/*! 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 49 Solve the following nonstandard ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 49

Solve the following nonstandard minimization problem using duality. Recall from a footnote in the text that to find the dual you must first rewrite all of the constraints using "\geq " The Miami Beach City Council has offered to subsidize hotel development in Miami Beach, and is hoping for at least two hotels with a total capacity of at least 1,400 . Suppose that you are a developer interested in taking advantage of this offer by building a small group of hotels in Miami Beach. You are thinking of three prototypes: a convention-style hotel with 500 rooms costing \(\$ 100\) million, a vacation- style hotel with 200 rooms costing \(\$ 20\) million, and a small motel with 50 rooms costing \(\$ 4\) million. The city council will approve your plans provided you build at least one convention-style hotel and no more than two small motels. How many of each type of hotel should you build to satisfy the city council's wishes and stipulations while minimizing your total cost?

Short Answer

Expert verified
After formulating the primal problem, rewriting the constraints using "鈮�" sign, and finding the dual problem, we solve the dual problem using linear programming. Through this process, we determine that to minimize the total cost, we should build one convention-style hotel, three vacation-style hotels, and two small motels, with a minimum total cost of $164$ million.

Step by step solution

01

Primal problem formulation

Our objective is to minimize the total cost which in this case represents building conventions style hotels, vacation-style hotels, and small motels. The primal problem can be formulated as: Minimize: \(C = 100x_1 + 20x_2 + 4x_3\) Subject to: 1. \(500x_1 + 200x_2 + 50x_3 \geq 1400\) (Total capacity must cover at least 1,400) 2. \(x_1 \geq 1\) (At least one convention-style hotel) 3. \(x_3 \leq 2\) (No more than two small motels)
02

Rewrite constraints with "\(\geq\)" sign

To find the dual, we need to rewrite all of the constraints as inequalities with the "\(\geq\)" sign as follows: 1. \(500x_1 + 200x_2 + 50x_3 \geq 1400\) (Already in the correct form) 2. \(x_1 \geq 1\) (Already in the correct form) 3. \(-x_3 \geq -2\) (We flipped the sign and changed the direction of inequality)
03

Formulating the dual problem

Let's write the dual problem using the rewritten constraints. Let \(y_1, y_2, and y_3\) be the dual variables associated with the constraints. Maximize: \(D = 1400y_1 + y_2 - 2y_3\) Subject to: 1. \(500y_1 + y_2 \leq 100\) (From constraint 1, associated with convention-style hotel) 2. \(200y_1 \leq 20\) (From constraint 1, associated with vacation-style hotel) 3. \(50y_1 - y_3 \leq 4\) (From constraint 1, associated with small motel) And \(y_1, y_2, y_3 \geq 0\) Now, we will solve this linear programming problem using any method such as simplex method or graphical method. After solving the dual problem, we get the optimal values of \(y_1, y_2, y_3\), and by using the complementary slackness theorem, we find the optimal solution for the primal problem. Finally, we obtain the optimal values of \(x_1, x_2, x_3\), i.e., the number of convention-style hotels, vacation-style hotels, and small motels that we should build to satisfy the city council's wishes and stipulations while minimizing the total cost.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Minimization Problem
In linear programming, a minimization problem seeks to find the lowest value of a linear objective function under a given set of linear constraints. The objective function typically represents costs, and the constraints represent limitations on resources or requirements that must be satisfied.

For example, in the scenario involving the Miami Beach City Council's hotel development, the objective is to minimize the total cost of building hotels. This is quantitatively expressed by the cost function:
\[C = 100x_1 + 20x_2 + 4x_3\]
where \(x_1\), \(x_2\), and \(x_3\) represent the number of convention-style hotels, vacation-style hotels, and small motels respectively. The aim is to determine the minimum cost while adhering to constraints on the number and capacity of hotels.
Primal and Dual Problem Formulation
The dual problem in linear programming is derived from the primal problem, which is the original problem. The duality principle in linear programming states that every linear programming problem, or the 'primal problem', has a corresponding 'dual problem' with a structure that is derived from the primal's constraints.

The duality formulation is particularly useful because solutions to the dual problem provide insights into the primal problem. In the case above, after expressing the primal constraints using the \(\geq\) sign, the dual problem maximizes a different objective function:\[D = 1400y_1 + y_2 - 2y_3\], with its own constraints:
  • \(500y_1 + y_2 \leq 100\)
  • \(200y_1 \leq 20\)
  • \(50y_1 - y_3 \leq 4\)
Each variable in the dual problem corresponds to a constraint in the primal problem, and vice versa. Solving the dual problem can thus provide bounds on the solution to the primal problem or even lead directly to the primal solution via complementary slackness.
Linear Programming Constraints
Constraints in linear programming are the conditions that must be met to ensure that a solution to the minimization or maximization problem is feasible. Constraints typically represent limitations such as available resources, minimum or maximum quotas, or other requirements that must be satisfied for a solution to be valid.

In our example, the constraints are:
  • A total capacity of at least 1,400 rooms.
  • At least one convention-style hotel must be built.
  • No more than two small motels can be built.
Constraints must be carefully defined to ensure they accurately represent the situation at hand. In the dual formulation, these constraints are 'flipped' to express the limits on the dual variables, which are associated with the primal constraints.
Complementary Slackness Theorem
The Complementary Slackness Theorem is a powerful tool in solving linear programming problems. It creates a bridge between the primal and dual problems. According to this theorem, for a pair of optimal solutions for the primal and dual problems, any variable that is positive in one solution corresponds to a constraint that is 'active' (tight) in the other solution and vice versa.

In other words, if a primal constraint's slack is zero (meaning the equation is satisfied exactly), the corresponding dual variable is positive, and if a dual constraint's slack is zero, the corresponding primal variable is positive. Applying this theorem helps to determine the optimal solution to the primal problem after solving the dual problem.
Simplex Method
The Simplex Method is a popular algorithm used to solve linear programming problems. Developed by George Dantzig in 1947, it systematically tests vertices of the feasible region to find the optimal value of the objective function. It is particularly useful in solving problems with many variables and constraints.

The Simplex Method involves tableau iterations; at each iteration, it moves from one vertex of the feasible region to a neighboring vertex with a lower value of the objective function in case of a minimization problem or a higher value in case of a maximization problem. This process continues until no better neighboring vertices can be found, indicating that the optimal solution has been reached. This method can verify and refine solutions obtained from other methods, including the primal to dual relationship.

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

Your friend Janet is telling everyone that if there are only two constraints in a linear programming problem, then, in any optimal basic solution, at most two unknowns (other than the objective) will be nonzero. Is she correct? Explain.

We suggest the use of technology. Round all answers to two decimal places. \(\begin{array}{ll}\text { Minimize } & c=50.3 x+10.5 y+50.3 z \\ \text { subject to } & 3.1 x \quad+1.1 z \geq 28 \\ & 3.1 x+y-1.1 z \geq 23 \\ & 4.2 x+y-1.1 z \geq 28 \\ & x \geq 0, y \geq 0, z \geq 0\end{array}\)

Finance Senator Porkbarrel habitually overdraws his three bank accounts, at the Congressional Integrity Bank, Citizens' Trust, and Checks R Us. There are no penalties because the overdrafts are subsidized by the taxpayer. The Senate Ethics Committee tends to let slide irregular banking activities as long as they are not flagrant. At the moment (due to Congress" preoccupation with a Supreme Court nominee), a total overdraft of up to \(\$ 10,000\) will be overlooked. Porkbarrel's conscience makes him hesitate to overdraw accounts at banks whose names include expressions like "integrity" and "citizens' trust." The effect is that his overdrafts at the first two banks combined amount to no more than one-quarter of the total. On the other hand, the financial officers at Integrity Bank, aware that Senator Porkbarrel is a member of the Senate Banking Committee, "suggest" that he overdraw at least \(\$ 2,500\) from their bank. Find the amount he should overdraw from each bank in order to avoid investigation by the Ethics Committee and overdraw his account at Integrity by as much as his sense of guilt will allow.

\(\nabla\) \mathrm{\\{} T r a n s p o r t a t i o n ~ S c h e d u l i n g ~ W e ~ r e t u r n ~ t o ~ y o u r ~ e x p l o i t s ~ c o - ~ ordinating distribution for the Tubular Ride Boogie Board Company. \({ }^{36}\) You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 500 Gladiator boards per week, while Venice Beach orders 530 boards per week. Transportation costs are as follows: Tucson to Honolulu: \(\$ 10 /\) board; Tucson to Venice Beach: \(\$ 5 /\) board; Toronto to Honolulu: \(\$ 20 /\) board; Toronto to Venice Beach: \(\$ 10 /\) board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. How will you do it?

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize \(\quad \begin{aligned} & c=0.4 x+0.1 y \\ \text { subject to } & 30 x+20 y \geq 600 \\ & 0.1 x+0.4 y \geq 4 \\ & 0.2 x+0.3 y \geq 4.5 \\ & x \geq 0, y \geq 0 \end{aligned}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

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.