Prakash Awasthy
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Finance domain application
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You are CFO of one of the biggest e-tail company in India
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You foresee that your firm would require substantial capital for technological advancement on AI, Drone and self-driven logistics
Capital for
AI
Drone
Self-driven logistics
When
After 1 year
After 3 years
After 5 years
How much (minimum) [in Cr.]
2000
2000
3000
2

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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
AI (1 Year)
Drone (3 years)
Self-driven logistics (5 years)
Total
Huge
0.5
1.5
2.5
4.5
Growth
1.5
3
1.5
6
1
1
Rapid
2.5
4.5
3

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What should be the investment mix?
Huge
AI (1 Year)
Drone (3 years)
0.5
1.5
Self-driven logistics (5 years)
2.5
Growth
1.5
3
1.5
Rapid
2.5
1
1
Minimum Capital
2000
2000
3000
4

HR domain application
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•
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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

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Every LPP has an associated dual problem
Resource
Labor (hrs)
Clean Room (hrs)
Testing Room (hrs)
Profit (Rs)
LQP
1
1
3
1000
DMP
1
2
1
500
Availability
(‘000)
10
16
24
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Primal: Maximize the profit such that resource(labor, cleaning, testing) usage is with in a limit
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Dual: Minimize the resource usage such that margin for each product is above a threshold
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Primal
𝑀𝑎𝑥 1000 𝐿 + 500 𝐷 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
1
1
𝐿 + 1 𝐷 ≤ 10
𝐿 + 2 𝐷 ≤ 16
3 𝐿 + 1 𝐷 ≤ 24
𝐿, 𝐷 ≥ 0
Dual
𝑀𝑖𝑛 10 𝑢1 + 16 𝑢2 + 24 𝑢3 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
1
1 𝑢1 + 1 𝑢2 + 3 𝑢3 ≥ 1000 𝑢1 + 2 𝑢2 + 1 𝑢3 ≥ 500 𝑢1, 𝑢2, 𝑢3 ≥ 0
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What are u1, u2 and u3?
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Shadow prices of resources (Labor, cleaning and testing, respectively)
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Practice problem: Solve primal and dual problems (on solver) for the product-mix problem and compare their results
9

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Shadow prices
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Computational reasons [Sometimes it is efficient to solve dual of a problem]
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Sometimes easier to do sensitivity analysis
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Effect of adding a new variable
10

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Suppose in addition to LQP and DMP, there is a third printer TQP
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You get a margin of INR 700 per TQP
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TQP takes 1.5 hrs each Labor, Cleaning and Testing
What would be the optimal product mix now?
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Let’s write the primal and dual of this modified problem
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Effect of adding a constraint
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Changing OFC (objective function coefficient)
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Individually
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Range of optimality
Simultaneously
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100% rule
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Same multiplier for both OFCs
Changing RHSs
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Individually
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Simultaneously (100% Rule)
Range of feasibility, dual prices
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Adding a new constraint
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Adding a new variable
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