Or Final Project[1]

PROJECT ON OPERATION RESEARCH

OR PROJECT ON TRANSPORTATION AND LPP

1


ACKNOWLEDGEMENT

We think if any of us honestly reflects on who we are, how we got here, what we think we

might do well, and so forth, we discover a debt to others that spans written history. The work

of some unknown person makes our lives easier every day. We believe it's appropriate to

acknowledge all of these unknown persons; but it is also necessary to acknowledge those

people we know have directly shaped our lives and our work.

Apart from the efforts of the group members, the success of this project depends largely on the

encouragement and guidelines of many others. I take this opportunity to express my gratitude

to the people who have been instrumental in the successful completion of this report.

I would like to show my greatest appreciation to Prof. Pankaj who acted as mentor through out

the duration of our project. I can’t say thank you enough for the tremendous support that Sir

has given during the entire project. Without his efficient management and skills we would not

be able to complete our project successfully. I feel motivated and encouraged every time.

Without his support and help this project would not have materialized.

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Index

S.No Particular Page No.

1 Transportation Problem Background 4

1.1 Objective 4-5

1.2 Problem Description 5-8

1.3 Analysis 1 10-18

1.4 Analysis 2 18-30

1.5 Final Table 30

2 LPP Problem Background 31

2.1 Objective 32

2.2 Problem Description 33-35

2.3 Analysis 38-47

2.4 Advantages and Limitations of LPP 47-49

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Transportation problem based on calculating expenditure while selecting

new location

Background

The Texago Corporation is a large, fully integrated petroleum company based in the United

States. The company produces most of its oil in its own oil fields and then imports the rest of

what it needs from the Middle East. An extensive distribution network is used to transport the

oil to the company’s refineries and then to transport the petroleum products from the refineries

to Texago’s distribution centers. The locations of these various facilities are given in Table 1.

Texago is continuing to increase market share for several of its major products. Therefore,

management has made the decision to expand output by building an additional refinery and

increasing imports of crude oil from the Middle East. The crucial remaining decision is where

to locate the new refinery. The addition of the new refinery will have a great impact on the

operation of the entire distribution system, including decisions on how much crude oil to

transport from each of its sources to each refinery (including the new one) and how much

finished product to ship from each refinery to each distribution center.

Objective

The three key factors for management’s decision on the location of the new refinery are –

1. The cost of transporting the oil from its sources to all the refineries, including the new one.

2. The cost of transporting finished product from all the refineries, including the new one, to

the distribution centers.

3. Operating costs for the new refinery, including labor costs, taxes, the cost of needed supplies

(other than crude oil), energy costs, the cost of insurance, the effect of financial incentives

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provided by the state or city, and so forth. (Capitol costs are not a factor since they would be

essentially the same at any of the potential sites.)

The situation the company was facing: According to this case study the Texago corporation has

to build a new refinery and for that purpose they have selected three locations.The locations

they have selected are –

Table 1

Potential sites for Texago’s new refineries and their main advantages

Potential Site Main Advantages

Near Los Angeles, California

1. Near California oil fields

2. Ready access from Alaska oil fields

3. Fairly near San Francisco distribution center

Near Galveston, Texas

1. Near Texas oil fields

2. Ready access from Middle East imports

3. Near corporate headquarters

Near St. Louis, Missouri

1. Low operating costs

2. Centrally located for distribution centers

3. Ready access to crude oil via Mississippi River

Table 2

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Location of Texago’s current facilities

The following table shows the annual demand of crude oil that is converted into petroleum and

the annual supply of crude oil by the various oil fields.

Table 3

Refinery Crude oil Needed Annually (Million

barrels)

Oil Field Crude Oil produced annually (Million

Barrels)

New Orleans 100 Texas 80

Charleston 60 California 60

Seattle 80 Alaska 100

New one 120 Total 240

Total 360 Needed Imports(360-240) 120

6

Type of Facility Locations

Oil fields

1. Texas

2. California

3. Alaska

Refineries

1. Near New Orleans, Louisiana

2. Near Charleston, South Carolina

3. Near Seattle, Washington

Distribution centers

1. Pittsburgh, Pennsylvania

2. Atlanta, Georgia

3. Kansas City, Missouri

4. San Francisco, California


According to the case study the of Texago Corporation the management wants all the

refineries, including the new one, to operate at full capacity. Therefore, the task force begins by

determining how much crude oil each refinery would need to receive annually under these

conditions. and this is mentioned in the above schedule.

As we can see ,the right side of the table 3 shows the current annual output of crude oil from

the various oil fields. These quantities are expected to remain stable for some years to come.

Since the refineries need a total of 360 million barrels of crude oil, and the oil fields will

produce a total of 240 million barrels, the difference of 120 million barrels will need to be

imported from the Middle East.

Since the amounts of crude oil produced or purchased will be the same regardless of which

location is chosen for the new refinery, the task force concludes that the associated production

or purchase costs (exclusive of shipping costs) are not relevant to the site selection decision.

On the other hand, the costs for transporting the crude oil from its source to a refinery are very

relevant. These costs are shown in Table 4 for both the three current refineries and the three

potential sites for the new refinery.

Also very relevant are the costs of shipping the finished product from a refinery to a

distribution center. Letting one unit of finished product correspond to the production of a

refinery from 1 million barrels of crude oil, these costs are given in Table 5. The bottom row of

the table shows the number of units of finished product needed by each distribution center. The

final key body of data involves the operating costs for a refinery at each potential site.

Estimating these costs requires site visits by several members of the task force to collect

detailed information about local labor costs, taxes, and so forth. Comparisons then are made

with the operating costs of the current refineries to help refine these data.

In addition, the task force gathers information on one-time site costs for land, construction, and

so forth, and amortizes these costs on an equivalent uniform annual cost basis. This process

leads to the estimates shown in Table 6.

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Analysis (Six Applications of a Transportation Problem)

Armed with these data, the task force now needs to develop the following key financial

information for management:

1. Total shipping cost for crude oil with each potential choice of a site for the new refinery.

2. Total shipping cost for finished product with each potential choice of a site for the new

refinery.

Table 4 Cost data for shipping crude oil to a Texago refinery

Cost per Unit Shipped (Millions of Dollars per Million Barrels) Refinery or Potential Refinery

New Orleans Charleston Seattle

Los Angeles Galveston

St. Louis

Sources

Texas 2 4 5 3 1 1

California 5 5 3 1 3 4

Alaska 5 7 3 4 5 7

Middle East 2 3 5 4 3 4

Table 5 Cost data for shipping finished product to a distribution center

Cost per Unit Shipped (Millions of Dollars) Distribution

8


Center

Pittsburgh Atlanta Kansas City San Francisco

Refinery

New Orleans 6.5 5.5 6 8

Charleston 7 5 4 7

Seattle 7 8 4 3

Potential Refinery

Los Angeles 8 6 3 2

Galveston 5 4 3 6

St. Louis 4 3 1 5

Number of units needed 100 80 80 100

Table 6 Estimated operating costs for a Texago refinery at each potential site

Site Annual Operating Cost (Millions of Dollars)

Los Angeles 620

Galveston 570

St. Louis 530

To calculate costs, once a site is selected, an optimal shipping plan will be determined and

then followed. Therefore, to find either type of cost with a potential choice of a site, it is

necessary to solve for the optimal shipping plan given that choice and then calculate the

corresponding cost.

Analysis 1:

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All these values in the following tables are calculated from the Table 4 and these are selected

on the basis of whichever site you have selected. These values will change according to the

location selected and the total cost is calculated on that basis. The optimal solution is obtained

for each location using using Vogel’s Approximation method.

1. Texago Corp. Site-Selection Problem (Shipping to Refineries, Including Los

Angeles)

The changing cells Shipment Quantity give Texago management an optimal plan for

shipping crude oil if Los Angeles is selected as the new site for the refinery

Unit Cost ( $ millions ) New Orleans Charleston Seattle Los Angeles

Oil Fields

Texas 2 4 5 3

California 5 5 3 1

Alaska 5 7 3 4

Middle East 2 3 5 4

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11


Total cost = (40x2) + (40x3) + (60x1) + (80x3) + (20x4) + (60x2) + (60x3)

= $880 million.

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2. Texago Corp. Site-Selection Problem (Shipping to Refineries, Including

Galveston)

The changing cells Shipment Quantity give Texago management an optimal plan for shipping

crude oil if Galveston is selected as the new site for a refinery

Unit Cost ( $ millions ) New

Orleans

Charleston Seattle Galveston

Oil Fields

Texas 2 4 5 1

California 5 5 3 3

Alaska 5 7 3 5

Middle East 2 3 5 3

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Total = (20x2) + (60x1) + (60x3) + (20x5) + (80x3) + (60x2) + (60x3)

= $ 920million

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3. Texago Corp. Site-Selection Problem (Shipping to Refineries, Including St. Louis)

The changing cells Shipment Quantity give Texago management an optimal plan for

shipping crude oil if St. Louis is selected as the new site for a refinery

Unit Cost ( $ millions ) New

Orleans

Charleston Seattle St. Louis

Oil Fields

Texas 2 4 5 1

California 5 5 3 4

Alaska 5 7 3 7

Middle East 2 3 5 4

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16


Total = (80x1) + (20x5) + (40x4) + (20x5) + (80x3) + (80x2) + (40x3)

= $ 960 million

If Los Angeles were to be chosen as the site for the new refinery (Fig. 2), the total annual cost

of shipping crude oil in the optimal manner would be $880 million. If Galveston were chosen

instead , this cost would be $920 million, whereas it would be $960 million if St. Louis were

chosen.

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So according to the calculations the cost for transportation will be minimum if Los Angeles is

selected as new site for refinery . But only this factor is not sufficient to for Texago

Corporation to finalize its location of new site therefore it considered other conditions like

what is the cost for shipping finished product from Los Angeles, Galveston and St.Louis to

distribution centers which are Pittsburgh ,Atlanta, Kansas City and San Francisco..

Analysis 2 :

The analysis of the cost of shipping finished product is similar. The data in all the figures

shows the Matrix model for this transportation problem, where rows come directly from the

first values of Table 5. The New Site row would be filled in from one of the next three rows of

Table 5, depending on which potential site for the new refinery is currently under evaluation.

Since the units for finished product leaving a refinery are equivalent to the units for crude oil

coming in, the data in Supply come from the left side of Table 3.

For each of the three alternative sites, three separate separate calculations have been used for

planning the shipping of crude oil and the shipping of finished product. However, another

option would have been to combine all this planning into a single spreadsheet model for each

site and then to simultaneously optimize the plans for the two types of shipments.

1. Texago Corp. Site-Selection Problem (Shipping to D.C.’s When Choose Los

Angeles)


finished product if Los Angeles is selected as the new site for a refinery

Unit Cost ( $ millions ) Pittsburgh Atlanta Kansas City San Francisco

Refineries


Charleston 7 5 4 7

Seattle 7 8 4 3

18


Los Angeles 8 6 3 2

19


20


Total = (80x6.5) + (20x5.5) + (60x5) + (20x7) + (60x3) + (80x3) + (40x2)

= $1.57 billion

2. Texago Corp. Site-Selection Problem (Shipping to D.C.’s When Choose Galveston)

21


The changing cells ShipmentQuantit give Texago management an optimal plan for shipping

finished product if Galveston is selected as the new site for a refinery


Refineries


Charleston 7 5 4 7

Seattle 7 8 4 3

St. Louis 5 4 3 6

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23


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Total = (6.5x20) + (5.5x80) + (40x4) + (40x4) + (20x7) + (80x3) + (80x5) + (40x3)

= $1.63 billion

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3.Texago Corp. Site-Selection Problem (Shipping to D.C.’s When Choose St. Louis)


finished product if St. Louis is selected as the new site for a refinery


Refineries


Charleston 7 5 4 7

Seattle 7 8 4 3

St. Louis 4 3 1 5

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Total(St. Louis) =(60x6.5) + (40x5.5) + (40x5) + (20x7) + (80x3) + (40x4) + (80x1)

= $1.43 Billion

From a purely financial viewpoint, St. Louis is the best site for the new refinery. This site

would save the company about $200 million annually as compared to the Galveston alternative

and about $150 million as compared to the Los Angeles alternative. As we can see from the

above calculation when Los Angeles is chosen as a location for refinery then the cost for

shipping to distribution center is coming $1570 million and if Galveston is chosen as the

location for the new refinery then the cost comes upto $1630 million whereas for St.Louis the

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cost for shipping to distribution center is $1430 million which a clear savings of $150 and $200

million which is mentioned above.

Final Table

Site

Total Cost of Shipping Crude Oil

Total Cost of Shipping Finished

Products

Operating Cost for New Refinery

Total Variable Cost

Los Angeles $880 million $1.57 billion $620 million $3.07 billion

Galveston $920 million $1.63 billion $570 million $3.12 billion

St. Louis $960 million $1.43 billion $530 million $2.92 billion

Table 7 shows the total the total variable cost of each prospective site that were selected for

refinery .

Total Variable Cost =Total Cost of shipping Crude Oil + Total Cost of Shipping Finished

Products + Operating Cost for New Refinery

As we can see from the table 7 that total variable cost for the location St. Louis comes out to

be minimum among the 3 location selected and the total variable cost for the St. Louis is $

2.92 billion and therefore St. Louis selected as the location for refinery by Texago Corporation.

Linear Programming based Effective Maintenance and Manpower Planning

Strategy: A Case Study

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In this study, maintenance related data of a cocoa processing industry in Akure, Ondo State of

Nigeria were collected, classified and analyzed statistically. Linear Programming (LP) model

was formulated based on the outcomes of the analyzed data. The data analyzed includes

maintenance budget, maintenance cycle, production capacity and waiting time of production

facilities in case of failure. Data were analyzed based on manpower cost, machine depreciation

cost and the spare part cost, which were assumed to be proportion to the number/magnitude of

the breakdowns. The generated LP model was solved using software named “TORA”. The

results of the model showed that four maintenance crews were needed to effectively carryout

maintenance jobs in the industry. The sensitivity analysis showed that the results have a wide

range of feasibility.

In any production firm there are two sub systems, human and technical. The two sub-systems

must be balanced and coordinated in order to function effectively. Many studies have been

carried out on how to make maintenance and manpower planning effective in a production

firm.

In previous studies, models adopted to analyze prevailing situation include:

1)Simulation model

2)Queue model

3)Utility model

4)Network analysis.

In this study, linear programming technique is used to analyze maintenance operations and

manpower planning in a production firm used to analyze maintenance operations and

manpower planning in a production firm that was used as a case study.

Objective

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In pursuing the objective of the study, data were collected, classified, analyzed and the linear

programming model was formulated based on the data analyzed. The department whose data

was studied is the maintenance section of a production firm (Cocoa Processing Industry) in

Akure, Ondo State of Nigeria, which is responsible for the keeping of the plant and machinery

used for cocoa processing in operable condition. The data were collected through the use of

questionnaires and oral interview among employees in the maintenance section of the firm.

The data that were collected include the following: number and list of all the machines; types

of maintenance applied; budget on the maintenance; factors affecting maintenance; present

level of manpower planning in maintenance department; maintenance cycle of each of the

machines; and the waiting time of each of the machines.

After the collection of the data a close monitoring of the maintenance operations of the

production section was done over a period of two weeks to make ensure reliability of the data.

The major machines on which scheduled or time based preventive maintenance was carried out

include machine one:

(1) Cleaning and destoner,

(2) Dryer

(3) Winnower

(4) Reactor

(5) Roaster

(6) Map mill

(7) Liquor press

(8) Butter press

(9) Boiler.

The factors affecting the maintenance operation of the firm include understaffing in the

maintenance section, mismanagement of budgetary allocation, inadequate tools, equipment and

spare part.

Table1: Budget for Maintenance Operation

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Manpower remuneration for the production section.

N 6,900,000:00

Total allocation for spare part for maintenance of the machine.

N 4,000,000:00

Total cost for depreciation of the machine. N52,000,000:00

Table2: Maintenance Cycle, Production Capacity and Waiting Time.

M/C Maintenance Cycle, (Hrs)

(Codes)

Waiting Time (H1)

Production Capacity (Tons/hr)

1 720 (i) 1 3 2 1384 (ii) 2 2 3 5760 (iii) 2.5 2 4 1800 (iv) 3.33 2 5 2000 (v) 5 2.5 6 720 (vi) 6 1.5 7 2000 (vii) 8 1.5 8 1600 (viii) 1 2.5 9 2000 (ix) 3 1.5

The variables used are:

X1 = Number of crew allocated to machine 1 X2 = Number of crew allocated to machine 2

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.

.

.

.

X3 = Number of crew allocated to machine 3 .

. X9 = Number of crew allocated to machine 9

Objective:

The objective is to maximize the percentage production hour available per maintenance cycle

of each machine. That is minimize the waiting time of each machine.

Table3: Maintenance Cost Analysis

Machine 1 2 3 4 5 6 7 8 9 Max. Availa

ble cost/hr

. Manpower cost/hr. 13.49 15.74 5.62 17.90 28.10 53.96 44.97 6.75 16.86 347.22

Spare part cost/hr. 20.99 24.50 8.75 27.99 43.73 83.96 69.96 10.50 26.24 850.69

Depreciation cost/hr 826.97 432.40 137.83 344.57 334.57 826.94 344.57 413.49 344.57 9027.78

Table4: Production and Maintenance Hour Analyses

Machine S/N Number of repairs in a year

Max. hrs. available for repair in a year

% Production hrs. Available/ production cycle.

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1 12 12 99.8

2 4 14 99.8

3 2 5 99.9

4 5 16 99.8

5 5 25 99.7

6 12 48 99.4

7 5 40 99.6

8 6 6 99.9

9 5 15 99.8

Conditions for application of simplex method

In order that the simplex method may be applied to a linear programming problem, the

following two conditions have to be satisfied.

1. The R.H.S of each of the constraint, bi should be non negative. If an LPP has a

constraint for which a negative resource value is given, it should be in first step,

converted into a positive value by multiplying both sides of the constraint by -1.

2. Each of the decision variables of the problem should be non negative.

The working of the simplex method proceeds by preparing a series of tables called simplex

tableaus.

Unbounded solution

If at any iteration no departing variable can be found corresponding to entering variable, the

value of the objective function can be increased indefinitely, i.e., the solution is unbounded.

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Multiple (infinite) solutions

If in the final tableau, one of the non-basic variables has a coefficient 0 in the Z-row, it

indicates that an alternative solution exists. This non-basic variable can be incorporated in the

basis to obtain another optimal solution. Once two such optimal solutions are obtained, infinite

number of optimal solutions can be obtained by taking a weighted sum of the two optimal

solutions.

Infeasible solution

If in the final tableau, at least one of the artificial variables still exists in the basis, the solution

is indefinite.

Minimization versus maximization problems

As discussed earlier, standard form of LP problems consist of a maximizing objective function.

Simplex method is described based on the standard form of LP problems, i.e., objective

function is of maximization type. However, if the objective function is of minimization type,

simplex method may still be applied with a small modification. The required modification can

be done in either of following two ways.

1. The objective function is multiplied by −1 so as to keep the problem identical and

‘minimization’ problem becomes ‘maximization’. This is because of the fact that minimizing a

function is equivalent to the maximization of its negative.

2. While selecting the entering nonbasic variable, the variable having the maximum coefficient

among all the cost coefficients is to be entered. In such cases, optimal solution would be

determined from the tableau having all the cost coefficients as nonpositive (≤0 ) Still one

difficulty remains in the minimization problem. Generally the minimization problems consist

of constraints with ‘greater-than-equal-to’ ( ≥) sign. For example, minimize the price (to

compete in the market); however, the profit should cross a minimum threshold. Whenever the

goal is to minimize some objective, lower bounded requirements play the leading role.

Constraints with ‘greater-than-equal-to’ ( ≥) sign are obvious in practical situations.

35


The Equations Formed(According to the Tables):

The general linear program is of the form.

Max. Z* = 0.998X1 + 0.998X2 + 0.999X3 +0.998X4 + 0.997X5 + 0.994X6 +0.996X7 +

0.999X8 + 0.998X9

Subject to:

13.49X1 + 15.74X2 +5.62X3 + 17.90X4 + 28.1X5 + 53.96X6 + 44.97X7 + 6.75X8 +0.998X9 ≤ 347.22

20.99X1 + 24.5X2 + 8.75X3 + 27.99X4 + 43.73X5 + 83.96X6 + 69.96X7 + 10.5X8 +26.24X9 ≤ 850.69

826.97X1 + 432.4X2 + 137.83X3 + 344.57X4 + 344.57X5 + 826.97X6 + 344.57X7 + 413.49X8 + 344.57X9 ≤ 9027.78

Constraints on the maximum hour available for maintenance in each maintenance cycle:

X1 ≤ 1, X2 ≤ 2, X3 ≤ 2.5, X4 ≤ 3.33, X5 ≤5, X6 ≤ 4, X7 ≤ 8, X8 ≤ 1, X9 ≤ 3

Non-negativity X1≥0, X2≥0, X3≥0, … X9≥0 `

Output

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37


38


39


40


Iteration 7

Sensitivity

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The simplex algorithm is an iterative procedure for finding, in an systematic manner, the

optimal solution to a linear programming problem. Simplex method selects the optimal

solution from among the set of feasible solutions to the problem. By using this technique we

42


can consider a minimum number of feasible solutions to obtain an optimal one. This technique

is also helpful in determining that whether a given solution is optimal or not.

For applying simplex method to the solution of an LPP, first of all, an approximately selected

set of variables is introduced in the problem. The iterative process begins by assigning values

only to these variables and the primary variables of the problem are all set equal to zero. The

algorithm then replaces one of the initial variables by another variable—the variable which

contributes most to the desired optimal enters in, while the variable creating the bottleneck to

the optimal solution goes out. This improves the value of the objective function. This

procedure of substitution of variables is repeated until no further improvement in the objective

function value is possible. The algorithm terminates there indicating that the optimal solution is

reached, or that the given problem has no solution.

Sensitivity Analysis

Sensitivity analysis is used for ascertaining the limits within which objective coefficients can

be changed, limits within which quantities can be changed and the limits within which

constraint coefficients can be changed without affecting the solution.

1. limits of cj’s

2. limits of bi’s

3. limits of aij’s

In other words, Sensivity analysis is used for ascertaining how sensitive the existing solution is

to the above mentioned three changes.

Limits can be fixed for each quantity within which it can be changed without affecting the

existing solution and the shadow price of each resource.

1. Changes in objective function coefficients, cj’s

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Under this we consider as to how the changes in the coefficients of the decision variables in the

objective function, shall influence the optimal solution.

We determine that whether the change of profit per unit of a product, that is currently being

produced, causes a change in the optimal solution to the problem. over a certain range, a

change, positive or negative, in the unit profit would not cause a change in the optimal

solution.

The least positive value provides the answer as to how much the profit could increase without

changing the solution. The least negative value is the maximum decrease in the profit that

would not cause a change in the profit.

2. Changes in the bi values: Right hand side ranging

Under this we determine the range over which each of the shadow price will remain valid.

The following two steps are used to determine the limits-:

a) Find the ratio of bi and the corresponding slack variable coefficient in the last table.

b) Find the least positive and the least negative ratio.

Least positive ratio gives the amount by which resource can be reduced and the least negative

ratio gives the amount by which given resources can be increased without affecting the shadow

price and the solution.

If the resource is not fully utilized, it can be reduced to the extend it is unutilized and it can be

increased to any extend without affecting the solution.

3. Change in the Technological coefficients,aij’s

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In context of production problems ,aij values are determined by the technological

consideration. Due to technological improvements ,the time required for processing a particular

product may be reduced, which might or might not affect the optimal product mix.

Now after conducting a sensivity analysis of the problem we have obtained the following

results.

The objective coefficient of variable X1 ranges between 0.30 to infinity which leads to

reducing the cost by 0.70(1-0.30).

The objective coefficient of variable X2 ranges between 0.34 to infinity which can reduce the

cost by 0.66


cost by 0.87

The objective coefficient of variable X4 ranges between 0.40 to infinity which reduces the cost

by 0.60


cost by 0.37

The objective coefficient of variable X6 ranges between –infinity to 1.20 which can reduce the

cost by 0.20

The objective coefficient of variable X7 ranges between 0.83 to 1.60 which can reduce the cost

by 0.00


cost by 0.85

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cost by 0.98

Advantages of Linear Programming

The linear programming technique helps to make the best possible use of available

productive resources (such as time, labour, machines etc.)

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In a production process, bottle necks may occur. For example in a factory some

machines may be in a great demand while others may lie idle for sometime. A

significant advantage of linear programming is highlighting of such bottle necks.

The quality of decision making is improved by this technique because the decisions are

made objectively and not subjectively.

By using this technique, wastage of resources like time and money may be avoided.

Limitations of Linear Programming

Linear programming is applicable only to problems where the constraints and the

objective function are linear i.e,where they can be expressed as equations which

represent straight lines.In real life situations,when constraints or objective functions are

not linear, this technique cannot be used.

Factors such as uncertainty, weather conditions etc. are not taken into consideration.

There may not be an integer as the solution, e.g., the number of men required may be a

fraction and the nearest integer may not be the optimal solution.

i.e., Linear programming technique may give practical valued answer which is not

desirable.

Only one single objective is dealt with while in real life situations, problems come with

multi-objectives.

Parameters are assumed to be constants but in reality they may not be so.

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Documents

Or Final Project[1]