1Transportation ModelsLesson 4 LECTURE FOUR Transportation Models

2Transportation ModelsLesson 4 Introduction –Many business problems lend themselves to a network formulation. –Network problems can be efficiently solved by compact algorithms due to there special mathematical structure, even for large scale models. The importance of network models The importance of network models

3Transportation ModelsLesson 4 NETWORK A NETWORK problem is one that can be represented by... Nodes Arcs Function on Arcs

4Transportation ModelsLesson 4 Network Terminology –Flow : the amount sent from node i to node j, over an arc that connects them. –The following notation is used: Xij = amount of flowXij = amount of flow Uij = upper bound of the flowUij = upper bound of the flow Lij = lower bound of the flowLij = lower bound of the flow

5Transportation ModelsLesson 4 –Directed/undirected arcs : when flow is allowed in one direction the arc is directed (marked by an arrow). When flow is allowed in two directions, the arc is undirected (no arrows). –Adjacent nodes : a node (j) is adjacent to another node (i) if an arc joins node i to node j. Network Terminology

6Transportation ModelsLesson 4 Path / Connected nodesPath / Connected nodes –Path is a collection of arcs formed by a series of adjacent nodes. –The nodes are said to be connected if there is a path between them. Network Terminology

7Transportation ModelsLesson 4 Cycles / Trees / Spanning TreesCycles / Trees / Spanning Trees –Cycle is a path starting at a certain node and returning to the same node without using any arc twice. –Tree is a series of nodes that contain no cycles. –Spanning tree is a tree that connects all the nodes in a network. Network Terminology

8Transportation ModelsLesson 4 The Transportation Problem Transportation problems arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods. Transportation problems arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods.

9Transportation ModelsLesson 4 Problem definition Problem definition – There are m sources. Source i has a supply capacity of Si. – There are n destinations. The demand at destination j is Dj. The Transportation Problem

10Transportation ModelsLesson 4 Objective:Objective: Minimize the total shipping cost of supplying the destinations with the required demand from the available supplies at the sources. The Transportation Problem

11Transportation ModelsLesson 4 CARLTON PHARMACEUTICALS Carlton Pharmaceuticals supplies drugs and other medical supplies. It has three plants in: Cleveland, Detroit, Greensboro. It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis. Management at Carlton would like to ship cases of a certain vaccine as economically as possible.

12Transportation ModelsLesson 4 DataData –Unit shipping cost, supply, and demand CARLTON PHARMACEUTICALS

13Transportation ModelsLesson 4 AssumptionsAssumptions –Unit shipping cost is constant. –All the shipping occurs simultaneously. –The only transportation considered is between sources and destinations. –Total supply equals total demand. CARLTON PHARMACEUTICALS

14Transportation ModelsLesson 4 NETWORK REPRESENTATION Boston Boston Richmond Atlanta St.Louis Destinations Sources Cleveland Detroit Greensboro S 1 =1200 S 2 =1000 S 3 = 800 D 1 =1100 D 2 =400 D 3 =750 D 4 =

15Transportation ModelsLesson 4 The Mathematical Model –Decision variables X ij = amount shipped from source i to destination j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

16Transportation ModelsLesson 4 Boston Boston Richmond Atlanta St.Louis D 1 =1100 D 2 =400 D 3 =750 D 4 =750 The Supply Constraints Cleveland S 1 =1200 S 1 =1200 X11 X12 X13 X14 Supply from Cleveland X11+X12+X13+X14 = 1200 Detroit S 2 =1000 X21 X22 X23 X24 Supply from Detroit X21+X22+X23+X24 = 1000 Greensboro S 3 = 800 X31 X32 X33 X34 Supply from Greensboro X31+X32+X33+X34 = 800

17Transportation ModelsLesson 4 ============== The Complete Mathematical Model

18Transportation ModelsLesson 4 Range of optimality WINQSB Sensitivity Analysis If this path is used, the total cost will increase by $5 per unit shipped along it

19Transportation ModelsLesson 4 Range of feasibility Shadow prices for plants (source)- the cost saving resulting from 1 case of vaccine that was NOT shipped out to the distribution centre (destination) Shadow prices for distribution centre (destination) - the cost incurred for 1 extra case of vaccine shipped to the distribution centre (destination)

20Transportation ModelsLesson 4 Sensitivity Analysis Reduced costs The amount of transportation cost reduction per unit that makes a given route economically attractive.The amount of transportation cost reduction per unit that makes a given route economically attractive. If the route is forced to be used under the current cost structure, for each item shipped along it, the total cost increases by an amount equal to the reduced cost.If the route is forced to be used under the current cost structure, for each item shipped along it, the total cost increases by an amount equal to the reduced cost.

21Transportation ModelsLesson 4 Shadow prices For the plants (source), shadow prices convey the cost savings realized for each extra case of vaccine available at plant.For the plants (source), shadow prices convey the cost savings realized for each extra case of vaccine available at plant. For the warehouses (destination), shadow prices convey the cost incurred from having an extra case demanded at the warehouse.For the warehouses (destination), shadow prices convey the cost incurred from having an extra case demanded at the warehouse. Sensitivity Analysis

22Transportation ModelsLesson 4 Special Cases of the Transportation Problem –Cases may arise that appear to violate the assumptions necessary to solve the transportation problem using standard methods. –Modifying the resulting models make it possible to use standard solution methods.

23Transportation ModelsLesson 4 Blocked routes - shipments along certain routes are prohibited.Blocked routes - shipments along certain routes are prohibited. Minimum shipment - the amount shipped along a certain route must not fall below a specified level.Minimum shipment - the amount shipped along a certain route must not fall below a specified level. Maximum shipment - an upper limit is placed on the amount shipped along a certain route.Maximum shipment - an upper limit is placed on the amount shipped along a certain route. Transshipment nodes - intermediate nodes that may have demand, supply, or no demand and no supply of their own.Transshipment nodes - intermediate nodes that may have demand, supply, or no demand and no supply of their own. Special Cases of the Transportation Problem

24Transportation ModelsLesson 4 Solution of transportation problems Two phases: 1 st phase: –Find an initial feasible solution 2 nd phase: –Check for optimality and improve the solution

25Transportation ModelsLesson 4 Vogel’s approximation method Operational steps: Step 1: For each column and row, determine its penalty cost by subtracting their two of their least cost Step 2: Select row/column that has the highest penalty cost in step 1 Step 3: Assign as much as allocation to the selected row/column that has the least cost Step 4: Block those cells that cannot be further allocated Step 5: Repeat above steps until all allocations have been assigned

26Transportation ModelsLesson 4 Medical Supply Transportation Problem A Medical Supply company produces catheters in packs at three productions facilities. The company ships the packs from the production facilities to four warehouses. The packs are distributed directly to hospitals from the warehouses. The table on the next slide shows the costs per pack to ship to the four warehouses.

Medical Supply Seattle New York Phoenix Miami FROM PLANT Juarez $19 $ 7 $ 3 $21 Seoul Tel Aviv TO WAREHOUSE Capacity Juarez100 Seoul300 Tel Aviv200 Demand Seattle150 New York100 Phoenix200 Miami150

28Transportation ModelsLesson 4 Medical Supply Transportation Problem JXjsXjnXjpXjm100 SNPM XssXsnXspXsm XtsXtnXtpXtm S300 T200 Warehouse Demand 600 TO WAREHOUSE Plant Capacity From Plant Number of constraints = number of rows + number of columns Total plant capacity must equal total warehouse demand. Although this may seem unrealistic in real world application, it is possible to construct any transportation problem using this model.

29Transportation ModelsLesson 4 SNPM J100 S300 T200 Demand 600 Capacity From To Vogel’s approximation method = = = = = =1 50

30Transportation ModelsLesson 4 SNPM J100 S300 T200 Demand 600 Capacity From To Vogel’s approximation method C =7,000

31Transportation ModelsLesson 4 Solving Transportation Networks Manually Worked example 5.10 (page 166 – 173)