AED Economics 702: Computational Economics

1
AED Economics 702 Computational
Economics Transportation, Transshipment
Assignment Models Winston Venkataramanan
Chapter 7.
2
Model Description

• A transportation problem basically deals with the
  problem of finding the best way to fulfill the
  demand of n demand points using the capacities of
  m supply points. The optimal solution will
  include a variable cost constraint for shipping
  the product from one supply point to a demand
  point.

3
Formulating Transportation Problems

• Example 1 PowerCo has three (3) electric power
  plants that supply the electric needs of four
  cities (4).
• The associated supply of each plant and demand of
  each city is given in the table 1.
• The cost of sending 1 million kwh of electricity
  from a plant to a city depends on the distance
  the electricity must travel.

4
Transportation tableau

• A transportation problem is specified by the
  supply points or nodes, the demand points or
  nodes, and the shipping or transport costs
  (generally taken as linear). So the relevant
  data can be summarized in a transportation
  tableau.

5
Table 1. Shipping costs, Supply, and Demand for
PowerCo Example
Transportation Tableau
6
Objective function

• Since we want to minimize the total cost of
  shipping from plants to cities
• Minimize Z ?(i1,3) ?(j1,4) c(i,j) X(i,j)
• Minimize Z 8X116X1210X139X14
• 9X2112X2213X237X24
• 14X319X3216X335X34

7
Supply Constraints

• Each supply point has a upper bound on production
  capacity
• Plant 1 X11 X12 X13 X14 lt 35
• Plant 2 X21 X22 X23 X24 lt 50
• Plant 3 X31 X32 X33 X34 lt 40

8
Demand Constraints

• Each demand point has a needed level of
  electricity in kwh
• City 1 X11 X21 X31 gt 45
• City 2 X12 X22 X32 gt 20
• City 3 X13 X23 X33 gt 30
• City 4 X14 X24 X34 gt 30

9
Sign Constraints

• Since a negative amount of electricity can not be
  shipped to a city all Xijs must be non negative
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

10
LP Formulation of PowerCos Problem

• Min Z 8X116X1210X139X149X2112X2213X237
  X24
• 14X319X3216X335X34
• S.T. X11X12X13X14 lt 35 (Supply Constraints)
• X21X22X23X24 lt 50
• X31X32X33X34 lt 40
• X11X21X31 gt 45 (Demand Constraints)
• X12X22X32 gt 20
• X13X23X33 gt 30
• X14X24X34 gt 30
• Xij gt 0 (i 1,2,3 j 1,2,3,4)

11
General Description of a Transportation Problem

• A set of m supply points from which a good is
  shipped. Supply point i can supply at most si
  units.
• A set of n demand points to which the good is
  shipped. Demand point j must receive at least di
  units of the shipped good.
• Each unit produced at supply point i and shipped
  to demand point j incurs a variable cost of cij.

12
General model notation

• Xij number of units shipped from supply point i
  to demand point j

13
Balanced Transportation Problem

• If Total Supply equals to Total Demand, the
  problem is said to be a balanced transportation
  problem

14
Balancing a TP if total supply exceeds total
demand

• Given total supply gt total demand
• Create a dummy demand point to absorb surplus
  supply,
• Cost of transport to the fictitious demand point
  is zero!

15
Balancing a transportation problem if total
supply is less than total demand

• The problem has no feasible solution,
• Generally in such situations a penalty cost is
  often associated with unmet demand. The total
  penalty cost is desired to be at a minimum at the
  optimal solution.
• The model is balanced by adding a dummy or
  shortage supply point. The cost of shipping from
  this point is the penalty.

16
Balancing a TP if total supply is less than total
demand
17
7
8
10
9
7
8
20
22
23
Table 5. Transportation Tableau for Reservoir
Example
18
Finding Basic Feasible Solution for TP

• A balanced TP with m supply points and n demand
  points is easy to solve, although it has m n
  equality constraints. The reason for that is, if
  a set of decision variables (xijs) satisfy all
  but one constraint, the values for xijs will
  satisfy that remaining constraint automatically.

19
Transportation Models in LINDO
LINDO can be used to model transportation
problems but they must be typed out in long form
20
Transportation Model in LINGO
LINGO is particularly suited to modeling
transportation problems as it can handle
multi-dimensional arrays
21
LINGO and Microsoft Excel
22
Linking LINGO with Excel through the _at_OLE command
LINGO can be linked using the _at_OLE command
directly to an Excel Spreadsheet. Data for the
model as well as model output can be passed back
and forth between LINGO and MS Excel.
23
Transshipment Problems

• A transportation problem allows only shipments
  that go directly from supply points to demand
  points.
• In many situations, shipments are allowed between
  supply points or between demand points. Sometimes
  there may also be points (called transshipment
  points) through which goods can be transshipped
  on their journey from a supply point to a demand
  point.
• Fortunately, the optimal solution to a
  transshipment problem can be found by solving a
  transportation problem.

24
Transshipment Problems

• The following steps describe how the optimal
  solution to a transshipment problem can be found
  by solving a transportation problem.
• Step1. If necessary, add a dummy demand point
  (with a supply of 0 and a demand equal to the
  problems excess supply) to balance the problem.
  Shipments to the dummy and from a point to itself
  will be zero. Let s total available supply.

25
Transshipment Problems

• Step2. Construct a transportation tableau as
  follows A row in the tableau will be needed for
  each supply point and transshipment point, and a
  column will be needed for each demand point and
  transshipment point.
• Each supply point will have a supply equal to
  its original supply
• Each demand point will have a demand to its
  original demand.

26
Transshipment Problems

• Let s total available supply. Then each
  transshipment point will have a supply equal to
  (points original supply) s and a demand equal
  to (points original demand) s.
• This ensures that any transshipment point that is
  a net supplier will have a net outflow equal to
  points original supply and a net demander will
  have a net inflow equal to points original
  demand.

27
Transshipment Problems

• Although we dont know how much will be shipped
  through each transshipment point, we can be sure
  that the total amount will not exceed s.

Lets review the transshipment model given in WV
and compare to the transportation model
28
Transshipment or Transportation?

• Transportation allows shipment from supply node
  to demand node only.
• Transshipment allows flows from supply node i, to
  supply node j.
• Transshipment allows flows from supply nodes to
  intermediate nodes before flowing to demand
  nodes.
• Supply nodes can send but not receive.
• Demand nodes can receive but not send.
• Transshipment nodes can both send and receive
  from other nodes.

29
WIDGETCO MANUFACTURING
Memphis, Denver (S) / N.Y., Chicago (TS), /
L.A., Boston (D)
30
WIDGETCO MANUFACTURING FLOW DIAGRAM
New York
Memphis 150 w/day
Los Angeles 130 w/day
Boston 130 w/day
Denver 200 w/day
Chicago
DEMAND NODES
SUPPLY NODES
TRANSSHIPMENT NODES
31
Convert the Transshipment model to a balanced
Transportation model and solve

• If total supply exceeds total demand, create a
  dummy demand node to balance the model.
• Shipments to the dummy node and from a node to
  itself incur zero shipping cost.
• Construct a transportation tableau.
• Each supply node and transshipment node require a
  row in the tableau.
• Each demand node and transshipment node require a
  column in the tableau.
• Each supply node and demand nodes have supplies
  and demands equal to their respective quantities.
• Transshipment points have a supply and demand
  equal to the that points supply or demand plus S
  total supply.

32
WIDGETCO MANUFACTURING FLOW DIAGRAM
130
130
New York
Memphis 150 w/day
Los Angeles 130 w/day
Boston 130 w/day
Denver 200 w/day
Chicago
130
DEMAND NODES
SUPPLY NODES
TRANSSHIPMENT NODES
33
What happens if shipments between Memphis and
Denver are allowed ?
8
8
13
25
28
0
12
0
26
15
25
6
17
0
0
16
6
0
14
16
0