intro to management science: linear programming

Introduction to Management Science, 10e (Taylor)

Introduction to Management Science, 10e (Taylor)

Chapter 4 Linear Programming: Modeling Examples

1) When formulating a linear programming problem constraint, strict inequality signs (i.e., less than < or, greater than >) are not allowed.

Answer: TRUE

Diff: 2Page Ref: Ch 2 review

Main Heading: Formulation and Computer Solution

Key words: formulation

2) When formulating a linear programming model on a spreadsheet, the measure of performance is located in the target cell.

Answer: TRUE



Key words: spreadsheet solution

3) The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign

Answer: FALSE



Key words: formulation, standard form

4) The standard form for the computer solution of a linear programming problem requires all variables to be on the left side, and all numerical values to be on the right side of the inequality or equality sign.

Answer: TRUE




5) Fractional relationships between variables are not permitted in the standard form of a linear program.

Answer: TRUE




6) A constraint for a linear programming problem can never have a zero as its right-hand-side value.

Answer: FALSE




7) The right hand side of constraints cannot be negative.

Answer: FALSE




8) A systematic approach to model formulation is to first define decision variables.

Answer: TRUE




9) A systematic approach to model formulation is to first construct the objective function before determining the decision variables.

Answer: FALSE




10) In a linear programming model, a resource constraint is a problem constraint with a greater-than-or-equal-to () sign.

Answer: FALSE




11) Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem.

Answer: TRUE

Diff: 2Page Ref: 111-116

Main Heading: A Product Mix Example

Key words: formulation, product mix problem

12) Product mix problems cannot have "greater than or equal to" () constraints.

Answer: FALSE



Key words: product mix

13) When using a linear programming model to solve the "diet" problem, the objective is generally to maximize profit.

Answer: FALSE


Main Heading: A Diet Example

Key words: objective function

14) When using a linear programming model to solve the "diet" problem, the objective is generally to maximize nutritional content.

Answer: FALSE



Key words: objective function

15) In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories.

Answer: FALSE



Key words: formulation, diet example

16) Solutions to diet problems in linear programming are always realistic.

Answer: FALSE



Key words: diet example

17) Diet problems usually maximize nutritional value.

Answer: FALSE



Key words: diet example

18) In most media selection decisions, the objective of the decision maker is to minimize cost.

Answer: FALSE


Main Heading: Marketing Example

Key words: marketing problem, media selection

19) 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: TRUE




20) Linear programming model of a media selection problem is used to determine the relative value of each advertising media.

Answer: FALSE




21) In a media selection problem, maximization of audience exposure may not result in maximization of total profit.

Answer: TRUE




22) In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.

Answer: TRUE


Main Heading: A Transportation Example

Key words: transportation problem, formulation

23) In an unbalanced transportation model, supply does not equal demand and supply constraints have signs.

Answer: TRUE




24) Transportation problems can have solution values that are non-integer and must be rounded.

Answer: FALSE



Key words: transportation problem, solution

25) 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: TRUE




26) 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: FALSE




27) 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: TRUE




28) 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: FALSE




29) Blending problems usually require algebraic manipulation in order to write the LP in "standard form."

Answer: TRUE


Main Heading: Data Envelopment Analysis

Key words: blending

30) Data Envelopment Analysis indicates which type of service unit makes the highest profit.

Answer: FALSE


Main Heading: A Blend Example

Key words: blending

31) Data Envelopment Analysis indicates the the relative _________ of a service unit compared with others.

Answer: efficiency or productivity


Main Heading: Data Envelopment Analysis

Key words: data envelopment analysis

32) __________ types of linear programming problems often result in fractional relations between variables which must be eliminated.

Answer: blending



Key words: blending

33) When formulating a linear programming model on a spreadsheet, the measure of performance is located in the __________ cell.

Answer: target

Diff: 2Page Ref: 113



34) When the __________ command is used in an Excel spreadsheet, all the values in a column (or row) are multiplied by the values in another column (or row) and then summed.

Answer: SUMPRODUCT




35) For product mix problems, the constraints are usually associated with __________.

Answer: resources or time



Key words: product mix

36) 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



Key words: formulation, constraint

37) The objective function of a diet problem is usually to __________ subject to nutritional requirements.

Answer: minimize costs



Key words: diet problem

38) Investment problems maximize __________.

Answer: return on investments


Main Heading: An Investment Example

Key words: investment

39) 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




40) In __________ problem, maximization of audience exposure may not result in maximization of total profit.

Answer: media selection




41) In a balanced transportation model, supply equals __________ .

Answer: demand




42) In a __________ transportation problem, supply exceeds demand.

Answer: unbalanced




The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 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.

43) What is the formulation for this problem?

Answer: MAX Z = 0. 4L + 0.5V

s.t.

2L + 3V 4800

6L + 8V 9600

1L + 2V 2000


Main Heading: Product Mix Example

Key words: computer solution

44) 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



Key words: slack, computer solution

45) 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




A croissant shop produces 2 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.

46) This represents what type of linear programming application?

Answer: product mix



Key words: computer solution

47) What is the formulation for this problem?

Answer: MAX Z = $.20B + $.30C

s.t.

6B + 3C 6600

1B + 1C 1400

2B + 4C 4800

C 400




48) For the production combination of 600 bear claws and 800 almond filled croissants, how much flour and almond paste is remaining?

Answer: flour = 0 ounces and almond paste = 0 ounces




49) If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company's 3 products in period 2 is equal to 400.

Answer: X12 + X22 + X32 400



Key words: transportation problem, supply constraint

50) Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be shipped to 3 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 3 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 = 200

x2A + x2B +x2C = 200

x3A + x3B +x3C = 300

x1A + x2A +x3A = 250

x1B + x2B +x3B = 150

x1C + x2C +x3C = 200



Key words: computer solution, transportation/distribution

51) 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 based paint and 5833 gallons of oil based paint



Key words: blending

Andy Tyre manages Tyre's Wheels, Inc. Andy has received an order for 1000 standard wheels and 1200 deluxe wheels 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 hours of production time and a deluxe wheel, .6 hours. The cost of carrying a wheel from one month to the next is $2.

52) 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 production

S1O = number of standard wheels produced in month 1 on overtime production

S2R = number of standard wheels produced in month 2 on regular time production

S2O = number of standard wheels produced in month 2 on overtime production

D1R = number of deluxe wheels produced in month 1 on regular time production

D1O = number of deluxe wheels produced in month 1 on overtime production

D2R = number of deluxe wheels produced in month 2 on regular time production

D2O = number of deluxe wheels produced in month 2 on overtime production

Y1 = number of standard wheels stored from month 1 to month 2.

Y2 = number of deluxe wheel s 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 Y2


Main Heading: Multiperiod Scheduling

Key words: linear program multiperiod scheduling

53) Write the constraints for this problem.

Answer: S1R + S1O - Y1 = 1000

D1R + D1O - Y2 = 1200

S2R + S2O + Y1 = 750

D2R + D2O + Y2 = 1000.5 S1R + .6 D1R 1000

.5 S1O + .6 D1O 500

.5 S2R + .6 D2R 1000

.5 S2O + .6 D12O 50


Main Heading: Multiperiod Scheduling

Key words: linear program multiperiod scheduling

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 4,000 yards of cotton and 3,000 yards of rayon per week.

The table below shows relevant manufacturing information:

ShirtTotal Yards of fabric per shirtFabric requirementMinimum weekly contractsMaximum DemandSelling Price

Athletic1.00at least 60% cotton500600$30

Varsity1.20no more than 30% rayon650850$40

Surfer0.90As much as 80% cotton300700$36

54) Assume that the decision variables are defined as follows:

A = total number of athletic shirts produced

V = total number of varsity shirts produced

S = total number of surfer shirts produced

C = yards of cotton purchased

R = yards of rayon purchased

Xij = 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 - 7R



Key words: objective function, model construction

55) Write the constraints for the fabric requirements.

Answer: Form of constraints: Total yards used is greater than (or less than) total yards required x (% fabric required) shirts produced

XCA 0.6 A

XVR 0.36V

XSC 0.72 S



Key words: blending

56) 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)/1

V =( XCV + XRV)/1.2

S =( XCS + XRS)/0.9

Standard form:

A -XCA - XRA) = 0

1.2 V - XCV - XRV =0

0.9 S - XCS - XRS=0



Key words: blending

57) 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 $3.50, which uses 16 cans of meow munch and 2 cans of feline fodder.



Key words: diet

58) 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:


Main Heading: Investment Example

Key words: investment

59) When systematically formulating a linear program, the first step is

A) Construct the objective function

B) Formulate the constraints

C) Identify the decision variables

D) Identify the parameter values

E) Identify a feasible solution

Answer: C


Main Heading: Formulation


60) The following types of constraints are ones that might be found in linear programming formulations:

1.

2. =

3. >

A) 1 and 2

B) 2 and 3

C) 1 and 3

D) all of the above

Answer: A

Diff: 2Page Ref: Review



61) 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 two". 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

E) -.4x2 + .6x7 + .6x8 0

Answer: C


Main Heading: An Investment Example


62) 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 APercent per pound in Feed BMinimum daily requirement (pounds)

1202430

2301050

303020

4241560

5102040

The constraint for ingredient 3 is:

A) .5A + .75B = 20

B) .3B = 20

C) .3 B 20

D) .3B 20

E) A + B = .3(20)

Answer: D



Key words: solution

The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 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.

63) 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

E) salt and herbs

Answer: B



Key words: solution, slack

64) What is the constraint for salt?

A) 6L + 8V 4800

B) 1L + 2V 4800

C) 3L + 2V 4800

D) 2L + 3V 4800

E) 2L + 1V 4800

Answer: D




65) 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: D



Key words: formulation, feasibility

66) 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,

A) X32 400

B) X21 + X22 + X23 400

C) X12 + X22 + X32 400

D) X12 + X22 + X32 400

E) X23 400

Answer: C




67) Balanced transportation problems have the following type of constraints:

A)

B)

C) =

D)

intro to management science: linear programming

Documents

model formulation

computer solution key

page ref

review main heading

false diff

true diff

linear programming model

product mix example