13. LPP-V

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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

AI

Drone

Self-driven logistics

When

After 1 year

After 3 years

After 5 years

How much (minimum) [in Cr.]

2000

2000

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

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

Where to invest?

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

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

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

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

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

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

Range of optimality

Simultaneously

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

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