FE501 – OMF1Ex

Questions Only

LP-2.3 ✅

Q: Write 2.3 in the standard form:

Exercise 2.1 ✅

_Write the following linear program in standard form _

Exercise 2.2

Exercise 2.3

Exercise 2.4

Duality 1 ✅

For a given optimization problem:

Here are a few alternative feasibile solutions:

Prove that (x1; x2; x3; x4) = (5, 2, 0, 0) is the optimal solution for the given LP.

Exercise 2.5

Weak Duality Theorem

Let x be any feasible solution to the primal LP and y be any feasible solution to the dual LP. Proof the following weak duality theory:

Exercise 2.9

Excercise 2.10

Exercise 2.11

Example 2.1 bond allocation problem

Exercise 2.17 Solve with simplex method

3.1 Short Term Financing

Exercise 3.2

Exercise 3.15 Work Force Planning

Q & A

LP-2.3 ✅

Q: Write 2.3 in the standard form:

A

Exercise 2.1 ✅

_Write the following linear program in standard form _

A

Exercise 2.2

Exercise 2.3

Exercise 2.4

Programmed Eq

Duality 1 ✅

For a given optimization problem:

Here are a few alternative feasibile solutions:

Prove that (x1; x2; x3; x4) = (5, 2, 0, 0) is the optimal solution for the given LP.

Extra: (This most possibily will not be in the exam 😀

Exercise 2.5

The objective value of any primal feasible

solution is at least as large as the objective value of any feasible dual solution.

This fact is known as the weak duality theorem:

Weak Duality Theorem

Let x be any feasible solution to the primal LP and y be any feasible solution to the dual LP. Proof the following weak duality theory:

Exercise 2.9

Excercise 2.10

The complementary slackness theorem is:

Exercise 2.11

a)

b)

Example 2.1 bond allocation problem

Excel File

Reduced cost :

• What happens if we force the decision variables from 0 to 1? (what happens if force the variable final value to some value, the objective value will be reduced)
• In excel we can do this by adding another constraint and making xi = VAL (restricting the case, reduce the value)
• Reduced cost is only non-zero for non-basic vairables (final value is 0)

Shadow Price:

• What will be the affect of increasing the resources (increasing the RHS of constraints)?
• shadow price = change in the optimal objective value for a unit increase in the RHS of the constraints

Interpretation:

1️⃣ if we increase 100 to 101, the objective value will become 352.

2️⃣ in the second constraint, 350 is smaller than 360 and is not used out, so increase in the RHS dose not change the objective value

3️⃣ increase in the RHS constraint from 150 to 151 (1 unit ) in crease in the objective value to 1 unit == 351

Exercise 2.17 Solve with simplex method

3.1 Short Term Financing

Excel File

Exercise 3.2

Exercise 3.15 Workforce Planning

Excel File