Deck 9: Linear Programming Applications

Question 1

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.

Answer 1

B

Question 2

Revenue management methodology was originally developed for the banking industry.

Answer 2

False

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 > 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

D

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

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 6

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 7

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 8

Modern revenue management systems maximize revenue potential for an organization by helping to manage

A)pricing strategies.
B)reservation policies.
C)short-term supply decisions.
D)All of the alternatives are correct.

Question 9

Double-subscript notation for decision variables should be avoided unless the number of decision variables exceeds nine.

Question 10

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

Question 11

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

Question 12

A company makes two products from steel;one requires 2 tons of steel and the other requires 3 tons.There are 100 tons of steel available daily.A constraint on daily production could be written as: 2x₁ + 3x₂ < 100.

Question 13

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 14

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

Question 15

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 < 5
B)A - .5B + C < 0
C).5A - B - .5C < 0
D)-.5A + .5B - .5C < 0

Question 16

Portfolio selection problems should acknowledge both risk and return.

Question 17

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

Question 18

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

Question 19

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 < 100;P₁₄ - P₁₃ < 100
B)P₁₃ - P₁₄ < 100;P₁₃ - P₁₄ > 100
C)P₁₃ - P₁₄ < 100;P₁₄ - P₁₃ > 100
D)P₁₃ - P₁₄ > 100;P₁₄ - P₁₃ > 100

Question 20

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 < 400
B)P₁₂ + P₂₂ + P₃₂ < 400
C)P₃₂ < 400
D)P₂₃ < 400

Question 21

The SMM Company,which is manufacturing a new instant salad machine,has $280,000 to spend on advertising.The product is only to be test marketed initially in the Dallas area.The money is to be spent on an advertising blitz during one weekend (Friday,Saturday,and Sunday)in January,and SMM is limited to television advertising.
The company has three options available: daytime advertising,evening news advertising and the Super Bowl.Even though the Super Bowl is a national telecast,the Dallas Cowboys will be playing in it,and hence,the viewing audience will be especially large in the Dallas area.A mixture of one-minute TV spots is desired.The table below gives pertinent data:
SMM has decided to take out at least one ad in each option.Further,there are only two Super Bowl ad spots available.There are 10 daytime spots and 6 evening news spots available daily.SMM wants to have at least 5 ads per day,but spend no more than $50,000 on Friday and no more than $75,000 on Saturday.Formulate and solve a linear program to help SMM decide how the company should advertise over the weekend.

Question 22

For the multiperiod production scheduling problem in the textbook,period n - 1's ending inventory variable was also used as period n's beginning inventory variable.

Question 23

The primary limitation of linear programming's applicability is the requirement that all decision variables be nonnegative.

Question 24

National Wing Company (NWC)is gearing up for the new B-48 contract.Currently NWC has 100 equally qualified workers.Over the next three months NWC has made the following commitments for wing production:
Each worker can either be placed in production or can train new recruits.A new recruit can be trained to be an apprentice in one month.The next month,he,himself,becomes a qualified worker (after two months from the start of training).Each trainer can train two recruits.The production rate and salary per employee is estimated below.
At the end of July,NWC wishes to have no recruits or apprentices but have at least 140 full-time workers.Formulate and solve a linear program for NWC to accomplish this at minimum total cost.

Question 25

Winslow Savings has $20 million available for investment.It wishes to invest over the next four months in such a way that it will maximize the total interest earned over the four month period as well as have at least $10 million available at the start of the fifth month for a high rise building venture in which it will be participating.
For the time being,Winslow wishes to invest only in 2-month government bonds (earning 2% over the 2-month period)and 3-month construction loans (earning 6% over the 3-month period).Each of these is available each month for investment.Funds not invested in these two investments are liquid and earn 3/4 of 1% per month when invested locally.
Formulate and solve a linear program that will help Winslow Savings determine how to invest over the next four months if at no time does it wish to have more than $8 million in either government bonds or construction loans.

Question 26

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.
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.
Develop a linear programming model that would tell Larkin how to fill the order for 1200 lock sets at the minimum cost.

Question 27

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.

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 28

Compared to the problems in the textbook,real-world problems generally require more variables and constraints.

Question 29

A company makes two products,A and B.A sells for $100 and B sells for $90.The variable production costs are $30 per unit for A and $25 for B.The company's objective could be written as: MAX 190x₁ - 55x₂.

Question 30

Information on a prospective investment for Wells Financial Services is given below.
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 31

A decision maker would be wise to not deviate from the optimal solution found by an LP model because it is the best solution.

Question 32

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.
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 33

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.
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 34

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.
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 35

Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the right-hand side is acceptable since both are a measure of time.

Question 36

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.
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 37

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.
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 38

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 39

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.

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 40

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.
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 41

John Sweeney is an investment advisor who is attempting to construct an "optimal portfolio" for a client who has $400,000 cash to invest.There are ten different investments,falling into four broad categories that John and his client have identified as potential candidates for this portfolio.
The following table lists the investments and their important characteristics.Note that Unidyde Equities (stocks)and Unidyde Debt (bonds)are two separate investments,whereas First General REIT is a single investment that is considered both an equities and a real estate investment.
Formulate and solve a linear program to accomplish John's objective as an investment advisor which is to construct a portfolio that maximizes his client's total expected after-tax return over the next year,subject to a number of constraints placed upon him by the client for the portfolio:1.Its (weighted)average liquidity factor must be at least 65.2.The (weighted)average risk factor must be no greater than 55.3.At most,$60,000 is to be invested in Unidyde stocks or bonds.4.No more than 40% of the investment can be in any one category except the money category.5.No more than 20% of the investment can be in any one investment except the money market fund.6.At least $1,000 must be invested in the money market fund.7.The maximum investment in All Saver's Certificates is $15,000.8.The minimum investment desired for debt is $90,000.9.At least $10,000 must be placed in a T-Bill account.

Question 42

BP Cola must decide how much money to allocate for new soda and traditional soda advertising over the coming year.The advertising budget is $10,000,000.Because BP wants to push its new sodas,at least one-half of the advertising budget is to be devoted to new soda advertising.However,at least $2,000,000 is to be spent on its traditional sodas.BP estimates that each dollar spent on traditional sodas will translate into 100 cans sold,whereas,because of the harder sell needed for new products,each dollar spent on new sodas will translate into 50 cans sold.
To attract new customers BP has lowered its profit margin on new sodas to 2 cents per can as compared to 4 cents per can for traditional sodas.How should BP allocate its advertising budget if it wants to maximize its profits while selling at least 750 million cans?