A network problem is one that can be represented by...

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Presentation transcript:

A network problem is one that can be represented by... 4.1 Introduction A network problem is one that can be represented by... 6 8 9 10 Nodes 7 Arcs 10 Function on Arcs

Network Terminology Flow Directed/undirected arcs the amount sent from node i to node j, over an arc that connects them. The following notation is used: Xij = amount of flow Uij = upper bound of the flow Lij = lower bound of the flow 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 Path / Connected nodes Path :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. Cycles / Trees / Spanning Trees Cycle : a path starting at a certain node and returning to the same node without using any arc twice. Tree : a series of nodes that contain no cycles. Spanning tree : a tree that connects all the nodes in a network ( it consists of n -1 arcs).

4.2 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.

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

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.

CARLTON PHARMACEUTICALS Data Unit shipping cost, supply, and demand Assumptions Unit shipping costs are constant. All the shipping occurs simultaneously. The only transportation considered is between sources and destinations. Total supply equals total demand. To From Boston Richmond Atlanta St. Louis Supply Cleveland $35 30 40 32 1200 Detroit 37 40 42 25 1000 Greensboro 40 15 20 28 800 Demand 1100 400 750 750

CARLTON PHARMACEUTICALS Network presentation

Destinations Boston Sources Cleveland Richmond Detroit Atlanta St.Louis Destinations Sources Cleveland Detroit Greensboro D1=1100 37 40 42 32 35 30 25 15 20 28 S1=1200 S2=1000 S3= 800 D2=400 D3=750 D4=750

CARLTON PHARMACEUTICALS – Linear Programming Model The structure of the model is: Minimize Total Shipping Cost ST [Amount shipped from a source] £ [Supply at that source] [Amount received at a destination] = [Demand at that destination] Decision variables Xij = the number of cases shipped from plant i to warehouse j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

The supply constraints Cleveland S1=1200 X11 X12 X13 X14 Supply from Cleveland X11+X12+X13+X14 = 1200 Detroit S2=1000 X21 X22 X23 X24 Supply from Detroit X21+X22+X23+X24 = 1000 Boston Greensboro S3= 800 X31 X32 X33 X34 Supply from Greensboro X31+X32+X33+X34 = 800 D1=1100 Richmond D2=400 Atlanta D3=750 St.Louis D4=750

CARLTON PHARMACEUTICAL – The complete mathematical model Total shipment out of a supply node cannot exceed the supply at the node. £ = Total shipment received at a destination node, must equal the demand at that node.