Deck 4: Linear Programming Applications in Marketing, Finance and Operations Management

Question 1

If a real-world problem is correctly formulated, it is not possible to have alternative optimal solutions.

Answer 1

False

Question 2

Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, then

A)B $\le$ 5
B)A - .5B + C $\le$ 0
C).5A - B - .5C $\le$ 0
D)-.5A + .5B - .5C $\le$ 0

Answer 2

-.5A + .5B - .5C $\le$ 0

Question 3

A marketing research application uses the variable HD to represent the number of homeowners interviewed during the day. The objective function minimizes the cost of interviewing this and other categories and there is a constraint that HD $\ge$ 100. The solution indicates that interviewing another homeowner during the day will increase costs by 10.00. What do you know?

A)the objective function coefficient of HD is 10.
B)the dual price for the HD constraint is 10.
C)the objective function coefficient of HD is -10.
D)the dual price for the HD constraint is -10.

Answer 3

the dual price for the HD constraint is -10.

Question 4

Media selection problems usually determine

A)how many times to use each media source.
B)the coverage provided by each media source.
C)the cost of each advertising exposure.
D)the relative value of each medium.

Question 5

The objective function for portfolio selection problems usually is maximization of expected return or

A)maximization of investment types
B)minimization of cost
C)minimization of risk
D)maximization of number of shares

Question 6

If an LP problem is not correctly formulated, the computer software will indicate it is infeasible when trying to solve it.

Question 7

For a portfolio selection problem with the objective of maximizing expected return, the dual price for the available funds constraint provides information about the

A)proportion of the portfolio that is invested in a particular investment type
B)return from additional investment funds
C)degree of portfolio diversification that is optimal
D)cost of an additional unit of a particular investment type

Question 8

Department 3 has 2500 hours. Transfers are allowed to departments 2 and 4, and from departments 1 and 2. If A_i measures the labor hours allocated to department i and T_ij the hours transferred from department i to department j, then

A)T₁₃ + T₂₃ - T₃₂ -T₃₄ - A₃ = 2500
B)T₃₁ + T₃₂ - T₂₃ -T₄₃ + A₃ = 2500
C)A₃ + T₁₃ + T₂₃ -T₃₂ - T₃₄ = 2500
D)A₃ - T₁₃ - T₂₃ + T₃₂ + T₃₄ = 2500

Question 9

To properly interpret dual prices, one must know how costs were allocated in the objective function.

Question 10

If P_ij = the production of product i in period j, then to indicate that the limit on production of the company's three products in period 2 is 400,

A)P₂₁ + P₂₂ + P₂₃ $\le$ 400
B)P₁₂ + P₂₂ + P₃₂ $\le$ 400
C)P₃₂ $\le$ 400
D)P₂₃ $\le$ 400

Question 11

Let P_ij = the production of product i in period j. To specify that production of product 1 in period 3 and in period 4 differs by no more than 100 units,

A)P₁₃ -P₁₄ $\le$ 100; P₁₄ -P₁₃ $\le$ 100
B)P₁₃ - P₁₄ $\le$ 100; P₁₃ -P₁₄ $\ge$ 100
C)P₁₃ -P₁₄ $\le$ 100; P₁₄ -P₁₃ $\ge$ 100
D)P₁₃ - P₁₄ $\ge$ 100; P₁₄ - P₁₃ $\ge$ 100

Question 12

It is improper to combine manufacturing costs and overtime costs in the same objective function.

Question 13

The dual price for a constraint that compares funds used with funds available is .058. This means that

A)the cost of additional funds is 5.8%.
B)if more funds can be obtained at a rate of 5.5%, some should be.
C)no more funds are needed.
D)the objective was to minimize.

Question 14

Media selection problems can maximize exposure quality and use number of customers reached as a constraint, or maximize the number of customers reached and use exposure quality as a constraint.

Question 15

For the marketing research problem presented in the textbook, the research firm's objective is to conduct the market survey so as to meet the client's needs at a minimum cost.

Question 16

Production constraints frequently take the form:
beginning inventory + sales-production = ending inventory

Question 17

Let M be the number of units to make and B be the number of units to buy. If it costs $2 to make a unit and $3 to buy a unit and 4000 units are needed, the objective function is

A)Max 2M + 3B
B)Min 4000 (M + B)
C)Max 8000M + 12000B
D)Min 2M + 3B

Question 18

To study consumer characteristics, attitudes, and preferences, a company would engage in

A)client satisfaction processing.
B)marketing research.
C)capital budgeting.
D)production planning.

Question 19

Portfolio selection problems should acknowledge both risk and return.

Question 20

An LP model for a large-scale production scheduling problem involving numerous products, machines, and time periods can require thousands of decision variables and constraints.

Question 21

The Tots Toys Company is trying to schedule production of two very popular toys for the next three months: a rocking horse and a scooter. Information about both toys is given below. $$\begin{array} { l c c c c c } \text { Toy } & \begin{array} { c } \text { Begin. Invty. } \\\text { June 1 }\end{array} & \begin{array} { c } \text { Required } \\\text { Plastic }\end{array} & \begin{array} { c } \text { Required } \\\text { Time }\end{array} & \begin{array} { c } \text { Production } \\\text { Cost }\end{array} & \begin{array} { c } \text { Production } \\\text { Cost }\end{array} \\\hline \text { Rocking Horse } & 25 & 5 & 2 & 12 & 1 \\\text { Scooter } & 55 & 4 & 3 & 14 & 1.2\end{array}$$ $$\begin{array} { l c c c c } & \text { Plastic } & \text { Time } & \text { Monthly Demand } & \text { Monthly Demand } \\\text { Summer Schedule } & \text { Available } & \text { Available } & \text { Horse } & \text { Scooter } \\\hline \text { June } & 3500 & 2100 & 220 & 450 \\\text { July } & 5000 & 3000 & 350 & 700 \\\text { August } & 4800 & 2500 & 600 & 520\end{array}$$ Develop a model that would tell the company how many of each toy to produce during each month. You are to minimize total cost. Inventory cost will be levied on any items in inventory on June 30, July 31, or August 31 after demand for the month has been satisfied. Your model should make use of the relationship
Beginning Inventory + Production - Demand = Ending Inventory
for each month. The company wants to end the summer with 150 rocking horses and 60 scooters as beginning inventory for Sept. 1. Don't forget to define your decision variables.

Question 22

FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below. $$\begin{array} { l c c c c } \text { Ingredient } & \text { Cost } & \text { Volume } & \text { Fat Grams } & \text { Calories } \\\hline \text { Dried Fruit } & .35 & 1 / 4 \mathrm { cup } & 0 & 150 \\\text { Nut Mix } & .50 & 3 / 8 \mathrm { cup } & 10 & 400 \\\text { Cereal Mix } & .20 & 1 \mathrm { cup } & 1 & 50\end{array}$$ The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The blend should contain no more than 1000 calories and no more than 25 grams of fat. Dried fruit must be at least 20% of the volume of the mixture, and nuts must be no more than 15% of the weight of the mixture. Develop a model that meets these restrictions and minimizes the cost of the blend.

Question 23

Give examples of how variations in the workforce assignment model presented in the textbook could be applied to other types of allocation problems.

Question 24

Super City Discount Department Store is open 24 hours a day. The number of cashiers need in each four hour period of a day is listed below. $\begin{array}{lc}\text { Period } & \text { Cashiers Needed }\\\hline10 \mathrm{p} . \mathrm{m} . \text { to } 2 \mathrm{am} . \mathrm{m} . & 8 \\2 \mathrm{a} . \mathrm{m} \text { to } 6 \mathrm{am} . & 4 \\6 \mathrm{a} . \mathrm{m} . \text { to } 10 \mathrm{am} . & 7 \\10 \mathrm{am} . \text { to } 2 \mathrm{p} . \mathrm{m} . & 12 \\2 \mathrm{p} . \mathrm{m} . \text { to } 6 \mathrm{p} . \mathrm{m} . & 10 \\6 \mathrm{p} . \mathrm{m} \text { to } 10 \mathrm{p} . \mathrm{m} . & 15\end{array}$ If cashiers work for eight consecutive hours, how many should be scheduled to begin working in each period in order to minimize the number of cashiers needed?

Question 25

Larkin Industries manufactures several lines of decorative and functional metal items. The most recent order has been for 1200 door lock units for an apartment complex developer. The sales and production departments must work together to determine delivery schedules. Each lock unit consists of three components: the knob and face plate, the actual lock itself, and a set of two keys. Although the processes used in the manufacture of the three components vary, there are three areas where the production manager is concerned about the availability of resources. These three areas, their usage by the three components, and their availability are detailed in the table. $$\begin{array} { l c c c c } \text { Resource } & \text { Knob and Plate } & \text { Lock } & \text { Key (each) } & \text { Available } \\\hline \text { Brass Alloy } & 12 & 5 & 1 & 15000 \text { units } \\\text { Machining } & 18 & 20 & 10 & 36000 \text { minutes } \\\text { Finishing } & 15 & 5 & 1 & 12000 \text { minutes }\end{array}$$ A quick look at the amounts available confirms that Larkin does not have the resources to fill this contract. A subcontractor, who can make an unlimited number of each of the three components, quotes the prices below. $$\begin{array} { l c c } \text { Component } & \text { Subcontractor Cost } & \text { Larkin Cost } \\\hline \text { Knob and Plate } & 10.00 & 6.00 \\\text { Lock } & 9.00 & 4.00 \\\text { Keys (set of 2) } & 1.00 & .50\end{array}$$ Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.

Question 26

Describe some common feature of multiperiod financial planning models.

Question 27

Discuss the need for the use of judgment or other subjective methods in mathematical modeling.

Question 28

A constraint with non-zero slack will have a positive dual price, and a constraint with non-zero surplus will have a negative dual price.

Question 29

Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles. Current schedules yield the following information. $$\begin{array} { l | c c c | c | c } \text { Component } & \begin{array} { c } \text { Requirements } \\\text { Plastic }\end{array} & \text { Time } & \text { Space } & \begin{array} { c } \text { Cost to } \\\text { Manufacture }\end{array} & \begin{array} { c } \text { Cost to } \\\text { Purchase }\end{array} \\\hline \text { Front } & 3 & 10 & 2 & 8 & 12 \\\text { Seat/Frame } & 4 & 6 & 2 & 6 & 9 \\\text { Rear wheel (each) } & .5 & 2 & 1 & 1 & 3\\\text { Available }&50000&160000&30000\end{array}$$ The company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles so has gathered purchase information for each component. Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased in order to provide 12000 fully completed tricycles at the minimum cost.

Question 30

Why should decision makers who are primarily concerned with marketing or finance or production know about linear programming?

Question 31

Evans Enterprises has bought a prime parcel of beachfront property and plans to build a luxury hotel. After meeting with the architectural team, the Evans family has drawn up some information to make preliminary plans for construction. Excluding the suites, which are not part of this decision, the hotel will have four kinds of rooms: beachfront non-smoking, beachfront smoking, lagoon view non-smoking, and lagoon view smoking. In order to decide how many of each of the four kinds of rooms to plan for, the Evans family will consider the following information.
a.After adjusting for expected occupancy, the average nightly revenue for a beachfront non-smoking room is $175.The average nightly revenue for a lagoon view non-smoking room is $130.Smokers will be charged an extra $15.
b.Construction costs vary.The cost estimate for a lagoon view room is $12,000 and for a beachfront room is $15,000.Air purifying systems and additional smoke detectors and sprinklers ad $3000 to the cost of any smoking room.Evans Enterprises has raised $6.3 million in construction guarantees for this portion of the building.
c.There will be at least 100 but no more than 180 beachfront rooms.
d.Design considerations require that the number of lagoon view rooms be at least 1.5 times the number of beachfront rooms, and no more than 2.5 times that number.
e.Industry trends recommend that the number of smoking rooms be no more than 50% of the number of non-smoking rooms.Develop the linear programming model to maximize revenue.

Question 32

G and P Manufacturing would like to minimize the labor cost of producing dishwasher motors for a major appliance manufacturer. Although two models of motors exist, the finished models are indistinguishable from one another; their cost difference is due to a different production sequence. The time in hours required for each model in each production area is tabled here, along with the labor cost. $$\begin{array} { c | c c } & \text { Model 1 } & \text { Model 2 } \\\hline \text { Area } A & 15 & 3 \\\text { Area B } & 4 & 10 \\\text { Area C } & 4 & 8 \\\hline \text { Cost } & 80 & 65\end{array}$$ Currently labor assignments provide for 10,000 hours in each of Areas A and B and 18000 hours in Area C. If 2000 hours are available to be transferred from area B to Area A, 3000 hours are available to be transferred from area C to either Areas A or B, develop the linear programming model whose solution would tell G&P how many of each model to produce and how to allocate the workforce.

Question 33

Information on a prospective investment for Wells Financial Services is given below. $$\begin{array} { l | c c c c } & { \text { Period } } \\& 1 & 2 & 3 & 4 \\\hline \text { Loan Funds Available } & 3000 & 7000 & 4000 & 5000 \\\text { Investment Income } & & & & \\\text { (\% of previous period's investment) } & & 110 \% & 112 \% & 113 \% \\\text { Maximum Investment } & 4500 & 8000 & 6000 & 7500 \\\text { Payroll Payment } & 100 & 120 & 150 & 100\end{array}$$ In each period, funds available for investment come from two sources: loan funds and income from the previous period's investment. Expenses, or cash outflows, in each period must include repayment of the previous period's loan plus 8.5% interest, and the current payroll payment. In addition, to end the planning horizon, investment income from period 4 (at 110% of the investment) must be sufficient to cover the loan plus interest from period 4. The difference in these two quantities represents net income, and is to be maximized. How much should be borrowed and how much should be invested each period?

Question 34

A&C Distributors is a company that represents many outdoor products companies and schedules deliveries to discount stores, garden centers, and hardware stores. Currently, scheduling needs to be done for two lawn sprinklers, the Water Wave and Spring Shower models. Requirements for shipment to a warehouse for a national chain of garden centers are shown below. $$\begin{array} { c | c | c | c | c | c } \text { Month } & \begin{array} { c } \text { Shipping } \\\text { Capacity }\end{array} & \text { Product } & \begin{array} { c } \text { Minimum } \\\text { Requirement }\end{array} & \begin{array} { c } \text { Unit Cost } \\\text { to Ship }\end{array} & \begin{array} { c } \text { Per Unit } \\\text { Inventory Cost }\end{array} \\\hline \text { March } & 8000 & \text { Water Wave } & 3000 & .30 & .06 \\& & \begin{array} { c } \text { Spring } \\\text { Shower }\end{array} & 1800 & .25 & .05 \\\hline \text { April } & 7000 & \begin{array} { c } \text { Water Wave } \\\text { Spring } \\\text { Shower }\end{array} & \begin{array} { c } 4000\\\\4000\\\end{array} & \begin{array} { c } .40\\\\.30\\\end{array} &\begin{array} { c } .09\\\\.06\\\end{array} \\\hline \text { May } & 6000 & \text { Water Wave } & 5000 & .50 & .12 \\& & \text { Spring shower } & 2000 & .35 & .07\end{array}$$ Let S_ij be the number of units of sprinkler i shipped in month j, where i = 1 or 2, and j = 1, 2, or 3. Let W_ij be the number of sprinklers that are at the warehouse at the end of a month, in excess of the minimum requirement.
a.Write the portion of the objective function that minimizes shipping costs.
b.An inventory cost is assessed against this ending inventory.Give the portion of the objective function that represents inventory cost.
c.There will be three constraints that guarantee, for each month, that the total number of sprinklers shipped will not exceed the shipping capacity.Write these three constraints.
d.There are six constraints that work with inventory and the number of units shipped, making sure that enough sprinklers are shipped to meet the minimum requirements.Write these six constraints.

Question 35

An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below. $$\begin{array} { c c c c } \text { Medium } & \text { Cost Per Ad } & \text { \# Reached } & \text { Exposure Quality } \\\hline \text { TV } & 500 & 10000 & 30 \\\text { Radio } & 200 & 3000 & 40 \\\text { Newspaper } & 400 & 5000 & 25\end{array}$$ If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000, develop the model that will maximize the number reached and achieve an exposure quality if at least 1000.

Question 36

The Meredith Ribbon Company produces paper and fabric decorative ribbon which it sells to paper products companies and craft stores. The demand for ribbon is seasonal. Information about projected demand and production for a particular type of ribbon is given. $\begin{array}{llll}&\text { Demand (yards) } & \text { Froduction Cost Per Yard} & \text { Froduction Capacity (yards) }\\\hline\text { Quarter 1 } & 10,000 & .03 & 30,000 \\\text { Quarter 2 } & 18,000 & .04 & 20,000 \\\text { Quarter 3 } & 16,000 & .06 & 20,000 \\\text { Quarter 4 } & 30,000 & .08 & 15,000\end{array}$ An inventory holding cost of $.005 is levied on every yard of ribbon carried over from one quarter to the next.
a.Define the decision variables needed to model this problem.
b.The objective is to minimize total cost, the sum of production and inventory holding cost.Give the objection function.
c.Write the production capacity constraints.
d.Write the constraints that balance inventory, production, and demand for each quarter.Assume there is no beginning inventory in quarter 1.
e.To attempt to balance the production and avoid large changes in the workforce, production in period 1 must be within 5000 yards of production in period 2.Write this constraint.

Question 37

Island Water Sports is a business that provides rental equipment and instruction for a variety of water sports in a resort town. On one particular morning, a decision must be made of how many Wildlife Raft Trips and how many Group Sailing Lessons should be scheduled. Each Wildlife Raft Trip requires one captain and one crew person, and can accommodate six passengers. The revenue per raft trip is $120. Ten rafts are available, and at least 30 people are on the list for reservations this morning. Each Group Sailing Lesson requires one captain and two crew people for instruction. Two boats are needed for each group. Four students form each group. There are 12 sailboats available, and at least 20 people are on the list for sailing instruction this morning. The revenue per group sailing lesson is $160. The company has 12 captains and 18 crew available this morning. The company would like to maximize the number of customers served while generating at least $1800 in revenue and honoring all reservations.

Question 38

What benefits exist in using linear programming for production scheduling problems?