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.
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
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.
Examples:-
Customers arrive at a bakery at an average rate of 16 per hour on week day mornings. The arrival distribution can be described by a Poisson distribution with a mean of 16. Each clerk can serve a customer in an average of three minutes; this time can be described by an exponential distribution with a mean of 3 minute.
a) What are the arrival and service rate?
b) Compute the average number of customer being served at any time?
c) Suppose it has been determined that the average number of customers waiting in line 3.2 Compute the average number of customers in the system ( i.e. waiting in line or being served), the average time customers wait in line, and the average time in the system.
d) Determine the system utilization for M=1, 2 and 3 servers.
======ANSWER ======
a) Arrival Rate = 16 customer per hour is given in the problem
Service Rate = is customer can serve an average of 3 minute that’s mean 60 /3 = 20 customer per hour.
b) Average Number of customer = Arrival rate / Service Rate
Average Number of customer = 16/20 = 0.8 customer
c) Average number in a system L system = L queue + Average number of customer
L system = 3.2 + 0.8 = 4 customer
Average Time customer Wait in line = (L queue /Arrival Rate) = (3.2/16) = 12 minute
Average Time customer Wait in the system = (L queue /Arrival Rate) + (1/Service rate)
= (3.2/16) + (1/20) = 15 minute
d) System Utilization = =
for M = 1 System utilization =
For M =2 System utilization =
For M=3 System utilization =