/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 718 Give an example of a problem tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 718

Give an example of a problem that is amenable to linear programming methods.

Short Answer

Expert verified
A transportation problem can be solved using linear programming methods. In this example, a company has two factories (F1 and F2) manufacturing products and three stores (S1, S2, and S3) selling them. The factories have different production capacities and the stores have varying demands. The objective is to minimize the total cost of transportation while meeting the demand and not exceeding the production capacities. We formulate the problem with a linear objective function, decision variables, constraints, and parameters. We minimize the cost function \(Z = c_{11}x_{11} + c_{12}x_{12} + c_{13}x_{13} + c_{21}x_{21} + c_{22}x_{22} + c_{23}x_{23}\), subject to constraints related to production capacities, demand fulfillment, and non-negativity.

Step by step solution

01

Problem Description

A company has two factories (F1 and F2) that manufacture products and three stores (S1, S2, and S3) that sell those products. The factories have different production capacities, and the stores have different demand for the products. The cost of shipping one unit of the product from each factory to each store is given. The company wants to minimize the total cost of transportation while satisfying all the demand and not exceeding the production capacity of each factory.
02

Objective Function

Our objective is to minimize the total cost of transportation. Let's denote the number of units shipped from factory i to store j as x_ij. The total cost can be represented as: \[ \text{Minimize} \quad Z = c_{11}x_{11} + c_{12}x_{12} + c_{13}x_{13} + c_{21}x_{21} + c_{22}x_{22} + c_{23}x_{23} \] Here, c_ij is the cost of shipping one unit from factory i to store j.
03

Variables and Parameters

We have the following decision variables and parameters: Decision variables: - \(x_{11}\): Number of units shipped from Factory 1 to Store 1 - \(x_{12}\): Number of units shipped from Factory 1 to Store 2 - \(x_{13}\): Number of units shipped from Factory 1 to Store 3 - \(x_{21}\): Number of units shipped from Factory 2 to Store 1 - \(x_{22}\): Number of units shipped from Factory 2 to Store 2 - \(x_{23}\): Number of units shipped from Factory 2 to Store 3 Parameters: - \(c_{ij}\): Cost of shipping one unit from Factory i to Store j - \(P_i\): Production capacity of Factory i - \(D_j\): Demand for products at Store j
04

Constraints

We need to consider the following constraints: 1. The supply from each factory must not exceed its production capacity: - \( x_{11} + x_{12} + x_{13} \leq P_1 \) (Factory 1) - \( x_{21} + x_{22} + x_{23} \leq P_2 \) (Factory 2) 2. The demand of each store must be satisfied: - \( x_{11} + x_{21} \geq D_1 \) (Store 1) - \( x_{12} + x_{22} \geq D_2 \) (Store 2) - \( x_{13} + x_{23} \geq D_3 \) (Store 3) 3. Non-negativity constraint: - \( x_{ij} \geq 0 \) for all i, j
05

Linear Programming Formulation

Putting everything together, our linear programming problem is: Minimize \[ Z = c_{11}x_{11} + c_{12}x_{12} + c_{13}x_{13} + c_{21}x_{21} + c_{22}x_{22} + c_{23}x_{23} \] Subject to \[ x_{11} + x_{12} + x_{13} \leq P_1 \\ x_{21} + x_{22} + x_{23} \leq P_2 \\ x_{11} + x_{21} \geq D_1 \\ x_{12} + x_{22} \geq D_2 \\ x_{13} + x_{23} \geq D_3 \\ x_{ij} \geq 0 \quad \text{for all} \quad i, j \] The given problem is a transportation problem, which is a classic example amenable to linear programming methods. After defining the problem, we can use linear programming solvers to find the optimal solution.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the problem $$ \begin{array}{ll} \operatorname{maximize} & \mathrm{x}_{0}=5 \mathrm{x}_{1}+12 \mathrm{x}_{2}+4 \mathrm{x}_{3} \\ \text { subject to } & \mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3} \leq 5 \\\ & 2 \mathrm{x}_{1}-\mathrm{x}_{2}+3 \mathrm{x}_{3}=2 \\ & \mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq 0 \end{array} $$ Solve the primal and dual problems by the Simplex method. Compare the results.

Solve the following linear programming problem: Maximize \(\quad 6 \mathrm{~L}_{1}+11 \mathrm{~L}_{2}\) subject to: $$ \begin{array}{r} 2 \mathrm{~L}_{1}+\mathrm{L}_{2} \leq 104 \\ \mathrm{~L}_{1}+2 \mathrm{~L}_{2} \leq \mathbf{7 6} \end{array} $$ and \(\quad L_{1} \geq 0, L_{2} \geq 0\)

Maximize \(\mathrm{x}_{0}=3 \mathrm{x}_{1}+9 \mathrm{x}_{2}\) subject to: $$ \begin{gathered} \mathrm{x}_{1}+4 \mathrm{x}_{2} \leq 8 \\ \mathrm{x}_{1}+2 \mathrm{x}_{2} \leq 4 \\ \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0 \end{gathered} $$ Use the simplex technique to solve.

Graph the solutions for the following system $$ \begin{array}{ll} & x+2 y \geq 8 \\ \text { and } & x-2 y \geq 2 \\ \text { and } & x \leq 9 \end{array} $$

Assume that two products \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) are manufactured on two machines 1 and \(2 .\) Product \(\mathrm{x}_{1}\) requires three hours on machine 1 and one-half hour on machine \(2 .\) Product \(\mathrm{x}_{2}\) requires two hours on machine 1 and 1 hour on machine \(2 .\) There are six hours of available capacity on machine 1 and four hours on machine \(2 .\) Finally, each unit of \(\mathrm{x}_{1}\) produces a net increase in profit of \(\$ 12.00\) and each unit of \(\mathrm{x}_{2}\) an incremental profit of \(\$ 4.00 .1\) ) Maximize the profit. 2 ) Obtain a solution to this problem using the simplex method. 3) Apply sensitivity analysis to the final tableau.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.