Primal and dual lp problems, Econometrics

Assignment Help:

As in the model solved initially, the following is the LP model

Maximize Z = $42.13*(x11 + x12 + x13 + x14) + $38.47*(x21 + x22 + x23 + x24) + $27.87*(x31 + x32 + x33 + x34)

With subject to constraints

Production Constraints

x11 + x21 + x31 = 3814

x12 + x22 + x32 = 2666

x13 + x23 + x33 = 4016

x14 + x24 + x34 = 1300

PN quality constraints

107x11 + 93x12 + 87x13 + 108x14 ≥ 100*(x11 + x12 + x13 + x14)

107x21 + 93x22 + 87x23 + 108x24 ≥ 91*(x21 + x22 + x23 + x24)

RVP quality constraints

5x11 + 8x12 + 4x13 + 21x14 ≤ 7*(x11 + x12 + x13 + x14)

5x21 + 8x22 + 4x23 + 21x24 ≤ 7*(x21 + x22 + x23 + x24)

Non-negativity constraint

xij  ≥ 0

where xij  is the amount of i, i = 1 (Alkylate), 2 (catalytic-cracked), 3 (straight-run), 4 (isopentane) to be mixed in j, j = 1 (Blend 1 or Avgas A), 2 (Blend 2 or Avgas B), 3 (Raw)

Dual problem

Associated with every LP problem is a related dual problem. If the objective in the original problem is maximization, then the objective in the dual is the minimization of a related (but different) function. Conversely, an original minimization problem has a related dual maximization problem. Thus in our case, since the original problem is a maximization, the dual is a minimization of a related function.

Associated with each constraint of the original problem is a dual variable. Since the original problem had eight constraints, the dual problem will have eight variables, namely w1, w2, w3, w4, w5, w6, w7 and w8. It is not required to have a variable for non-negativity constraint of the original problem. As we see the last four constraints (quality constraints) in the original problem can be still solved as follows

107x11 + 93x12 + 87x13 + 108x14 ≥ 100x11 + 100x12 + 100x13 + 100x14

7x11 - 7x12 - 13x13 + 8x14 ≥ 0

Similarly,

107x21 + 93x22 + 87x23 + 108x24 ≥ 91x21 + 91x22 + 91x23 + 91x24

16x21 + 2x22 - 4x23 + 17x24 ≥ 0

5x11 + 8x12 + 4x13 + 21x14 ≤ 7x11 + 7x12 + 7x13 + 7x14

-2x11 + 1x12 - 3x13 + 14x14 ≤ 0

-2x21 + 1x22 - 3x23 + 14x24 ≤ 0

With the quality constraints solved as above, the objective of the dual problem is minimization as below. It is to be noted that if the constraints in the original problem had greater than or equal to sign, they will be taken as negative values in the minimization problem. The constraints with equal to and less than or equal to signs will be taken as positive values or as such in the original problem. Since there are 12 decision variables in the original problem, the dual will have 12 constraints as below, excluding the non-negativity constraint.

Minimize Z = 3814w1 + 2666w2 + 4016w3+ 1300w4 - 0w5 - 0w6 + 0w7 + 0w8

With subject to constraints

w1 - 7w5 - 2w7 ≥ 42.13

w2 + 7w5 + w7 ≥ 42.13

w3 + 13w5 - 3w7 ≥ 42.13

w4 - 8w5 + 14w7 ≥ 42.13

w1 - 16w6 - 2w8 ≥ 38.47

w2 - 2w6 + w8 ≥ 38.47

w3 + 4w6 - 3w8 ≥ 38.47

w4 - 17w6 + 14w8 ≥ 38.47

w1 ≥ 27.87

w2 ≥ 27.87

w3 ≥ 27.87

w4 ≥ 27.87

wi ≥ 0

The dual variables wi, i = 1, 2, 3, 4 represent the marginal value of production constraints, i = 5, 6 represent the marginal value of PN quality constraints and i = 7, 8 represent the marginal value of RVP quality constraints. On solving this using excel solver, we get the same objective as $481,742.9. If we look more closely into the sensitivity report of the original problem, we can notice that the values of decision variables obtained in the dual problem will be none other than the shadow prices of the constraints in the original problem.


Related Discussions:- Primal and dual lp problems

Firm''s marginal revenue function, A firm has the following inverse demand ...

A firm has the following inverse demand function:       where Q  is Quantity and P  is Price   (a) Find the firm's marginal revenue function. (b) Find the level of out

Goal for development, How can a person achieve his goal for development?Exp...

How can a person achieve his goal for development?Explain it with 5 examples.

find the weighted average cost of capital , PrivateJets (PJ) is considerin...

PrivateJets (PJ) is considering expanding its operations in the corporate travel market. Currently, PJ has a capital structure with a 25% debt-equity ratio. Their levered equity

Role , what is role of education in economic development?

what is role of education in economic development?

Methodology, What methodology will be suitable to use for a doctoral resea...

What methodology will be suitable to use for a doctoral research proposal thesis(The impact of persistent poverty on rural urban migration in Nigeria)?

Wooldridge text book, The following table contains the ACT scores and the G...

The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to

The south African government is not providing, Ask question #are there a...

Ask question #are there any welfare or subsidy payment that should be reviewed or added?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd