Formulation of Linear Programming Models

Definition of TermsLinear Programming (LP) is a deterministic mathematical technique designed to assist an organization in allocating scarce resources or in choosing from among many alternatives.Programming refers to planning or selecting the best solution to a problem using mathematical techniques.

Formulation is the process of developing the mathematical model

Requirements of an LP ProblemThere must be be process or decision variables from which the decision maker can operate at different levels All problems seek to maximize or minimize some quantityLP problems have constraints, that limit the degree to which an objective can be pursuedThe objectives and constraints must be expressed in terms of linear equations and inequalities

Basic Assumptions of LPProportionality exists in the objective and constraintsAdditivity dictates that the total of all activities equals the sum of the individual activitiesDivisibility states that solutions need not be whole numbers.Certainty assumes all parameters are known and certain.

Typical LP ModelsProduct Mix Feed Mix Production SchedulingInvestment PortfolioBlendingTrim LossOthers

The General LP ModelMax/Min Z = c1 x1 + c2x2 + . . . + cnxn

subject to: A11x1 + A12x2 + . . . + A1nxn

A Product Mix ProblemThe Opti Mize Company manufactures two products that compete for the same (limited) resources. Relevant information is:ProductABAvailable resources

Labor-hrs/unit1220 hrs/dayMachine hrs/unit2230 hrs/dayCost/unit$6$20$180/day

Profit/unit$5$15

The ModelLet X1 = number of units of product A to manufacture X2 = number of units of product B to manufacture

Max Profit = z = 5 X1 + 15 X2

subject to: X1 + 2 X2

ShirtStop makes logo tees and sells them in its chain of retail stores. It contracts with two different plantsone in Lyon and one in Rennes. The tees from the Lyon plant cost 0.46 apiece, and 9% of them are defective and cannot be sold. The tees from the Rennes plant cost only 0.35 each, but they have an 18% defective rate. ShirtStop needs 3,500 tees. To retain its relationship with the two plants, it wants to order at least 1,000 shirts from each. It would also like at least 88% of the shirts it receives to be salable.

Suppose ShirtStop decides it wants to minimize the defective tees while keeping costs below 2,000.

A canning company produces two sizes of cansregular and large. The cans are produced in 10,000-can lots. The cans are processed through a stamping operation and a coating operation. The company has 30 days available for each stamping and coating. A lot of regular-size cans requires 2 days to stamp and 4 days to coat, whereas a lot of large cans requires 4 days to stamp and 2 days to coat. A lot of regular-size cans earns $800 profit, and a lot of large-size cans earns $900 profit. In order to fulfill its obligations under a shipping contract, the company must produce at least nine lots altogether. The company want to maximize profit.

A Blending ProblemThe B. A. Nutt Company sells mixed nuts of two quality levels. The expensive mix should not contain more than 25% peanuts nor less than 40% cashews. The cheap mix should not have more than 60% peanuts and no less than 20% cashews. Cashews cost 50 cents a pound and peanuts cost 20 cents a pound. The expensive mix sells for 80 cents a pound and the cheap mix for 40 cents a pound. What should the blend of each mix be in order to maximize profit. The company has $100 a day with which to purchase nuts.

A Marketing ExampleThe I. B. Adman Advertising Company is planning a large media blitz covering television, radio, and magazines to sell management science to the public. The companys objective is to reach as many people as possible. Results of a market survey show:TelevisionDay time Prime Time Radio Magazines

cost per unit$40,000 $75,000 $30,000 $15,000# people400,000 900,000 500,000 200,000# business300,000 400,000 200,000 100,000

The company has a budget of $800,000 to spend on the campaign. It requires at least two million exposures among the business community. Television must be limited to $500,000, at 3 units of day time and 2 units of prime time must be purchased. Advertising units on both radio and magazines should be between 5 and 10.

Trim-Loss ProblemThe I. M. Torn Paper Company produces rolls of paper 12 inches wide by 1000 feet in length. These standard rolls are purchased by many of their customers. However, some customers prefer to receive special sizes, namely 2-inch, 3.5 inch, and 5-inch rolls, all 1000 feet long. The special sized rolls are cut from the standard 12-inch roll.Alternative cutssize of rolls2-in3.5-in5-inwaste16000210203220140210530116410.5required/mo50020001500

Job-TrainingThe Never-Say-Die Life Insurance Company hires and trains a large number of salespersons each month to replacethose who have departed. Trained salespersons must be usedto train new salespersons. Training takes one month and thereis a 20 percent attrition rate by the end of the month. While asalesperson is training a new employee, that person cannot beused in the field selling insurance. The monthly demand for experienced salespeople is:MonthDemand (in the field)January100February150March200April225May175

Trainees receive $400 per month while it costs the company $850 permonth for an experienced salesperson. Ten percent of all experiencedsalespeople will leave the company by the end of each month.