Transportation Problem

A Network Model and Linear Programming Formulation

Introduction

• The transportation problem arises frequently in planning for the distribution of goods and services from several supply locations to several demand locations.

• The quantity of goods available at each supply location (origin) is limited and the quantity of goods needed at each of several demand locations (destination) is known.

• The objective in a transportation problem is to minimize the cost of shipping goods from the origin to the destinations.

Formulation

• Linear programming model of the transportation problem is

• i = index for origins, i = 1, 2, .... , m• j = index for destinations, j = 1, 2, .... , n

• xij = = number of units shipped from origin i to destination j

• cij = cost per unit of shipping from origin i to destination j

• si = supply or capacity in units at origin i

• dj = demand in units at destination j

Formulation

• Min

• Subject to

i = 1, 2, ..... , m Supply

j = 1, 2, ..... , n Demand

• xij ≥ 0 for all i and j

• Add constraints of the form xij ≤ Lij if the route from origin i to destination j has capacity Lij is called a capacitated transportation problem.

• Add rout minimum constraints of the form xij ≥ Mij if the route from origin i to destination j must handle at least Mij units.

m

i

n

jijijxc

1 1

n

1jiij sx

m

1ijij dx

Foster Generators

• Foster Generators has plants in Cleveland, Bedford and York. The production capacities over the next three month planning period for one particular type of generator are as follows:

Origin PlantThree Month Production

Capacity (units)1 Cleveland 50002 Bedford 60003 York 2500  Total 13500

Foster Generators

• The firm distributers its generators through four regional distribution centres located in Boston, Chicago, St. Louis and Lexington; the tree month forecast of demand for the distribution centres is follows:

DestinationDistribution Centre

Three Month Demand Forecast (units)

1 Boston 60002 Chicago 40003 St. Louis 20004 Lexington 1500  Total 13500

Foster Generators

• Management would like to determine how much of its production should be shipped from each plant to each distribution centre. Figure shows that the 12 distribution routes Foster can used. Such a graph is called network; the circle is referred to the nodes and the line connecting the nodes as arcs. Each origin or destination is represented by a node and each possible shipping route is represented by an arc. The amount of supply is written next to each origin node and the amount of demand is written next to each destination node. The goods shipped from origins to destinations represent the flow in network. The direction of flow from origins to destinations is indicated by the arrows.

Foster Generators

• For Foster’s transportation problem, the objective is to determine the route to be used and the quantity to be shipped via each route that will provide the minimum total transportation cost. The cost for each unit shipped on each route is given in the Table and is shown on each arc in Figure.

  DestinationOrigin Boston Chicago St. Louis LexingtonCleveland 3 2 7 6Bedford 7 5 2 3York 2 5 4 5

Foster Generators Formulation

• There are 12 variables and 7 constraints for Foster Generators transportation problem:

• Min 3x11 + 2x12 + 7x13 + 6x14 + 7x21 + 5x22 + 2x23 + 3x24 + 2x31 + 5x32 + 4x33 + 5x34

• x11 + x12 + x13 + x14 ≤ 5000

• x21 + x22 + x23 + x24 ≤ 6000

• x31 + x32 + x33 + x34 ≤ 2500

• x11 + x21 + x31 = 6000

• x12 + x22 + x32 = 4000

• x13 + x23 + x33 = 2000

• x14 + x24 + x34 = 1500

• xij ≥ 0 for i = 1, 2, 3; j = 1, 2, 3, 4

Transhipment Problem

• Transhipment problem is an extension of transportation problem in which intermediate nodes referred to as transhipment nodes.

• Transhipment problem permits shipments of goods from origin to transhipment nodes and on to destination, from one origin to another origin, from one transhipment location to another, from one destination location to another and directly from origins to destinations.

• The supply available at each origin is limited and the demand at each destination is specified. Objective of the transhipment problem is to determine how many units should be shipped over each arc in the network so that all destination demands are satisfied with the minimum possible transportation cost.

Ryan Electronics

• Ryan is an electronics company with production facilities in Denver and Atlanta. Components produced at either facility may be shipped to either of the firm’s regional warehouses, which are located in Kansas City and Louisville. From the regional warehouse the firm supplies retail outlets in Detroit, Miami, Dallas and New Orleans. The transportation cost per unit for each distribution route is shown in the Table given bellow and the arcs of the network model depicted in the Figure. The key features of the problem are shown in the network model in Figure. The supply at each origin and demand at each destination are shown in the left and right margins, respectively. Nodes 1 and 2 are the origin nodes; nodes 3 and 4 are the transhipment nodes and nodes 5, 6, 7 and 8 are the destination nodes.

Ryan Electronics

Table: Transportation costs per unit  Warehouse    Plant Kansas City Louisville    Denver 2 3    Atlanta 3 1      Retail OutletWarehouse Detroit Miami Dallas New OrleansKansas City 2 6 3 6Louisville 4 4 6 5

Ryan Electronics

• Min 2x13 + 3x14 + 3x23 + 1x24 + 2x35 + 6x36 + 3x37 + 6x38 + 4x45 + 4x46 + 6x47 + 5x48

• Subject to

• x13 + x14 ≤ 600

• x23 + x24 ≤ 400

• - x13 - x23 + x35 + x36 + x37 + x38 = 0

• - x14 - x24 + x45 + x46 + x47 + x48 = 0

• x35 + x45 = 200

• x36 + x46 = 150

• x37 + x47 = 350

• x38 + x48 = 300

• xij ≥ 0 for all i and j

• Example: Two automobile plants, Pl and P2, are linked to three dealers, Dl, D2, and D3, by way of two transit centers T1 and T2. The supply amounts at plants P1 and P2 are 1000 and 1200 cars and the demand amounts at dealers Dl, D2, and D3, are 800, 900, and 300 cars. The shipping cost per car (in hundreds of dollars) between pairs of nodes are shown on the connecting links (or arcs) of the network in the figure.

• Solution: Transshipment occurs in the network in figure, because entire supply amount of 2200 (= 1000 + 1200) cars at nodes P1 and P2 could pass through any node of network before ultimately reaching their destinations at nodes Dl, D2, and D3.

• The nodes of the network with both input and output arcs (i.e. T1, T2, Dl, and D2) act as both sources and destinations and are referred to as transshipment nodes.

• Remaining nodes are either pure supply nodes (i.e. Pl and P2) or pure demand nodes (i.e. D3).

• Transshipment model can be converted into a regular transportation model with six sources (P1, P2, T1, T2, Dl and D2) and five destinations (T1, T2, Dl, D2, and D3).

• Amounts of supply and demand at different nodes are

• Supply at a pure supply node = Original supply• Supply at a transshipment node = Original supply + Buffer• Demand at a pure demand node = Original demand• Demand at a transshipment node = Original demand + Buffer• Buffer amount should be sufficiently large to allow all original

supply (or demand) units to pass through any of transshipment nodes.

• Let B be desired buffer amount, then B = Total supply (or demand) = 1000 + 1200 (or 800 + 900 + 500) = 2200 cars

• Using the information and unit shipping costs given on network the equivalent regular transportation model is constructed

• The solution of resulting transportation model (determined by TORA) is shown in figure.

• Dealer D2 receives 1400 cars keeps to satisfy its demand and sends the remaining 500 cars to dealer D3.

T1 T2 D1 D2 D3 Supply

P1 3 4 M M M 1000

P2 2 5 M M M 1200

T1 0 7 8 6 M B

T2 M 0 M 4 9 B

D1 M M 0 5 M B

D2 M M M 0 3 B

Demand B B 800+B 900+B 500