Or Final Project[1]....

PROJECT ON OPERATION RESEARCH

OR PROJECT ON TRANSPORTATION AND LPP

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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 1

Particular Transportation Problem Background

Page No. 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

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

2

Transportation problem based on calculating expenditure while selecting new location

Background 3

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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 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 –

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Table 1 Potential sites for Texago’s new refineries and their main advantages

Potential Site

Main Advantages 1. Near California oil fields

Near Los Angeles, California

2. Ready access from Alaska oil fields 3. Fairly near San Francisco distribution center 1. Near Texas oil fields

Near Galveston, Texas

2. Ready access from Middle East imports 3. Near corporate headquarters 1. Low operating costs

Near St. Louis, Missouri

2. Centrally located for distribution centers 3. Ready access to crude oil via Mississippi River

Table 2 Location of Texago’s current facilities

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Type of Facility

Locations 1. Texas

Oil fields

2. California 3. Alaska 1. Near New Orleans, Louisiana

Refineries

2. Near Charleston, South Carolina 3. Near Seattle, Washington 1. Pittsburgh, Pennsylvania 2. Atlanta, Georgia

Distribution centers

3. Kansas City, Missouri 4. San Francisco, California

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. 6

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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

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

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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. 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

Sources

Charleston

Seattle

Los Angeles Galveston

St. Louis

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

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Table 5

Cost data for shipping finished product to a distribution center

Cost per Unit Shipped (Millions of Dollars) Distribution Center Pittsburgh New Orleans

Refinery

Potential Refinery

Kansas City

6.5

5.5

6

8

Charleston

7

5

4

7

Seattle

7

8

4

3

Los Angeles

8

6

3

2

Galveston

5

4

3

6

St. Louis

4

3

1

5

100

80

80

100

Number of units needed

Table 6

Atlanta

San Francisco

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

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

Analysis 1: 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 )

Oil Fields

New Orleans

Charleston

Seattle

Los Angeles

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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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

Charleston

Seattle

Galveston

Orleans

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

Charleston

Seattle

St. Louis

Orleans

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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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) The changing cells Shipment Quantity give Texago management an optimal plan for shipping finished product if Los Angeles is selected as the new site for a refinery Unit Cost ( $ millions )

Pittsburgh

Atlanta

Kansas City

San Francisco

6.5

5.5

6

8

Charleston

7

5

4

7

Seattle

7

8

4

3

Los Angeles

8

6

3

2

New Orleans Refineries

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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

PROJECT ON OPERATION RESEARCH

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

Unit Cost ( $ millions )

Pittsburgh

Atlanta

Kansas City

San Francisco

6.5

5.5

6

8

Charleston

7

5

4

7

Seattle

7

8

4

3

St. Louis

5

4

3

6

New Orleans Refineries

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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) The changing cells Shipment Quantity give Texago management an optimal plan for shipping finished product if St. Louis is selected as the new site for a refinery Unit Cost ( $ millions )

Pittsburgh

Atlanta

Kansas City

San Francisco

6.5

5.5

6

8

Charleston

7

5

4

7

Seattle

7

8

4

3

St. Louis

4

3

1

5

New Orleans Refineries

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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 Total Cost of Shipping Finished Products

Operating Cost for New Refinery

Total Variable Cost

Site

Total Cost of Shipping Crude Oil

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 maintenance of the machine.

N 4,000,000:00

part

for

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 2 3 4 5 6 7 8 9

720 (i) 1384 (ii) 5760 (iii) 1800 (iv) 2000 (v) 720 (vi) 2000 (vii) 1600 (viii) 2000 (ix)

1 2 2.5 3.33 5 6 8 1 3

3 2 2 2 2.5 1.5 1.5 2.5 1.5

The variables used are: X1 = Number of crew allocated to machine 1 X2 = Number of crew allocated to machine 2 . . 32

PROJECT ON OPERATION RESEARCH

. . 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

Manpower cost/hr.

1

13.49

2

3

4

5

6

7

8

9

15.74

5.62

17.90

28.10

53.96

44.97

6.75

16.86

Max. Availa ble cost/hr. 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

9027.7 8

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.

1

12

12

99.8

2

4

14

99.8

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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. Multiple (infinite) solutions

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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.

The Equations Formed(According to the Tables):

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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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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

PROJECT ON OPERATION RESEARCH

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 The objective coefficient of variable X3 ranges between 0.12 to infinity which can reduce the cost by 0.87 The objective coefficient of variable X4 ranges between 0.40 to infinity which reduces the cost by 0.60 The objective coefficient of variable X5 ranges between 0.62 to infinity which can reduce the 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 The objective coefficient of variable X8 ranges between 0.15 to infinity which can reduce the cost by 0.85

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The objective coefficient of variable X9 ranges between 0.02 to infinity which can reduce the 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.)

46

PROJECT ON OPERATION RESEARCH  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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PROJECT ON OPERATION RESEARCH

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