LP slide

1. Application of Linear
Programming in Profit
Maximization
By:
Roshan Koirala (357/074)
T.U. Regd. No: 5-2-37-2043-2017
Tri- Chandra Multiple Campus, Tribhuvan University
On the supervision of:
Chet Nath Tiwari
AssistantProfessor, DoM
Tri-Chandra Multiple Campus, TU

2. Outlines
❑Introduction
❑Background of study
❑Literature Review
❑Methodology and Model
❑Case Study
❑Conclusion
❑References
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3. Introduction
Linear Programming is a branch of applied mathematics that deals with
the management or distribution problems for optimization
(minimization and maximization)
Minimization :- Transportation cost problem, managing time
Maximization :- Production capacity, return of invested amount
In early 1940's, George Dantzig developed linear
programming problem working with US Air Force
to solve military related problem.
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4. Background of the Study
❑ Simplex method started by Dantzig is highly efficient.
❑ First introduce to solve the transportation problem of arms and
army In WWII.
❑ Now adopted worldwide.
❑ Area of application: food, agriculture, transportation,
manufacturing.
❑ Using modern computers, LPP with thousands of variable can be
solved by simplex method.
❑ Students or researcher can use software like LINDO, MAPLE to
solve LPP with limited no. of variable.
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5. LiteratureReview
❑Amit Kumar Jain (2019) optimized the net return of Central Bank of
India in area of interest for loan allocated in different sectors.
❑P. N. Ezra (2020) used LP in production and selling of different sizes of
bottles for Nigeria Bottle Company.
❑Nur Anith Syahirah (2021) solves the daily requirement nutrition in
the KFC’s fast food by minimizing the cost of production.
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6. Methodology and Model
Simplex method is the most common way to solve the linear programming
problem. Beside that if there are only two variable we can use graphical method. In
case of greater or equal to constrains we add artificial variable and use Big M
method instead of simplex method.
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7. Case Study
A financial institution lends money from central bank(NCB) and public as
deposit and allocates loan to different sector with different interest rate.
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8. Case Study: Introduction of problem and Variable
Now the financial institution needs to allocate the funds to different
sector in a such way that net return of interest is maximum.
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Let x1, x2, x3, x4 be the funds (in Lakhs) allocated for the Women S, Agriculture S, Business, and
Personal loan with the following conditions.
CONSTRAINS
▪ The totalamount allocatedto different loanproducts is not more than the total funds collected.
▪ The totalamount for Women Subsidy and Agriculture Subsidy loan must be at least 15% and 20% of the
total funds collected respectively.
▪ The totalamount for Agriculture Subsidy loan and Home loan must be in between of 30% to 40% of the
total funds collected.
▪ The totalamount for Home loan and Personal loan must not exceed 50% of the totalfunds collected.

9. Case Study: Data representation
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Now interpretating the problem in mathematicalform:
Left one is general form and right one is standard form with artificial variable.
where s1, s4, and s5 are slack variables, s2, s3 are surplus variables, and Y1, Y2, Y3 are artificial variables.

10. Case Study: Solution via Big M method
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The LPP is solved via Big M method and the screenshot of the initial
table with multiple negative value of Z and the final table with all
value of Z is positive is given below.

11. Case Study: Solution
Solving by Big M method we get the optimal value as:
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12. Case Study: Verification of result
Screenshot of problem and solution using LINDO
Input Output
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13. Case Study: Result and conclusion
❑Maximum Z = 86.75 lakhs
❑Allocation on women subsidy, agriculture subsidy and personal loan
is 150, 350 and 500 lakhs respectively.
❑No allocation for business loan.
❑Overall return is 8.675% of circulated amount.
❑Average borrowing rate is 4.5%
❑Max profit is 4.675%
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14. Conclusion
❑LPP introduced from arm management, now used optimize business
profit.
❑From large scale industry to small business.
❑It is easy to find the solution once the problem is identified and
analyzed mathematically.
❑The discipline is still underdeveloped in Nepal.
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15. References
❑Jain, A.K. (2020) Application of Linear Programming for Profit Maximization of
Pharma Company, School of Science, Rajasthan, India.
❑G.B. Dantzi (1963), Linear Programming and Extensions, Princeton University Press,
Princeton, NJ.
❑Akpan, N.P. and Iwok, I.A. (2016), Application of linear programming for optimal use
of raw materials in bakery. International Journal of Mathematics and Statistics
Invention, 4(8): Pg. 51-57.
❑Ahmed, M.M., Khan, A.R., Uddin Md. S. and Ahmed, F. (2016), A New Approach to
Solve Transportation Problem. Open Journal of Optimization, 5: Pg. 22-30.
❑Yahya, W.B., Garba, M.K. Ige, S.O., Adeyosoy, A. E. (2012) Profit maximization in a
product mix company using linear programming. European Journal of Business and
Management, 4(17):Pg. 34-35.
❑Cooper, W. W. (2002). Abraham Charnes and W. W. Cooper (Et al.): A brief history of
a long collaboration in developing industrial uses of linear programming. Operations
Research, 50(1), 35-41.
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16. References
❑Haider-Manager, Zaman & Tariq, Muhammad. (2016). Application of Linear
Programming for Profit Maximization: A Case of Paints Company, Pakistan,
International Journal of Management Sciences and Business Research. Pg. 12-14.
❑Amit Kumar Jain, Ramakant Bhardwaj, Hemlata Saxena, Anurag Choubey. (2019).
Application of Linear Programming for Profit Maximization of the Bank and the
Investor, International Journal of Engineering and Advanced Technology (IJEAT), Pg. 1.
❑P. N. Ezra, A. V. Oladugba, F. O. Ohanuba, N. O. Igwe, C. A. Okonta, Application of
Linear Programming on the Profit Maximization of Nigerian Bottling
Company, American Journal of Operational Research, Pg. 39-43.
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17. THANK YOU
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