linear programming on computer_ch78

To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna

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CHAPTER 7 & 8

LINEAR PROGRAMMING MODELS: COMPUTER

METHODSDownload Excel Solver for windows from LMS or:http://www.solver.com/dwnxls.php


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Learning ObjectivesYOU will be able to:1. Understand the basic assumptions and

properties of linear programming (LP).2. Solve any LP problem using Excel Solver.3. Understand special issues in LP such as

infeasibility, un-boundedness, redundancy, and alternative optimal solutions.


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IntroductionLinear programming (LP) is a widely used mathematical modeling technique designed to help managers in planning and decision

making relative to resource allocation. LP is a technique that helps in resource allocation

decisions.Programming refers to modeling and solving a problem mathematically.


7-4 © 2006 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

Examples of Successful LP Applications

1. Development of a production schedule that will satisfy future demands for a firm’s production while minimizing total production and inventory

costs2. Selection of product mix in a factory to

make best use of machine-hours and labor-hours available

while maximizing the firm’s products



Examples of Successful LP Applications (continued)

3.Determination of grades of petroleum products to yield the maximum profit

4. Selection of different blends of raw materials to feed mills to produce finished feed combinations at minimum cost

5. Determination of a distribution system that will minimize total shipping cost from several warehouses to various market locations



LP Problem Statement- ExampleHal has enough clay to make 24small vases or 6 large vases.

He only has enough of a special glazing compound to glaze 16 of the small vases or 8 of the large vases.

Let X1 the number of small vases and X2 the number of large vases.

The smaller vases sell for $3 each, while the larger vases would bring $9 each.


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4 Properties of LPAll LP problems have 4 properties in common:

All problems seek to maximize or minimize some quantity (the objective function).

The presence of restrictions or constraints limits the degree to which we can pursue our objective.

There must be alternative courses of action to choose from.

The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities. 2A + 5B = 10

2A2 + 5B3 + 3AB = 10


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Formulating Linear Programming Problems Formulating a linear program involves

developing a mathematical model to represent the managerial problem.

Once the managerial problem is understood, begin to develop the mathematical statement of the problem.

The steps in formulating a linear program follow on the next slide.


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Steps in LP Formulations1. Completely understand the managerial

problem being faced.2. Identify the objective and the

constraints.3. Define the decision variables.4. Use the decision variables to write

mathematical expressions for the objective function and the constraints.

Formulating Linear Programming Problems (continued)

Applications we will be studyingMarketing ApplicationsManufacturing ApplicationsEmployee Scheduling ApplicationsFinancial ApplicationsTransportation ApplicationsIngredient Blending Applications



Marketing Applications Linear programming models have been used in the

advertising field as a decision aid in selecting an effective media mix.

Media selection problems can be approached with LP from two perspectives. - The objective can be to maximize audience exposure or to

minimize advertising costs.

Marketing Applications

The Win Big Gambling Club Their Marketing Budget per week is 8000$ The table on the following slide presents the number

of potential gamblers reached by making use of an advertisement in each of four media.

It also provides the cost per advertisement placed and the maximum number of ads that can be purchased per week.


Media Selection: Win Big Gambling Club

Medium AudienceReached

Per Ad

CostPer

Ad($)

MaximumAds Purchased Per Week

TV spot (1 minute) 5,000 800 12

Daily newspaper(full page ad)

8,500 925 5

Radio spot (30 seconds, prime time)

2,400 290 25

Radio spot (1 minute, afternoon)

2,800 380 20

Marketing ApplicationsThe Win Big Gambling Club The contractual arrangements require that at least five radio

spots be placed each week. Management insists that no more than $1,800 be spent on

radio advertising every week. The problem can be stated as follows LetX1 = # of 1-minute TV spots each weekX2 = # of daily paper ads each weekX3 = # of 30-second radio spots each weekX4 = # of 1-minute radio spots each week


Win Big Gambling Club LP EquationsTIP: All the science lies here, so practice formulating these from word problems

Win Big Gambling Club – Solver in ExcelStep-1: Write Variable in

each column

Step 4: Write Constraint name here in each row of first column & add constraints

Step 2: Objective

(Minimize or Maximize)

Solution is displayed

here in bold.Step 3: start by putting in

0 for each required Variable

Step 5: Put Sumproduct formula here & insert Sign

Step 6: Input in Solver &

Solve

Win Big Gambling Club – Solver in ExcelSolution is

displayed here in bold.

Step 6: Input in Solver &

Solve

Win Big Gambling Club – Solver in Excel

The parameter solution produces an audience exposure of 67,240 contacts.

Because X1 and X3 are fractional, Win Big would probably round them to 2 and 6, respectively.

Problems that demand all integer solutions are discussed in detail in Chapter 11.

X1 = 1.97 TV spotsX2 = 5 newspaper adsX3 = 6.2 30-second radio spotsX4 = 0 1-minute radio spots



Marketing Applications Linear programming has also been applied to

marketing research problems and the area of consumer research.

The next example illustrates how statistical pollsters can reach strategy decisions with LP.


Management Sciences Associates (MSA) The MSA determines that it must fulfill several requirements in

order to draw statistically valid conclusions:1. Survey at least 2,300 U.S. households.2. Survey at least 1,000 households whose heads are 30 years of age

or younger.3. Survey at least 600 households whose heads are between 31 and

50 years of age.4. Ensure that at least 15% of total surveyed live in a state that

borders on Mexico.5. Ensure that max 20% 51 Years old live in a state that borders on

Mexico out of all those surveyed who are 51 years of age.6. MSA decides that all surveys should be conducted in person.

MSA’s goal is to meet the sampling requirements at the least possible cost.



Marketing ApplicationsManagement Sciences Associates (MSA) MSA estimates that the costs of reaching people in each

age and region category are as follows:

Let:X1 = # of 30 or younger & in a border stateX2 = # of 31-50 & in a border stateX3 = # 51 or older & in a border stateX4 = # 30 or younger & not in a border stateX5 = # of 31-50 & not in a border stateX6 = # 51 or older & not in a border state


Note that the variables in the constraints are moved to the left-hand

side of the inequality.

Answer: The computer solution to MSA’s problem costs $15,166 and below:

Manufacturing Applications

Production Mix

A fertile field for the use of LP is in planning for the optimal mix of products to manufacture.

A company must meet a myriad of constraints, ranging from financial concerns to sales demand to material contracts to union labor demands.

Its primary goal is to generate the largest profit possible.



Manufacturing ApplicationsFifth Avenue Industries

Four varieties of ties produced:1. one is an expensive, all-silk tie, 2. one is an all-polyester tie, and 3. two are blends of polyester and cotton.

The table on the following slide illustrates the cost and availability (per monthly production planning period) of the three materials used in the production process




The table on the next slide summarizes the contract demand for each of o the four styles of ties, o the selling price per tie, and o the fabric requirements of each variety.

Monthly cost & availability of material




Fifth Avenue’s goal is to maximize its monthly profit. It must decide upon a policy for product mix. LetX1 = # of all-silk ties produced per monthX2 = # polyester tiesX3 = # of blend 1 poly-cotton tiesX4 = # of blend 2 poly-cotton ties



Manufacturing ApplicationsCalculate profit for each tie:Profit = Sales price – Cost per yard x Yards per tie

Manufacturing Applications: Product Mix

3 Types of Constraints in this

case:Material Availability

Meeting ContractMaximum Demand

Manufacturing Applications

3 Types of Constraints:Material Availability

Meeting ContractMaximum Demand



Manufacturing Applications Fifth Avenue



Manufacturing ApplicationsProduction Scheduling This type of problem resembles the product mix model for

each period in the future. The objective is either to maximize profit or to minimize the

total cost (production plus inventory) of carrying out the task. Production scheduling is well suited to solution by LP because

it is a problem that must be solved on a regular basis. When the objective function and constraints for a firm are

established, the inputs can easily be changed each month to provide an updated schedule.



Manufacturing ApplicationsProduction Scheduling Setting a low-cost production schedule over a period of weeks

or months is a difficult and important management problem. The production manager has to consider many factors:

o labor capacity, o inventory and storage costs, o space limitations, o product demand, and o labor relations.

Because most companies produce more than one product, the scheduling process is often quite complex.

Manufacturing Applications – Greenberg Motors – Production Scheduling

• Greenberg Motors, Inc., manufactures two different electrical motors for sale under contract to Drexel Corp., a well-known producer of small kitchen appliances.

• Its model GM3A is found in many Drexel food processors, and its model GM3B is used in the assembly of blenders.

• Three times each year, the procurement officer at Drexel contracts Irwin Greenberg, the founder of Greenberg Motors, to place a monthly order for each of the coming four months.

• Drexel’s demand for motors varies each month based on its own sales forecasts, production capacity, and financial position. Greenberg has just received the January–April order and must begin his own four-month production plan. The demand for motors is shown


Meet Demand 8 plus 2

Inventory Cost


Goal is to minimize Costs.

Man-Hours, 4 Max plus 4 Min

Warehouse Storage Capacity, 4 (1 each month)

Increase in Production Costs

Production costs are currently $20 per unit for the GM3A and $15 per unit for the GM3B. However, each of these is due to increase by 10% on March 1 as a new labor agreement goes into effect.


To start with: We have 8 Variables:

A1 A2A3 A4

B1 B2B3 B4


Objective is to minimize CostsSo what are our Costs?


Production CostsInventory CostsSo Let’s represent Inventory Costs.

iA1 iA2iA3 iA4iB1 iB2

iB3 iB4


Total Variables now are:

iA1 iA2iA3 iA4iB1 iB2

iB3 iB4

A1 A2A3 A4

B1 B2B3 B4

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Solution:



Financial ApplicationsPortfolio Selection The selection of specific investments from a wide variety

of alternatives is a problem frequently encountered byo managers of banks, o mutual funds, o investment services, and o insurance companies.

The manager’s overall objective is usually to maximize expected return on investment o given a set of legal, policy, or risk restraints.

Financial Applications Portfolio Selection

International City Trust (ICT) ICT has $5 million available for immediate investment

and wishes to do two things:(1) maximize the interest earned on the investments made

over the next six months, and (2) satisfy the diversification requirements as set by the board

of directors.


International City Trust (ICT)The specifics of the investment possibilities are as follows:

Also, the board specifies that at least 55% of the funds invested must be in gold stocks and

construction loans, and that no less than 15% be invested in trade credit.


International City Trust (ICT)Formulate ICT’s decision as an LP problem.Let

X1 = dollars invested in trade creditX2 = dollars invested in corporate bondsX3 = dollars invested in gold stocksX4 = dollars invested in construction loans

The objective function formulation is presented on the next slide.

Financial Applications Portfolio Selection International City Trust (ICT)

Employee Scheduling ApplicationsAssignment Problems Assignment problems involve finding the most efficient

assignment of o people to jobs, o machines to tasks, o police cars to city sectors, o salespeople to territories, and so on.

The objective might be o to minimize travel times or costs or o to maximize assignment effectiveness.

Employee Scheduling ApplicationsHSBC Problem

Hong Kong Bank of Commerce and Industry is a busy bank that has requirements for between 10 and 18 tellers, depending on the time of day. The lunch time, from noon to 2 PM is usually heaviest.•The bank now employs 12 full-time tellers, but many people are on its roster of available part-time employees.

•A part-time employee must put in exactly four hours per day but can start anytime between 9 AM and 1 PM.

Employee Scheduling Applications•Part-timers are a fairly inexpensive labor pool, since no retirement or lunch benefits are provided for them.

•Full-timers, on the other hand, work from 9 to 5 but are allowed 1 hour for lunch. (Half of the full-timers eat at 11AM, the other half at noon.)

•Full-timers thus provide 35 hours per week of productive labor time.

Employee Scheduling Applications•By corporate policy, the bank limits part-time hours to a maximum of 50% of the day’s total requirement.

•Part-timers earn $8 per hour (or $32 per day) on average, and full-timers earn $100 per day in salary and benefits, on average.

•The bank would like to set a schedule that would minimize its total personnel costs.

•It will release one or more of its full-time tellers if it is profitable to do so.

•Another constraint will limit the total number of full-time workers to be no more than 12. In formulating this as a linear

program, the objective is to minimize cost.

Employee Scheduling Applications

•There is a constraint for each hour of the day, stating that the number of people working at the bank during that hour should be at least the minimum number shown in Table, so there are eight of these constraints.

•The last constraint will specify that the number of part-time hours must not exceed 50% of the total hours.


The bank must decide how many full-time tellers to use, so there will be one decision variable for that.

Similarly, the bank must decide about using part-time tellers, but this is more complex as the part-time workers can start at different times of the day while all full-time workers start at the beginning of the day

Thus, there must be a variable indicating the number of part-time workers starting at each hour of the day from 9AM until 1PM. Any worker who starts at 1PM will work until closing, so there is no need to consider having any part-time workers start after that.

IMPORTANT:Alternate Optimal Solutions

Alternate optimal solutions are common in many LP problems.

The sequence in which you enter the constraints into QM for Windows can affect the solution found.

No matter the final variable values, every optimal solution will have the same objective function value.

Truck Loading ProblemThe truck loading problem involves deciding which items to load on a truck so as to maximize the value of a load shipped. As an example, we consider Goodman Shipping, an Orlando firm owned by Steven Goodman. One of his trucks, with a capacity of 10,000 pounds, is about to be loaded.* Awaiting shipment are the following items:

Truck Loading ProblemThe objective is to maximize the total value of the items loaded onto the truck without exceeding the truck’s weight capacity.

Truck Loading Problem

Truck Loading Problem – Integer Programming

Diet Problems

The diet problem, one of the earliest applications of LP, was originally used by hospitals to determine the most economical diet for patients.

Known in agricultural applications as the feed mix problem, the diet problem involves specifying a food or feed ingredient combination that satisfies stated nutritional requirements at a minimum cost level.

Diet ProblemsWhole Food Nutrition

The Whole Food Nutrition Center uses three bulk grains to blend a natural cereal that it sells by the pound. The cost of each bulk grain and the protein, riboflavin, phosphorus, and magnesium units per pound


On the packaging for each of its products, Whole Food indicates the nutritional content in each bowl of cereal when eaten with 0.5 cup of milk.

The USRDA (U.S. Recommended Dietary Allowance) and the more recent DRI (Dietary Reference Intake) were consulted to establish recommended amounts of certain vitamins and minerals for an average adult.

Based on these figures and the desired amounts for labeling on the package, Whole Food has determined that each 2-ounce serving of the cereal should contain 3 units of protein, 2 units of riboflavin, 1 unit of phosphorous, and 0.425 units of magnesium.


In modeling this as a linear program, the objective is to minimize cost.

There will be four constraints (One each for) stipulating that the number of units must be at least the minimum amount specified.

1. protein

2. riboflavin

3. phosphorous

4. magnesium

Since these requirements are for each 2 ounce serving, the last constraint must say that the total amount of the grains used will be 2 ounces, or 0.125 pounds.


In defining the variables, notice that the cost is expressed per pound of the three grains.

Thus, in order to calculate the total cost, we must know the number of pounds of the grains used in one serving of the cereal.

Also, the numbers in Data Table are expressed in units per pound of grain, so defining variables as the number of pounds of the grains makes it easier to calculate the amounts of protein, riboflavin, phosphorous, and magnesium. Let

XC = pounds of grain C in 0.125 pounds or 2-Oz serving of cereal

XB = pounds of grain B in 0.125 pounds or 2-Oz serving of cereal

XA = pounds of grain A in 0.125 pounds or 2-Oz serving of cereal

Ingredient Mix and Blending Problems

Diet and feed mix problems are actually special cases of a more general class of LP problems known as ingredient or blending problems.

Blending problems arise when a decision must be made regarding the blending of two or more resources to produce one or more products.

Resources, in this case, contain one or more essential ingredients that must be blended so that each final product contains specific percentages of each ingredient.

Ingredient Mix and Blending ProblemsLow Knock Oil Company

The Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution.

The grades, regular and economy, are produced by refining a blend of two types of crude oil, type X100 and type X220.

Each crude oil differs not only in cost per barrel, but in composition as well.

The following table indicates the percentage of crucial ingredients found in each of the crude oils and the cost per barrel for each:


Weekly demand for the regular grade of Low Knock gasoline is at least 25,000 barrels, and demand for the economy is at least 32,000 barrels per week. At least 45% of each barrel of regular must be ingredient A.At most 50% of each barrel of economy should contain ingredient B.While the gasoline yield from one barrel of crude depends on the type of crude and the type of processing used, we will assume for the sake of this example that one barrel of crude oil will yield one barrel of gasoline.

Low Knock Oil Company

Shipping Applications

• The transportation or shipping problem involves determining the amount of goods or items to be transported from a number of origins (or sources) to a number of destinations.

• The objective usually is to minimize total shipping costs or distances. Constraints in this type of problem deal with capacities at each origin and requirements at each destination.

Shipping ApplicationsTop Speed Bicycle Company

• The Top Speed Bicycle Co. manufactures and markets a line of 10-speed bicycles nationwide.

• The firm has final assembly plants in two cities in which labor costs are low, New Orleans and Omaha.

• Its three major warehouses are located near the large market areas of New York, Chicago, and Los Angeles.

Shipping ApplicationsTop Speed Bicycle Company

• The sales requirements for the next year at the New York warehouse are 10,000 bicycles, at the Chicago warehouse 8,000 bicycles, and at the Los Angeles warehouse 15,000 bicycles.

• The factory capacity at each location is limited. New Orleans can assemble and ship 20,000 bicycles; the Omaha plant can produce 15,000 bicycles per year.

• The cost of shipping one bicycle from each factory to each warehouse differs, and these unit shipping costs are

Top Speed Bicycle Company

a network representation of this problem


• The company wishes to develop a shipping schedule that will minimize its total annual transportation costs.

• Each circle or node represents a source or a destination in this problem.

• Each source and destination is numbered to facilitate the variable definitions.

• Each arrow indicates a possible shipping route, and the cost per unit to ship along each of these is shown also in Figure below


• When formulating a linear program for this, the objective is to minimize the transportation costs.

• There are two supply constraints (one for each source): – (1) No more than 20,000 can be shipped from New Orleans. – (2) No more than 15,000 can be shipped from Omaha.

• There are three demand constraints (one for each destination): – (1) The total number shipped to New York must equal 10,000. – (2) The total number shipped to Chicago must equal 8,000.– (3) The total number shipped to Los Angeles must equal 15,000.

• The company must decide how many bicycles to ship along each of the shipping routes (i.e., from each source to each destination).


• In Figure the arrows indicate where the items originate and where they are delivered, and the shipping cost per unit is shown next to the arrow. The decisions are the number of units to ship along each of these shipping routes.

• In defining the variables, it is easiest to use double subscripts. Let• Each source and destination is numbered to facilitate the

variable definitions. • Each arrow indicates a possible shipping route, and the cost

per unit to ship along each of these is shown also in Figure below

Application Types covered in this Chapter

1. Marketing Applications2. Manufacturing Applications3. Employee Scheduling Applications4. Financial Applications5. Transportation Applications6. Ingredient Blending Applications


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5 Basic Assumptions of Linear Programming

1.Certainty: numbers in the objective and constraints are known with

certainty and do not change during the period being studied

2.Proportionality: exists in the objective and constraints constancy between

production increases and resource utilization if production of 1 unit of a product uses 3 hours,

production of 10 units would use 30 hours.


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5 Basic Assumptions of Linear Programming (continued)

3. Additivity: the total of all activities equals the sum of the individual activities If the production of one product generated $3 profit and the

production of another product generated $8 profit, the total profit would be the sum of these two, which would be $11.

4. Divisibility: solutions need not be in whole numbers (integers) solutions are divisible, and may take any fractional value5. Non-negativity: all answers or variables are greater than or equal to (≥) zero negative values of physical quantities are impossible


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TIPS

Practice making constraints!Practice solving on excel!

Practice & Practice!

linear programming on computer_ch78

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