Exam 9: Linear Programming Applications

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.

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

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,

Media selection problems usually determine

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

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

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.

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

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.

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

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.

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.

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.

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.

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

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.

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.

Revenue management methodology was originally developed for the banking industry.

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.

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