Presentation on theme: "Managerial Decision Modeling with Spreadsheets"— Presentation transcript:
1 Managerial Decision Modeling with Spreadsheets Chapter 3Linear Programming Modeling (Selected) Applications: With Computer Analyses in Excel
2 Learning ObjectivesModel wide variety of linear programming (LP) problems.Understand major business application areas for LP problems: manufacturing, marketing, labor scheduling, blending, transportation, finance, and multi-period planning.Gain experience in setting up and solving LP problems using Excel’s Solver.
3 Production Mix Problem Fifth Avenue IndustriesNationally known menswear manufacturer.Produces four varieties of neckties.All-silk tie.All-polyester tie.Two different polyester and cotton blends.Has fixed contracts with major department stores.Table 3.1 summarizes contract demand for products.
4 Production Mix Problem MaterialCost per yardMaterial available per month (yards)Silk$201,000Polyestar$63,000Cotton$91,600Fifth Avenue uses a standard labor cost of $0.75 per tie (for any variety)
6 Production Mix Problem Fifth Avenue IndustriesEach all-silk tie requires -Cost per tie = yards of silk x $20 per yard = $2.5Revenue per tie = $6.70 selling price per silk tie.Profit per tie = Revenue per tie - Cost per tie =$ $2.5 – = $3.45.Profit for other three products -Profit per all-polyester tie = $2.32.Profit per Blend - 1 poly-cotton tie = $2.81profit per Blend - 2 poly-cotton tie = $3.25
7 Production Mix Problem Fifth Avenue IndustriesObjective: maximize profit menswear ties.$3.45 S + $2.32P + $2.81 B $3.25 B2Where:S = number of all-silk ties produced per month.P = number of polyester ties.B1 = number of Blend - 1 poly-cotton ties.B2 = number of Blend - 2 poly-cotton ties.
8 Production Mix Problem Objective: maximize profit =$3.45S + $2.32 P + $2.81 B1 + $3.25 B2Subject to:(Yards of silk)(Yards of polyester)(Yards of cotton)(Contract minimum for all silk)(Contract minimum)(Contract minimum for all polyester)(Contract maximum)
9 Production Mix Problem Objective: maximize profit =$4.075 S + $3.07 P + $3.56 B1 + $4.00 B2Subject toConstraints - Continued(Contract minimum Blend 1)(Contract maximum)(Contract minimum Blend 2)
10 Media SelectionWin Big Gambling Club promotes gambling junkets from a large Midwestern city to casinos in the Bahamas.Club has budgeted up to $8,000 per week for local advertising.Money is to be allocated among four promotional media:TV spots,Newspaper ads, andTwo types of radio advertisements.Win Big’s goal - reach largest possible high-potential audience through various media.
11 Media SelectionContract arrangements require at least five radio spots be placed each week.Management insists no more than $1,800 be spent on radio advertising each week.
12 Media Selection Objective: maximize audience coverage= 5000T N P +2800AT = number of 1-minute TV spots taken eachweek.N = number of full-page daily newspaper adstaken each week.P = number of 30-second prime-time radio spotsA = number of 1-minute afternoon radio spots
13 Media Selection Objective: maximize audience coverage = 5000 T N P ASubject to
14 Media Selection The optimal solution found to be: T = television spots.N = newspaper ads.P = second prime time radio spots.A = minute afternoon radio spots.This produces an audience exposure of 67,240 contacts.
15 Marketing Research Problem Management Sciences Associates (MSA) handles consumer surveys. MSA has to determine, for a client, that it must fulfill several requirements in order to draw statistically valid conclusions on sensitive issue of new U.S. immigration laws:1. Survey at least 2,300 U.S. households.2. Survey at least 1,000 households whose headsare 30 years of age or younger.
16 Marketing Research Problem 3. Survey at least 600 households whose heads arebetween 31 and 50 years of age.4. Ensure that at least 15% of those surveyed livein a state that borders on Mexico.5. Ensure that at least 50 % of those surveyed who are 30 years of age or younger live in a state that does not border Mexico *(missing)5. Ensure that no more than 20% of those surveyedwho are 51 years of age or over live in a statethat borders on Mexico.
17 Marketing Research Problem Objective: minimize total interview costs =$7.50 B1 + $6.80 B2 + $5.50 B3 +$6.90 N1 + $7.25 N2 + $6.10 N3B1 = number 30 years or younger and live in border state.B2 = number years and live in border state.B3 = number 51 years or older and live in border state.N1 = number 30 years or younger and do not live in borderstate.N2 = number years and do not live in border state.N3 = number 51 years or older and do not live in border state.
18 Marketing Research Problem Objective: minimize total interview costs =$7.50 B1 + $6.80 B2 + $5.50 B3 +$6.90 N1 + $7.25 N2 + $6.10 N3Subject to
19 Marketing Research Problem B1 + B2 + B3 0.15(B1 + B2 + B3 + N1 + N2 + N3)Rewritten as: B1 + B2 + B (B1 + B2 + B3 + N1 + N2 + N3) 0Simplifies to:0.85B B B N N N3 0AndB3 ≤ 0.2(B3 + N3)0.8B N3 < 0
20 Marketing Research Problem Optimal solution shows that it costs $15,166 and requires one to survey households as follows: State borders Mexico and years =State borders Mexico and 51 years =State not borders Mexico and 30 years = 1,000State not borders Mexico and 51 years =
21 Employee Scheduling Application Hong Kong Bank now employs 12 full-time tellers. Part-time employees (four hours per day) are available. They can start work anytime between 9 A.M. and 1 P.M.Tellers requirements:
22 Employee Scheduling Application Hong Kong BankLabor Constraints:Full-timers work from 9 A.M. to 5 P.M.Allowed 1 hour for lunch.Half of full-timers eat at 11 A.M. and other half at noon.Full-timers thus provide 35 hours per week of productive labor time.Part-time hours limited to a maximum of 50% of day’s total requirement.Costs:Part-timers earn $7 per hour (or $28 per day) on average.Full-timers earn $90 per day in salary and benefits, on average.
23 Employee Scheduling Application Hong Kong BankDecision Variables:F = number of. full-time tellersP1 = part-timers starting at 9 A.M. (leaving at 1 P.M.)P2 = part-timers starting at 10 A.M. (leaving at 2 P.M.)P3 = part-timers starting at 11 A.M. (leaving at 3 P.M.)P4 = part-timers starting at noon (leaving at 4 P.M.)P5 = part-timers starting at 1 P.M. (leaving at 5 P.M.)
25 Employee Scheduling Application Constraints (Continued):Part-time worker hours cannot exceed 50% total hours required each day, which is sum of tellers needed each hour.Simplifying yields,
26 Employee Scheduling Application Excel entries for model reveal optimal solution.Employ 10 full-time tellers.7 part-time tellers at 10 A.M.2 part-time tellers at 11 A.M.5 part-time tellers at noon.Total cost of $1,292 per day.There are several alternate optimal solutions.
27 Employee Scheduling Application There are several alternate optimal solutions.In practice sequence in which constraints are listed in model may affect specific solution found.One alternate solution.Employ 10 full-time tellers.6 part-time tellers at 9 A.M.1 part-time teller at 10 A.M.2 part-time teller at 11 A.M.5 part-time tellers at noon.Total cost of this policy is also $1,292.
28 Ingredient Blending Applications Diet ProblemsDiet problem involves specifying a food or foodingredient combination that satisfies stated nutritionalrequirements at minimum cost.Whole Food Nutrition Center uses three bulk grains to blend natural cereal that sells by the pound.Each 2-ounce serving of cereal, when taken with 1.2 cup of whole milk, meets an average adult’s minimum daily requirement for protein, riboflavin, phosphorus, and magnesium.
29 Ingredient Blending Applications Whole Food Nutrition CenterDiet ProblemsMinimum adult daily requirement:Protein 3 units.Riboflavin 2 units.Phosphorus 1 unit.Magnesium unit.Select blend of grains to meet USRDA at minimum cost.
30 Ingredient Blending Applications Decision Variables:A = pounds of grain in one 2-ounce cereal serving.B = pounds of grain in one 2-ounce cereal serving.C = pounds of grain in one 2-ounce cereal serving.
31 Ingredient Blending Applications Objective: minimize total cost of mixing 2-ounce serving =$0.33 A + $0.47 B + $0.38 CSubject to(Protein units)(Riboflavin units)(Phosphorous units)(Magnesium units)(Total mix 2 ounces or pound)
32 Multi-period Applications Most challenging application of LP is modeling multi-period scenarios.Situations where decision maker has to determine optimal decisions for several periods (weeks, months, etc.).These problems especially difficult because decision choices in later periods are directly dependent on decisions made in earlier periods.
33 Multi-period Applications Production SchedulingGreenberg Motors, Inc., manufactures two differentelectrical motors for sale under contract to Drexel Corp.Model GM3A is found in many Drexel foodprocessors, and model GM3B is used in assemblyof blenders.Production planning must consider four factors:Desirability of producing same number of each motor each month.Necessity to reduce inventory carrying, or holding, costs.Warehouse limitations cannot be exceeded.Company’s no-layoff policy.
34 Multi-period Applications: Production Scheduling Greenberg Motors, Inc.Scheduled Orders.Decision Variables.PAi = number of model GM3A motors produced inmonth i (i =1,2,3,4 for January–April).PBi = number of model GM3B motors produced inmonth i (i=1,2,3,4 for January–April).
35 Multi-period Applications: Production Scheduling Greenberg Motors, Inc.Associated Costs.Production costs now -GM3A is $10.GM3B is $6.Prices will increase in March to $11 and $6.60, respectively.Production Cost =$10 PA1 + $10 PA2 + $11 PA3 + $11 PA4 + $6 PB1 + $6 PB2 + $6.60 PB3 + $6.60 PB4
36 Multi-period Applications: Production Scheduling Greenberg Motors, Inc.IAi = level of on-hand inventory for GM3A at end ofmonth i (i=1,2,3,4 for January–April)PBi = level of on-hand inventory for GM3B at end ofCarrying CostGM3A is $0.18 per month.GM3B is $0.13 per month.Cost of carrying inventory =$0.18 IA1 + $0.18 IA2 + $0.18 IA3 + $0.18 IA4 +$0.13 IB1 + $0.13 IB2 + $0.13 IB3 + $0.13 IB4
38 Multi-period Applications:Production Scheduling Inventory at end of month is:January’s demand for GM3As is 800 and for GM3Bs is 1,000, write relation as:IA1 = PA IB1 = PBRewrite as:PA1 - IA1 =PB1 - IB1 = 1000
39 Multi-period Applications: Production Scheduling Greenberg Motors, Inc.If Greenberg wants additional 450 GM3As and 300 GM3Bs at end of April, add constraints:February GM3A demandFebruary GM3B demandMarch GM3A demandMarch GM3B demandApril GM3A demandApril GM3B demandand
40 Multi-period Applications:Production Scheduling Warehouse space constraints:
41 Multi-period Applications: Production Scheduling Labor Constraints:(January min. hours)(January max. hours)(February min. hours)(February max. hours)(March min. hours)(March max. hours)(April min. hours)(April max. hours)
43 Summary Continued discussion of LP models. More experience in formulating and solving problems from variety of disciplines and applications:Marketing, manufacturing, employee scheduling,Finance, transportation, ingredient blending, andMulti-period planning.Illustrated setup and solution of models using Excel’s Solver add-in.