AMA INTERNATIONAL UNIVERSITY
Management Science-1
ABI-201
Final Period Handout Summary
Part-1: Network Models
Many managerial problems in areas such as transportation system design, information
systems design, project scheduling can be solved using the network mathematical
models. In this chapter we are going to discuss three types of network modes: The
shortest Route Problems, the minimal spanning tree problem and the maximum flow
problem. In each case, we will show how a network model can be developed and solved
in order to provide an optimal solution to the problem.
i.
The Shortest-Route Problem
In this type of problem the main objective is to determine the shortest route between
any pair of nodes in the network. Nodes are any point in the network and can
represent by station or house or factory or destinations the Arc (the line connected
the nodes) represent the distance between any two nodes.
ii.
The Minimal Spanning Tree
In network terminology the minimal spanning Tree problem involves using the Arcs
of the network to reach all nodes of the network in such a fashion that the total
length of all the Arcs used in minimized. These methods could be used in cable
connections, roads reclamation and so on.
iii.
The Maximum Flow Problem
Consider a network with one input or source node (where flow is generated) and
one output or Sink node (a node that absorb flow). The flow capacity is the
maximum flow for an arc of the network. The maximum flow technique that
determine flow of any quantity or substances through a network. The maximum
flow problem asks: what is the maximum amounts of flow (vehicles, messages,
water, flow) that can enter and exit the network at any period of time. In this
problem we attempt to transmit flow through all arcs of the network as efficiently as
possible. The amount of flow is limited due to capacity restrictions on the various
arcs of the network.
Theoretical background (Network & Queuing model)
 Shortest Route: is the shortest path between two nodes in a network.
 Spanning Tree: is a set of N-1 arcs that connect every node in the network with
all other nodes where N is the number of Nodes.
 Minimum spanning Tree: is the Spanning Tree with the minimum length.
 Arc Capacity: is the Maximum flow for an arc of the network. The arc capacity is
one direction may not equal the arc capacity in the reverse direction.
 Source: A node that generates flow, flow that only can flow away from it and
never into it.
 Sink: A node that absorbs flow, flow that can only move into it and never away
from it.
 Maximum Flow: The maximum amount of flow that can enter and exit a
network system during a given period of time.
 Queue : A waiting line
 Queuing Theory: The body of knowledge dealing with waiting lines.
 Single Channel: A waiting Line with only one service Facility.
 Poisson probability distribution: the probability distribution used to describe
the arrival pattern for some waiting line models.
 Exponential Probability Distribution: the probability distribution use to describe
the time for some waiting line model.
 FCFS: The queue discipline that’s serves waiting unit on a first come first serves
basis.
 Transient Period: The start-up period for a waiting line before the waiting line
reaches a normal or steady state operations.
 Steady state: The normal operation of the waiting line after it has gone through
a start up or transient period. General operating characteristics of waiting line
are computed for steady state condition.
 Mean Arrival Rate: The average number of customer or units arriving in a given
period.
 Mean Service Rate: The average number of customer or units that can be served
by one service facility in a given period.
 Multiple Channels: A waiting line with two or more parallel service facility.
 Blocking: when arriving units cannot enter the waiting line because the system is
full. Blocking can occur when waiting lines are not allowed or waiting line has a
finite capacity.
 Infinite calling population: the population of customers or unit who may seek
service has no specified upper limit.
 Finite calling population: the population of customers or unit who may seek
service has a fixed and finite value.
 Waiting line system or queuing system: Includes the customer population
source as well as the process or service system.
 Balking: The customer decides not to enter the waiting line.
 Reneging: The customer enters the line but decides to exit before being served.
 Jockeying: The customer enters one line and then switches to a different line in
an effort to reduce the waiting time
NETWORK PROBLEMS
Problem#1
In problem 1 above what is the shortest route that can be attend when we want to transfer
from Point 1 to point 6.
Problem#2
In problem 2 above what is the shortest route that can be attend when we want to transfer
from Point 1 to point 6.
Problem#3
Problem#4
Problem# 5
The state of Ohio recently purchased land for a new state park, and park planners have identified
the ideal locations for the lodge, cabins, picnic groves, boat dock, and scenic points of interest.
These locations are represented by the nodes of the following network. The arc of the network
represents possible road alternatives in the park. If the state park designers want to minimize the
total road miles that must be constructed in the park and still permit access to all facilities
(nodes), which road alternatives should be constructed?
Problem# 6
Morgan trucking company operates a special fast services pickup and delivery services between
Chicago and 10 other cities located in a four state area. When Morgan receives a request for
services, it dispatches a truck from Chicago to the city requesting service as soon as possible.
Since both fast service and minimum travel cost are objective for Morgan. It is important that the
dispatched truck take the shortest route from Chicago to the specified city. Assume that the
following network with distance given in miles represent the high network for this problem and
find the shortest-route distance from Chicago to all 10 cities?
Part-1: Waiting Line Models
“Queue or waiting line is commonly found where ever customers arrive randomly
for service. Some examples of waiting lines we encounter in our daily lives include the
lines at super market checkouts, fast food restaurants airport ticket counter etc. Queue
discipline refers to the order in which customer are processed. The capacity of queuing
systems is a function of the capacity of each server and the number of servers being used.
The term server and channel have the same meaning. Queuing theory is a mathematical
models or mathematical approach to the analysis of waiting line, which balance between
the cost of customer dissatisfaction cost and the operational cost. The single channel is a
waiting line with only one service facility and the System utilization it represents the
percentage of capacity utilized.
Equations to be used
Examples:-
Example#2
Example
answer
D:
utilization at M= 1 = 16/(1*20) =
utilization at M= 2 = 16/(2*20) =
utilization at M= 3 = 16/(3*20) =
EXAM IDENTIFICATION
Minimum spanning Tree
Queuing Theory
Mean Arrival Rate
Shortest Route
Infinite calling population
Transient Period:
Transient Period
Source
Queue
Sink
Maximum Flow:
FCFS
Mean Arrival Rate
Multiple Channels
Arc Capacity
channel
 ---------- (1) -------- is the shortest path between two nodes in a network.
 ---------- (2) -------- is the Spanning Tree with the minimum length.
 ---------- (3) -------- is the Maximum flow for an arc of the network. The arc
capacity is one direction may not equal the arc capacity in the reverse direction.
 ---------- (4) -------- A node that generates flow, flow that only can flow away from
it and never into it.
 ---------- (5) -------- A node that absorbs flow, flow that can only move into it and
never away from it.
 ---------- (6) --------: The maximum amount of flow that can enter and exit a
network system during a given period of time.
 ----------(7) -------- A waiting line
 ----------(8) --------: The body of knowledge dealing with waiting lines.
 ---------- (9) --------: The queue discipline that’s serves waiting unit on a first come
first serves basis.
 ---------- (10) -------- The start-up period for a waiting line before the waiting line
reaches a normal or steady state operations.