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Applications of Linear Programming and Integer Linear Programming in Business Optimization

This work by Tran Van Hoai explores the evolution and applications of Linear Programming (LP) and Integer Linear Programming (ILP) models in business and government, yielding millions in savings. It delves into constructing effective LP/ILP models, emphasizing clarity and appropriate simplification to reflect real-life scenarios. Through practical examples such as TV production optimization and financial portfolio modeling, it showcases the effective use of LP/ILP in minimizing risk and maximizing returns while adhering to constraints. This pivotal study aids professionals in resource management and decision-making.

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Applications of Linear Programming and Integer Linear Programming in Business Optimization

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  1. Applications of LP/ILP Models Tran Van HoaiFaculty of Computer Science & Engineering HCMC University of Technology Tran Van Hoai

  2. Evolution of LP/ILP • Millions USD saved from applying LP/ILP to business/government • Motivated by • Simplex • Digital computers • Applied to Tran Van Hoai

  3. Building good LP/ILP models • Familiarity • Limited resources • Overall (tradeoff) objective • Different perspectives • Simplification • Models always simplify real-life, but which is simplified is important • Clarity • Model must be clear What constitutes the proper simplification is subject to individual judgment and experience (George Dantzig) Tran Van Hoai

  4. Summation variables/constraints • Introduce new variables to be easier to understand/debug • Summation of variables/constraints • Production of 3 TV models • resource: 7000 pounds plastic • 2 pounds/TV1, 3 pounds/TV2, 4 pounds/TV3 • profit: • $23/TV1, $34/TV2, $45/TV3 • management constraint: not any TV model exceed 40% total production Tran Van Hoai

  5. First model Not management constraint anymore No meaning as natural input (especially on spreadsheet) Tran Van Hoai

  6. Revised model • Define summation variable X4 = total production of TVs • Add summation constraint X1 + X2 + X3 - X4 = 0 Clarity (although more variables/constraints) Summation constraint Summation variable Tran Van Hoai

  7. Applications of LP/ILP • More realistic example • Reduced version in different practical applications • Portfolio model Tran Van Hoai

  8. Financial portfolio model • Consider return projections of investment • Measure of risk, volatility, liquidity, short/long term • Highly nonlinear in nature, but we consider a linear case Tran Van Hoai

  9. Jones investment service(advise clients on investment) • Problem summary • Determine amount to be placed in each investment • Minimize total risk • Invest all $100,000 • Meet the goals developed with client • annual return at least 7% • at least 50% in A-rated investments • at least 40% in immediately liquid investments • no more $30,000 in savings and deposit Tran Van Hoai

  10. LP model Tran Van Hoai

  11. Analysis (1) • Binding constraints • What exceed minimum requirements • Investment Tran Van Hoai

  12. Analysis (2) • Reduced costs: in order to be included, risk factor must be lowered • Optimality range • Shadow price: risk increased by Tran Van Hoai

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