ??? = 502 22 19 M 19 19 20 23 M ???= ?5 ?? ??? = 103 22 19 23 19 19 20 23 ???= ?29 ??? = 404 19 23 19 19 20 23 ???= ?26 ??? = 305 19 23 19 19 23 ???= ?24* ??? = 306 Irrelevant ??? = 0

??? = 20??? = 30? = 2,570

*Tie with???? ?24 broken arbitrarily.Table 4 - Initial BF solution from Russells approximation method.

The initial transportation simplex tableau is shown in Table 5.


Berdoo(min)

Berdoo(extra)

LosDevils

SanGo

Hollyglass Supply ui

Source

ColomboRiver

16 16 1340

22 1710

50

SacronRiver

1430

14 1330

19 15 60

CalorieRiver

190

1920

20 2330

M 50

DummyM 0 M 0 0

5050

Demand 30 20 70 30 60 Z=2,570vj

Table 5 Initial transportation simplex tableau.

We may now apply the optimality test. Since row 3 has the largest number of allocations (3 basicvariables), we set ?? = 0 (we have ?? ? ? 1 basic variables and ?? ? unknowns, so we mayarbitrate a value). Solve each equation that corresponds to a basic variable in the initial BFsolution:

???: 19 = ?? ? ??. Set ?? = 0, so ?? = 19,???: 19 = ?? ? ??. ?? = 19,???: 23 = ?? ? ??. ?? = 23.???: 14 = ?? ? ??. Know ?? = 19, so ?? ? ?5.???: 13 = ?? ? ??. Know ?? =?5, so ?? = 18.???: 13 = ?? ? ??. Know ?? = 18, so ?? ? ?5.???: 17 = ?? ? ??. Know ?? ? ?5, so ?? = 22.???: 0 = ?? ? ??. Know ?? = 22, so ?? ? ?22.


We can also perform these operations directly in the transportation simplex tableau (see Table 6):


Berdoo(min)

Berdoo(extra)

LosDevils

SanGo


Source

ColomboRiver

16 16 1340

22 1710

50 -5

SacronRiver

1430

14 1330

19 15 60 -5

CalorieRiver

190

1920

20 2330

M 50 0

DummyM 0 M 0 0

5050 -22

Demand 30 20 70 30 60 Z=2,570vj 19 19 18 23 22

Table 6 Complete initial transportation simplex tableau.

We are now in a position to apply the optimality test by checking the values of ??? ? ?? ? ?? of thenonbasic variables given in Table 6:

Nonbasic variable ??? ? ?? ? ?? Nonbasic variable ??? ? ?? ? ????? 2 ??? 2??? 2 ??? M-22??? 4 ??? M+3??? 0 ??? 3??? 1 ??? M+4??? -2 ??? -1

Because two of these values are negative (for ??? and ???), we conclude that the current BFsolution is not optimal. Therefore, the transportation simplex method must next go to aniteration to find a better BF solution.

The entering basic variable is ??? because it has the largest negative value of ??? ? ?? ? ??. When??? is increased from 0 by any particular amount, a chain reaction is set off that requiresalternately decreasing and increasing current basic variables by the same amount in order tocontinue satisfying the supply and demand constraints. This chain reaction is depicted in the nexttable, where the + sign inside a box in cell (2, 5) indicates that the entering basic variable is beingincreased there and the + or ? sign next to other circles indicate that a basic variable is beingincreased or decreased there.


Berdoo(min)

Berdoo(extra)

LosDevils

SanGo


Source

ColomboRiver

16 16 1340

22 1710

50 -5

SacronRiver

1430

14 1330

19 15

+

60 -5

CalorieRiver

190

1920

20 2330

M 50 0

DummyM 0 M 0 0

5050 -22

Demand 30 20 70 30 60 Z=2,570vj 19 19 18 23 22

+

-

-


Each donor cell (indicated by a minus sign) decreases its allocation by exactly the same amount asthe entering basic variable and each recipient cell (indicated by a plus sign) is increased. Theentering basic variable will be increased as far as possible until the allocation for one of the donorcells drops all the way down to 0. Since the original allocations for the donor cells are ??? = 10and ??? = 30, ??? will be the one that drops to 0 as ??? is increased (by 10). Therefore, ??? is theleaving basic variable.

Since each of the basic variables is being increased or decreased by 10, the values of the basicvariables in the new BF solution are ??? = 50, ??? = 30,??? = 20,??? = 10,??? = 0,??? =20, ??? = 30 and ??? = 50.Apply the optimality test and the necessary (plus 2) iterations to obtain the optimal solution??? = 50,??? = 20, ??? = 40, ??? = 30, ??? = 20,??? = 30 and ??? = 20 with? = 2,460.

3. Solving with Excel Solver we obtain the same optimal solution (verify!):

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OD3 PL Transport Assignment