kaushik pmor

8/7/2019 kaushik pmor

1/18

Project Management and Operational Research Page 1

TABLE OF CONTENTS

CHAPTER TOPIC PAGE

01 Background theory Linear programming 2

1.1 Methodology Explanation 3

1.2 Limitations of linear programming 4

02 Analysis of case

2.1 Range of Feasibility 8

2.2 Range of optimality 9

2.3 Reduced cost 10

2.4 Shadow price 10

2.5 Slack or Surplus 11

03 Discussion of three options 12

04 Recommendations 16

05 Reference 17


2/18


1.Introduction to Linear programmingIntroduction:

Linear programming or mathematical programming is the branch of management science that deals

with solving optimization problem, in which we want to maximize a function such as profit or

expected return or minimize a function such as cost, time, distance.In a decision-making embroilment,

model formulation is important because it represents the essence of business decision problem. The

term formulation is used to mean the process of converting the verbal description and numerical data

into mathematical expressions, which represents the relevant relationship among decision factors,

objectives and restrictions on the use of resources. Linear Programming (LP) is a particular type of

technique used for economic allocation of 'scarce' or 'limited' resources, such as labor, material,

machine, time, warehouse space, capital, energy, etc. to several competing activities, such as products,

services, jobs, new equipment, projects, etc. on the basis of a given criterion of optimally. The phrase

scarceresources mean resources that are not in unlimited in availability during the planning period.

The criterion of optimality generally is either performance, return on investment, profit, cost, utility,

time, distance, etc.

George B Dentzing develops this technique when he was working with US air force during world war

2, Which is primarily used for solving military logistic problem. But in todays world, the technique is

being used in all functional areas of management, hospitals, airlines, agriculture, military operations,

oil refining, education, energy planning, pollution control, transportation planning and scheduling,

research and development.The most important of application of linear programming has been the allocation problem, allocation

of scare resource for optimal results. Usually there are many jobs which share the common resources and

the variable resources are not adequate enough to allow each job to be carried out to the fullest extent.

Therefore, the objective in such a situation is to allot available resources to the job in such way as to

either maximize the total revenue or to minimize the total cost subject to resource constraints.

Linear programming can be applied to various fields of study. Most extensively it is used in

business and economic situations, but can also be utilized for some engineering problems. Some

industries that use linear programming models include transportation, energy, telecommunications,

and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing,scheduling, assignment, and design.


3/18


Methodology

It is a model that does seek to maximize or minimize a linear objectives function subject to a set of

linear constraints. But all the means is that the objective function and constraints contain onlymathematical term involving variables (X1, X2, X2)that are raised to the first power (e.g 5X1).

Mathematical model:

The general mathematical model of the linear-programming problem

Maximize (or Minimize) the Objective Function:

C1X1 + C2X2 + + CjXj + + CnXn

Subject to the conditions (constraints)

A11X1 + A12X2 + + A1jXj + + A1nXn {=} B1

A21X2 + A22X2 + + A2jXj + + A2nXn {=} B2

.

Ai1X2 + Ai2X2 + + A ijXj + + AinXn{=} Bi

.

Am1X2 + Am2X2 + + AmjXj + + AmnXn{=} Bm

And

X1, X2,..Xj, ,Xn 0 (non-negativity)

Where:

Cj for j= 1,2,,n; Bi for i= 1,2,,n and Aij are all constants

A is a matrix of known coefficients

Cj for j= 1,2,,n are called cost coefficients

Xj for j= 1,2,,n are variables to be solved for. Finding out X j we will know the optimal

solution.

(Source: Gauss 2003, p.6)


4/18


Applications ofLinear Programming

Applications Objective Constraints

Manufacturing Determine production quantities that

maximize profit

yLabor availabilityyResource availability

Finance Allocate funds to maximize

expected return

yDiversificationyAcceptable risk levels

Advertising Select a media mix that maximizes

exposure to a target population

yBudgetyLength of advertising campaign

Worker

Training

Assign workers to production and

training activities to maximize profit

while building a workforce

yProduction quotasyNumber of qualified instructors

and trainees available

Construction Plan tasks and assign labor to met a

production schedule

yOrdering of tasksyProject deadline

Oil Refining Blend raw crude oils into different

grades of gasoline

ySupply of raw crude oil anddemand for different grades of

gasoline

yRequired characteristics of thedifferent grades of gasoline

Transportation Assign delivery of resources to

minimize transportation costs

ySupply/ demand of productyShipping capacities

Agriculture Determine a plant rotation plan to

maximize long term profit

yAnticipated demand for cropsyRotation restrictions

Military

Operations

Assign troops and material to

accomplish a military mission

yTroop availability/ trainingyTransportation of resources

(Source: Lawrence &Pasternack 2002, p.50)

LIMITATION OF LINEAR PROGRAMMING IN REAL LIFE SITUATIONS

1. Forecasting is based on the past data hence it may not always accurate and it can givewrong results.

2. Defining the specific objective function is not easy for all linear programs.3. The data available may be conflicting in nature. This may cause confusion during analysis.4. LP is based upon relative relations between input and output. This means that input and output

are multiplicative, divisible and additive, but the relation between input and output are not

always linear.


5/18


BAY CITY MOVERS

Bay City Movers is a local company that specializes in intercity moves. In the business plan

submitted to its backers, Bay City has committed itself to a total trucking capacity of at least 36 tons.

The company is in the process of replacing its entire fleet of trucks with 1 ton picks up trucks and

2.5-ton moving van type trucks. The 1 ton pick up trucks will be manned by one worker, whereas

the large vans will utilize a total of four personnel for larger moves.

Bay City Movers currently employs 48 workers and has facilities for 40 trucks. Pick up trucks cost

the company $24,000 and the moving vans cost $60,000. The company wishes to make a minimum

investment in trucks that will provide a trucking capacity of at least 36 tons while not requiring any

new hires or trucking facilities.

Although the continuity assumption is violated (since the number of each truck purchased must be

integer), use a linear programming model to determine the optimal purchase of pick up trucks and

vans for Bay City Movers. You will find that alternative optimal solutions are possible.

Prepare a report detailing several of these options and discuss the pros and cons of each. Among the

alternatives, you should present in your report are the following:

5. Purchasing only one type of truck.1. Purchasing the same number of pick up trucks as moving vans.

Purchasing the minimum total number of trucks.


6/18


Mathematical model for Bay City movers:

The company is in the process of replacing their entire fleets of trucks with one ton pick up trucks and2.5-ton moving van type trucks.

There are two variables in the case. They are:

Let X1be 1-ton pick up trucks.

X2 be 2.5-ton moving van trucks.

The three constraints in the case are:

Let C1 which represents the capacity of the trucks

C2 that represents the number of workers currently employed for the job

C3 represents the facilities provided to the trucks.

The objective of the case is to reduce the investment-involved n the replacement of the fleet of trucks.

The objective function:

DECISION VARIABLES

Minimize 2400X1+60000X2

Subject to:

X1+4X2= 0 (Non negativity)


7/18


WinQSB Printout:

INPUT-

OUTPUT-


8/18


Range of Feasibility

The range of feasibility of Bay City movers:

In case of Capacity:

The range of feasibility for the company is 30 to 44 where the minimum ranges

being the 30 and the maximum range being 44. The actual capacity utilized are 36units, 8 more units

can be increased, by paying a certain price if the company feels that they can improve the output by

increasing the capacity.

In case of workers:

The actual workers utilized by the company are 48 and the range, which is inside

the range, which is from 36 to 57.6. So the company can increase by their workers by hiring 10 more

workers.

In case of Facility:

The range of feasibility varies from 24 to M (Infinity) so there is no maximum

given to which the company can definitely figure on when they come to the increasing their facility.

The current facility constraint utilized by the company is 24 we know that there is a scope to increase

but due to lack of the maximum range the company does not get a clear idea of which extent they

increase.


9/18


RANGE OF OPTIMALITY

In case of Pick up truck:

The actual cost of one unit of pick up truck is 24,000. The range from whichthe range of optimality varies is from 24,000 to M (infinity) this shows that the cost can be increased

to infinity but the cost cannot be reduced than the minimum range which is 24,000 by doing so the

optimum solution will tend to change.

In case of Moving van:

The cost involved in one moving van is 60,000. The range of optimality, which

has the minimum range of M (Infinity) and the maximum range being the 60,000 the company

cannot exceed its cost or the maximum range but try and reduce its cost if it wants to.


10/18


REDUCED COST

The reduced cost for both the pick up truck and the moving van are zero. So there is no requirement

for the company to reduce in their costs. Thus 24,000 being the reduced cost of Pick up truck and

60,000 being the reduced cost of Moving van remain constant.

SHADOW PRICE


11/18


Capacity-

Every unit of capacity over 36 units will cost the company 24,00 per unit.

Workers and Facility-

Since the shadow price of both workers and facility are zero the company need

not increase on the number of units, which they already have.

SLACK OR SURPLUS

Capacity and Workers:

The slack or surplus for both the capacity and workers being zero,it means that

the company is utilizing their resources related to the workers and capacity to its optimum thus there

is no surplus.

Facility-


12/18


The surplus for facility are 16 units, thus the company is not utilizing its resources efficiently

when it comes to facility.

Discussion of three optionsOption 1

Purchasing only one type of truck

Purchasing only Pick up truck:

Input:


13/18


Output:

This is the purchase of only pick up truck. The optimum solution over here is feasible. The maximum

capacity of the pick up truck is 40. The minimum purchase of the pick up trucks is 36. The number of

workers utilized totally is 36. The maximum allowable trucks that can be purchased are 8. Thus the

values are feasible enough for the company to work on and achieve their objectives.

Purchase only Moving vans:

Input


14/18


Output

Since the minimum and maximum range of capacity is negative and the company, which has a huge

difference of range when it comes to the number of workers, and high shadow price for every extra

unit of worker, thus the solution is infeasible.

OPTIONS 2

Purchasing equal number of pick up trucks as moving vans:


15/18


Input:

Output:

The capacity is being negative and a huge shadow price for purchase of a new unit of worker being

high the optimum solution is infeasible.

OPTION 3


16/18


PURCHASING THE MINIMUM TOTAL NUMBEROF TRUCKS

INPUT

OUTPUT

The solution arrived for purchasing minimum total number of trucks is, for Pickup trucks it is 16 and

for van it is 8. The unit cost for each truck is one. The unit cost being 1 for both the cases indicate that

the company can reduce its cost. Through this the objective function 24 is achieved. The total

investment the company has to use is 864,000.Thus the solution is also feasible.

RECOMMENDATIONS AND CONCLUSION


17/18


In the above problem of Bay City Movers, the main objective function is to minimize the cost of

purchase of moving vans and pickup trucks (i.e., 24000 X 1 + 60000 X2)

When we analyze the option 1, purchasing only one truck, it was feasible because, the solution given

in the WinQsb is very effective for the company to achieve its objective function.

When we analyze the option 3, purchasing minimum total number of trucks, the solution is feasible, as

it is given that the company can purchase 16 pickup trucks and 8 moving vans, which ultimately come

up to 24, which is the objective function of the company and hence it is feasible.

The most recommended option for the company to achieve the objective function; minimum

purchasing of trucks is option 3. As it says the cost for each truck is only 1 and the minimum trucks

and vans they can purchase is 16 and 8, the minimum total trucks are 24. It is recommendable that if

the company chooses option three, it can attain the objective of getting more profits, by reducing the

cost of purchase.

Reference


18/18


Dantzig, G. (1999).Linear Programming-Theory and Extensions.Berlin:Springer

Karloff, H.(1991) Linear Programming, Boston: Ashton Press

Gass.S. (1985), LinearProgramming Method and Application

Loomba, N (1964). Paul. Linear Programming: An Introductory Analysis.New York: McGraw-Hill.

kaushik pmor

Documents

Transcript of kaushik pmor