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LPP- V Prakash Awasthy 1 Where to invest? Finance domain application • You are CFO of one of the biggest e-tail company in India • You foresee that your firm would require substantial capital for technological advancement on AI, Drone and self-driven logistics Capital for When How much (minimum) [in Cr.] AI After 1 year 2000 Drone After 3 years 2000 Self-driven logistics After 5 years 3000 2 Where to invest? • There are three types of assets where you can invest in order to fulfil the requirements of capital in future years • Here are per unit returns for each of the terms for three investment options Huge Growth Rapid AI (1 Year) 0.5 1.5 2.5 Drone (3 years) 1.5 3 1 Self-driven logistics (5 years) Total 2.5 1.5 1 4.5 6 4.5 3 Where to invest? • What should be the investment mix? Huge Growth Rapid Minimum Capital AI (1 Year) 0.5 1.5 2.5 2000 Drone (3 years) 1.5 3 1 2000 1.5 1 3000 Self-driven logistics (5 years) 2.5 4 Hiring and Training Problem HR domain application • The Feeder-Service Airlines Company must decide how many new flight attendants to hire and train over the next six months. The requirements expressed as the number of attendant-flight-hours needed are 8000 in January; 9000 in February; 8000 in March; 10,000 in April; 9000 in May; and 12,000 in June • It takes one month of training before a flight attendant can be put on a regular flight; so a person must be hired at least a month before (s)he is actually needed. Also, a trainee requires 100 hours of supervision by experienced flight attendants during the month of training so that 100 less hours are available for flight service by regular flight attendants • Each experienced flight attendant can work up to 150 hours in a month, and Feeder-Service has 60 regular flight attendants available at the beginning of January. Each month, approximately 10% of the experienced flight attendants quit their jobs • An experienced flight attendant costs the company INR 2.4 lac and a trainee INR 1 lac a month in salary and other benefits 5 Duality • Every LPP has an associated dual problem Resource LQP DMP Availability (‘000) Labor (hrs) Clean Room (hrs) Testing Room (hrs) Profit (Rs) 1 1 3 1000 1 2 1 500 10 16 24 6 Product-mix: Primal and dual • Primal: Maximize the profit such that resource(labor, cleaning, testing) usage is with in a limit • Dual: Minimize the resource usage such that margin for each product is above a threshold 7 Product-mix: Primal and dual Primal 1000 + 500 1 + 1 ≤ 10 1 + 2 ≤ 16 3 + 1 ≤ 24 , ≥ 0 Dual 10 1 + 16 2 + 24 3 1 1 + 1 2 + 3 3 ≥ 1000 1 1 + 2 2 + 1 3 ≥ 500 1, 2, 3 ≥ 0 8 Product-mix: Primal and dual • What are u1, u2 and u3? • Shadow prices of resources (Labor, cleaning and testing, respectively) • Practice problem: Solve primal and dual problems (on solver) for the product-mix problem and compare their results 9 Why find dual of a problem? • Shadow prices • Computational reasons [Sometimes it is efficient to solve dual of a problem] • Sometimes easier to do sensitivity analysis • Effect of adding a new variable 10 Adding a new variable • Suppose in addition to LQP and DMP, there is a third printer TQP • You get a margin of INR 700 per TQP • TQP takes 1.5 hrs each Labor, Cleaning and Testing • What would be the optimal product mix now? • Let’s write the primal and dual of this modified problem • Effect of adding a constraint 11 Sensitivity analysis recap • Changing OFC (objective function coefficient) • Individually • Simultaneously Range of optimality • 100% rule • Same multiplier for both OFCs • Changing RHSs • Individually • Simultaneously (100% Rule) Range of feasibility, dual prices • Adding a new constraint • Adding a new variable 12