Managerial decision modeling with spreadsheets by stair render chapter 05

Network Flow ModelsConsist of a network that can be represented with nodes and arcs 1.. Unbalanced Transportation Model• If Total Supply > Total Demand, then for each supply node: outflo

Chapter 5:

Transportation, Assignment

and Network Models

Network Flow Models

Consist of a network that can be represented with nodes and arcs

1 Transportation Model

2 Transshipment Model

3 Assignment Model

4 Maximal Flow Model

5 Shortest Path Model

6 Minimal Spanning Tree Model

Characteristics of Network Models

• A node is a specific location

• An arc connects 2 nodes

• Arcs can be 1-way or 2-way

Flow Balance at Each Node

(total inflow) – (total outflow) = Net flow

Node Type Net Flow

Destination > 0 Transshipment = 0

The Transportation Model

Decision: How much to ship from each origin to each destination?

Objective: Minimize shipping cost

Decision Variables Xij = number of desks shipped from factory i to warehouse j

Objective Function: (in $ of transportation cost)

Min 5XDA + 4XDB + 3XDC + 8XEA + 4XEB + 3XEC + 9XFA + 7XFB + 5XFCSubject to the constraints:

Flow Balance For Each Supply Node

(inflow) - (outflow) = Net flow

- (XDA + XDB + XDC) = -100 (Des Moines)

OR XDA + XDB + XDC = 100 (Des Moines)

Other Supply Nodes

XEA + XEB + XEC = 300 (Evansville)

XFA + XFB + XFC = 300 (Fort Lauderdale)

Flow Balance For Each Demand Node

XDA + XEA + XFA = 300 (Albuquerque)

XDB + XEB + XFB = 200 (Boston)

XDC + XEC + XFC = 200 (Cleveland)

Go to File 5-1.xls

Unbalanced Transportation Model

• If (Total Supply) > (Total Demand), then for each supply node:

(outflow) < (supply)

• If (Total Supply) < (Total Demand), then for each demand node:

(inflow) < (demand)

Transportation Models With Max-Min and Min-Max Objectives

• Max-Min means maximize the smallest decision variable

• Min-Max mean to minimize the largest decision variable

• Both reduce the variability among the Xij values

Go to File 5-3.xls

The Transshipment Model

• Similar to a transportation model

• Have “Transshipment” nodes with both inflow and outflow

Node Type Flow Balance Net Flow (RHS) Supply inflow < outflow Negative Demand inflow > outflow Positive Transshipment inflow = outflow Zero

Revised Transportation Cost Data

Note: Evansville is both an origin and a destination

Objective Function: (in $ of transportation cost)

Min 5XDA + 4XDB + 3XDC + 2XDE + 3XEA + 2XEB + 1XEC + 9XFA + 7XFB + 5XFC + 2XFESubject to the constraints:

Supply Nodes (with outflow only)

- (XDA + XDB + XDC + XDE) = -100 (Des Moines)

- (XFA + XFB + XFC + XFE) = -300 (Ft Lauderdale)

Evansville (a supply node with inflow)

(XDE + XFE) – (XEA + XEB + XEC) = -300

Fit-it Shop Assignment Example

Have 3 workers and 3 repair projects

Decision: Which worker to assign to which project?

Objective: Minimize cost in wages to get all 3 projects done

Estimated Wages Cost

of Possible Assignments

Can be Represented

as a Network Model

Decision Variables

Xij = 1 if worker i is assigned to project j

0 otherwise

Objective Function (in $ of wage cost)

Min 11XA1 + 14XA2 + 6XA3 + 8XB1 + 10XB2 + 11XB3 + 9XC1 + 12XC2 + 7XC3Subject to the constraints:

(see next slide)

One Project Per Worker (supply nodes)

- (XA1 + XA2 + XA3) = -1 (Adams)

The Maximal-Flow Model

Where networks have arcs with limited capacity, such as roads or pipelines

Decision: How much flow on each arc?

Objective: Maximize flow through the network from an origin to a destination

Road Network Example

Modified Road Network

Objective Function

Max X61

Subject to the constraints:

Flow Balance At Each Node Node

Flow Capacity Limit On Each Arc

Xij < capacity of arc ij

Go to File 5-6.xls

The Shortest Path Model

For determining the shortest distance to travel through a network to go from an origin to a destination

Decision: Which arcs to travel on?

Objective: Minimize the distance (or time) from the origin to the destination

Ray Design Inc Example

• Want to find the shortest path from the factory to the warehouse

• Supply of 1 at factory

• Demand of 1 at warehouse

Decision Variables

Xij = flow from node i to node j

Note: “flow” on arc ij will be 1 if arc ij is used, and 0 if not usedRoads are bi-directional, so the 9 roads require 18 decision variables

Objective Function (in distance)

Min 100X12 + 200X13 + 100X21 + 50X23 + 200X24 + 100X25 + 200X31 + 50X32 + 40X35 + 200X42 + 150X45 + 100X46 + 40X53 + 100X52 + 150X54 + 100X56 + 100X64 + 100X65

Subject to the constraints:

(see next slide)

Flow Balance For Each Node Node

(X21 + X31) – (X12 + X13) = -1 1(X12+X32+X42+X52)–(X21+X23+X24+X25)=0 2

Minimal Spanning Tree

For connecting all nodes with a minimum total distance

Decision: Which arcs to choose to connect all nodes?

Objective: Minimize the total distance of the arcs chosen

Lauderdale Construction Example

Building a network of water pipes to supply water to 8 houses (distance in hundreds of feet)

Characteristics of Minimal Spanning Tree Problems

• Nodes are not pre-specified as origins or destinations

• So we do not formulate as LP model

• Instead there is a solution procedure

Steps for Solving Minimal Spanning Tree

1. Select any node

2. Connect this node to its nearest node

3. Find the nearest unconnected node and connect it to the tree (if there is a tie, select one

arbitrarily)

4. Repeat step 3 until all nodes are connected

Steps 1 and 2

Starting arbitrarily with node (house) 1, the closest node is node 3

Second and Third Iterations

Fourth and Fifth Iterations

Sixth and Seventh Iterations

After all nodes (homes) are connected the total distance is 16 or 1,600 feet of water pipe