The Transportation Model –

Formulations

The Transportation Model

The transportation model is a special class of LPPs that deals with transporting(=shipping) a commodity from sources (e.g. factories) to destinations (e.g. warehouses). The objective is to determine the shipping schedule that minimizes the total shipping cost while satisfying supply and demand limits. We assume that the shipping cost is proportional to the number of units shipped on a given route.

We assume that there are m sources 1,2, …, m and n destinations 1, 2, …, n. The cost of shipping one unit from Source i to Destination j is cij.

We assume that the availability at source i is ai (i=1, 2, …, m) and the demand at the destination j is bj (j=1, 2, …, n). We make an important assumption: the problem is a balanced one. That is

n

jj

m

ii ba

11

That is, total availability equals total demand.

We can always meet this condition by introducing a dummy source (if the total demand is more than the total supply) or a dummy destination (if the total supply is more than the total demand).

Let xij be the amount of commodity to be shipped from the source i to the destination j.

Thus the problem becomes the LPP

n

jijij

m

i

xcz11

Minimize

subject to

),...,2,1(

),...,2,1(

1

1

njbx

miax

j

m

iij

i

n

jij

0ijx

Thus there are mn decision variables xij and m+n constraints. Since the sum of the first m constraints equals the sum of the last n constraints (because the problem is a balanced one), one of the constraints is redundant and we can show that the other m+n-1 constraints are LI. Thus any BFS will have only m+n-1 nonzero variables.

Though we can solve the above LPP by Simplex method, we solve it by a special algorithm called the transportation algorithm. We present the data in an mn tableau as explained below.

cc1111 cc1212 cc1n1n aa11

cc2121 cc2222 cc2n2n aa22

ccm1m1 ccm2m2 ccmnmn aamm

bb11 bb22 bbnn

Source

1

2

.

.

m

Destination

1 2 . . n Supply

Demand

Formulation of Transportation Models

Example 5.1-2

MG Auto has three plants in Los Angeles, Detroit, and New Orleans, and two major distribution centers in Denver and Miami. The capacities of the three plants during the next quarter are 1000, 1300 and 1200 cars. The quarterly demands at the two distribution centers are 2300 and 1400 cars. The transportation cost per car from Los Angeles to Denver and Miami are $80 and $215 respectively. The corresponding figures from Detroit and New Orleans are 100, 108 and 102, 68 respectively.

Formulate the transportation Model.

Since the total demand = 3700 > 3500 (Total supply) we introduce a dummy supply with availability 3700-3500=200 units to make the problem a balanced one. If a destination receives u units from the dummy source, it means that that destination gets u units less than what it demanded. We usually put the cost per unit of transporting from a dummy source as zero (unless some restrictions are there). Thus we get the transportation tableau

80 80 215215 10001000

100100 108108 13001300

102102 6868 12001200

0 0 00 200200

23002300 14001400

Source

Destination

Denver Miami Supply

Demand

Los Angeles

Detroit

New Orleans

Dummy

We write inside the (i,j) cell the amount to be shipped from source i to destination j. A blank inside a cell indicates no amount was shipped.

Problem 5 Problem Set 5.1A Page 169

In the previous problem, penalty costs are levied at the rate of $200 and $300 for each undelivered car at Denver and Miami respectively. Additionally no deliveries are made from the Los Angeles plant to the Miami distribution center. Set up the transportation model.

The above imply that the "cost" of transporting a car from the dummy source to Denver and Miami are respectively 200 and 300. The second condition means we put a "high" transportation cost from Los Angeles to Miami. We thus get the tableau

80 80 MM 10001000

100100 108108 13001300

102102 6868 12001200

200200 300300 200200

23002300 14001400

Source

Destination

Denver Miami Supply

Demand

Los Angeles

Detroit

New Orleans

Dummy

Note: M indicates a very "big" positive number. In TORA it is denoted by "infinity".

Problem 8 Problem Set 5.1A Page 170

Three refineries with daily capacities of 6,5, and 8 million gallons, respectively, supply three distribution areas with daily demands of 4,8, and 7 million gallons, respectively.Gasoline is distributed to the three distribution areas through a network of pipelines. The transportation cost is 10 cents per 1000 gallons per pipeline mile. The table below gives the mileage between the refineries and the distribution areas. Refinery 1 is not connected to the distribution area 3.

Distribution Area1 2 3

Refinery

1 120 180 -

2 300 100 80 3 200 250 120

Construct the associated transportation model.

(Solution in the next slide)

12 12 1818 MM 66

3030 1010 88 55

2020 2525 1212 88

44 88 77

Source

Destination

1 2 3 Supply

1

2

3

Demand

The problem is a balanced one. M indicates a very "big" positive number.

Refinery

Distribution Area

The total cost will be 1000* iji j

ijxc

3

1

3

1

Problem 10 Problem Set 5.1A Page 170

In the previous problem, suppose that the daily demand at area 3 drops to 4 million gallons. Surplus production at refineries 1 and 2 is diverted to other distribution areas by truck. The transportation cost per 100 gallons is $1.50 from refinery 1 and $2.20 from refinery 2. Refinery 3 can divert its surplus production to other chemical processes within the plant.

Formulate the problem as a transportation model.

We introduce a dummy destination. Solution follows.

12 12 1818 MM 1515 66

3030 1010 88 2222 55

2020 2525 1212 00 88

44 88 44 33

Source

Destination

1 2 3 Dummy Supply 1 2 3

Demand

M indicates a very "big" positive number.

Refinery

Distribution Area

The total cost will be 1000* iji j

ijxc

3

1

3

1

Problem 11 Problem Set 5.1 A Pages 170-171

Three orchards supply crates of oranges to four retailers. The daily demand at the four retailers is 150,150,400, and 100 crates, respectively. Supply at the three orchards is dictated by available regular labor and is estimated at 150, 200, and 250 crates daily. However, both orchards 1 and 2 have indicated that they could supply more crates, if necessary by using overtime labor. Orchard 3 does not offer this option. The transportation costs (in dollars) per crate from the orchards to the retailers are given in Table below.

Retailer1 2 3 4

Orchard1 1 2 3 2 2 2 4 1 2 3 1 3 5 3

Formulate the problem as a transportation model.

Since the orchards 1, 2 can supply more crates with overtime labor, we increase their capacities to 150+200=350 and 200+200=400 respectively (as initially the total supply fell short by 200). But then to balance the problem we add a dummy destination. The tableau follows.

11 22 33 22 00 350350

22 44 11 22 00 400400

11 33 55 33 MM 250250

150150 150150 400400 100100 200200

Source

Destination

1 2 3 4 Dummy Supply

1

2

3

Demand

Orchard

Retailer

Problem 8.1-9 from Hillier and Lieberman (Introduction to Operations Research, 7th Edition)

The Build-Em-Fast Company has agreed to supply its best customer with three widgets during each of the next 3 weeks, even though producing them will require some overtime work. The relevant production data are as follows:

Week Max Production Max Production Prod Cost / unit Regular Time Overtime Regular Time

1 2 2 $300 2 3 2 $500 3 1 2 $400

The cost / unit produced overtime for each week is $100 more than for regular time. The cost of storage is $50 / unit for each week it is stored. There is already an inventory of 2 widgets on hand currently, but the company does not want to retain any widgets in inventory after the 3 weeks.

Formulate the problem as a transportation problem.

There are 6 “sources” namely widgets produced regular time and overtime for the three weeks. Also there are 3 “destinations” viz. demand for the three weeks.

We let xij as the number of units produced regular time in week (i+1)/2 for use in week j (i=1,3,5; j=1,2,3). We let xij as the number of units produced overtime in week i/2 for use in week j (i=2,4,6; j=1,2,3). Thus

2

1

2

3

2

2

636261

535251

434241

333231

232221

131211

xxx

xxx

xxx

xxx

xxx

xxx

To make these equalities we add a dummy destination and let xi4 as the amount “transported” from Source i to this dummy. Thus the availabilities at the 6 sources are 2,2,3,2,1,2 respectively.

The demands at the three destinations (=demand for the three weeks) are1,3,3 respectively (as the initial inventory is 2 widgets).

To make the problem balanced we add demand 12 – 7 = 5 at the dummy destination.

The cost per unit, cij are as follows:

0400350300 14131211 cccc

0500450400 24232221 cccc

0550500 34333231 cccMc

0650600 44434241 cccMc

0400 54535251 ccMcMc

0500 64636261 ccMcMc

These are written in a transportation tableau.

300 300

350350 400400 00 22

400400 450450 500500 00 22

MM 500500 550550 0 0 33

MM 600600 650650 0 0 22

MM MM 400400 00 11

MM MM 500500 00 22

11 33 33 55

Source

Destination

1 2 3 Dummy Supply 1 2 3

Demand

Prod.week1 Reg time

4

5

6

Prod.week1 Over timeProd.week2 Reg time

Prod.week3 Reg time

Prod.week2 Over time

Prod.week3 Over time

Demand for Week