Chapter 8 LP Modeling Applications with Computer Analyses in Excel and QM for Windows

Chapter 8 LP Modeling Applications with Computer Analyses in Excel and QM for Windows Learning Objectives After completing this chapter, students will be able to: 1. Model a wide variety of medium to large LP problems 2. Understand major application areas, including marketing, production, labor scheduling, fuel blending, transportation, and finance 3. Gain experience in solving LP problems with QM for Windows and Excel Solver software 28 Prentice-Hall, Inc. 29 Prentice-Hall, Inc. 29 Prentice-Hall, Inc. 8 2 Chapter Outline 8.1 Introduction 8.2 Marketing Applications 8.3 8.4 8.5 Financial Applications 8.6 8.7 Transshipment Applications 8.8 Introduction The graphical method of LP is useful for understanding how to formulate and solve small LP problems There are many types of problems that can be solved using LP The principles developed here are applicable to larger problems 29 Prentice-Hall, Inc. 8 3 29 Prentice-Hall, Inc. 8 4 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 Maximize audience exposure Minimize advertising costs Marketing Applications The Win Big Gambling Club promotes gambling junkets to the Bahamas They have $8, per week to spend on advertising Their goal is to reach the largest possible highpotential audience Media types and audience figures are shown in the following table They need to place at least five radio spots per week No more than $1,8 can be spent on radio advertising each week 29 Prentice-Hall, Inc. 8 5 29 Prentice-Hall, Inc. 8 6 1

Marketing Applications Win Big Gambling Club advertising options MEDIUM TV spot (1 minute) Daily newspaper (fullpage ad) Radio spot (3 seconds, prime time) Radio spot (1 minute, afternoon) AUDIENCE REACHED PER AD 5, 8,5 2,4 2,8 COST PER AD ($) 8 925 29 38 MAXIMUM ADS PER WEEK 12 5 25 2 29 Prentice-Hall, Inc. 8 7 Win Big Gambling Club The problem formulation is X 1 = number of 1-minute TV spots each week X 2 = number of daily paper ads each week X 3 = number of 3-second radio spots each week X 4 = number of 1-minute radio spots each week Objective: Maximize audience coverage = 5,X 1 + 8,5X 2 + 2,4X 3 + 2,8X 4 Subject to X 1 12 (max TV spots/wk) X 2 5 (max newspaper ads/wk) X 3 25 (max 3-sec radio spots ads/wk) X 4 2 (max newspaper ads/wk) 8X 1 + 925X 2 + 29X 3 + 38X 4 $8, (weekly advertising budget) X 3 + X 4 5 (min radio spots contracted) 29X 3 + 38X 4 $1,8 (max dollars spent on radio) X 1, X 2, X 3, X 4 29 Prentice-Hall, Inc. 8 8 Win Big Gambling Club Win Big Gambling Club The problem solution Program 8.1B Program 8.1A 29 Prentice-Hall, Inc. 8 9 29 Prentice-Hall, Inc. 8 1 Marketing Research Linear programming has also been applied to marketing research problems and the area of consumer research Statistical pollsters can use LP to help make strategy decisions 29 Prentice-Hall, Inc. 8 11 Marketing Research Management Sciences Associates (MSA) is a marketing research firm MSA determines that it must fulfill several requirements in order to draw statistically valid conclusions Survey at least 2,3 U.S. households Survey at least 1, households whose heads are 3 years of age or younger Survey at least 6 households whose heads are between 31 and 5 years of age Ensure that at least 15% of those surveyed live in a state that borders on Mexico Ensure that no more than 2% of those surveyed who are 51 years of age or over live in a state that borders on Mexico 29 Prentice-Hall, Inc. 8 12 2

Marketing Research MSA decides that all surveys should be conducted in person It estimates the costs of reaching people in each age and region category are as follows REGION State bordering Mexico State not bordering Mexico COST PER PERSON SURVEYED ($) AGE 3 $7.5 $6.9 AGE 31-5 $6.8 $7.25 AGE 51 $5.5 $6.1 Marketing Research MSA s goal is to meet the sampling requirements at the least possible cost The decision variables are X 1 = number of 3 or younger and in a border state X 2 = number of 31-5 and in a border state X 3 = number 51 or older and in a border state X 4 = number 3 or younger and not in a border state X 5 = number of 31-5 and not in a border state X 6 = number 51 or older and not in a border state 29 Prentice-Hall, Inc. 8 13 29 Prentice-Hall, Inc. 8 14 Objective function subject to Marketing Research Minimize total interview costs = $7.5X 1 + $6.8X 2 + $5.5X 3 + $6.9X 4 + $7.25X 5 + $6.1X 6 X 1 + X 2 + X 3 + X 4 + X 5 + X 6 2,3 (total households) X 1 + X 4 1, (households 3 or younger) X 2 + X 5 6 (households 31-5) X 1 + X 2 + X 3.15(X 1 + X 2 + X 3 + X 4 + X 5 + X 6 ) (border states) X 3.2(X 3 + X 6 ) (limit on age group 51+ who can live in border state) X 1, X 2, X 3, X 4, X 5, X 6 Marketing Research Computer solution in QM for Windows Notice the variables in the constraints have all been moved to the left side of the equations Program 8.2 29 Prentice-Hall, Inc. 8 15 29 Prentice-Hall, Inc. 8 16 Marketing Research The following table summarizes the results of the MSA analysis It will cost MSA $15,166 to conduct this research REGION State bordering Mexico State not bordering Mexico AGE 3 1, AGE 31-5 6 AGE 51 14 56 Production Mix LP can be used to plan the optimal mix of products to manufacture 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 29 Prentice-Hall, Inc. 8 17 29 Prentice-Hall, Inc. 8 18 3

Fifth Avenue Industries produces four varieties of ties One is expensive all-silk One is all-polyester Two are polyester and cotton blends The table on the below shows the cost and availability of the three materials used in the production process MATERIAL Silk Polyester Cotton COST PER YARD ($) 21 6 9 MATERIAL AVAILABLE PER MONTH (YARDS) 8 3, 1,6 The firm has contracts with several major department store chains to supply ties Contracts require a minimum number of ties but may be increased if demand increases Fifth Avenue s goal is to maximize monthly profit given the following decision variables X 1 = number of all-silk ties produced per month X 2 = number polyester ties X 3 = number of blend 1 poly-cotton ties X 4 = number of blend 2 poly-cotton ties 29 Prentice-Hall, Inc. 8 19 29 Prentice-Hall, Inc. 8 2 Contract data for Fifth Avenue Industries VARIETY OF TIE All silk All polyester Poly-cotton blend 1 Poly-cotton blend 2 Table 8.1 SELLING PRICE PER TIE ($) 6.7 3.55 4.31 4.81 MONTHLY CONTRACT MINIMUM 6, 1, 16, 8,5 MONTHLY DEMAND 7, 14, 16, 8,5 MATERIAL REQUIRED PER TIE (YARDS).125.8.1.1 MATERIAL REQUIREMENTS 1% silk 1% polyester 5% polyester- 5% cotton 3% polyester- 7% cotton Fifth Avenue also has to calculate profit per tie for the objective function VARIETY OF TIE All silk All polyester Poly-cotton blend 1 Poly-cotton blend 2 SELLING PRICE PER TIE ($) $6.7 $3.55 $4.31 $4.81 MATERIAL REQUIRED PER TIE (YARDS).125.8.5.5.3.7 MATERIAL COST PER YARD ($) $21 $6 $6 $9 $6 $9 COST PER TIE ($) $2.62 $.48 $.3 $.45 $.18 $.63 PROFIT PER TIE ($) $4.8 $3.7 $3.56 $4. 29 Prentice-Hall, Inc. 8 21 29 Prentice-Hall, Inc. 8 22 The complete Fifth Avenue Industries model Objective function Maximize profit = $4.8X 1 + $3.7X 2 + $3.56X 3 + $4.X 4 Subject to.125x 1 8 (yds of silk).8x 2 +.5X 3 +.3X 4 3, (yds of polyester).5x 3 +.7X 4 1,6 (yds of cotton) X 1 6, (contract min for silk) X 1 7, (contract min) X 2 1, (contract min for all polyester) X 2 14, (contract max) X 3 13, (contract mini for blend 1) X 3 16, (contract max) X 4 6, (contract mini for blend 2) X 4 8,5 (contract max) X 1, X 2, X 3, X 4 29 Prentice-Hall, Inc. 8 23 Excel formulation for Fifth Avenue LP problem Program 8.3A 29 Prentice-Hall, Inc. 8 24 4

Solution for Fifth Avenue Industries LP model Production Scheduling Setting a low-cost production schedule over a period of weeks or months is a difficult and important management task Important factors include labor capacity, inventory and storage costs, space limitations, product demand, and labor relations When more than one product is produced, the scheduling process can be quite complex The problem resembles the product mix model for each time period in the future Program 8.3B 29 Prentice-Hall, Inc. 8 25 29 Prentice-Hall, Inc. 8 26 Greenberg Motors, Inc. manufactures two different electric motors for sale under contract to Drexel Corp. Drexel places orders three times a year for four months at a time Demand varies month to month as shown below Greenberg wants to develop its production plan for the next four months MODEL GM3A GM3B JANUARY 8 1, FEBRUARY 7 1,2 MARCH 1, 1,4 APRIL 1,1 1,4 Production planning at Greenberg must consider four factors Desirability of producing the same number of motors each month to simplify planning and scheduling Necessity to inventory carrying costs down Warehouse limitations The no-lay-off policy LP is a useful tool for creating a minimum total cost schedule the resolves conflicts between these factors Table 8.2 29 Prentice-Hall, Inc. 8 27 29 Prentice-Hall, Inc. 8 28 Double subscripted variables are used in this problem to denote motor type and month of production X A,i = Number of model GM3A motors produced in month i (i = 1, 2, 3, 4 for January April) X B,i = Number of model GM3B motors produced in month i It costs $1 to produce a GM3A motor and $6 to produce a GM3B Both costs increase by 1% on March 1, thus Cost of production = $1X A1 + $1X A2 + $11X A3 + 11X A4 + $6X B1 + $6X B2 + $6.6X B3 + $6.6X B4 We can use the same approach to create the portion of the objective function dealing with inventory carrying costs I A,i = Level of on-hand inventory for GM3A motors at the end of month i (i = 1, 2, 3, 4 for January April) I B,i = Level of on-hand inventory for GM3B motors at the end of month i The carrying cost for GM3A motors is $.18 per month and the GM3B costs $.13 per month Monthly ending inventory levels are used for the average inventory level Cost of carrying inventory = $.18X A1 + $.18X A2 + $.18X A3 +.18X A4 + $.13X B1 + $.13X B2 + $.13X B3 + $.13 B4 29 Prentice-Hall, Inc. 8 29 29 Prentice-Hall, Inc. 8 3 5

We combine these two for the objective function Minimize total cost = $1X A1 + $1X A2 + $11X A3 + 11X A4 + $6X B1 + $6X B2 + $6.6X B3 + $6.6X B4 + $.18X A1 + $.18X A2 + $.18X A3 +.18X A4 + $.13X B1 + $.13X B2 + $.13X B3 + $.13X B4 End of month inventory is calculated using this relationship Inventory at the end of this month Inventory at the end of last month Current month s production = + Sales to Drexel this month Greenberg is starting a new four-month production cycle with a change in design specification that left no old motors in stock on January 1 Given January demand for both motors I A1 = + X A1 8 I B1 = + X B1 1, Rewritten as January s constraints X A1 I A1 = 8 X B1 I B1 = 1, 29 Prentice-Hall, Inc. 8 31 29 Prentice-Hall, Inc. 8 32 Constraints for February, March, and April X A2 + I A1 I A2 = 7 February GM3A demand X B2 + I B1 I B2 = 1,2 February GM3B demand X A3 + I A2 I A3 = 1, March GM3A demand X B3 + I B2 I B3 = 1,4 March GM3B demand X A4 + I A3 I A4 = 1,1 April GM3A demand X B4 + I B3 I B4 = 1,4 April GM3B demand And constraints for April s ending inventory I A4 = 45 I B4 = 3 29 Prentice-Hall, Inc. 8 33 We also need constraints for warehouse space I A1 + I B1 3,3 I A2 + I B2 3,3 I A3 + I B3 3,3 I A4 + I B4 3,3 No worker is ever laid off so Greenberg has a base employment level of 2,24 labor hours per month By adding temporary workers, available labor hours can be increased to 2,56 hours per month Each GM3A motor requires 1.3 labor hours and each GM3B requires.9 hours 29 Prentice-Hall, Inc. 8 34 Labor hour constraints 1.3X A1 +.9X B1 2,24 (January min hrs/month) 1.3X A1 +.9X B1 2,56 (January max hrs/month) 1.3X A2 +.9X B2 2,24 (February labor min) 1.3X A2 +.9X B2 2,56 (February labor max) 1.3X A3 +.9X B3 2,24 (March labor min) 1.3X A3 +.9X B3 2,56 (March labor max) 1.3X A4 +.9X B4 2,24 (April labor min) 1.3X A4 +.9X B4 2,56 (April labor max) All variables Nonnegativity constraints Greenberg Motors solution PRODUCTION SCHEDULE Units GM3A produced Units GM3B produced Inventory GM3A carried Inventory GM3B carried Labor hours required JANUARY 1,277 1, 477 2,56 FEBRUARY 1,138 1,2 915 2,56 MARCH 842 1,4 758 2,355 APRIL 792 1,7 45 3 2,56 Total cost for this four month period is $76,31.61 Complete model has 16 variables and 22 constraints 29 Prentice-Hall, Inc. 8 35 29 Prentice-Hall, Inc. 8 36 6

Assignment Problems Involve determining the most efficient way to assign resources to tasks Objective may be to minimize travel times or maximize assignment effectiveness Assignment problems are unique because they have a coefficient of or 1 associated with each variable in the LP constraints and the right-hand side of each constraint is always equal to 1 Ivan and Ivan law firm maintains a large staff of young attorneys Ivan wants to make lawyer-to-client assignments in the most effective manner He identifies four lawyers who could possibly be assigned new cases Each lawyer can handle one new client The lawyers have different skills and special interests The following table summarizes the lawyers estimated effectiveness on new cases 29 Prentice-Hall, Inc. 8 37 29 Prentice-Hall, Inc. 8 38 Effectiveness ratings LAWYER Adams Brooks Carter Darwin DIVORCE 6 9 4 6 Let X ij = where CORPORATE MERGER 2 3 8 7 CLIENT S CASE EMBEZZLEMENT 1 if attorney i is assigned to case j otherwise 8 5 3 6 EXHIBITIONISM i = 1, 2, 3, 4 stands for Adams, Brooks, Carter, and Darwin respectively j = 1, 2, 3, 4 stands for divorce, merger, embezzlement, and exhibitionism 5 8 4 4 29 Prentice-Hall, Inc. 8 39 The LP formulation is Maximize effectiveness = 6X 11 + 2X 12 + 8X 13 + 5X 14 + 9X 21 + 3X 22 + 5X 23 + 8X 24 + 4X 31 + 8X 32 + 3X 33 + 4X 34 + 6X 41 + 7X 42 + 6X 43 + 4X 44 subject to X 11 + X 21 + X 31 + X 41 = 1 (divorce case) X 12 + X 22 + X 32 + X 42 = 1 (merger) X 13 + X 23 + X 33 + X 43 = 1 (embezzlement) X 14 + X 24 + X 34 + X 44 = 1 (exhibitionism) X 11 + X 12 + X 13 + X 14 = 1 (Adams) X 21 + X 22 + X 23 + X 24 = 1 (Brook) X 31 + X 32 + X 33 + X 34 = 1 (Carter) X 41 + X 42 + X 43 + X 44 = 1 (Darwin) 29 Prentice-Hall, Inc. 8 4 Solving Ivan and Ivan s assignment scheduling LP problem using QM for Windows Labor Planning Addresses staffing needs over a particular time Especially useful when there is some flexibility in assigning workers that require overlapping or interchangeable talents Program 8.4 29 Prentice-Hall, Inc. 8 41 29 Prentice-Hall, Inc. 8 42 7

Hong Kong Bank of Commerce and Industry has requirements for between 1 and 18 tellers depending on the time of day Lunch time from noon to 2 pm is generally the busiest The bank employs 12 full-time tellers but has many part-time workers available Part-time workers must put in exactly four hours per day, can start anytime between 9 am and 1 pm, and are inexpensive Full-time workers work from 9 am to 3 pm and have 1 hour for lunch Labor requirements for Hong Kong Bank of Commerce and Industry TIME PERIOD 9 am 1 am 1 am 11 am 11 am Noon Noon 1 pm 1 pm 2 pm 2 pm 3 pm 3 pm 4 pm 4 pm 5 pm NUMBER OF TELLERS REQUIRED 1 12 14 16 18 17 15 1 29 Prentice-Hall, Inc. 8 43 29 Prentice-Hall, Inc. 8 44 Part-time hours are limited to a maximum of 5% of the day s total requirements Part-timers earn $8 per hour on average Full-timers earn $1 per day on average The bank wants a schedule that will minimize total personnel costs It will release one or more of its part-time tellers if it is profitable to do so We let F = full-time tellers P 1 = part-timers starting at 9 am (leaving at 1 pm) P 2 = part-timers starting at 1 am (leaving at 2 pm) P 3 = part-timers starting at 11 am (leaving at 3 pm) P 4 = part-timers starting at noon (leaving at 4 pm) P 5 = part-timers starting at 1 pm (leaving at 5 pm) 29 Prentice-Hall, Inc. 8 45 29 Prentice-Hall, Inc. 8 46 Objective function Minimize total daily personnel cost = $1F + $32(P 1 + P 2 + P 3 + P 4 + P 5 ) subject to F + P 1 1 (9 am 1 am needs) F + P 1 + P 2 12 (1 am 11 am needs).5f + P 1 + P 2 + P 3 14 (11 am noon needs).5f + P 1 + P 2 + P 3 + P 4 16 (noon 1 pm needs) F + P 2 + P 3 + P 4 + P 5 18 (1 pm 2 pm needs) F + P 3 + P 4 + P 5 17 (2 pm 3 pm needs) F + P 4 + P 5 15 (3 pm 4 pm needs) F + P 5 1 (4 pm 5 pm needs) F 12 (12 full-time tellers) 4P 1 + 4P 2 + 4P 3 + 4P 4 + 4P 5.5(112) (max 5% part-timers) P 1, P 2, P 3, P 4, P 5 29 Prentice-Hall, Inc. 8 47 There are several alternate optimal schedules Hong Kong Bank can follow F = 1, P 2 = 2, P 3 = 7, P 4 = 5, P 1, P 5 = F = 1, P 1 = 6, P 2 = 1, P 3 = 2, P 4 = 5, P 5 = The cost of either of these two policies is $1,448 per day 29 Prentice-Hall, Inc. 8 48 8

Financial Applications Financial Applications Portfolio Selection Bank, investment funds, and insurance companies often have to select specific investments from a variety of alternatives The manager s overall objective is generally to maximize the potential return on the investment given a set of legal, policy, or risk restraints International City Trust (ICT) invests in short-term trade credits, corporate bonds, gold stocks, and construction loans The board of directors has placed limits on how much can be invested in each area INVESTMENT Trade credit Corporate bonds Gold stocks Construction loans INTEREST EARNED (%) 7 11 19 15 MAXIMUM INVESTMENT ($ MILLIONS) 1. 2.5 1.5 1.8 29 Prentice-Hall, Inc. 8 49 29 Prentice-Hall, Inc. 8 5 Financial Applications Financial Applications ICT has $5 million to invest and wants to accomplish two things Maximize the return on investment over the next six months Satisfy the diversification requirements set by the board The board has also decided that at least 55% of the funds must be invested in gold stocks and construction loans and no less than 15% be invested in trade credit The variables in the model are X 1 = dollars invested in trade credit X 2 = dollars invested in corporate bonds X 3 = dollars invested in gold stocks X 4 = dollars invested in construction loans 29 Prentice-Hall, Inc. 8 51 29 Prentice-Hall, Inc. 8 52 Financial Applications Objective function Maximize dollars of interest earned =.7X 1 +.11X 2 +.19X 3 +.15X 4 subject to X 1 1,, X 2 2,5, X 3 1,5, X 4 1,8, X 3 + X 4.55(X 1 + X 2 + X 3 + X 4 ) X 1.15(X 1 + X 2 + X 3 + X 4 ) X 1 + X 2 + X 3 + X 4 5,, X 1, X 2, X 3, X 4 Financial Applications The optimal solution to the ICT is to make the following investments X 1 = $75, X 2 = $95, X 3 = $1,5, X 4 = $1,8, The total interest earned with this plan is $712, 29 Prentice-Hall, Inc. 8 53 29 Prentice-Hall, Inc. 8 54 9

Shipping Problem The transportation or shipping problem involves determining the amount of goods or items to be transported from a number of origins to a number of destinations The objective usually is to minimize total shipping costs or distances This is a specific case of LP and a special algorithm has been developed to solve it The Top Speed Bicycle Co. manufactures and markets a line of 1-speed bicycles The firm has final assembly plants in two cities where labor costs are low It has three major warehouses near large markets The sales requirements for the next year are New York 1, bicycles Chicago 8, bicycles Los Angeles 15, bicycles The factory capacities are New Orleans 2, bicycles Omaha 15, bicycles 29 Prentice-Hall, Inc. 8 55 29 Prentice-Hall, Inc. 8 56 The cost of shipping bicycles from the plants to the warehouses is different for each plant and warehouse FROM New Orleans Omaha TO NEW YORK $2 $3 CHICAGO $3 $1 LOS ANGELES The company wants to develop a shipping schedule that will minimize its total annual cost $5 $4 The double subscript variables will represent the origin factory and the destination warehouse X ij = bicycles shipped from factory i to warehouse j So X 11 = number of bicycles shipped from New Orleans to New York X 12 = number of bicycles shipped from New Orleans to Chicago X 13 = number of bicycles shipped from New Orleans to Los Angeles X 21 = number of bicycles shipped from Omaha to New York X 22 = number of bicycles shipped from Omaha to Chicago X 23 = number of bicycles shipped from Omaha to Los Angeles 29 Prentice-Hall, Inc. 8 57 29 Prentice-Hall, Inc. 8 58 Objective function Formulation for Excel s Solver Minimize total shipping costs = 2X 11 + 3X 12 + 5X 13 + 3X 21 + 1X 22 + 4X 23 subject to X 11 + X 21 = 1, (New York demand) X 12 + X 22 = 8, (Chicago demand) X 13 + X 23 = 15, (Los Angeles demand) X 11 + X 12 + X 13 2, (New Orleans factory supply) X 21 + X 22 + X 23 15, (Omaha factory supply) All variables Program 8.5A 29 Prentice-Hall, Inc. 8 59 29 Prentice-Hall, Inc. 8 6 1

Solution from Excel s Solver Top Speed Bicycle solution TO FROM New Orleans NEW YORK 1, CHICAGO LOS ANGELES 8, Omaha 8, 7, Program 8.5A 29 Prentice-Hall, Inc. 8 61 Total shipping cost equals $96, Transportation problems are a special case of LP as the coefficients for every variable in the constraint equations equal 1 This situation exists in assignment problems as well as they are a special case of the transportation problem 29 Prentice-Hall, Inc. 8 62 Truck Loading Problem The truck loading problem involves deciding which items to load on a truck so as to maximize the value of a load shipped Goodman Shipping has to ship the following six items The objective is to maximize the value of items loaded into the truck The truck has a capacity of 1, pounds The decision variable is X i = proportion of each item i loaded on the truck ITEM 1 2 3 4 5 6 VALUE ($) 22,5 24, 8, 9,5 11,5 9,75 WEIGHT (POUNDS) 7,5 7,5 3, 3,5 4, 3,5 29 Prentice-Hall, Inc. 8 63 29 Prentice-Hall, Inc. 8 64 Objective function Excel Solver formulation for Goodman Shipping subject to Maximize load value = $22,5X 1 + $24,X 2 + $8,X 3 + $9,5X 4 + $11,5X 5 + $9,75X 6 7,5X 1 + 7,5X 2 + 3,X 3 + 3,5X 4 + 4,X 5 + 3,5X 6 1, lb capacity X 1 1 X 2 1 X 3 1 X 4 1 X 5 1 X 6 1 X 1, X 2, X 3, X 4, X 5, X 6 Program 8.6A 29 Prentice-Hall, Inc. 8 65 29 Prentice-Hall, Inc. 8 66 11

Solver solution for Goodman Shipping Program 8.6B The Goodman Shipping problem has an interesting issue The solution calls for one third of Item 1 to be loaded on the truck What if Item 1 can not be divided into smaller pieces? Rounding down leaves unused capacity on the truck and results in a value of $24, Rounding up is not possible since this would exceed the capacity of the truck Using integer programming, the solution is to load one unit of Items 3, 4, and 6 for a value of $27,25 29 Prentice-Hall, Inc. 8 67 29 Prentice-Hall, Inc. 8 68 Transshipment Applications The transportation problem is a special case of the transshipment problem When the items are being moved from a source to a destination through an intermediate point (a transshipment point), the problem is called a transshipment problem Transshipment Applications Distribution Centers Frosty Machines manufactures snowblowers in Toronto and Detroit These are shipped to regional distribution centers in Chicago and Buffalo From there they are shipped to supply houses in New York, Philadelphia, and St Louis Shipping costs vary by location and destination Snowblowers can not be shipped directly from the factories to the supply houses 29 Prentice-Hall, Inc. 8 69 29 Prentice-Hall, Inc. 8 7 Transshipment Applications Frosty Machines network Transshipment Applications Frosty Machines data Source Toronto Transshipment Point Chicago Destination New York City FROM Toronto Detroit CHICAGO $4 $5 BUFFALO $7 $7 TO NEW YORK CITY PHILADELPHIA ST LOUIS SUPPLY 8 7 Detroit Buffalo Philadelphia Chicago Buffalo Demand $6 $2 45 $4 $3 35 $5 $4 3 Figure 8.1 St Louis 29 Prentice-Hall, Inc. 8 71 Frosty would like to minimize the transportation costs associated with shipping snowblowers to meet the demands at the supply centers given the supplies available 29 Prentice-Hall, Inc. 8 72 12

Transshipment Applications A description of the problem would be to minimize cost subject to 1. The number of units shipped from Toronto is not more than 8 2. The number of units shipped from Detroit is not more than 7 3. The number of units shipped to New York is 45 4. The number of units shipped to Philadelphia is 35 5. The number of units shipped to St Louis is 3 6. The number of units shipped out of Chicago is equal to the number of units shipped into Chicago 7. The number of units shipped out of Buffalo is equal to the number of units shipped into Buffalo 29 Prentice-Hall, Inc. 8 73 Transshipment Applications The decision variables should represent the number of units shipped from each source to the transshipment points and from there to the final destinations T 1 = the number of units shipped from Toronto to Chicago T 2 = the number of units shipped from Toronto to Buffalo D 1 = the number of units shipped from Detroit to Chicago D 2 = the number of units shipped from Detroit to Chicago C 1 = the number of units shipped from Chicago to New York C 2 = the number of units shipped from Chicago to Philadelphia C 3 = the number of units shipped from Chicago to St Louis B 1 = the number of units shipped from Buffalo to New York B 2 = the number of units shipped from Buffalo to Philadelphia B 3 = the number of units shipped from Buffalo to St Louis 29 Prentice-Hall, Inc. 8 74 Transshipment Applications The linear program is Minimize cost = 4T 1 + 7T 2 + 5D 1 + 7D 2 + 6C 1 + 4C 2 + 5C 3 + 2B 1 + 3B 2 + 4B 3 Transshipment Applications The solution from QM for Windows is subject to T 1 + T 2 8 D 1 + D 2 7 C 1 + B 1 = 45 C 2 + B 2 = 35 C 3 + B 3 = 3 T 1 + D 1 = C 1 + C 2 + C 3 T 2 + D 2 = B 1 + B 2 + B 3 T 1, T 2, D 1, D 2, C 1, C 2, C 3, B 1, B 2, B 3 (supply at Toronto) (supply at Detroit) (demand at New York) (demand at Philadelphia) (demand at St Louis) (shipping through Chicago) (shipping through Buffalo) (nonnegativity) Program 8.7 29 Prentice-Hall, Inc. 8 75 29 Prentice-Hall, Inc. 8 76 Diet Problems One of the earliest LP applications Used to determine the most economical diet for hospital patients Also known as the feed mix problem The Whole Food Nutrition Center uses three bulk grains to blend a natural cereal They advertise the cereal meets the U.S. Recommended Daily Allowance (USRDA) for four key nutrients They want to select the blend that will meet the requirements at the minimum cost NUTRIENT Protein Riboflavin Phosphorus Magnesium USRDA 3 units 2 units 1 unit.425 units 29 Prentice-Hall, Inc. 8 77 29 Prentice-Hall, Inc. 8 78 13

We let X A = pounds of grain A in one 2-ounce serving of cereal X B = pounds of grain B in one 2-ounce serving of cereal X C = pounds of grain C in one 2-ounce serving of cereal Whole Foods Natural Cereal requirements GRAIN A B C Table 8.5 COST PER POUND (CENTS) 33 47 38 PROTEIN (UNITS/LB) 22 28 21 RIBOFLAVIN (UNITS/LB) 16 14 25 PHOSPHOROUS (UNITS/LB) 8 7 9 MAGNESIUM (UNITS/LB) 5 6 The objective function is Minimize total cost of mixing a 2-ounce serving = $.33X A + $.47X B + $.38X C subject to 22X A + 28X B +21X C 3 (protein units) 16X A + 14X B +25X C 2 (riboflavin units) 8X A + 7X B + 9X C 1 (phosphorous units) 5X A + X B + 6X C.425 (magnesium units) X A + X B + X C =.125 (total mix) X A, X B, X C 29 Prentice-Hall, Inc. 8 79 29 Prentice-Hall, Inc. 8 8 Whole Food solution using QM for Windows Program 8.8 Ingredient Mix and Blending Problems Diet and feed mix problems are special cases of a more general class of problems known as ingredient or blending problems Blending problems arise when decisions must be made regarding the blending of two or more resources to produce one or more product Resources may contain essential ingredients that must be blended so that a specified percentage is in the final mix 29 Prentice-Hall, Inc. 8 81 29 Prentice-Hall, Inc. 8 82 The Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution The two grades, regular and economy, are created by blending two different types of crude oil The crude oil differs in cost and in its content of crucial ingredients The firm lets X 1 = barrels of crude X1 blended to produce the refined regular X 2 = barrels of crude X1 blended to produce the refined economy X 3 = barrels of crude X22 blended to produce the refined regular X 4 = barrels of crude X22 blended to produce the refined economy CRUDE OIL TYPE INGREDIENT A (%) INGREDIENT B (%) COST/BARREL ($) The objective function is X1 X22 35 6 55 25 3. 34.8 Minimize cost = $3X 1 + $3X 2 + $34.8X 3 + $34.8X 4 29 Prentice-Hall, Inc. 8 83 29 Prentice-Hall, Inc. 8 84 14

Problem formulation At least 45% of each barrel of regular must be ingredient A (X 1 + X 3 ) = total amount of crude blended to produce the refined regular gasoline demand Thus,.45(X 1 + X 3 ) = amount of ingredient A required But.35X 1 +.6X 3 = amount of ingredient A in refined regular gas Problem formulation Minimize cost = 3X 1 + 3X 2 + 34.8X 3 + 34.8X 4 subject to X 1 + X 3 25, X 2 + X 4 32,.1X 1 +.15X 3.5X 2.25X 4 X 1, X 2, X 3, X 4 So.35X 1 +.6X 3.45X 1 +.45X 3 or.1x 1 +.15X 3 (ingredient A in regular constraint) 29 Prentice-Hall, Inc. 8 85 29 Prentice-Hall, Inc. 8 86 Solution from QM for Windows Program 8.9 29 Prentice-Hall, Inc. 8 87 15