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

Evolution of LP/ILP Millions USD saved from applying LP/ILP to business/government Motivated by – Simplex – Digital computers Applied to Tran Van Hoai2 Aircraft fleet assignment Health careFire protectionDiary production Telecom network expansion Bank portfolio selection Defense/aerospa ce contracting Military deployment Air pollution control AgricultureLand use planning

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 Tran Van Hoai3 What constitutes the proper simplification is subject to individual judgment and experience (George Dantzig) What constitutes the proper simplification is subject to individual judgment and experience (George Dantzig)

Summation variables/constraints Introduce new variables to be easier to understand/debug – Summation of variables/constraints Tran Van Hoai4 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 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

First model MAX23X 1 +34X 2 +45X 3 S.T.2X 1 +3X 2 +4X 3 ≤7000 X1X1 ≤.4(X 1 +X 2 +X 3 ) X2X2 ≤ X3X3 ≤ X 1,X 2,X 3 ≥ Tran Van Hoai5 MAX23X 1 +34X 2 +45X 3 S.T.2X 1 +3X 2 +4X 3 ≤7000.6X 1 -.4X 2 -.4X 3 ≤0 -.4X 1 +.6X 2 -.4X 3 ≤0 -.4X 1 -.4X 2 +.6X 3 ≤0 X 1,X 2,X 3 ≥0 No meaning as natural input (especially on spreadsheet) Not management constraint anymore

Define summation variable X 4 = total production of TVs Add summation constraint X 1 + X 2 + X 3 - X 4 = 0 MAX23X 1 +34X 2 +45X 3 S.T.2X 1 +3X 2 +4X 3 ≤7000 X1X1 +X2X2 +X3X3 -X4X4 =0 X1X1 -.4X 4 ≤0 X2X2 - ≤0 X3X3 - ≤0 X 1,X 2,X 3 ≥0 Revised model Tran Van Hoai6 Summation variable Summation constraint Clarity (although more variables/constraints) Clarity (although more variables/constraints)

Applications of LP/ILP More realistic example Reduced version in different practical applications – Portfolio model Tran Van Hoai7

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 Hoai8

Jones investment service (advise clients on investment) Tran Van Hoai9 Potential investmentExpecte d return John’s rating Liquidity analysis Risk factor Savings account4.0%AImmediate0 Certificate of deposit5.2%A5-year0 Atlantic Lighting7.1%B+Immediate25 Arkansas REIT10.0%BImmediate30 Bedrock insurance annuity8.2%A1-year20 Nocal mining bond6.5%B+1-year15 Minicomp systems20.0%AImmediate65 Antony hotels12.5%CImmediate40 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 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

LP model MIN25X 1 +30X 4 +20X 5 +15X 6 +65X 7 +40X 8 S.T.X1X1 +X2X2 +X3X3 +X4X4 +X5X5 +X6X6 +X7X7 +X8X8 = X X X 3 +.1X X X 6 +.2X X 8 ≥7500 X1X1 +X2X2 +X5X5 +X7X7 ≥50000 X1X1 +X3X3 +X4X4 +X7X7 +X8X8 ≥40000 X1X1 +X2X2 ≤30000 All X i ≥ Tran Van Hoai10

Analysis (1) Binding constraints What exceed minimum requirements Investment Tran Van Hoai11 Expected annual return$7500 Liquid investment$40000 Savings & certificate deposit$30000 A-rated investment$50000$77333 Savings$17333Arkansas$22666 Certificate deposit $12666Bedrock insurance $47333

Analysis (2) Reduced costs: in order to be included, risk factor must be lowered Optimality range Shadow price: risk increased by Tran Van Hoai12 Atlantic4.67Minicomp1.67 Nocal0.67Antony1.67 Bedrock19.5 (20-0.5) -> ( ) Savings0 -> 1.17 $1 extra to $ $1 extra liquid investments4