Managerial Decision Modeling with Spreadsheets Chapter 3 Linear Programming Modeling Applications: With Computer Analyses in Excel

Learning Objectives Model 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.

3.1 Introduction Delta Airlines is example of use of LP model for solving real-world scheduling problems resulting in significant cost reductions for company. Similar examples exist in other managerial decision making areas, such as: production mix, labor scheduling, job assignment, production scheduling, marketing research, media selection, shipping and transportation, ingredient mix, and financial portfolio selection. Purpose is to show how one can use LP to modeling for decision-making in these areas.

3.2 Marketing Application Media Selection - Win 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, and Two types of radio advertisements. Win Big’s goal - reach largest possible high-potential audience through various media.

Audience Reached Per Ad Media Selection Data Audience Reached Per Ad Cost Per Ad(s) Maximum Ads Per Week TV spot (1minute) 5,000 800 12 Daily newspaper (full page) 8,500 925 5 Radio spot (30 sec, prime time) 2,400 290 25 Radio spot (1 min, afternoon) 2,800 380 20 Contract 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.

Media Selection Data LP Formulation Objective: maximize audience coverage = 5000T + 8500N + 2400P +2800A T = number of 1-minute TV spots taken each week. N = number of full-page daily newspaper ads taken each week. P = number of 30-second prime-time radio spots taken each week. A = number of 1-minute afternoon radio spots taken each week.

Media Selection Data LP Formulation Objective: maximize audience coverage = 5000 T + 8500 N + 2400 P +2800 A Subject to

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 heads are 30 years of age or younger.

Marketing Research Problem 3. Survey at least 600 households whose heads are between 31 and 50 years of age. 4. Ensure that at least 15% of those surveyed live in a state that borders on Mexico. 5. Ensure that no more than 20% of those surveyed who are 51 years of age or over live in a state that borders on Mexico.

MSA’s Goal: Meet Sampling Requirements With Minimum Cost Objective: minimize total interview costs = $7.50 B1 + $6.80 B2 + $5.50 B3 + $6.90 N1 + $7.25 N2 + $6.10 N3 B1 = number 30 years or younger and live in border state. B2 = number 31-50 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 border state. N2 = number 31-50 years and do not live in border state. N3 = number 51 years or older and do not live in border state.

MSA’s Goal LP Formulation Objective: minimize total interview costs = $7.50 B1 + $6.80 B2 + $5.50 B3 + $6.90 N1 + $7.25 N2 + $6.10 N3 Subject to

Rewriting Last Two Constraints B1 + B2 + B3  0.15(B1 + B2 + B3 + N1 + N2 + N3) Rewritten as:  B1 + B2 + B3 - 0.15(B1 + B2 + B3 + N1 + N2 + N3)  0 Simplifies to: 0.85B1 + 0.85B2 + 0.85B3 - 0.15N1 - 0.15N2 - 0.15N3  0 And B3 ≤ 0.2(B3 + N3) 0.8B3 - 0.2N3 < 0

Optimal Solution to MSA’s Marketing Research Problem Optimal solution shows that it costs $15,166 and requires one to survey households as follows:  State borders Mexico and 31-50 years = 600 State borders Mexico and  51 years = 140 State not borders Mexico and  30 years = 1,000 State not borders Mexico and  51 years = 560

3.3 Manufacturing Applications Production Mix Problem Fifth Avenue Industries Nationally 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.

3.3 Manufacturing Applications Fifth Avenue Industries Table 3.1 Data for Fifth Avenue Industries Tie Price Monthly Contract Monthly Demand Material Required Material Silk 6.70 6,000 7,000 0.125 Polyester 3.55 10,000 14,000 0.08 Poly-Cotton Blend 1 4.31 13,000 16,000 0.10 50%-50% Blend 2 4.81 8,500 30%-70%

Profit Per Unit Fifth Avenue Industries For each all-silk tie - Cost per tie = 0.125 yards of silk x $21 per yard = $2.625. Revenue per tie = $6.70 selling price per silk tie. Profit per tie = Revenue per tie - Cost per tie = $6.70 - $2.625 = $4.075. Profit for other three products - Profit per all-polyester tie = $3.07. Profit per Blend - 1 poly-cotton tie = $3.56. Profit per Blend - 2 poly-cotton tie = $4.00.

Objective Function Objective: maximize profit menswear ties. Fifth Avenue Industries Objective: maximize profit menswear ties. $4.075 S + $3.07 P + $3.56 B1 + $4.00 B2 Where: 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.

Objective Function and Constraints Objective: maximize profit = $4.075 S + $3.07 P + $3.56 B1 + $4.00 B2 Subject to (Yards of silk) (Yards of polyester) (Yards of cotton) (Contract minimum for all silk) (Market maximum) (Contract minimum for all polyester)

Objective Function and Constraints Objective: maximize profit = $4.075 S + $3.07 P + $3.56 B1 + $4.00 B2 Subject to Constraints - Continued (Contract minimum Blend 1) (Market maximum) (Contract minimum Blend 2)

3.4 Employee Scheduling Application Time period # of Tellers Required 9 a.m. – 10a.m. 10 10 a.m.–11a.m. 12 11 a.m. – noon 14 Noon – 1 p.m. 16 1 p.m. – 2 p.m. 18 2 p.m. – 3 p.m. 17 3 p.m. – 4 p.m. 15 4 p.m. – 5 p.m. Labor Planning Problem Hong Kong Bank now employs 12 full-time tellers. Part-time employee (four hours per day) are available. Tellers requirements:

Employee Scheduling Application Hong Kong Bank Labor 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 $4 per hour (or $16 per day) on average. Full-timers earn $50 per day in salary and benefits, on average.

Employee Scheduling Application Hong Kong Bank Decision Variables: F = full-time tellers P1 = 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.)

Hong Kong Bank LP Formulation Objective: minimize total daily labor cost $50 F + $16 ( P1 + P2 + P3 + P4 ) Subject to (9 A.M. - 10 A.M. needs) (10 A.M. - 11 A.M. needs) (11 A.M. - noon needs) (noon - 1 P.M. needs) (1 P.M. - 2 P.M. needs) (2 P.M. - 3 P.M. needs) (3 P.M. - 4 P.M. needs) (4 P.M. - 5 P.M. needs) (full-time tellers available)

Hong Kong Bank LP Formulation Constraints (Continued): Part-time worker hours cannot exceed 50% total hours required each day, which is sum of tellers needed each hour. Simplifying yields,

Hong Kong Bank Solution 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 $724 per day. There are several alternate optimal solutions.

Hong Kong Bank Solution 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 $724.

3.5 Financial Applications Portfolio Selection International City Trust (ICT) invests in short-term trade credits, corporate bonds, gold stocks, and construction loans. ICT has $5 million available for immediate investment and wishes to do two things: maximize interest earned on investments made over next six months and satisfy diversification requirements as set by board of directors.

Portfolio Specification International City Trust Investment Possibilities: Investment Interest Earned (%) Maximum Investment ($millions) Trade credit 7 1 Corporate Bonds 11 2.5 Gold stocks 19 1.5 Construction Loan 15 1.8 Board specifies at least 55% of funds invested must be in gold stocks and construction loans. No less than 15% be invested in trade credit.

Investment Formulation International City Trust Decision Variables: T = dollars invested in trade credit B = dollars invested in corporate bonds G = dollars invested in gold stocks C = dollars invested in construction loans

Investment Formulation International City Trust Objective: maximize investment interest dollars earned. 0.07 T + 0.11 B + 0.19 G + 0.15 C Subject to

Rewriting Last Two Constraints G + C > 0.55(T + B + G +C ) Rewritten as: -0.55T - 0.55B + 0.45G + 0.45C  0 Gold stock And T > 0.15 (T + B + G +C ) 0.85T - 0.15B - 0.15G - 0.15C  0 Trade credit

3.6 Transportation Applications Truck Loading Problem Truck loading problem involves deciding which items to load on a truck so as to maximize value of a load shipped. Consider Goodman Shipping. One truck with a capacity of 10,000 pounds is next to be loaded. Several other items are awaiting shipment. Each items awaiting shipment has associated dollar value and weight. Objective - maximize total value of items loaded on truck without exceeding truck’s weight capacity.

Transportation Applications Goodman Shipping Items Awaiting Shipment: Item Total Value ($) Weight (Pounds) 1 22,500 7,500 2 24,000 3 8,000 3,000 4 9,500 3,500 5 11,500 4,000 6 9,750

Goodman Shipping LP Formulation Objective: maximize load value = $22,500 P1+ $24,000 P2 + $8,000 P3 + $9,500 P4 + $11,500 P5 + $9,7500 P6 Subject to 7,500 P1 + 7,500 P2+ 3,000 P3+ 3,500 P4 + 4,000 P5 + 3,000 P6 < 10,000 P1 < 1 P2 < 1 P3 < 1 P4 < 1 P5 < 1 P6 < 1 P1, P2, P3, P4, P5, P6 > 0 Where: Pi is proportion of each item i loaded on truck.

Goodman Shipping Problem Using Pounds – Not Proportions Formulate alternate model for problem. Decision variables in model are weights in pounds shipped, rather than proportion. Layout for model is identical to model shown previously. Solution to model shows maximum load value is $31,500. Load value achieved by shipping 2,500 pounds (0.33 of 7,500 pounds available item 1) and 7,500 pounds (all 7,500 pounds available item 2).

3.7 Ingredient Blending Applications Diet Problems Diet problem involves specifying a food or food ingredient combination that satisfies stated nutritional requirements 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.

3.7 Ingredient Blending Applications Whole Food Nutrition Center Diet Problems Minimum adult daily requirement: Protein 3 units. Riboflavin 2 units. Phosphorus 1 unit. Magnesium 0.425 unit. Select blend of grains to meet USRDA at minimum cost.

Whole Food’s Natural Cereal Requirements Grain Cost per pound (cents) Protein (unit/lb) Riboflavin Phosphorus Magnesium A 33 22 16 8 5 B 47 28 14 7 C 38 21 25 9 6 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.

Whole Food’s LP Formulation Objective: minimize total cost of mixing 2-ounce serving = $0.33 A + $0.47 B + $0.38 C Subject to (Protein units) (Riboflavin units) (Phosphorous units) (Magnesium units) (Total mix 2 ounces or 0.125 pound)

Ingredient Blending Applications Ingredient Mix and Blending Problems Blending problems arise when decision must be made regarding blending of two or more products to produce one or more products. Resources contain one or more essential ingredients that must be blended so each final product contains specific percentages of each ingredient. Example - Deals with application frequently seen in petroleum industry. Blending crude oils to produce refinable gasoline.

Blending Problem Example Low Knock Oil Company Low Knock Oil Company produces two grades of cut-rate gasoline for industrial distribution. Regular. Economy. Produced by refining a blend of two types of crude oil. Type X100. Type X220.

Blending Problem Example Weekly demand for Regular at least 25,000 barrels. Weekly demand for Economy grade at least 32,000 barrels. At least 45% of each barrel of regular must be ingredient A. At most 50% of each barrel of economy should contain ingredient B.

Blending Problem Example Decision Variables: R1 = barrels crude oil X100 blended to produce refined Regular. E1 = barrels crude oil X100 blended to produce refined Economy. R2 = barrels crude oil X220 blended to produce refined Regular. E2 = barrels crude oil X100 blended to produce refined

Blending Problem Example Low Knock Oil Company Product Type Ingredients and Costs Crude Oil Type Ingredient A (%) Ingredient B (%) Cost/Barrel ($) X100 35 55 30.00 X220 60 25 34.80

Low Knock Oil Company LP Formulation Two More Requirements - FIRST requirement: At least 45% of each barrel of Regular must be ingredient A. (R1+R2) is amount of crude blended to produce refined Regular gasoline demanded. 0.45 (R1+R2) is minimum amount of A required. 0.35 R1+0.60 R2 is amount of A in regular gas. 0.35 R1+0.60 R2 > 0.45 (R1+R2) Recalculate as: -0.10 R1+0.15 R2 > 0

Low Knock Oil Company LP Formulation Two More Requirements - SECOND requirement: At most 50% of each barrel of economy must be ingredient B. (E1 + E2) is amount of crude blended to produce refined Economy gasoline demanded. 0.50 (E1+E2) is maximum amount of ingredient B allowed. 0.55 E1+ 0.25 E2 = amount of ingredient B in Economy gas 0.55E1+0.25E2 < 0.50(E1+E2) Recalculating yields: 0.05E1 – 0.25E2 < 0

LP Problem Formulation Low Knock Oil Company Objective: minimize cost = $30 R1 + $30 E1 + $34.80 R2 + $34.80 E2 Subject to (Demand for Regular) (Demand for Economy) (Ingredient A in Regular) (Ingredient B in Economy)

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. Illustrated setup and solution of models using Excel’s Solver add-in.