/*! 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 40 Deluxe River Cruises operates a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 40

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 -day cruise in May. It costs $$\$ 44,000$$ to operate a type-A vessel and $$\$ 54,000$$ to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

Short Answer

Expert verified
To minimize operating costs, Deluxe River Cruises should use 4 type-A vessels and 3 type-B vessels. The minimum cost is $$\$300,000$$.

Step by step solution

01

Define the variables

Let x be the number of type-A vessels, and y be the number of type-B vessels.
02

Write the constraints involving deluxe and standard cabins

Deluxe cabins constraint: \(60x + 80y \ge 360\) Standard cabins constraint: \(160x + 120y \ge 680\)
03

Create the objective function (total cost)

The objective function represents the total cost, C, to operate the vessels, and we want to minimize it: C = 44,000x + 54,000y
04

Solve the system considering constraints and the objective function

We'll start by dividing the constraint equations to get simpler coefficients: Deluxe cabins constraint: \(3x + 4y \ge 18\) Standard cabins constraint: \(4x + 3y \ge 17\) Now, graph these constraint inequalities and find the feasible region where both conditions are met. The coordinates of the vertices of the feasible region are the points where the lines intersect. The vertices are (0,9), (4,3), and (17,0). Next, we substitute these vertices into the objective function to find the minimum cost: C(0,9) = 44,000(0) + 54,000(9) = 486,000 C(4,3) = 44,000(4) + 54,000(3) = 300,000 C(17,0) = 44,000(17) + 54,000(0) = 748,000 The minimum cost is $$\$300,000$$ at the point (4,3), meaning that Deluxe River Cruises should use 4 type-A vessels and 3 type-B vessels to minimize operating costs in order to meet the stated requirements of the Odyssey Travel Agency.

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.

Mathematical Optimization
Mathematical optimization is the selection of the best element from some set of available alternatives. In the context of linear programming, it refers to finding the maximum or minimum value of a linear objective function, a mathematical expression that may relate to costs, profits, or other quantities that need to be maximized or minimized. In the Deluxe River Cruises example, the objective function is the total cost of operating the vessels, which the company is aiming to minimize.

This process involves deciding how best to use limited resources, which could include anything from raw materials and money to time and manpower, in order to achieve a desired outcome while adhering to a set of predefined constraints.
Inequality Constraints
Inequality constraints define the conditions that solutions to an optimization problem must satisfy. These take the form of inequalities that represent limits or requirements that dictate feasible solutions to the problem. In the scenario given, we have two constraints: one for the number of deluxe cabins and one for the number of standard cabins. Each constraint consists of a linear inequality that must be met or exceeded, ensuring that the number of cabins provided meets the minimum demanded by the charter agreement with Odyssey Travel Agency.

Constraints are crucial in defining the 'feasible region', which is the set of all possible points that satisfy all inequality constraints. When graphed, these constraints help in visualizing and identifying the feasible region.
Objective Function
An objective function, in linear programming, is a formula used to assess the quality of any potential solution. Specifically, it defines what is being optimized and sets the criteria for evaluating the decisions made within the constraints of the problem. For Deluxe River Cruises, the objective function is the total cost of operating the fleet, which they seek to minimize. This function is expressed as a linear equation: the cost of operating each vessel (type-A and type-B) is multiplied by the number of vessels of that type in operation, and the sum of these products gives the total cost.

The goal is to find the values for the variables representing the number of vessels that will result in the lowest possible total cost within the feasible region.
Feasible Region
The feasible region in a linear programming problem is the set of all possible solutions that satisfy all the constraints of a problem. It is the graphical representation of the area where all inequality constraints overlap. Solutions outside this region are called 'infeasible' and do not meet the criteria defined by the problem.

In the Deluxe River Cruises problem, the feasible region is depicted on a graph where each axis represents the number of type-A or type-B vessels. Coordinates within this region satisfy both the deluxe and standard cabin constraints. The vertices of this polygonal region are especially important as they represent potential solutions, and for linear programming problems, the optimal solution will always reside at one of the vertices of the feasible region.
Operations Research
Operations research is a discipline that deals with the application of advanced analytical methods to help make better decisions. It involves techniques like mathematical modeling, statistical analysis, and mathematical optimization, all of which help in optimizing complex scenarios such as in resource allocation, inventory control, scheduling, and logistics.

Operations research strives to provide a framework to model the complexity of operational systems and synthesize decisions that will enhance performance. Linear programming is an essential part of operations research and is often used to solve problems involving maximization of profit or minimization of cost with a given set of resources and constraints, just like the problem faced by Deluxe River Cruises.

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

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=x+4 y-2 z \\ \text { subject to } & 3 x+y-z \leq 80 \\ & 2 x+y-z \leq 40 \\ & -x+y+z \leq 80 \\ x & \geq 0, y \geq 0, z \geq 0 \end{array} $$

Custom Office Furniture is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and \(\mathrm{C}\) -are to be marketed. Each model-A desk requires \(1 \frac{1}{4} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each model-B desk requires \(1 \frac{1}{2} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each modelC desk requires \(1 \frac{1}{2} \mathrm{hr}, \frac{3}{4} \mathrm{hr}\), and \(\frac{1}{2} \mathrm{hr}\) for fabrication, assembly, and finishing, respectively. The profit on each model-A desk is $$\$ 26$$, the profit on each model-B desk is $$\$ 28$$, and the profit on each model-C desk is $$\$ 24$$. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is \(310 \mathrm{hr}, 205 \mathrm{hr}\), and \(190 \mathrm{hr}\), respectively. To maximize Custom's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?

Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in newspaper I is seen by 70,000 group-A customers, 40,000 group-B customers, and 20,000 group-C customers. Each advertisement in newspaper II is seen by 10,000 group-A, 20,000 group-B, and 40,000 group-C customers. Each advertisement in newspaper I costs $$\$ 1000$$, and each advertisement in newspaper II costs $$\$ 800$$. Everest would like their advertisements to be read by at least 2 million people from group A. \(1.4\) million people from group \(\mathrm{B}\), and 1 million people from group C. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost? What is the minimum cost?

Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=-2 x+y \\ \text { subject to } & x+2 y \leq 6 \\ & 3 x+2 y \leq 12 \\ & x \geq 0, y \geq 0 \end{aligned} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{rr} \text { Maximize } & P=15 x+12 y \\ \text { subject to } & x+y \leq 12 \\ & 3 x+y \leq 30 \\ & 10 x+7 y \leq 70 \\ x & \geq 0, y \geq 0 \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.