Section 3.1 Linear Programming Problems Section 3.2 Graphing Linear...

Section 3.1Linear Programming Problems

Section 3.2Graphing Linear Inequalities

Section 3.3Solving Linear Programming Problems

Tekin Karadag

Texas A&M University

Department of MathematicsTexas A&M University, College Station

24-26 September, 2019

Tekin Karadag (TAMU) Mathematics for Business and Social Sciences Week5 1 / 27

1 LINEAR PROGRAMMING PROBLEMS

2 GRAPHING SYSTEM OF LINEAR INEQUALITIES

3 GRAPHICAL SOLUTION OF LINEAR PROGRAMMING PROBLEMS


Linear Programming Problems

In many business and economic problems, we are asked to optimize(maximize or minimize) a function subject to a system of equalities orinequalities. The function to be optimized is called objective function.The system of equalities or inequalities to which the objective function issubjected reflects the constraints (eg. limitations on resources such asmaterials and labor) imposed on the solution(s) to the problem. Problemsof this nature are called mathematical programming problems.We are interested in a specific type of programming problems which arebasically the linear programming problems.

Remark

A linear programming problem consists of a linear objective function tobe maximized or minimized subject to certain constraints in the form oflinear equations or inequalities.


HOW TO SET UP A LINEAR PROGRAMMINGPROBLEMS

To solve linear programming problems, we first need to set up problems.We want to translate the sentences into the objective function (what we’reoptimizing) and the constraints (the system of linear inequalities).

STEPS FOR SETTING UP A LINEAR PROGRAMMINGPROBLEMS

(1) Identify the variables (x and y).

(2) Identify the objective function (NOT AN INEQUALITY).

(3) Identify constraints in the paragraph.

(4) Identify hidden constraints (usually, x ≥ 0 and y ≥ 0).


A Maximization Problem

Example

Formulate but do not solve problemAce Novelty wishes to produce two types of souvenirs: Types A and B. Each TypeA souvenir will result in a profit of $1, and each Type B souvenir will result in aprofit of $1.20. To manufacture a Type A souvenir requires 2 minutes on MachineI, and 1 minute on Machine II. A Type B souvenir requires 1 minute on MachineI, 3 minutes on Machine II. There are 3 hours available on Machine 1 and 5 hoursavailable on Machine II for processing the order. How many souvenirs of eachtype should Ace Novelty make to maximize its profit?

Solution:


A Minimization Problem

Example

Formulate but do not solve problemA nutritionist advises an individual who is suffering from iron and vitamin Bdeficiency to take at least 2400 miligrams(mg) of iron, 2100 mg of vitamin B1,and 1500 mg of vitamin B2 over a period of time. Two vitamin pills are suitable,Brand A and Brand B. Each Brand A pills costs 6 cents and contains 40 mg ofiron, 10 mg of vitamin B1, and 5 mg of vitamin B2. Each brand B pill costs 8cents and contains 10 mg of iron and 15 mg each of vitamins B1 and B2. Whatcombination of pills should the individual purchase to meet the minimum iron andvitamin requirements at the lowest cost?

Solution:


Example

Formulate but do not solve problemDeluxe River Cruises operates a fleet of river vessels. The fleet has two types ofvessels. The type A vessel has 60 deluxe cabins and 160 standard cabins, whereasthe type B vessel has 80 deluxe cabins and 120 standard cabins. Under a charteragreement with Odyssey Travel Agency, Deluxe River Cruises is to provideOdyssey with a minimum of 360 deluxe and 680 standard cabins for their 15-daycruise in May. It costs $44,000 to operate a type A vessel and $54,000 to operatea type B vessel for that period. How many of each type vessel should be used tokeep the operating costs to a minimum?

Solution:


Example

Formulate but do not solve problemA company manufactures x units of product A, y units of product B, andz units of product C. Each product is processed in three departments: I, II,and III. The total available labor-hours per week for departments I, II, andIII is 860, 1240, and 1100, respectively. The time requirements (in hoursper unit) and profit per unit for each product are given in the table below.How many units of each product should the company produce in order tomaximize its profit, P?

Product A Product B Product C

Dept. I 2 2 3

Dept. II 1 3 3

Dept. III 1 2 4

Profit 14 16 20

Solution:


GRAPHING SYSTEM OF LINEAR INEQUALITIES

Previously, we saw that a linear equation in two variables x and y

ax + by + c = 0 a,b not both equal to zero

has a solution set that may be exhibited graphically as points on a straightline in the xy -plane. We now show that there is also a simple graphicalrepresentation for linear inequalities in two variables:

ax + by + c < 0 ax + by + c ≤ 0

ax + by + c > 0 ax + by + c ≥ 0

Before turning to a general procedure for graphing such inequalities, let’sconsider a specific example. Suppose we wish to graph

2x + 3y < 6.


Procedure for Graphing Linear Inequalities

(1) Draw the graph of the equation obtained for the given inequalityby replacing the inequality sign with an equal sign. Use a dashed ordotted line if the problem involves a strict inequality. Otherwise, use asolid line to indicate that the line itself constitutes part of the solution.

(2) Pick a test point (a, b) lying in one of the half-planes determinedby the line sketched in Step 1 and substitute the numbers a and b forthe values of x and y in the given inequality. For simplicity, use theorigin whenever possible.

(3) If the inequality is satisfied, the graph of the solution to theinequality is the half-plane containing the test point. Otherwise, thesolution is the half-plane not containing the test point.


Example

Determine the solution set for the inequality 2x + 3y ≥ 6.

Solution:


Example

Determine the solution set for the inequality x − 2y < 0.

Solution:


Technology Option

Remark

A graphing utility can be used to plot the graph of a linear equality. Forexample, to plot the solution set for the first example we saw, first rewritethe equation 2x + 3y = 6 in the form y = 2 − 2

3x . Next, enter thisexpression for Y1 in the calculator, and move the cursor to the left of Y1.Then press ENTER repeatedly, and select the icon that indicates theshading option desired. The required graph follows.


Graphing Systems of Linear Inequalities

By the solution set of a system of linear inequalities in the twovariables x and y , we mean the set of all points (x , y) satisfying eachinequality of the system. The graphical solution of such a system may beobtained by graphing the solution set for each inequality independentlyand then determining the region in common with each solution set.


Example

Sketch the solution set for the system

x ≥ 0

y ≥ 0

x + y − 6 ≤ 0

2x + y − 8 ≤ 0

Solution:


DefinitionThe solution set of a system of linear inequalities is bounded if it can be enclosedby a circle. Otherwise, it is unbounded.

Example

Determine the graphical solution set for the following system of linear inequalities:

2x + y ≥ 50

x + 2y ≥ 40

x ≥ 0

y ≥ 0

Solution:


GRAPHICAL SOLUTION OF LINEAR PROGRAMMINGPROBLEMS

Graphical MethodLinear programming problems in two variables have relatively simplegeometric interpretations. For example, the system of linear constraintsassociated with a two-dimensional linear programming problem, unless it isinconsistent, defines a planar region or a line segment whose boundary iscomposed of straight-line segments and/or half-lines. Such problems aretherefore amenable to graphical analysis.


Consider the following two-dimensional linear programming problem:

Maximize P = 3x + 2y

subject to 2x + 3y ≤ 12

2x + y ≤ 8

x ≥ 0, y ≥ 0

The system of linear inequalities above defines the planar region S . Each point inS is a candidate for the solution of the problem at hand and is referred to as afeasible solution. The set itself is referred to as a feasible set.

Our goal is to find, from among all the points in the set S , the point(s) that

optimizes the objective function P. Such a feasible solution is called an optimal

solution and constitutes the solution to the linear programming problem under

consideration.


Theorem

Solution(s) of Linear Programming Problems: If a linear programming has asolution, then it must occur at a vertex, or corner point, of the feasible set Sassociated with the problem

TheoremExistence of a Solution: Suppose we are given a linear programming problemwith a feasible set S and an objective function P = ax + by .

(a) If S is bounded, then P has both a maximum and a minimum value onS .

(b) If S is unbounded and both a and b are nonnegative, then P has aminimum value on S provided that the constraints defining S include theinequalities x ≥ 0 and y ≥ 0.

(c) If S is the empty set, then the linear programming problem has nosolution; that is, P has neither a maximum nor a minimum values.


The Method of Corners

1. Graph the feasible set.

2. Find the coordinates of all corner points(vertices) of the feasibleset.

3. Evaluate the objective function at each corner point.

4. Find the vertex that renders the objective function a maximum(minimum). If there is only one such vertex, then this vertexconstitutes a unique solution to the problem. If the objective functionis maximized (minimized) at two adjacent corner points of S , thereare infinitely many optimal solutions given by the points on the linesegment determined by these two vertices.


Example

Maximization Problem Solve the linear programming problem by the method ofcorners.

Maximize P = x + 1.2y

subject to 2x + y ≤ 180

x + 3y ≤ 300

x ≥ 0, y ≥ 0

Solution:


Example

Minimization Problem Solve the linear programming problem by the method ofcorners.

Minimize C = 6x + 8y

subject to 40x + 10y ≥ 2400

10x + 15y ≥ 2100

5x + 15y ≥ 1500

x ≥ 0, y ≥ 0

Solution:


Example

A Linear Programming Problem with Multiple Solutions Find the maximumand minimum of P = 2x + 3y subject to the following system of linearinequalities:

2x + 3y ≤ 30

−x + y ≤ 5

x + y ≥ 5

x ≤ 10

x ≥ 0, y ≥ 0

Solution:


Example

An Unbounded Linear Programming Problem with No Solution Solve thelinear programming problem by the method of corners.

Maximize P = x + 2y

subject to − 2x + y ≤ 4

x − 3y ≤ 3

x ≥ 0, y ≥ 0

Solution:


Example

An Infeasible Linear Programming Problem Solve the linear programmingproblem by the method of corners.

Maximize P = x + 2y

subject to x + 2y ≤ 4

2x + 3y ≥ 12

x ≥ 0, y ≥ 0

Solution:


Example

Beta Aerobics manufactures two models of steppers used for aerobic exercises.Manufacturing each luxury model requires 10 lb of plastic and 10 min of labor.Manufacturing each standard model requires 16 lb of plastic and 8 min of labor.The profit for each luxury model is $40, and the profit for each standard model is$30. If 6000 lb of plastic and 60 labor-hours are available for the production ofthe steppers per day, how many steppers per day, how many steppers of eachmodel should Beta produce each day to maximize its profit?

Solution:


Example

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types ofvessels. The type A vessel has 60 deluxe cabins and 160 standard cabins, whereasthe type B vessel has 80 deluxe cabins and 120 standard cabins. Under a charteragreement with Odyssey Travel Agency, Deluxe River Cruises is to provideOdyssey with a minimum of 360 deluxe and 680 standard cabins for their 15-daycruise in May. It costs $44,000 to operate a type A vessel and $54,000 to operatea type B vessel for that period. How many of each type vessel should be used tokeep the operating costs to a minimum?

Solution:


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