Assignment of FIN-3104: Operations Research

Transcription

1 Assignment of FIN-3104: Operations Research

2 A Report On Maximizing the profits, Minimizing work hour & the transportation cost for BEXTEX using LP

3 Submitted to Suborna Barua Course Instructor/ Lecturer, Department of Finance, Faculty of Business Studies Jagannath University, Dhaka Submitted by Sultan Ahmed Khan Representative of the group Epimetheus BBA 3 rd Batch Department of Finance, Faculty of Business Studies Jagannath University, Dhaka.

4 Group Name: Epimetheus Group No: Name of the members of the group: Serial No: Name of the members of the group Roll Number 01 Sultan Ahmed Khan Md. Mynul Islam Mamunur Rashid Md. Anik Mahmud Md. Mofazzal Hossen Md. Mehedi Hassan Sharjil Ahmed Protiva Talukder Mohammad Didarul Islam Khan Mohammad Mahmudul Hasan Group Representative: Sultan Ahmed Khan. Group Coordinator Contact : Md. Mynul Islam. : epimetheus.jnu@gmail.com

5 December 7, 2011 The Course Instructor, Suborna Barua, Lecturer, Department of Finance, Jagannath University, Dhaka. Sub: Thanks giving letter to the respective faculty member. Sir, We are the student of Department of Finance (3rd batch) of Jagannath University, Dhaka & also from the group named Epimetheus. We are very much enthusiastic about our presentation. We are really happy to have such a presentation of challenging and interesting like this presentation & also thanks to you for making us worthy for corporate. Our presentation topic is Maximizing the profits, Minimizing work hours & the transportation cost for BEXTEX using LP. We have learned many things from this topic which will help us in future to conduct as an analyst in the organization. There were some obstacles we have faced at the time of collecting data about our topic. But we have overcome all the obstacles by the endeavor effort by each member of our group and tried our best to give an overview of our topic. We the group Epimetheus tried our best to make this presentation attractive, impeccable, interesting, informative and enjoyable by the help of electronic and print media in association with our honorable teacher, mentor, counselor, instructor and advocate Suborna Barua. We are really grateful to him. We had limitations at the time preparing presentation. So mistakes may occur in our demonstration of our presentation. We hope that, you will exempt our mistakes. Thanking in anticipation, Yours Fidel, Sultan Ahmed Khan Group Representative, Group- Epimetheus BBA 3 rd Batch Department of Finance Jagannath University,Dhaka.

6 First of all we would like to thank the Almighty for giving us the strength, and the aptitude to complete this report within due time. We are deeply indebted to our course teacher, mentor, and counselor, Suborna Barua for assigning us such an interesting topic named Maximizing the profits, Minimizing work hours & the transportation cost for BEXTEX using LP. We also express the depth of my appreciation to our honorable course teacher for his suggestion and guidelines, which helped us in completing this report.

7 The most important task for an organization is to minimize cost and maximize production. Every firm tries their level best to maintain such a production schedule which minimize their cost including their inventory cost and maximize their production or output. To find out the optimal level of output and cost specialist use Linear Programming techniques. Linear programming (LP or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming (mathematical optimization). We work on BEXTEX Ltd. BEXTEX Ltd. is a full service vendor with strong vertically integrated production facilities as well as creative & analytical capabilities which clearly sets us apart from most other South Asian vendors. We have tried to find out the optimal solution for maximizing the profit of BEXTEX LTD. Moreover, the route costs less. By applying the matrix method we found out that the optimal solution for maximizing the profit of the BEXTEX Limited is BDT. This is the ultimate output by following this method the firm can increase their production the most. Again, the firm use 3 ports which are the Chittagong seaport, Mongla seaport, and Benapole Landport of the country. We have improved our cost by BDT This is the least cost capturing route. And we have improved this cost by applying the transportation cost method of LP. Now, it gets easier for the authority to maximize the production rate and maintain the cost of routes. Without maintaining this schedule no firm can get their mission and vision such as Gain market leadership in high value added apparel in USA & Europe, Use Innovation & Speed as prime drivers, rather than cotton & cheap labor, dominate these markets in high quality.

8 NAME Page no Executive Summary Introduction Introduction 01 Chapter- 01 Rational of the study 01 Objective of the study 02 Scope of the study 02 Methodology of the study 02 Limitations of the study 02 Body of the term paper Linear Programming 03 Uses of LP 03 Types of Linear Programming 04 Chapter- 02 BEXIMCO 05 Mission, Vision of BEXIMCO 06 Product of BEXIMCO 07 Maximizing the profit 08 Minimizing work hour 12 Minimizing the transportation cost 16 Findings of the study 16 Chapter-03 Conclusion 27 Bibliography 27

9 Chapter- 01 Introduction Linear programming refers to mathematical programming. It is a technique of allocating limited resources in an optimum manner so as to satisfy the laws of supply and demand for the firm s products. It refers to a planning process that allocates resources- labor, materials, and machines, capital- in the best possible (optimal) way so that costs are minimized or profits are maximized. In LP, these resources are known as decision variables. Linear programming (LP or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming (mathematical optimization). Rationale of the study The case study is assigned by our course teacher Suborna Barua as a part of our Operations Research course. The topic of our report is Maximizing the profits & Minimizing the transportation cost for BEXTEX using LP. By conducting this report we can enhance our knowledge and skill to apply various research methods in professional life on higher educational life. The report has given us a chance to raise our quality in developing research instrument and its applications. By doing so, we can boost our acceptability in job market and develop our real life knowledge. Objective of the Case Study Primary objective The main objective of the study is to know about the uses of LP module in real business world. Secondary objective: The case study has the following objectives: The practice of LP in real business world. Use & types of LP. Maximizing the profits for BEXTEX.

10 Minimizing working hour & the transportation cost for BEXTEX Scope There were huge scopes to work in the area of this Report. Considering the dead line, and exposure of the paper has been wide-ranging. The study Maximizing the profits, Minimizing working hour & the transportation cost for BEXTEX using LP has covered overall scenario of real business world of Bangladesh. It deals with the employees & measures their quality & cost. We got a chance to work on the one of the top most research departments in the organization which supplies the forecast to the organization. By doing the assignment, we are able to know that the importance of LP to assess how the analysts of the organization working with this for minimizing the cost. In the report we have solved to problem. Methodology We have used the concept of the course, information of the report. Sources of Data Here the secondary sources of information were used. The secondary sources are: Books. Website. Direct interviews. Limitations While conducting the report on Maximizing the profits, Minimizing work hour & the transportation cost for BEXTEX using LP, some limitations were yet present there: Because of time shortage many related area can t be focused in depth. Website of the organization contains poor information. Recent data and information on different activities conducted by authority are restricted Non sharing tendency of the officials.

11 Chapter-2 Linear Programming Linear programming refers to mathematical programming. It is a technique of allocating limited resources in an optimum manner so as to satisfy the laws of supply and demand for the firm s products. It refers to a planning process that allocates resources- labor, materials, and machines, capital- in the best possible (optimal) way so that costs are minimized or profits are maximized. In LP, these resources are known as decision variables. Linear programming (LP or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming (mathematical optimization). Linear programming can be applied to various fields of study. It is used most extensively in business and economics, but can also be utilized for some engineering problems. 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 Use of Linear Programming The use of LP is made in regard to the problems of allocation, assignment, transportation, etc. But the most important of these is that allocation of scarce resources on which we shall concentrate. Some allocation problems are as follows: 1. Devising of a production schedule that could satisfy future demands for the firm s product and at the same time minimize production (including inventory) costs. 2. Choice of investment from a variety of shares and debentures so as to maximize return on investment. 3. Allocation of a limited publicity budget on various heads in order to maximize its effectiveness. 4. Selection of the product-mix to make the best use of machines, man-hours with a view to maximize profits. 5. Selecting the advertising mix that will maximize the benefit subject to the total advertising budget, Linear Programming can be effectively applied. 6. Determine the distribution system to minimize transport costs from several warehouses to various market places.

12 Types of Linear programming The Diet Problem: It is the problem of deciding how much of n different foods to include in a diet, given the cost of each food, and the particular combination of nutrient each food contains. The object is to minimize the cost of diet such that it contains a certain minimum amount of each nutrient Optimal Product Lines Problem: How much of n different products a firm should produce and sell, when each product requires a particular combination of labor, machine time and warehouse space per unit of output and where there are fixed limits on the amounts of labor, machine time and warehouse space available? Transportation Problem: It is a problem of determining a shipping schedule for a commodity say still or oil from each of the number of plants at different location to each of a number of a markets at different location in such a way as to minimize the total shipping cost subject to the constraints that the demand at each market will be satisfied and the supply add the plant will not be exceed. Linear Programming The Diet problem Optimal Product Line Problem Transportation Problem

13 BEXIMCO Bextex Ltd. (the "Company") was incorporated in Bangladesh as a Public Limited Company with limited liability on 8 March 1994 and commenced commercial operation in 1995 and also went into the public issue of shares and debentures in the same year. The shares of the Company are listed in the Dhaka and Chittagong Stock Exchanges of Bangladesh. Bextex Ltd. is the most modern composite mill in the region. Bextex Ltd. has an installed capacity of 288 high-speed air-jet looms in its weaving section and a high-tech dyeing and finishing section with a capacity of 100,000 yards of finished fabric per day. This company is located at the Beximco Industrial Park Bextex Ltd. has a state of the art composite knit fabric production mill, which serves the growing needs of high-quality knit garments exporters in Bangladesh. The project was set up as a state of the art knit fabric knitting, dyeing and finishing facility. During the year the Company produced and sold high quality of knit fabrics and bringing forth all the latest in hard and soft technologies in knitting, dyeing and finishing of knit fabric. Bextex Ltd. also has cotton and polyester blended yarn-spinning mill, with 122,000 spindles is one of the largest spinning mills of the country. The mill was set up to feed the country's export oriented industries. Bextex Ltd. produces specialized finishes of denim cloth for export in finished as well as cloth only form.

14 Mission BEXTEX Ltd. is a full service vendor with strong vertically integrated production facilities as well as creative & analytical capabilities which clearly sets us apart from most other South Asian vendors. Vision Gain market leadership in high value added apparel in USA & Europe. Use Innovation & Speed as prime drivers, rather than cotton & cheap labor. Dominate these markets in high quality : Men's, Women's, Children Shirts ( Dress & Casual ) Blouses ( formal & casual ), Skirts, Jackets Jeans & Casual non - denim bottoms Knitted tops & bottoms Commitment to the Environment Our company is very committed to preserve a healthy and pollution-free environment. It has a very efficient waste collection and disposal system. In order to reduce air pollution by exhaust of gas from engine-generators, it maintains a costly plant that uses the exhaust gas to generate steam for chilling unit. Above measures not only help keep the water & air free from pollution but also help save cost of water treatment & air conditioning. Your company uses only AZO-free dyes and is dedicated to ensure a healthy and eco-friendly environment.

15 Products of BEXIMCO Yarn Products Fabric Products Fabric Products Denim Products Special Yarn Products Unique Wrinkle-Free product

16 Maximizing the profit using LP model The company s profit of a Unit of yarn is BDT 1.08, Unit of fabric BDT 0.4 & Unit of knit BDT 2. The company wants to sell approximately units if yarn, unit of fabric, and unit of knit. Company s available hours monthly are 4368 hours for manufacturing (10 machines 14.5 hours daily throughout the month), 520 hours for printing (2 machines 8.5 hours daily throughout the month), 208 hours for washing ( 1 machine 7 hours daily throughout the month), 780 hours for packaging ( 3 machines 8.5 hours daily throughout the month) That is like the followings one:- Department Yarn,x1 (in hour) Fabric,x2 (in hour) Knit, x3 (in hour) Monthly Working hours Manufacturing Printing Washing Profit (BDT) What is the optimal solution for maximizing the profit of BEXTEX LTD.?

17 Maximizing the profits Given, Department Yarn,x1 (in hour) Fabric,x2 (in hour) Knit, x3 (in hour) Monthly Working hours Manufacturing Printing Washing Profit (BDT) Maximum the objective function Z= 1.08X X 2 +2X 3 Subject to constraint, 0.056X X X X X X X X X Where, X , X , X We know that AX=B X=B/A X=BA -1

18 According to the formula X X 2 = X D = (0.004* *0.001) (0.0067* *0.0033) (0.0067* * ) = = NX 1 = NX 1, D = 4368 (0.004* *0.001) (520* *208) (520* *208) = = NX 2 = NX 2, D = (520* *208) 4368 (0.0067* *0.0033) (0.0067* *0.0033) = =

19 NX 3 = NX 3, D = (0.004* *0.001) (0.0067* *0.0033) (0.0067* *0.0033) = = X 1 = = X 2 = = (This isn t mathematically applicable but theorically possible) X 3 = = For maximizing the problem we have to put the regarded value in the following equation (objective function) Z= 1.08X X 2 +2X 3 Thus the profit will maximize up to the following level Max Z= 1.08( ) + 0.4(0) + 2( ) [Because the value of X cannot be negative, thus we put X 2 = 0] Z= BDT Thus the optimal solution for maximizing the profit of the BEXTEX Limited is BDT. [Ans.]

20 Minimizing the work hour using LP model The company s profit of a Unit of yarn is BDT 1.08, Unit of fabric BDT 0.4 & Unit of knit BDT 2. The company wants to sell approximately units if yarn, unit of fabric, and unit of knit. Company s available hours monthly are 4368 hours for manufacturing (10 machines 14.5 hours daily throughout the month), 520 hours for printing (2 machines 8.5 hours daily throughout the month), 208 hours for washing ( 1 machine 7 hours daily throughout the month), 780 hours for packaging ( 3 machines 8.5 hours daily throughout the month) That is like the followings one:- Department Yarn,x1 Fabric,x2 Knit, x3 Monthly (in hour) (in hour) (in hour) Working hours Manufacturing Printing Washing Profit (BDT) Now the BEXTEX ltd. has decided that it will keep its production level unchanged to have that much profit. But if it can reduce its working hours then the profitability may increase. What will the minimum work hour for BEXTEX LTD.?

21 Minimizing the work hour Given, Department Yarn,x1 (in hour) Fabric,x2 (in hour) Knit, x3 (in hour) Monthly Working hours Manufacturing Printing Washing Profit (BDT) Minimum, Z= 4368X X X 3 Subject to constraint, 0.056X X X X X X X X X 3 2 Where, X 1 0, X 2 0, X 3 0 We know that AX=B X=B/A X=BA -1

22 According to the formula X X 2 = X 3 2 D = (0.004* *0.017) (0.025* *0.1) (0.025* * 0.1) = = NX 1 = NX 1,D = 1.08 (0.004* *0.017) (0.04* *2) (0.01* *2) = = NX 2 = NX 2, D = (0.04* *2) 1.08 (0.025* *0.1) (0.025*2 0.04*0.1) = =

23 NX 3 = NX 3, D = (0.004*2 0.04*0.017) (0.025*2 0.04*0.1) (0.025* *0.1) = = X 1 = = (This is not mathematically applicable but theorically possible) X 2 = = X 3 = = For minimizing the problem we have to put the regarded value in the following equation (objective function) Z= 4368X X X 3 Thus the profit will maximize up to the following level Min Z= 4368(0) + 520( ) + 208( ) [Because the value of X cannot be negative, thus we put X 1 = 0] Z= hour Thus the optimal solution for minimizing the profit of the BEXTEX Limited is hour. [Ans.]

24 Minimizing the transportation cost using LP model BEXTEX ltd. wants to spread its market all over the world. So it exports its products by 3 ports which are the Chittagong seaport, Mongla seaport, and Benapole Landport of the country. The company has three warehouse marked as A, B, C. There transportation table is Warehouse Chittagong Mongla Benapole Landport Available container career A BDT 8000 BDT 9000 BDT B BDT 7000 BDT BDT C BDT 7500 BDT BDT Demand Now find it out which route costs less?

25 Minimizing the route cost We know that the number of constraints must equal the number of rows & number of column when we set up BEXTEX transportation problem. The requirement can be express in the following way: R=S+D 1 The required route no is R=3+3-1=5 and we have five routes. We follow the northwest corner method for solving the problem. As the method implies we start work in the northwest corner or the upper left cell Warehouse A, Chittagong port. We have made an allocation to this cell that will use either all the demand for that row or all the supply for that column (smallest one). The allocation will be like the following ways:- Warehouse v j Chittagong Mongla Benapole Landport Available container career A (1) B (1) C Demand u i BEXTEX initial basic feasible solution is: 8000X X X X X (1) (1) = BDT 72500

26 We can introduce two quantities, u i & v j, where u i is the dual variable associated with row i nad v j is the dual variable associated with column j. For sensitivity test we use duality theory equation. That is X ij = u i +v j A common method is to chose the row with the largest number of allocation. We arbitrarily chose row warehouse B & set u 2 =0 Using substitutions, we calculate: u 2 =0 X 21 = u 2 +v =0+v 1 v 1 = 7000 X 11 = u 1 +v = u 1 u 1 = 1000 X 22 = u 2 +v =0+v 2 v 2 =10500 X 32 = u 3 +v = u u 3 =500 X 33 = u 3 +v = 500+ v 3 v 3 = 8500 Warehouse Chittagong Mongla Benapole Landport Available container career A (1) B (1) C Demand v j u i

27 Check for optimal solution (Non-basic cell) X ij u i v j 0 For cell X 12 : False with 2500 For cell X 13 : False with 2700 For cell X 23 : False with 500 For cell X 31 : True Now we use close loop path for further iterations thus we found: Warehouse Chittagong Mongla Benapole Landport Available container career A (1) + B (1) C Demand v j u i Now the new schedule is Warehouse v j Chittagong Mongla Benapole Landport Available container career A (0) (1) B (3) C Demand u i

28 For avoiding degeneracy problem, we made a route in the very first cell arbitrarily. Thus BEXTEX new initial basic feasible solution is : 8000X X X X X (0) (1) (3) = BDT Thus we have improved our cost by BDT Now we have to find out where this is optimal or mot. For that we use common method is to chose the row with the largest number of allocation. We arbitrarily chose row warehouse A & set u 1 =0 Using substitutions, we calculate: u 1 =0 X 11 = u 1 +v =0+v 1 v 1 = 8000 X 21 = u 2 +v = u 2 u 2 = 1000 X 12 = u 1 +v = 0+v 2 v 2 =9000 X 32 = u 3 +v = u u 3 =2000 X 33 = u 3 +v = v 3 v 3 = 7000 We found the following table: Warehouse Chittagong Mongla Benapole Landport Available container career A (0) (1) B (3) C Demand v j u i

29 Check for optimal solution (Non-basic cell) X ij u i v j 0 For cell X 13 : False with 200 For cell X 22 : ( 1000) True For cell X 23 : 8000 ( 1000) True For cell X 31 : False with 2500 Now we use close loop path for further iterations thus we found: Warehouse Chittagong Mongla Benapole Landport Available container career A (0)- (1)+ B (3) C Demand v j u i Now the new schedule is Warehouse v j Chittagong Mongla Benapole Landport Available container career A (1) B (3) C (0) Demand u i

30 For avoiding degeneracy problem, we made a route in the table arbitrarily. Thus BEXTEX new initial basic feasible solution is: 7000X X X X X (3) (1) (0) = BDT Thus we have same as before BDT Now we have to find out where this is optimal or mot. For that we use common method is to chose the row with the largest number of allocation. We arbitrarily chose row warehouse C & set u 3 =0 Using substitutions, we calculate: U 3 =0 X 31 = u 3 +v =0+v 1 v 1 = 7500 X 32 = u 3 +v = 0+ v 2 v 2 = X 33 = u 3 +v = 0+v 3 v 3 =9000 X 12 = u 1 +v = u u 1 = 2000 X 21 = u 2 +v = u 2 u 2 = 500 We found the following table: Warehouse Chittagong Mongla Benapole Landport Available container career A (1) B (3) C (0) Demand v j u i

31 Check for optimal solution (Non-basic cell) X ij u i v j 0 For cell X 11 : 8000 ( 2000) True For cell X 13 : 6800 ( 2000) False with 200 For cell X 22 : ( 500) True For cell X 23 : 8000 ( 500) False with 500 Now we use close loop path for further iterations thus we found Warehouse Chittagong Mongla Benapole Landport Available container career A (1) B (3)- + C (0)+ - Demand v j u i Now the new schedule is Warehouse v j Chittagong Mongla Benapole Landport Available container career A (1) B (1) C Demand u i

32 Thus BEXTEX new initial basic feasible solution is : 7000X X X X X (1) (1) = BDT Thus we have improved our cost by BDT Now we have to find out where this is optimal or mot. For that we use common method is to chose the row with the largest number of allocation. We arbitrarily chose row warehouse B & set u 2 =0 Using substitutions, we calculate: U 2 =0 X 21 = u 2 +v =0+v 1 v 1 = 7000 X 31 = u 3 +v = u 3 u 3 = 500 X 32 = u 3 +v = 500+v 2 v 2 =10500 X 12 = u 1 +v = u u 1 = 1500 X 23 = u 2 +v = 0+ v 3 v 3 = 8000 We found the following table: Warehouse Chittagong Mongla Benapole Landport Available container career A (1) B (1) C Demand v j u i

33 Check for optimal solution (Non-basic cell) X ij u i v j 0 For cell X 11 : 8000 ( 1500) True For cell X 13 : 6800 ( 1500) True For cell X 22 : True For cell X 23 : True All statements are true. So we cannot improve the solution further. Thus this is the optimal solution for the transport cost of BEXTEX Ltd. Thus the less route cost will be Z = 7000X X X X X 23 = 7000(1) (1) = BDT [Ans.]

34 Chapter-03 Findings and conclusion Findings of the study The intension of this study is to know about Human Resource Management & HR Department. The major findings of the overall study are discussed below: Importance of Linear programming The practice of LP in real business world. Use & types of LP. Maximizing the profits for BEXTEX. Minimizing the transportation cost for BEXTEX Find out the optimal solution for BEXTEX

35 Conclusion After completing this study we have learnt that LP is one of the most discussed & influential mathematical module uses in real business world. Authority cannot overlook this to solve out the optimal solution for maximizing the profits, minimizing the transportation cost & so on. But, authority is working with a very impressive way in software in recent days. LP are used to improve and develop the working skill of the employees in a minimum cost with at maximum profits (in a very suitable manner). Authority faces a lot of problems but they are solving it with a help of LP. In our country, the quality of the production sector is immensely depending on LP module. Bibliography Books Levin, Rubin, Stinson, Gardner; Quantitative Approaches to Management ; 8 th Edition. (McGraw-Hill INC Book Company, ). Web Sites