Introduction to Management Science, 11e (Taylor)Chapter 4 Linear Programming: Modeling Examples

1) Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem. Answer: TRUEDiff: 2 Page Ref: 112-116Section Heading: A Product Mix ExampleKeywords: formulation, product mix problemAACSB: Analytic skills

2) Product mix problems cannot have greater-than-or-equal-to (≥) constraints. Answer: FALSEDiff: 2 Page Ref: 112-116Section Heading: A Product Mix ExampleKeywords: product mixAACSB: Analytic skills

3) When using a linear programming model to solve the diet problem, the objective is generally to maximize profit. Answer: FALSEDiff: 2 Page Ref: 117-119Section Heading: A Diet ExampleKeywords: objective functionAACSB: Analytic skills

4) When using a linear programming model to solve the diet problem, the objective is generally to maximize nutritional content.Answer: FALSEDiff: 2 Page Ref: 117-119Section Heading: A Diet ExampleKeywords: objective functionAACSB: Analytic skills

5) In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories. Answer: FALSEDiff: 2 Page Ref: 117-119Section Heading: A Diet ExampleKeywords: formulation, diet exampleAACSB: Analytic skills

6) Solutions to diet problems in linear programming are always realistic. Answer: FALSEDiff: 2 Page Ref: 117-119Section Heading: A Diet ExampleKeywords: diet example

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7) Diet problems usually maximize nutritional value. Answer: FALSEDiff: 2 Page Ref: 117-119Section Heading: A Diet ExampleKeywords: diet example

8) In most media selection decisions, the objective of the decision maker is to minimize cost. Answer: FALSEDiff: 2 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selection

9) In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure. Answer: TRUEDiff: 2 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selection

10) A linear programming model of a media selection problem is used to determine the relative value of each advertising media. Answer: FALSEDiff: 3 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selection

11) In a media selection problem, maximization of audience exposure may not result in maximization of total profit. Answer: TRUEDiff: 2 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selection

12) In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities. Answer: TRUEDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulationAACSB: Analytic skills

13) The constraint x + y = z is written in standard form.Answer: FALSEDiff: 2 Page Ref: 121Section Heading: An Investment ExampleKeywords: standard form

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14) In an unbalanced transportation model, supply does not equal demand, and supply constraints have ≤ signs. Answer: TRUEDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulationAACSB: Analytic skills

15) Transportation problems can have solution values that are non-integer and must be rounded. Answer: FALSEDiff: 3 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, solutionAACSB: Analytic skills

16) In a transportation problem, the supply constraint represents the maximum amount of product available for shipment or distribution at a given source (plant, warehouse, mill). Answer: TRUEDiff: 1 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulation

17) In a transportation problem, a supply constraint (the maximum amount of product available for shipment or distribution at a given source) is a greater-than-or equal-to constraint (≥). Answer: FALSEDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulationAACSB: Analytic skills

18) In a transportation problem, a demand constraint for a specific destination represents the amount of product demanded by a given destination (customer, retail outlet, store). Answer: TRUEDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulation

19) A conservative approach to a balanced transportation model would be to make all constraints less-than-or-equal-to constraints.Answer: FALSEDiff: 2 Page Ref: 131Section Heading: A Transportation ExampleKeywords: balanced transportation model, transportation model formulation

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20) In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤). Answer: FALSEDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulation

21) Blending problems usually require algebraic manipulation in order to write the LP in "standard form."Answer: TRUEDiff: 1 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blending

22) Fractional relationships among variables are considered standard form in a blending problem.Answer: FALSEDiff: 1 Page Ref: 134Section Heading: A Blend ExampleKeywords: blending

23) Data envelopment analysis indicates which type of service unit makes the highest profit. Answer: FALSEDiff: 1 Page Ref: 142-146Section Heading: A Data Envelopment Analysis ExampleKeywords: blendingAACSB: Analytic skills

24) In a classic blending problem, revenue is maximized by subtracting cost from profit.Answer: FALSEDiff: 1 Page Ref: 133Section Heading: A Blend ExampleKeywords: blendingAACSB: Analytic skills

25) Data envelopment analysis problems are usually maximization problems.Answer: TRUEDiff: 1 Page Ref: 143Section Heading: A Data Envelopment Analysis ExampleKeywords: data envelopment analysis

26) Data envelopment analysis indicates the relative ________ of a service unit compared with others. Answer: efficiency or productivityDiff: 2 Page Ref: 142-146Section Heading: A Data Envelopment Analysis ExampleKeywords: data envelopment analysis

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27) ________ types of linear programming problems often result in fractional relations between variables which must be eliminated. Answer: Blending Diff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blending

28) Cranky Jerry's Furniture Factory makes tables and chairs. If he is interested in a profit maximizing level of production, he should probably follow the example for the ________ problem found in this chapter.Answer: product mixDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: product mix

29) When the ________ command is used in an Excel spreadsheet, all the values in a column (or row) are mutliplied by the values in another column (or row) and then summed. Answer: SUMPRODUCT Diff: 2 Page Ref: 119Section Heading: A Diet ExampleKeywords: spreadsheet solution

30) Cranky Jerry's Day Care wants to minimize their food cost while meeting the minimum (and I mean bare minimum) guidelines for nutrition as set forth by the state. The best approach would be to follow the example in this chapter for a(n) ________ problem.Answer: dietDiff: 2 Page Ref: 117-120Section Heading: A Diet ExampleKeywords: diet example

31) For product mix problems, the constraints are usually associated with ________. Answer: resources or time Diff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: product mix

32) The ________ for the computer solution of a linear programming problem requires all variables on the left side, and all numerical values on the right side of the inequality or equality sign. Answer: standard form Diff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulation, constraints

33) The objective function of a diet problem is usually to ________ subject to nutritional requirements. Answer: minimize costsDiff: 1 Page Ref: 117-120Section Heading: A Diet ExampleKeywords: diet problemAACSB: Analytic skills

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34) Investment problems maximize ________. Answer: return on investments Diff: 1 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: investment

35) In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the ________. Answer: audience exposure Diff: 2 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selectionAACSB: Analytic skills

36) In a(n) ________ problem, maximization of audience exposure may not result in maximization of total profit. Answer: media selection Diff: 3 Page Ref: 126-129Section Heading: A Marketing ExampleKeywords: marketing problem, media selection

37) In a balanced transportation model, supply equals ________. Answer: demandDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulation

38) An unbalanced transportation model must have at least one ________ constraint.Answer: less-than-or-equal-toDiff: 2 Page Ref: 131Section Heading: A Transportation ExampleKeywords: transportation problem, formulation

39) A balanced transportation model should have ________ constraints.Answer: = or "equal to"Diff: 2 Page Ref: 131Section Heading: A Transportation ExampleKeywords: transportation problem, formulationAACSB: Analytic skills

40) There are three plants scattered across the United States that manufacture Dull computers. These plants assemble products for customers throughout the United States, Canada, and Mexico. If Dull wishes to maximize profit by choosing the most economical pair of factory and customer for each order, they would be well-advised to follow the ________ model presented in this chapter.Answer: transportation problemDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation model

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41) In a(n) ________ transportation problem, supply exceeds demand.Answer: unbalancedDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation problem, formulationAACSB: Analytic skills

The owner of Chips etc. produces two kinds of chips: lime (L) and vinegar (V). He has a limited amount of the three ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of lime chips are $0.40, and for a bag of vinegar chips $0.50.

42) What is the formulation for this problem?Answer: MAX Z = 0. 4L + 0.5Vs.t.

2L + 3V ≤ 48006L + 8V ≤ 96001L + 2V ≤ 2000

Diff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: computer solutionAACSB: Analytic skills

43) For the production combination of 800 bags of lime and 600 bags of vinegar, which resource is not completely used up and how much is remaining? Answer: salt only, 1400 ounces remaining Diff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: slack, computer solutionAACSB: Analytic skills

44) The production combination of 450 bags of lime and 800 bags of vinegar is not feasible because one resource is exceeded. Which resource is exceeded and how much more is needed to produce this combination? Answer: herbs only, 50 ounces are needed Diff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: slack, computer solutionAACSB: Analytic skills

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A croissant shop produces two products: bear claws (B) and almond-filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS (tablespoons) of almond paste. An almond-filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. The shop must produce at least 400 almond-filled croissants due to customer demand. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each.

45) This represents what type of linear programming application?Answer: product mixDiff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: computer solution

46) What is the formulation for this problem? Answer: MAX Z = $.20B + $.30Cs.t.

6B + 3C ≤ 66001B + 1C ≤ 14002B + 4C ≤ 4800

C ≥ 400Diff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulation, constraintsAACSB: Analytic skills

47) For the production combination of 600 bear claws and 800 almond-filled croissants, how much flour and almond paste are remaining? Answer: flour = 600 ounces, almond paste = 400 TS Diff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: slack, computer solutionAACSB: Analytic skills

48) If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company's three products in period 2 is 400.Answer: X12 + X22 + X32 ≤ 400 Diff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: transportation problem, supply constraintAACSB: Analytic skills

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49) Small motors for garden equipment are produced at four manufacturing facilities and need to be shipped to three plants that produce different garden items (lawn mowers, rototillers, leaf blowers). The company wants to minimize the cost of transporting items between the facilities, taking into account the demand at the three different plants, and the supply at each manufacturing site. The table below shows the cost to ship one unit between each manufacturing facility and each plant, as well as the demand at each plant and the supply at each manufacturing facility.

Write the formulation for this problem.Answer: MIN Z = 4x1A + 4.5x1B + 3.2x1C + 3.5x2A + 3x2B + 4x2C + 4x3A + 3.5x3B + 4.25x3Cs.t.

x1A + x1B +x1C = 200x2A + x2B +x2C = 200x3A + x3B +x3C = 300x1A + x2A +x3A = 250x1B + x2B +x3B = 150x1C + x2C +x3C = 200

Diff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: computer solution, transportation/distributionAACSB: Analytic skills

50) Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How much oil-based and water-based paint should the Quickbrush make? Answer: 9167 gallons of water-base paint and 5833 gallons of oil-base paint Diff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blendingAACSB: Analytic skills

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Andy Tyre manages Tyre's Wheels, Inc. Andy has received an order for 1000 standard wheels and 1200 deluxe wheels for next month, and for 750 standard wheels and 1000 deluxe wheels the following months. He must fill all the orders. The cost of regular time production for standard wheels is $25 and for deluxe wheels, $40. Overtime production costs 50% more. For each of the next two months there are 1000 hours of regular time production and 500 hours of overtime production available. A standard wheel requires .5 hour of production time and a deluxe wheel, .6 hour. The cost of carrying a wheel from one month to the next is $2.

51) Define the decision variables and objective function for this problem.Answer: Define the decision variables:S1R = number of standard wheels produced in month 1 on regular time productionS1O = number of standard wheels produced in month 1 on overtime productionS2R = number of standard wheels produced in month 2 on regular time productionS2O = number of standard wheels produced in month 2 on overtime productionD1R = number of deluxe wheels produced in month 1 on regular time productionD1O = number of deluxe wheels produced in month 1 on overtime productionD2R = number of deluxe wheels produced in month 2 on regular time productionD2O = number of deluxe wheels produced in month 2 on overtime productionY1 = number of standard wheels stored from month 1 to month 2.Y2 = number of deluxe wheels stored from month 1 to month 2.MIN 25 S1R + 37.5 S1O + 40 D1R + 60 D1O + 25 S2R + 37.5 S2O + 40 D2R + 60 D2O + 2 Y1 + 2 Y2Diff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: linear programming multiperiod schedulingAACSB: Analytic skills

52) Write the constraints for this problem.Answer: S1R + S1O - Y1 = 1000D1R + D1O - Y2 = 1200S2R + S2O + Y1 = 750D2R + D2O + Y2 = 1000

.5 S1R + .6 D1R ≤ 1000

.5 S1O + .6 D1O ≤ 500

.5 S2R + .6 D2R ≤ 1000

.5 S2O + .6 D12O ≤ 500

Diff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: linear programming multiperiod schedulingAACSB: Analytic skills

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Bullseye Shirt Company makes three types of shirts: athletic, varsity, and surfer. The shirts are made from different combinations of cotton and rayon. The cost per yard of cotton is $5 and the cost for rayon is $7. Bullseye can receive up to 4000 yards of cotton and 3000 yards of rayon per week.

The table below shows relevant manufacturing information:

Shirt

Total Yards of Fabric per Shirt

Fabric Requirement

Minimum Weekly Contracts

Maximum Demand

SellingPrice

Athletic 1.00at least 60% cotton 500 600 $30

Varsity 1.20no more than 30% rayon 650 850 $40

Surfer 0.90as much as 80% cotton 300 700 $36

53) Assume that the decision variables are defined as follows:A = total number of athletic shirts producedV = total number of varsity shirts producedS = total number of surfer shirts producedC = yards of cotton purchasedR = yards of rayon purchasedXij = yards of fabric i (C or R) blended into shirt j (A, V or S)

Write the objective function.Answer: Max 30 A + 40 V + 36 S - 5C - 7RDiff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: objective function, model constructionAACSB: Analytic skills

54) Write the constraints for the fabric requirements.Answer: Form of constraints: Total yards used is greater than (or less than) total yards required × (% fabric required) shirts produced.XCA ≥ 0.6 AXVR ≤ 0.36VXSC ≤ 0.72 SDiff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blendingAACSB: Analytic skills

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55) Write the constraints for the total number of shirts of each style produced.Answer: Form of constraint: number of shirts produced = (total yards used to make the shirt)/(yards/shirt)A = (XCA + XRA)/1V = (XCV + XRV)/1.2S = (XCS + XRS)/0.9

Standard form:A - XCA - XRA = 01.2 V - XCV - XRV = 00.9 S - XCS - XRS = 0Diff: 3 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blendingAACSB: Analytic skills

56) Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel's cat food is made by mixing two types of cat food to obtain the "nutritionally balanced cat diet." The data for the two cat foods are as follows:

Kitty Kennels wants to be sure that the cats receive at least 5 ounces of protein and at least 3 ounces of fat per day. What is the cost of this plan, and how much fat and protein do the cats receive? Answer: Cost is $3.60, which uses 14 ounces of Meow Munch and 5.33 ounces of Feline Feed. Diff: 2 Page Ref: 117-120Section Heading: A Diet ExampleKeywords: dietAACSB: Analytic skills

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Spring is right around the corner and that can mean only one thing: it's time to incubate some eggs for a fresh crop of chickens to supplement the existing flock for the year. There are three breeds that are popular: Leghorns, Buff Orpingtons, and Cochins, and each has its own strengths and weaknesses. The Leghorns, for example, are superb layers but easily excitable. The Cochins, on the other hand, are very even-tempered but not the best layers. They do have marvelous plumage, with feathers that extend down to their feet. The Buff Orpingtons are good layers, have interesting plumage, and are mid-range in their temperament.

The chicken farmer would like this crop of chickens to produce as many eggs as possible while keeping the noise to a dull roar and having a nice array of birds free-ranging on his lawn during those lazy summer days. He has put the relevant data in table form. Plumage numbers are on a scale from 1-10, with 10 being the most desirable. The egg output is not on a scale, but is instead the average output for the breed, based on years of collecting eggs.

Bird Leghorn Cochin Buff OrpingtonEggs/week 5 2 4.5Plumage 3 9 6Temperment 2 8 5Fertilizer 3 2 4Appetite 5 4 8

Temperament is actually measured by the average volume of cackling, clucking, and crowing and is measured in decibels per bird. Appetite is measured in ounces of layer pellets per week consumed by each of the breeds, while fertilizer is measured as the output in ounces per week.

57) How should the objective function read?Answer: Max Z = 5L + 2C + 4.5BDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: objective functionAACSB: Analytic skills

58) What is a full set of constraints if the farmer wants this flock to produce less than 100 decibels of noise and more than 5 pounds of fertilizer, consume less than 10 pounds of layer pellets, and achieve a total plumage score of at least 75?Answer: 3L + 9C+ 6B >= 752L + 8C + 5B <= 1003L + 2C + 4B >= 805L + 4C + 8B <= 160Diff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: constraints, formulationAACSB: Analytic skills

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59) Formulate the objective function and constraints for this scenario.Answer: Max Z = 5L + 2C + 4.5B3L + 9C+ 6B >= 75Subject to:2L + 8C + 5B <= 1003L + 2C + 4B >= 805L + 4C + 8B <= 160Diff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulationAACSB: Analytic skills

60) The poultry farmer discovers, much to his dismay, that the product mix that satisfies all of his constraints isn't really a mix of birds--it's a flock consisting of nothing but 32 Leghorns. (His constraints were a plumage score greater than 75, fertilizer production greater than 80 ounces per week, temperment less than 100 decibels, and an appetite less than 160 ounces of layer pellets per week.) Not only is his flock going to be a very one-note mix in terms of color, he is afraid that the gleaming white feathers and deafening noise will attract predators, or worse yet, his neighbors. He immediately embarks on a breeding program to develop a Super-Cochin that can lay 4.5 eggs per week with all other performance measures the same. His optimal solution is now 27.5 Leghorns and 5.625 Super-Cochins. Assuming that he is willing to settle for a flock that has 27 Leghorns and 5 Super Cochins, what is the weekly output of eggs and performance with respect to the constraints?Answer: He should get 27∗5 + 4.5∗5 = 157.5 eggs per weekThe plumage score is now 126.The fertilizer output is 91 ounces per week,The temperment score is 94.The appetite is 155 ounces of layer pellets per week.Diff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: product mixAACSB: Analytic skills

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61) A credit union wants to make investments in the following:

The firm will have $2,500,000 available for investment during the coming year. The following restrictions apply: ∙ Risk-free securities may not exceed 30% of the total funds, but must comprise at least 5% of the total. ∙ Signature loans may not exceed 12% of the funds invested in all loans (vehicle, consumer, other secured loans, and signature loans) ∙ Consumer loans plus other secured loans may not exceed the vehicle loans ∙ Other secured loans plus signature loans may not exceed the funds invested in risk-free securities.

How should the $2,500,000 be allocated to each alternative to maximize annual return? What is the annual return?

Answer:

Diff: 3 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: investmentAACSB: Analytic skills

62) When systematically formulating a linear program, the first step is toA) construct the objective function. B) formulate the constraints. C) identify the decision variables. D) identify the parameter values. Answer: CDiff: 2 Page Ref: 113Section Heading: A Product Mix ExampleKeywords: formulation

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63) Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York Stock Exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in "stock 2". The constraint for this requirement can be written as: A) x2 ≥ .60 B) x2 ≥ .60 (x2 + x7 + x8) C) .4x2 - .6x7 - .6x8 ≤ 0 D) .4x2 - .6x7 - .6x8 ≥ 0 Answer: CDiff: 3 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: formulationAACSB: Analytic skills

64) The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B, which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient.

IngredientPercent per pound in Feed A

Percent per pound in Feed B

Minimum daily requirement (pounds)

1 20 24 302 30 10 503 0 30 204 24 15 605 10 20 40

The constraint for ingredient 3 is:A) .5A + .75B = 20 B) .3B = 20C) .3B≤ 20D) .3B ≥ 20Answer: DDiff: 2 Page Ref: 117-120Section Heading: A Diet ExampleKeywords: solutionAACSB: Analytic skills

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The owner of Chips etc. produces two kinds of chips: lime (L) and vinegar (V). He has a limited amount of the three ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of lime chips are $0.40, and for a bag of vinegar chips $0.50.

65) For the production combination of 800 bags of lime and 600 bags of vinegar, which of the three resources is (are) not completely used?A) flour only B) salt only C) herbs only D) salt and flour Answer: BDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: solution, slackAACSB: Analytic skills

66) What is the constraint for salt? A) 6L + 8V ≤ 4800 B) 1L + 2V ≤ 4800 C) 3L + 2V ≤ 4800 D) 2L + 3V ≤ 4800 Answer: DDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulation, constraintsAACSB: Analytic skills

67) Which of the following is not a feasible production combination? A) 0L and 0V B) 0L and 1000V C) 1000L and 0V D) 0L and 1200V Answer: DDiff: 1 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulation, feasibilityAACSB: Analytic skills

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68) If Xab = the production of product a in period b, then to indicate that the limit on production of the company's 3 products in period 2 is 400, we write:A) X32 ≤ 400 B) X21 + X22 + X23 ≤ 400 C) X12 + X22 + X32 ≤ 400 D) X12 + X22 + X32 ≥ 400 Answer: CDiff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: formulation, constraintsAACSB: Analytic skills

69) Balanced transportation problems have the following type of constraints: A) ≥B) ≤C) =D) <Answer: CDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: formulation, constraints

70) Compared to blending and product mix problems, transportation problems are unique because A) they maximize profit.B) the constraints are all equality constraints with no "≤" or "≥" constraints.C) they contain fewer variables.D) the solution values are always integers.Answer: DDiff: 2 Page Ref: 130-132Section Heading: A Transportation ExampleKeywords: transportation

71) The production manager for the Softy soft drink company is considering the production of two kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint? A) 2R + 4D ≤ 480 B) 2D + 4R ≤ 480 C) 2R + 3D ≤ 480 D) 3R + 2D ≤ 480 Answer: ADiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: formulation, constraintsAACSB: Analytic skills

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72) A croissant shop produces two products: bear claws (B) and almond-filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond-filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today's production run. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each. What is the optimal daily profit? A) $380 B) $400 C) $420 D) $440 Answer: ADiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: computer solutionAACSB: Analytic skills

73) The production manager for the Softy soft drink company is considering the production of two kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients), limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit?A) $270 B) $320 C) $420 D) $520 Answer: CDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: computer solutionAACSB: Analytic skills

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Spring is right around the corner and that can mean only one thing: it's time to incubate some eggs for a fresh crop of chickens to supplement the existing flock for the year. There are three breeds that are popular: Leghorns, Buff Orpingtons, and Cochins, and each has its own strengths and weaknesses. The Leghorns, for example, are superb layers but easily excitable. The Cochins, on the other hand, are very even-tempered but not the best layers. They do have marvelous plumage, with feathers that extend down to their feet. The Buff Orpingtons are good layers, have interesting plumage, and are mid-range in their temperament.

The chicken farmer would like this crop of chickens to produce as many eggs as possible while keeping the noise to a dull roar and having a nice array of birds free-ranging on his lawn during those lazy summer days. He has put the relevant data in table form. Plumage numbers are on a scale from 1-10, with 10 being the most desirable. The egg output is not on a scale, but is instead the average output for the breed, based on years of collecting eggs.

Bird Leghorn Cochin Buff OrpingtonEggs/week 5 2 4.5Plumage 3 9 6Temperment 2 8 5Fertilizer 3 2 4Appetite 5 4 8

Temperament is actually measured by the average volume of cackling, clucking, and crowing and is measured in decibels per bird. Appetite is measured in ounces of layer pellets per week consumed by each of the breeds, while fertilizer is measured as the output in ounces per week.

74) Which of the following is an appropriate objective function?A) Max Z = 5L + 2C + 4.5BB) Max Z = 10L + 10C + 10BC) Min Z = 5L + 2C + 4.5BD) Min Z = 10L + 10C + 10B

Answer: ADiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: objective, cost minimization, profit maximizationAACSB: Analytic skills

75) The poultry farmer would like to generate 80 ounces of fertilizer per week while feeding a maximum of 160 ounces of layer pellets each week. Which of these constraints is correct?A) 3L + 2C + 4C ≤ 80B) 5L + 4C + 8B ≤ 160C) 5L + 4C + 8B ≥ 160D) 5L + 4C +8B ≥ 80Answer: BDiff: 2 Page Ref: 112-115Section Heading: A Product Mix ExampleKeywords: constraint, model formulationAACSB: Analytic skills

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76) The poultry farmer has in mind the following levels for each of his metrics of interest: a plumage score greater than 75, fertilizer production greater than 80 ounces per week, temperment less than 100 decibels, and an appetite less than 160 ounces of layer pellets per week. When he runs his linear programming model, he discovers that his flock will consist entirely of Leghorn birds. He can picture the sad little faces of his children when he tells them that there will be no variety of birds gracing their front lawn this summer. Help him avoid the embarrassment by selecting a constraint that will ensure that there is some variety in his flock.A) 5L + 4C + 8B ≤ 160B) 5L + 4C + 8B ≥ 160C) C + B ≥ 5D) L + C + B ≥ 5 Answer: CDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: constraints, formulationAACSB: Analytic skills

77) When the poultry farmer runs his linear programming model, he discovers that his flock will consist entirely of Leghorn birds. He studies his sensitivity report (copied below) and decides to write a constraint that requires two Cochin chickens to be selected. If the current optimal mix of breeds results in 160 eggs per week, which of the following statements is best?

CellName

Final Value Reduced Cost

Objective Coefficient

Allowable Increase

Allowable Decrease

$B$1 Leghorn 32 0 5 1E+30 2.1875$C$1 Cochin 0 -2 2 3 1E+30

$D$1Buff Orpington 0 -3.5 4.5 3.5 1E+30

A) The new weekly egg output will be 169.B) The new weekly egg output will be 162.C) The new weekly egg output will be 158.D) The new weekly egg output will be 153.Answer: DDiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: constraint, model formulationAACSB: Analytic skills

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78) The poultry farmer has in mind the following levels for each of his metrics of interest: a plumage score greater than 75, fertilizer production greater than 80 ounces per week, temperment less than 100 decibels, and an appetite less than 160 ounces of layer pellets per week. Use this portion of the sensitivity report to evaluate the following statements:

CellName

Final Value Reduced Cost

Objective Coefficient

Allowable Increase

Allowable Decrease

$B$1 Leghorn 32 0 5 1E+30 2.1875$C$1 Cochin 0 -2 2 3 1E+30

$D$1Buff Orpington 0 -3.5 4.5 3.5 1E+30

A) The noise level around the farm should be over 60 decibels.B) The fertilizer generated will be 160 ounces per week.C) Layer pellets will be consumed at the rate of 96 ounces per week.D) The plumage score will be less than 40.Answer: ADiff: 2 Page Ref: 112-117Section Heading: A Product Mix ExampleKeywords: sensitivity analysis, right-hand-side valueAACSB: Analytic skills

79) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the supply constraint for component 1. A) x21 + x22 ≤ 8000 B) x12 + x22 ≥ 8000 C) x11 + x12 ≤ 8000 D) x21 + x22 ≥ 8000 Answer: CDiff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: formulationAACSB: Analytic skills

80) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the demand constraint for gasoline type 1. A) x21 + x22 = 11000 B) x12 + x22 = 11000 C) x11 + x21 ≤ 11000 D) x11 + x21 = 11000 Answer: DDiff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: formulationAACSB: Analytic skills

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81) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons, respectively. Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1. A) x11 + x12 (.35)(x11 + x21) B) x11 + .35(x11 + x12) C) -.65x11 + .35x21 ≤ 0 D) .65x11 - .35x21 ≤ 0 Answer: DDiff: 3 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: formulationAACSB: Analytic skills

82) Quickbrush Paint Company is developing a linear program to determine the optimal quantities of ingredient A and ingredient B to blend together to make oil-base and water-base paint. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. Assuming that x represents the number of gallons of oil-base paint, and y represents the gallons of water-base paint, which constraint is correctly represents the constraint on ingredient A? A) .9A + .1B ≤ 10,000 B) .9x + .1y ≤ 10,000 C) .3x + .7y ≤ 10,000 D) .9x + .3y ≤ 10,000 Answer: DDiff: 2 Page Ref: 133-136Section Heading: A Blend ExampleKeywords: blendAACSB: Analytic skills

83) A systematic approach to model formulation is to first A) construct the objective function .B) develop each constraint separately. C) define decision variables. D) determine the right hand side of each constraint. Answer: CDiff: 2 Page Ref: 113Section Heading: A Product Mix ExampleKeywords: model formulation

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84) Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ij = inventory quantity in period j, and dj = demand quantity in period j. Correct formulation of the demand constraint for a multiperiod scheduling problem is:A) rj + oj + i2 - i1 ≥ dj B) rj + oj + i1 - i2 ≥ dj C) rj + oj + i1 - i2 ≤ dj D) rj - oj - i1 + i2 ≥ dj Answer: ADiff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: formulation, constraintsAACSB: Analytic skills

85) In a multiperiod scheduling problem, the production constraint usually takes the form of: A) beginning inventory + demand - production = ending inventory B) beginning inventory - demand + production = ending inventory C) beginning inventory - ending inventory + demand = production D) beginning inventory + demand + production = ending inventory Answer: BDiff: 2 Page Ref: 137-142Section Heading: A Multiperiod Scheduling ExampleKeywords: model formulation, multi-period scheduling problemAACSB: Analytic skills

86) The type of linear program that compares services to indicate which one is less productive or inefficient is called A) product mix B) data envelopment analysis C) marketing D) blending Answer: BDiff: 2 Page Ref: 142Section Heading: A Data Envelopment Analysis ExampleKeywords: formulation

In a portfolio problem, X1, X2, and X3 represent the number of shares purchased of stocks 1, 2, and 3, which have selling prices of $15, $47.25, and $110, respectively. The investor has up to $50,000 to invest.

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87) The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulated as a linear programming constraint?A) X2 ≤ 10000

X2 + X3 ≤350B) 10,000 X2 ≤ 350X2 + 350X3C) 47.25X2 ≤10,000

X2 + X3 ≤ 350D) 47.25X2 ≤10,000

47.25 X2 + 110X3 ≤ 350Answer: CDiff: 2 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: investmentAACSB: Analytic skills

88) An appropriate part of the model would be:A) 15X1 + 47.25X2 +110 X3 ≤ 50,000B) MAX 15X1 + 47.25X2 + 110X3C) X1 + X2 +X3 ≤ 50,000D) MAX 50(15)X1 + 50 (47.25)X2 + 50 (110)X3Answer: ADiff: 2 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: investmentAACSB: Analytic skills

89) The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is:A) MAX .06X1 +.08X2 +.11X3B) MAX .06(15)X1 +.08(47.25)X2 +.11(110)X3C) MAX 15X1 + 47.25X2 +.110X3D) MAX (1/.06)X1 +.(1/08)X2 + (1/.11)X3Answer: BDiff: 2 Page Ref: 120-125Section Heading: An Investment ExampleKeywords: investmentAACSB: Analytic skills

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90) The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct?A) X1 ≤ 0.35B) X1 = 0.35 (50000)C) X1 ≤ 0.35(X1 + X2 + X3)D) X1 = 0.35(X1 + X2 + X3)Answer: CDiff: 2 Page Ref: 119-124Section Heading: An Investment ExampleKeywords: investmentAACSB: Analytic skills

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