Lp applications
Upcoming SlideShare
Loading in...5
×
 

Lp applications

on

  • 412 views

 

Statistics

Views

Total Views
412
Views on SlideShare
412
Embed Views
0

Actions

Likes
1
Downloads
3
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Lp applications Lp applications Presentation Transcript

  • © 2008 Prentice-Hall, Inc.LP Modeling Applicationswith Computer Analyses inExcel and QM for Windows
  • © 2009 Prentice-Hall, Inc. 8 – 2Learning Objectives1. Model a wide variety of medium to large LPproblems2. Understand major application areas,including marketing, production, laborscheduling, fuel blending, transportation, andfinance3. Gain experience in solving LP problems withQM for Windows and Excel Solver softwareAfter completing this chapter, students will be able to:After completing this chapter, students will be able to:
  • © 2009 Prentice-Hall, Inc. 8 – 3Chapter Outline3.13.1 Introduction3.23.2 Marketing Applications3.33.3 Manufacturing Applications3.43.4 Employee Scheduling Applications3.53.5 Financial Applications3.63.6 Transportation Applications3.73.7 Transshipment Applications3.83.8 Ingredient Blending Applications
  • © 2009 Prentice-Hall, Inc. 8 – 4Introduction The graphical method of LP is usefulfor understanding how to formulateand solve small LP problems There are many types of problems thatcan be solved using LP The principles developed here areapplicable to larger problems
  • © 2009 Prentice-Hall, Inc. 8 – 5Marketing Applications Linear programming models have beenused in the advertising field as a decisionaid in selecting an effective media mix Media selection problems can beapproached with LP from two perspectives Maximize audience exposure Minimize advertising costs
  • © 2009 Prentice-Hall, Inc. 8 – 6Marketing Applications The Win Big Gambling Club promotes gamblingjunkets to the Bahamas They have $8,000 per week to spend onadvertising Their goal is to reach the largest possible high-potential audience Media types and audience figures are shown inthe following table They need to place at least five radio spots perweek No more than $1,800 can be spent on radioadvertising each week
  • © 2009 Prentice-Hall, Inc. 8 – 7Marketing ApplicationsMEDIUMAUDIENCEREACHED PER ADCOST PERAD ($)MAXIMUM ADSPER WEEKTV spot (1 minute) 5,000 800 12Daily newspaper (full-page ad)8,500 925 5Radio spot (30seconds, prime time)2,400 290 25Radio spot (1 minute,afternoon)2,800 380 20 Win Big Gambling Club advertising options
  • © 2009 Prentice-Hall, Inc. 8 – 8Win Big Gambling Club The problem formulation isX1 = number of 1-minute TV spots each weekX2 = number of daily paper ads each weekX3 = number of 30-second radio spots each weekX4 = number of 1-minute radio spots each weekObjective:Maximize audience coverage= 5,000X1 + 8,500X2 + 2,400X3 + 2,800X4Subject to X1 ≤ 12 (max TV spots/wk)X2 ≤ 5 (max newspaper ads/wk)X3 ≤ 25 (max 30-sec radio spots ads/wk)X4 ≤ 20 (max newspaper ads/wk)800X1 + 925X2 + 290X3 + 380X4 ≤ $8,000 (weekly advertising budget)X3 + X4 ≥ 5 (min radio spots contracted)290X3 + 380X4 ≤ $1,800 (max dollars spent on radio)X1, X2, X3, X4 ≥ 0
  • © 2009 Prentice-Hall, Inc. 8 – 9Win Big Gambling ClubProgram 8.1A
  • © 2009 Prentice-Hall, Inc. 8 – 10Win Big Gambling ClubProgram 8.1B The problem solution
  • © 2009 Prentice-Hall, Inc. 8 – 11Marketing Research Linear programming has also been appliedto marketing research problems and thearea of consumer research Statistical pollsters can use LP to helpmake strategy decisions
  • © 2009 Prentice-Hall, Inc. 8 – 12Marketing Research Management Sciences Associates (MSA) is amarketing research firm MSA determines that it must fulfill severalrequirements in order to draw statistically validconclusions Survey at least 2,300 U.S. households Survey at least 1,000 households whose heads are 30years of age or younger Survey at least 600 households whose heads arebetween 31 and 50 years of age Ensure that at least 15% of those surveyed live in astate that borders on Mexico Ensure that no more than 20% of those surveyed whoare 51 years of age or over live in a state that borderson Mexico
  • © 2009 Prentice-Hall, Inc. 8 – 13Marketing Research MSA decides that all surveys should beconducted in person It estimates the costs of reaching people in eachage and region category are as followsCOST PER PERSON SURVEYED ($)REGION AGE ≤ 30 AGE 31-50 AGE ≥ 51State bordering Mexico $7.50 $6.80 $5.50State not bordering Mexico $6.90 $7.25 $6.10
  • © 2009 Prentice-Hall, Inc. 8 – 14Marketing ResearchX1 = number of 30 or younger and in a border stateX2 = number of 31-50 and in a border stateX3 = number 51 or older and in a border stateX4 = number 30 or younger and not in a border stateX5 = number of 31-50 and not in a border stateX6 = number 51 or older and not in a border state MSA’s goal is to meet the sampling requirementsat the least possible cost The decision variables are
  • © 2009 Prentice-Hall, Inc. 8 – 15Marketing ResearchObjective functionsubject toX1 + X2 + X3 + X4 + X5 + X6 ≥ 2,300 (total households)X1 + X4 ≥ 1,000 (households 30 or younger)X2 + X5 ≥ 600 (households 31-50)X1 + X2 + X3 ≥ 0.15(X1 + X2+ X3 + X4 + X5 + X6) (border states)X3 ≤ 0.20(X3 + X6) (limit on age group 51+ who can live inborder state)X1, X2, X3, X4, X5, X6 ≥ 0Minimize totalinterviewcosts= $7.50X1 + $6.80X2 + $5.50X3+ $6.90X4 + $7.25X5 + $6.10X6
  • © 2009 Prentice-Hall, Inc. 8 – 16Marketing Research Computer solution in QM for Windows Notice the variables in the constraints have allbeen moved to the left side of the equationsProgram 8.2
  • © 2009 Prentice-Hall, Inc. 8 – 17Marketing Research The following table summarizes the results of theMSA analysis It will cost MSA $15,166 to conduct this researchREGION AGE ≤ 30 AGE 31-50 AGE ≥ 51State bordering Mexico 0 600 140State not bordering Mexico 1,000 0 560
  • © 2009 Prentice-Hall, Inc. 8 – 18Manufacturing Applications Production Mix LP can be used to plan the optimal mix ofproducts to manufacture Company must meet a myriad of constraints,ranging from financial concerns to salesdemand to material contracts to union labordemands Its primary goal is to generate the largest profitpossible
  • © 2009 Prentice-Hall, Inc. 8 – 19Manufacturing Applications Fifth Avenue Industries produces four varieties ofties One is expensive all-silk One is all-polyester Two are polyester and cotton blends The table on the below shows the cost andavailability of the three materials used in theproduction processMATERIAL COST PER YARD ($)MATERIAL AVAILABLE PERMONTH (YARDS)Silk 21 800Polyester 6 3,000Cotton 9 1,600
  • © 2009 Prentice-Hall, Inc. 8 – 20Manufacturing Applications The firm has contracts with several majordepartment store chains to supply ties Contracts require a minimum number of ties butmay be increased if demand increases Fifth Avenue’s goal is to maximize monthly profitgiven the following decision variablesX1 = number of all-silk ties produced per monthX2 = number polyester tiesX3 = number of blend 1 poly-cotton tiesX4 = number of blend 2 poly-cotton ties
  • © 2009 Prentice-Hall, Inc. 8 – 21Manufacturing Applications Contract data for Fifth Avenue IndustriesVARIETY OFTIESELLINGPRICE PERTIE ($)MONTHLYCONTRACTMINIMUMMONTHLYDEMANDMATERIALREQUIREDPER TIE(YARDS)MATERIALREQUIREMENTSAll silk 6.70 6,000 7,000 0.125 100% silkAll polyester 3.55 10,000 14,000 0.08 100% polyesterPoly-cottonblend 14.31 13,000 16,000 0.1050% polyester-50% cottonPoly-cottonblend 24.81 6,000 8,500 0.1030% polyester-70% cottonTable 8.1
  • © 2009 Prentice-Hall, Inc. 8 – 22Manufacturing Applications Fifth Avenue also has to calculate profit per tiefor the objective functionVARIETY OF TIESELLINGPRICEPER TIE ($)MATERIALREQUIRED PERTIE (YARDS)MATERIALCOST PERYARD ($)COST PERTIE ($)PROFITPER TIE ($)All silk $6.70 0.125 $21 $2.62 $4.08All polyester $3.55 0.08 $6 $0.48 $3.07Poly-cottonblend 1$4.31 0.05 $6 $0.300.05 $9 $0.45 $3.56Poly-cottonblend 2$4.81 0.03 $6 $0.180.07 $9 $0.63 $4.00
  • © 2009 Prentice-Hall, Inc. 8 – 23Manufacturing Applications The complete Fifth Avenue Industries modelObjective functionMaximize profit = $4.08X1 + $3.07X2 + $3.56X3 + $4.00X4Subject to 0.125X1 ≤ 800 (yds of silk)0.08X2 + 0.05X3 + 0.03X4 ≤ 3,000 (yds of polyester)0.05X3 + 0.07X4 ≤ 1,600 (yds of cotton)X1 ≥ 6,000 (contract min for silk)X1 ≤ 7,000 (contract max)X2 ≥ 10,000 (contract min for all polyester)X2 ≤ 14,000 (contract max)X3 ≥ 13,000 (contract mini for blend 1)X3 ≤ 16,000 (contract max)X4 ≥ 6,000 (contract mini for blend 2)X4 ≤ 8,500 (contract max)X , X , X , X ≥ 0
  • © 2009 Prentice-Hall, Inc. 8 – 24Manufacturing Applications Excel formulation for Fifth Avenue LP problemProgram 8.3A
  • © 2009 Prentice-Hall, Inc. 8 – 25Manufacturing Applications Solution for Fifth Avenue Industries LP modelProgram 8.3B
  • © 2009 Prentice-Hall, Inc. 8 – 26Manufacturing Applications Production Scheduling Setting a low-cost production schedule over aperiod of weeks or months is a difficult andimportant 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, thescheduling process can be quite complex The problem resembles the product mix modelfor each time period in the future
  • © 2009 Prentice-Hall, Inc. 8 – 27Manufacturing Applications Greenberg Motors, Inc. manufactures twodifferent electric motors for sale under contractto Drexel Corp. Drexel places orders three times a year for fourmonths at a time Demand varies month to month as shown below Greenberg wants to develop its production planfor the next four monthsMODEL JANUARY FEBRUARY MARCH APRILGM3A 800 700 1,000 1,100GM3B 1,000 1,200 1,400 1,400Table 8.2
  • © 2009 Prentice-Hall, Inc. 8 – 28Manufacturing Applications Production planning at Greenberg must considerfour factors Desirability of producing the same number of motorseach 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 totalcost schedule the resolves conflicts betweenthese factors
  • © 2009 Prentice-Hall, Inc. 8 – 29Manufacturing Applications Double subscripted variables are used in thisproblem to denote motor type and month ofproductionXA,i = Number of model GM3A motors produced in month i(i = 1, 2, 3, 4 for January – April)XB,i = Number of model GM3B motors produced in month i It costs $10 to produce a GM3A motor and $6 toproduce a GM3B Both costs increase by 10% on March 1, thusduction = $10XA1 + $10XA2 + $11XA3 + 11XA4+ $6XB1 + $6XB2 + $6.60XB3 + $6.60XB4
  • © 2009 Prentice-Hall, Inc. 8 – 30Manufacturing Applications We can use the same approach to create theportion of the objective function dealing withinventory carrying costsIA,i = Level of on-hand inventory for GM3A motors at theend of month i (i = 1, 2, 3, 4 for January – April)IB,i = Level of on-hand inventory for GM3B motors at theend of month i The carrying cost for GM3A motors is $0.18 permonth and the GM3B costs $0.13 per month Monthly ending inventory levels are used for theaverage inventory levely = $0.18XA1 + $0.18XA2 + $0.18XA3 + 0.18XA4+ $0.13XB1 + $0.13XB2 + $0.13XB3 + $0.13B4
  • © 2009 Prentice-Hall, Inc. 8 – 31Manufacturing Applications We combine these two for the objective functioncost = $10XA1 + $10XA2 + $11XA3 + 11XA4+ $6XB1 + $6XB2 + $6.60XB3 + $6.60XB4+ $0.18XA1 + $0.18XA2 + $0.18XA3 + 0.18XA4+ $0.13XB1 + $0.13XB2 + $0.13XB3 + $0.13XB4 End of month inventory is calculated using thisrelationshipInventoryat the endof thismonthInventoryat the endof lastmonthSales toDrexel thismonthCurrentmonth’sproduction= + –
  • © 2009 Prentice-Hall, Inc. 8 – 32Manufacturing Applications Greenberg is starting a new four-monthproduction cycle with a change in designspecification that left no old motors in stock onJanuary 1 Given January demand for both motorsIA1 = 0 + XA1 – 800IB1 = 0 + XB1 – 1,000 Rewritten as January’s constraintsXA1 – IA1 = 800XB1 – IB1 = 1,000
  • © 2009 Prentice-Hall, Inc. 8 – 33Manufacturing Applications Constraints for February, March, and AprilXA2 + IA1 – IA2 = 700 February GM3A demandXB2 + IB1 – IB2 = 1,200 February GM3B demandXA3 + IA2 – IA3 = 1,000 March GM3A demandXB3 + IB2 – IB3 = 1,400 March GM3B demandXA4 + IA3 – IA4 = 1,100 April GM3A demandXB4 + IB3 – IB4 = 1,400 April GM3B demand And constraints for April’s ending inventoryIA4 = 450IB4 = 300
  • © 2009 Prentice-Hall, Inc. 8 – 34Manufacturing Applications We also need constraints for warehouse spaceIA1 + IB1 ≤ 3,300IA2 + IB2 ≤ 3,300IA3 + IB3 ≤ 3,300IA4 + IB4 ≤ 3,300 No worker is ever laid off so Greenberg has abase employment level of 2,240 labor hours permonth By adding temporary workers, available laborhours can be increased to 2,560 hours per month Each GM3A motor requires 1.3 labor hours andeach GM3B requires 0.9 hours
  • © 2009 Prentice-Hall, Inc. 8 – 35Manufacturing Applications Labor hour constraints1.3XA1 + 0.9XB1 ≥ 2,240 (January min hrs/month)1.3XA1 + 0.9XB1 ≤ 2,560 (January max hrs/month)1.3XA2 + 0.9XB2 ≥ 2,240 (February labor min)1.3XA2 + 0.9XB2 ≤ 2,560 (February labor max)1.3XA3 + 0.9XB3 ≥ 2,240 (March labor min)1.3XA3 + 0.9XB3 ≤ 2,560 (March labor max)1.3XA4 + 0.9XB4 ≥ 2,240 (April labor min)1.3XA4 + 0.9XB4 ≤ 2,560 (April labor max)All variables ≥ 0 Nonnegativity constraints
  • © 2009 Prentice-Hall, Inc. 8 – 36Manufacturing Applications Greenberg Motors solutionPRODUCTION SCHEDULE JANUARY FEBRUARY MARCH APRILUnits GM3A produced 1,277 1,138 842 792Units GM3B produced 1,000 1,200 1,400 1,700Inventory GM3A carried 477 915 758 450Inventory GM3B carried 0 0 0 300Labor hours required 2,560 2,560 2,355 2,560 Total cost for this four month period is $76,301.61 Complete model has 16 variables and 22constraints
  • © 2009 Prentice-Hall, Inc. 8 – 37 Assignment Problems Involve determining the most efficient way toassign resources to tasks Objective may be to minimize travel times ormaximize assignment effectiveness Assignment problems are unique because theyhave a coefficient of 0 or 1 associated witheach variable in the LP constraints and theright-hand side of each constraint is alwaysequal to 1Employee Scheduling Applications
  • © 2009 Prentice-Hall, Inc. 8 – 38 Ivan and Ivan law firm maintains a large staff ofyoung attorneys Ivan wants to make lawyer-to-client assignmentsin the most effective manner He identifies four lawyers who could possibly beassigned new cases Each lawyer can handle one new client The lawyers have different skills and specialinterests The following table summarizes the lawyersestimated effectiveness on new casesEmployee Scheduling Applications
  • © 2009 Prentice-Hall, Inc. 8 – 39 Effectiveness ratingsEmployee Scheduling ApplicationsCLIENT’S CASELAWYER DIVORCECORPORATEMERGER EMBEZZLEMENT EXHIBITIONISMAdams 6 2 8 5Brooks 9 3 5 8Carter 4 8 3 4Darwin 6 7 6 4Let Xij =1 if attorney i is assigned to case j0 otherwisewhere i = 1, 2, 3, 4 stands for Adams, Brooks, Carter,and Darwin respectivelyj = 1, 2, 3, 4 stands for divorce, merger,embezzlement, and exhibitionism
  • © 2009 Prentice-Hall, Inc. 8 – 40 The LP formulation isEmployee Scheduling ApplicationsMaximize effectiveness = 6X11 + 2X12 + 8X13 + 5X14 + 9X21 + 3X22+ 5X23 + 8X24 + 4X31 + 8X32 + 3X33 + 4X34+ 6X41 + 7X42 + 6X43 + 4X44subject to X11 + X21 + X31 + X41 = 1 (divorce case)X12 + X22 + X32 + X42 = 1 (merger)X13 + X23 + X33 + X43 = 1 (embezzlement)X14 + X24 + X34 + X44 = 1 (exhibitionism)X11 + X12 + X13 + X14 = 1 (Adams)X21 + X22 + X23 + X24 = 1 (Brook)X31 + X32 + X33 + X34 = 1 (Carter)X41 + X42 + X43 + X44 = 1 (Darwin)
  • © 2009 Prentice-Hall, Inc. 8 – 41 Solving Ivan and Ivan’s assignment schedulingLP problem using QM for WindowsEmployee Scheduling ApplicationsProgram 8.4
  • © 2009 Prentice-Hall, Inc. 8 – 42 Labor Planning Addresses staffing needs over a particulartime Especially useful when there is some flexibilityin assigning workers that require overlappingor interchangeable talentsEmployee Scheduling Applications
  • © 2009 Prentice-Hall, Inc. 8 – 43 Hong Kong Bank of Commerce and Industry hasrequirements for between 10 and 18 tellersdepending on the time of day Lunch time from noon to 2 pm is generally thebusiest The bank employs 12 full-time tellers but hasmany part-time workers available Part-time workers must put in exactly four hoursper day, can start anytime between 9 am and 1pm, and are inexpensive Full-time workers work from 9 am to 5 pm andhave 1 hour for lunchEmployee Scheduling Applications
  • © 2009 Prentice-Hall, Inc. 8 – 44 Labor requirements for Hong Kong Bank ofCommerce and IndustryEmployee Scheduling ApplicationsTIME PERIOD NUMBER OF TELLERS REQUIRED9 am – 10 am 1010 am – 11 am 1211 am – Noon 14Noon – 1 pm 161 pm – 2 pm 182 pm – 3 pm 173 pm – 4 pm 154 pm – 5 pm 10
  • © 2009 Prentice-Hall, Inc. 8 – 45 Part-time hours are limited to a maximum of 50%of the day’s total requirements Part-timers earn $8 per hour on average Full-timers earn $100 per day on average The bank wants a schedule that will minimizetotal personnel costs It will release one or more of its full-time tellers ifit is profitable to do soEmployee Scheduling Applications
  • © 2009 Prentice-Hall, Inc. 8 – 46Employee Scheduling Applications We letF = full-time tellersP1 = part-timers starting at 9 am (leaving at 1 pm)P2 = part-timers starting at 10 am (leaving at 2 pm)P3 = part-timers starting at 11 am (leaving at 3 pm)P4 = part-timers starting at noon (leaving at 4 pm)P5 = part-timers starting at 1 pm (leaving at 5 pm)
  • © 2009 Prentice-Hall, Inc. 8 – 47Employee Scheduling Applicationssubject toF + P1 ≥ 10 (9 am – 10 am needs)F + P1 + P2 ≥ 12 (10 am – 11 am needs)0.5F + P1 + P2 + P3 ≥ 14 (11 am – noon needs)0.5F + P1 + P2 + P3 + P4 ≥ 16 (noon – 1 pm needs)F + P2 + P3 + P4 + P5 ≥ 18 (1 pm – 2 pm needs)F + P3 + P4 + P5 ≥ 17 (2 pm – 3 pm needs)F + P4 + P5 ≥ 15 (3 pm – 4 pm needs)F + P5 ≥ 10 (4 pm – 5 pm needs)F ≤ 12 (12 full-time tellers)4P1 + 4P2 + 4P3 + 4P4 + 4P5 ≤ 0.50(112) (max 50% part-timers) Objective functionMinimize total dailypersonnel cost = $100F + $32(P1 + P2 + P3 + P4 + P5)
  • © 2009 Prentice-Hall, Inc. 8 – 48Employee Scheduling Applications There are several alternate optimal schedulesHong Kong Bank can follow F = 10, P2 = 2, P3 = 7, P4 = 5, P1, P5 = 0 F = 10, P1 = 6, P2 = 1, P3 = 2, P4 = 5, P5 = 0 The cost of either of these two policies is $1,448per day
  • © 2009 Prentice-Hall, Inc. 8 – 49Financial Applications Portfolio Selection Bank, investment funds, and insurancecompanies often have to select specificinvestments from a variety of alternatives The manager’s overall objective is generally tomaximize the potential return on theinvestment given a set of legal, policy, or riskrestraints
  • © 2009 Prentice-Hall, Inc. 8 – 50Financial Applications International City Trust (ICT) invests in short-termtrade credits, corporate bonds, gold stocks, andconstruction loans The board of directors has placed limits on howmuch can be invested in each areaINVESTMENTINTERESTEARNED (%)MAXIMUM INVESTMENT($ MILLIONS)Trade credit 7 1.0Corporate bonds 11 2.5Gold stocks 19 1.5Construction loans 15 1.8
  • © 2009 Prentice-Hall, Inc. 8 – 51Financial Applications ICT has $5 million to invest and wants toaccomplish two things Maximize the return on investment over the nextsix months Satisfy the diversification requirements set by theboard The board has also decided that at least 55% ofthe funds must be invested in gold stocks andconstruction loans and no less than 15% beinvested in trade credit
  • © 2009 Prentice-Hall, Inc. 8 – 52Financial Applications The variables in the model areX1 = dollars invested in trade creditX2 = dollars invested in corporate bondsX3 = dollars invested in gold stocksX4 = dollars invested in construction loans
  • © 2009 Prentice-Hall, Inc. 8 – 53Financial Applications Objective functionMaximizedollars ofinterestearned= 0.07X1 + 0.11X2 + 0.19X3 + 0.15X4subject to X1 ≤ 1,000,000X2 ≤ 2,500,000X3 ≤ 1,500,000X4 ≤ 1,800,000X3 + X4 ≥ 0.55(X1 + X2 + X3 + X4)X1 ≥ 0.15(X1 + X2 + X3 + X4)X1 + X2 + X3 + X4 ≤ 5,000,000X1, X2, X3, X4 ≥ 0
  • © 2009 Prentice-Hall, Inc. 8 – 54Financial Applications The optimal solution to the ICT is to make thefollowing investmentsX1 = $750,000X2 = $950,000X3 = $1,500,000X4 = $1,800,000 The total interest earned with this plan is$712,000
  • © 2009 Prentice-Hall, Inc. 8 – 55Transportation Applications Shipping Problem The transportation or shipping probleminvolves determining the amount of goods oritems to be transported from a number oforigins to a number of destinations The objective usually is to minimize totalshipping costs or distances This is a specific case of LP and a specialalgorithm has been developed to solve it
  • © 2009 Prentice-Hall, Inc. 8 – 56Transportation Applications The Top Speed Bicycle Co. manufactures andmarkets a line of 10-speed bicycles The firm has final assembly plants in two citieswhere labor costs are low It has three major warehouses near large markets The sales requirements for the next year are New York – 10,000 bicycles Chicago – 8,000 bicycles Los Angeles – 15,000 bicycles The factory capacities are New Orleans – 20,000 bicycles Omaha – 15,000 bicycles
  • © 2009 Prentice-Hall, Inc. 8 – 57Transportation Applications The cost of shipping bicycles from the plants tothe warehouses is different for each plant andwarehouseTOFROM NEW YORK CHICAGO LOS ANGELESNew Orleans $2 $3 $5Omaha $3 $1 $4 The company wants to develop a shippingschedule that will minimize its total annual cost
  • © 2009 Prentice-Hall, Inc. 8 – 58Transportation Applications The double subscript variables will represent theorigin factory and the destination warehouseXij = bicycles shipped from factory i to warehouse j SoX11 = number of bicycles shipped from New Orleans to New YorkX12 = number of bicycles shipped from New Orleans to ChicagoX13 = number of bicycles shipped from New Orleans to Los AngelesX21 = number of bicycles shipped from Omaha to New YorkX22 = number of bicycles shipped from Omaha to ChicagoX23 = number of bicycles shipped from Omaha to Los Angeles
  • © 2009 Prentice-Hall, Inc. 8 – 59Transportation Applications Objective functionMinimizetotalshippingcosts= 2X11 + 3X12 + 5X13 + 3X21 + 1X22 + 4X23subject to X11 + X21 = 10,000 (New York demand)X12 + X22 = 8,000 (Chicago demand)X13 + X23 = 15,000 (Los Angeles demand)X11 + X12 + X13 ≤ 20,000 (New Orleans factory supply)X21 + X22 + X23 ≤ 15,000 (Omaha factory supply)All variables ≥ 0
  • © 2009 Prentice-Hall, Inc. 8 – 60Transportation Applications Formulation for Excel’sSolverProgram 8.5A
  • © 2009 Prentice-Hall, Inc. 8 – 61Transportation Applications Solution from Excel’s SolverProgram 8.5A
  • © 2009 Prentice-Hall, Inc. 8 – 62Transportation Applications Total shipping cost equals $96,000 Transportation problems are a special case of LPas the coefficients for every variable in theconstraint equations equal 1 This situation exists in assignment problems aswell as they are a special case of thetransportation problem Top Speed Bicycle solutionTOFROM NEW YORK CHICAGO LOS ANGELESNew Orleans 10,000 0 8,000Omaha 0 8,000 7,000
  • © 2009 Prentice-Hall, Inc. 8 – 63Transportation Applications Truck Loading Problem The truck loading problem involves decidingwhich items to load on a truck so as tomaximize the value of a load shipped Goodman Shipping has to ship the followingsix itemsITEM VALUE ($) WEIGHT (POUNDS)1 22,500 7,5002 24,000 7,5003 8,000 3,0004 9,500 3,5005 11,500 4,0006 9,750 3,500
  • © 2009 Prentice-Hall, Inc. 8 – 64Transportation Applications The objective is to maximize the value of itemsloaded into the truck The truck has a capacity of 10,000 pounds The decision variable isXi = proportion of each item i loaded on the truck
  • © 2009 Prentice-Hall, Inc. 8 – 65Transportation ApplicationsMaximizeload value$22,500X1 + $24,000X2 + $8,000X3+ $9,500X4 + $11,500X5 + $9,750X6= Objective functionsubject to7,500X1 + 7,500X2 + 3,000X3+ 3,500X4 + 4,000X5 + 3,500X6 ≤ 10,000 lb capacityX1 ≤ 1X2 ≤ 1X3 ≤ 1X4 ≤ 1X5 ≤ 1X6 ≤ 1X1, X2, X3, X4, X5, X6 ≥ 0
  • © 2009 Prentice-Hall, Inc. 8 – 66Transportation Applications Excel Solver formulation for Goodman ShippingProgram 8.6A
  • © 2009 Prentice-Hall, Inc. 8 – 67Transportation Applications Solver solution for Goodman ShippingProgram 8.6B
  • © 2009 Prentice-Hall, Inc. 8 – 68Transportation Applications The Goodman Shipping problem has aninteresting issue The solution calls for one third of Item 1 to beloaded on the truck What if Item 1 can not be divided into smallerpieces? Rounding down leaves unused capacity on thetruck and results in a value of $24,000 Rounding up is not possible since this wouldexceed the capacity of the truck Using integer programminginteger programming, the solution is toload one unit of Items 3, 4, and 6 for a value of$27,250
  • © 2009 Prentice-Hall, Inc. 8 – 69Transshipment Applications The transportation problem is a specialcase of the transshipment problem When the items are being moved from asource to a destination through anintermediate point (a transshipment pointtransshipment point),the problem is called a transshipmenttransshipmentproblemproblem
  • © 2009 Prentice-Hall, Inc. 8 – 70Transshipment Applications Distribution Centers Frosty Machines manufactures snowblowersin Toronto and Detroit These are shipped to regional distributioncenters in Chicago and Buffalo From there they are shipped to supply housesin New York, Philadelphia, and St Louis Shipping costs vary by location anddestination Snowblowers can not be shipped directly fromthe factories to the supply houses
  • © 2009 Prentice-Hall, Inc. 8 – 71New York CityPhiladelphiaSt LouisDestinationChicagoBuffaloTransshipmentPointTransshipment Applications Frosty Machines networkTorontoDetroitSourceFigure 8.1
  • © 2009 Prentice-Hall, Inc. 8 – 72Transshipment Applications Frosty Machines dataTOFROM CHICAGO BUFFALONEW YORKCITY PHILADELPHIA ST LOUIS SUPPLYToronto $4 $7 — — — 800Detroit $5 $7 — — — 700Chicago — — $6 $4 $5 —Buffalo — — $2 $3 $4 —Demand — — 450 350 300 Frosty would like to minimize the transportationcosts associated with shipping snowblowers tomeet the demands at the supply centers given thesupplies available
  • © 2009 Prentice-Hall, Inc. 8 – 73Transshipment Applications A description of the problem would be tominimize cost subject to1. The number of units shipped from Toronto is not morethan 8002. The number of units shipped from Detroit is not morethan 7003. The number of units shipped to New York is 4504. The number of units shipped to Philadelphia is 3505. The number of units shipped to St Louis is 3006. The number of units shipped out of Chicago is equal tothe number of units shipped into Chicago7. The number of units shipped out of Buffalo is equal tothe number of units shipped into Buffalo
  • © 2009 Prentice-Hall, Inc. 8 – 74Transshipment Applications The decision variables should represent thenumber of units shipped from each source to thetransshipment points and from there to the finaldestinationsT1 = the number of units shipped from Toronto to ChicagoT2 = the number of units shipped from Toronto to BuffaloD1 = the number of units shipped from Detroit to ChicagoD2 = the number of units shipped from Detroit to ChicagoC1 = the number of units shipped from Chicago to New YorkC2 = the number of units shipped from Chicago to PhiladelphiaC3 = the number of units shipped from Chicago to St LouisB1 = the number of units shipped from Buffalo to New YorkB2 = the number of units shipped from Buffalo to PhiladelphiaB3 = the number of units shipped from Buffalo to St Louis
  • © 2009 Prentice-Hall, Inc. 8 – 75Transshipment Applications The linear program isMinimize cost =4T1 + 7T2 + 5D1 + 7D2 + 6C1 + 4C2 + 5C3 + 2B1 + 3B2 + 4B3subject toT1 + T2 ≤ 800 (supply at Toronto)D1 + D2 ≤ 700 (supply at Detroit)C1 + B1 = 450 (demand at New York)C2 + B2 = 350 (demand at Philadelphia)C3 + B3 = 300 (demand at St Louis)T1 + D1 = C1 + C2 + C3 (shipping through Chicago)T2 + D2 = B1 + B2 + B3 (shipping through Buffalo)T1, T2, D1, D2, C1, C2, C3, B1, B2, B3 ≥ 0 (nonnegativity)
  • © 2009 Prentice-Hall, Inc. 8 – 76Transshipment Applications The solution from QM for Windows isProgram 8.7