I figured out the dual but still need an economic interpretation please. The given image presents a linear programming problem with the objective function and constraints. Here's the transcription:**Objective Function:**Maximize $ z = 3x_1 + 2x_2 $**Subject to Constraints:**\[\frac{1}{40}x_1 + \frac{1}{60}x_2 \leq 1\]\[\frac{1}{50}x_1 + \frac{1}{50}x_2 \leq 1\]\[x_1, x_2 \geq 0\]**Explanation:**The objective is to maximize $ z $, which is a function of two decision variables $ x_1 $ and $ x_2 $. The constraints limit the possible values of $ x_1 $ and $ x_2 $ based on resource restrictions or other requirements.---**Dual Problem:**For the dual of this problem, define dual variables $ y_1 $ and $ y_2 $ corresponding to each of the constraints. The dual linear program will have:**Objective Function:**Minimize $ w = y_1 + y_2 $**Subject to Constraints:**\[\frac{1}{40}y_1 + \frac{1}{50}y_2 \geq 3\]\[\frac{1}{60}y_1 + \frac{1}{50}y_2 \geq 2\]\[y_1, y_2 \geq 0\]---**Economic Interpretation:**The primal problem represents optimizing the allocation of two types of resources (denoted by $ x_1 $ and $ x_2 $) to maximize profit $ z $. The constraints can be thought of as resource availability.The dual problem gives insights into the value of resources, suggesting how much the objective function $ w $ (in this case, costs) will change with the availability or utilization of resources $ y_1 $ and $ y_2 $. This economic interpretation aids in understanding resource pricing and scarcity.

Solution Max Z=3x1+2x2subject to1. 0.025x1+0.0167x2≤12. 0.02x1+0.02x2≤13. x1,x2≥0 The problem is…

Answered: I just need the dual of the problem and…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

See similar textbooks

Related questions

Question

I figured out the dual but still need an economic interpretation please.

The given image presents a linear programming problem with the objective function and constraints. Here's the transcription:

**Objective Function:**

Maximize $ z = 3x_1 + 2x_2 $

**Subject to Constraints:**

\[
\frac{1}{40}x_1 + \frac{1}{60}x_2 \leq 1
\]

\[
\frac{1}{50}x_1 + \frac{1}{50}x_2 \leq 1
\]

\[
x_1, x_2 \geq 0
\]

**Explanation:**

The objective is to maximize $ z $, which is a function of two decision variables $ x_1 $ and $ x_2 $. The constraints limit the possible values of $ x_1 $ and $ x_2 $ based on resource restrictions or other requirements.

---

**Dual Problem:**

For the dual of this problem, define dual variables $ y_1 $ and $ y_2 $ corresponding to each of the constraints. The dual linear program will have:

**Objective Function:**

Minimize $ w = y_1 + y_2 $

**Subject to Constraints:**

\[
\frac{1}{40}y_1 + \frac{1}{50}y_2 \geq 3
\]

\[
\frac{1}{60}y_1 + \frac{1}{50}y_2 \geq 2
\]

\[
y_1, y_2 \geq 0
\]

---

**Economic Interpretation:**

The primal problem represents optimizing the allocation of two types of resources (denoted by $ x_1 $ and $ x_2 $) to maximize profit $ z $. The constraints can be thought of as resource availability.

The dual problem gives insights into the value of resources, suggesting how much the objective function $ w $ (in this case, costs) will change with the availability or utilization of resources $ y_1 $ and $ y_2 $. This economic interpretation aids in understanding resource pricing and scarcity.

Expert Solution

Step 1

Solution

$\begin{array}{rcl} Max Z & = & 3 x_{1} + 2 x_{2} \\ subject to \\ 1 . 0.025 x_{1} + 0.0167 x_{2} & \leq & 1 \\ 2 . 0.02 x_{1} + 0.02 x_{2} & \leq & 1 \\ 3 . x_{1}, x_{2} & \geq & 0 \end{array}$

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '' we should add slack variable
2. As the constraint-2 is of type '' we should add slack variable

After introducing slack variables

$\begin{array}{rcl} Max Z & = & 3 x_{1} + 2 x_{2} + 0 S 1 + 0 S 2 \\ subject to \\ 1 . 0.025 x_{1} + 0.0167 x_{2} + S 1 & = & 1 \\ 2 . 0.02 x_{1} + 0.02 x_{2} + S 2 & = & 1 \\ 3 . x_{2}, x_{2}, S 1, S 2 & \geq & 0 \end{array}$

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, operations-management and related others by exploring similar questions and additional content below.

Similar questions