Deck 21: Modeling Using Linear Programming

Question 1

Which of the following is generally correct regarding the transportation problem of linear programming?

A)Supply = demand
B)Objective is maximizing profit contribution
C)Shipping cost from each source to each destination is the same
D)Exponential loading/unloading distribution is assumed

Answer 1

A

Question 2

If R_m = increase in the total production level during Month 'm' compared to Month m-1 and D_m = decrease in the total production level during Month 'm' compared to Month m-1, which of the following statements can be correct?

A)R_m = D_m = 0
B)Both R_m and D_m positive
C)Both R_m and D_m negative
D)One variable can be positive and the other negative

Answer 2

R_m = D_m = 0

Question 3

Deciding on how much of each grade of gasoline to produce is an example of the linear programming model for blending applications.

Answer 3

True

Question 4

Excel solver can handle basic linear programming but not the special transportation problem.

Question 5

If for a given (month overtime units O_t) = under-(time units U_t),

A)Non-negativity would not allow this
B)Only possible with subcontracting
C)There was no over- or under-time
D)Will work if one is positive and the other is negative in the same magnitude

Question 6

The transportation problem is a special type of linear programming that arises in planning the distribution of goods and services from only one supply point to several demand locations.

Question 7

The constraint that requires the beginning inventory plus production minus sales to equal the ending inventory is called material-balance equation.

Question 8

Since price is usually set by market conditions, the blending problem use of linear programming attempts to meet demand at minimum cost.

Question 9

The term 'programming' is used in linear programming because these models find the best 'program' or course of action to follow.

Question 10

Which of the following factors would generally not be part of a linear programming model for production scheduling?

A)Fluctuations in production
B)Inventory levels
C)Waiting time probability distributions
D)Storage capacity

Question 11

For linear programming to work for project crashing decisions, a dummy activity is needed at the beginning of the project, with a duration of zero 0) time.

Question 12

X₁, X₂ $\ge$ 0 refers to

A)Feasibility
B)Maximization
C)Production-rate changes
D)Non-negativity

Question 13

(a month's production) + (beginning inventory) - (ending inventory) - (number of lost sales for the month) =

A)Overtime limits
B)Demand in the month
C)Increase in product
D)Decrease in production

Question 14

Solutions to a linear programming model that satisfy all constraints are referred to as optimal.

Question 15

Which of the following factors would generally not be part of a linear programming model for blending?

A)Revenue
B)Cost
C)Product/component specifications
D)Supply = demand

Question 16

Any particular combination of decision variables is referred to as an) ____.

A)Objective
B)Solution
C)Constraint
D)Optimization

Question 17

The normal-time project cost does not depend on what crashing decisions are made. As a result, total crash costs can be minimized for a linear programming objective.

Question 18

Constant terms in the objective function are called object function ____.

A)Variables
B)Decision solutions
C)Coefficients
D)Constraints

Question 19

Ending inventory from the previous month) + current production) - ending inventory this month) =

A)Production rate change
B)This month's demand
C)Amount of overtime/under-time
D)Next month's inventory

Question 20

Neither the R_m, increase in the total production levels during Month 'm' compared to Month m-1, or the D_m, decrease in the total production level during Month 'm' compared to Month m-1, can be negative because only positive changes would be permitted due to the non-negativity requirement.

Question 21

The Pacific Computer Company makes two models of notebook personal computers: Model 410 with CD-ROM drive, and Model 540 with DVD drive. Profits on each model are $100 and $150, respectively. Weekly manufacturing data in minutes) are given below:
$$\begin{array} { l c c c } \text { Notebodk } & \text { Dept. A } & \text { Dept. B } \\\text { Model } & \text { Mamuf. Time } & \text { Manuf. Time } & \text { Assembly Time } \\\hline 410 & 5 & 14 & 25 \\540 & 10 & 7 & 40\end{array}$$
Manufacturing time available in department A for the coming week is 300 minutes, and for department B it is 420 minutes. Total time available during the week to assemble the fabricated units is 1600 minutes.
a. What is the objective function if the company wants to maximize profits?
b. What is the constraint corresponding to Department A assuming X1 corresponds to Model 410?
c. What is the optimum solution point for this problem?
d. What is the maximum possible total profit?

Question 22

A Singapore company manufactures 50-inch and 75-inch rear projection television sets. Each 50-inch set contributes $200 to profits and each 75-inch set contributes $475 to profits. The company has purchase commitments for 500 50-inch sets and 200 75-inch sets for the next month so they want to make at least that many. Although they think they can sell all the 50-inch sets that they could currently make, they do not think they can sell more than 375 75-inch sets. Their factory capacity allows them to make only 975 sets of both sizes total. They want to know how many of each type to make so as to maximize profits.
a. What is the objective function for this LP problem?
b. What are the constraints involving X1, assuming that X1 corresponds to 50-inch TV sets?
c. What is the optimal solution point for this problem?
d. What is the optimal value of the objective function?

Question 23

If a company has demand of 1,600 units for the month of May and a beginning of 900 units, and if X_m = production in May and L_m = the number of lost sales in May, which of the following represent the material-balance constraint?

A)X_m  I_m + 900 + L_m = 1,600
B)X_m + I_m  900 + L_m = 1,600
C)X_m  I_m + 900  L_m = 1,600
D)X_m  I_m + 900  L_m = 1,600

Question 24

A clothing distributor has four warehouses which serve four large cities. Each warehouse has a monthly capacity of 5,000 blue jeans. They are considering using a transportation LP approach to match demand and capacity. The following table provides data on their shipping cost, capacity, and demand constraints on a per-month basis:
$$\begin{array} { c c c c c } \text { Warehouse } & \text { City E } & \text { City F } & \text { City G } & \text { City H } \\\hline \text { A } & .53 & .21 & .52 & .41 \\\text { B } & .31 & .38 & .41 & .29 \\\text { C } & .56 & .32 & .54 & .33 \\\text { D } & .42 & .55 & .34 & .52 \\\text { City demand } & 2,000 & 3,000 & 3,500 & 5,500\end{array}$$
a. How many variables are there in this formulation?
b. How many constraints are involved in this problem?
c. What is the constraint corresponding to City F?

Question 25

A company is considering a rate change, either an increase or a decrease in production, and X_t = production in a time period
R_t = increase in production rate from Period t1 to Period t
D_t = decrease in production rate from Period t1 to Period t
Which of the following is correct?

A)X_t  X_{t - 1} = D_t  R_t
B)X_{t - 1}  X_t = D_t  R_t
C)X_{t - 1}  X_t = R_t  D_t
D)X_t  X_{t - 1} = R_t  D_t

Question 26

ABC Products produces three products (A, B and C) on three machines. Machines 1 and 2 are available for 40 hours a week and Machine 3 is available for 60 hours a week. Profit contribution and standard production time in hours are given in the following table:
$\begin{array}{cccc}&&\text {Product}\\& \underline{\text {A}}& \underline{\text {B}}&\underline{\text {C}}\\\text { Profit Contribution/Unit } & \$ 40 & \$ 60 & \$ 30 \\\text { Machine } \# 1 \text { time/unit) } & 0.6 & 2.2 & 1.8 \\\text { Machine } \# 2 \text { time/unit) } & 1.2 & 0.8 & 0.7 \\\text { Machine } \# 3 \text { time/unit) } & 2.4 & 2.8 & 2.0\end{array}$

Only one operator per machine is required on Machines #1 and #2. Two operators are required for Machine #3. Therefore, two hours of labor must be scheduled for each hour of Machine #3's time. To restate this requirement, two operators must be scheduled for each hour of Machine #3's operation, as well as one operator for each hour of Machine #1's operation and one operator for each hour of Machine #2's operation. A total of 110 labor hours is available for assignment to the three machines during the coming week. Other production requirements are that Product A cannot account for more than 40% of the units produced and that Product C must account for at least 25% of the units produced.
a. Develop the constraint for the capacity limit of Machine #1.
b. Develop the constraint for the capacity limit of Machine #3.
c. Develop the constraint for the labor capacity limit.
d. Develop the constraint for limiting Product A to no more than 40% of the units produced.
e. Develop the constraint that ensures Product C accounts for at least 25% of the units produced.

Question 27

Explain the essence of a blending problem.

Question 28

If Activity C directly precedes both D and E on a project network and C can be crashed three( 3) times in using linear programming,

A)Crash time affects only C
B)Crash time affects only C and D
C)Crash time affects only C and E
D)Crash time affects both D and E

Question 29

Power Fuels is developing a new additive for rocket fuel. The additive is a mixture of liquid ingredients A, B and C. For proper performance, the total amount of additive must be at least 18 ounces per gallon of fuel. For safety reasons, the total amount of additive must not exceed 22 ounces per gallon. At least 2 ounces of A must be used for every ounce of C. The amount of B must be greater than one-half the amount of A.
a. The cost per ounce for ingredients A, B and C is $50, $40 and $70, respectively. What is the objective function?
b. Develop the constraint for limiting the additives for safety reasons.
c. Develop the constraint for ensuring the performance requirement is met.
d. Develop the constraint that the relationship between Ingredient A and Ingredient C.
e. Develop the constraint that states the relationship between Ingredient B and Ingredient C.

Question 30

The Northwest Flower Company owns a greenhouse, which furnishes roses and carnations to florists in Oregon, Washington, and Idaho. The greenhouse can grow any combination of the two flowers. They sell the flowers in "bunches" with 25 blooms to a bunch. They have 10,000 square feet available for planting this year. Each bunch of roses takes about 4 square feet and each bunch of carnations about 5 square feet. Special fertilizer is required for flowers: roses need 5 pounds and carnations 2 pounds. The availability of the fertilizer is limited to 5000 pounds. Sales commitments require the company to grow at least 500 bunches of roses. Profit contributions are $6 per bunch of roses and $8 per bunch of carnations.
a. What is the objective function if the company wants to maximize its profits?
b. What is the constraint for the square footage assuming X1 corresponds to roses?
c. What is the optimal solution point for this problem?
d. What is the optimal value of the objective function?

Question 31

Explain the steps necessary to solve a linear programming problem using Microsoft Excel's Solver.

Question 32

Explain the essence of a transportation problem.

Question 33

Differentiate between a feasible solution and an optimal solution.

Question 34

A small lumber company in the Southeast produces two types of pine boards used in home construction: 2x4s and 2x6s dimensions in inches). They are attempting to determine how many of each to produce so as to minimize their costs on a per-minute basis. They have sales commitments to produce four 2x4s and two 2x6s per minute, but they think they shouldn't produce any more than eight 2x6s because of market demand. They are also trying to support the community by employing people. Thus they want to keep at least 12 men employed, but only need 2 men to produce each 2x4 and 1 person to produce each 2x6 per minute. It costs them $.50 to produce 2x4 and $.80 to produce a 2x6 per minute.
a. What is the objective function for this LP problem?
b. What is the constraint for the employment issue, assuming X1 corresponds to 2x4s?
c. What is the optimal solution point for this problem?
d. What is the optimal value of the objective function?

Question 35

What characterizes a linear function?

Question 36

Discuss the concept of constrained optimization. Include some examples.

Question 37

A computer manufacturing company wants to develop a monthly plan for shipping finished products from three of its manufacturing facilities to three regional warehouses. It is thinking about using a transportation LP formulation to exactly match capacities and requirements. Data on transportation costs in dollars per unit), capacities, and requirements are given below:
$$\begin{array} { c c c c c } & \text { Plant } & \text { Plant } & \text { Plant } & \\\underline { \text { Warehouse} } & 1 & \underline { 2 } & \underline { 3 } & \underline { \text { Requirements } }\\\mathbf { x } & 2 . \overline { 4 } 1 & 1 . \overline { 6 } 3 & \text { 2.09 } & 8,000 \\Y & 3.18 & 5.62 & 1.74 & 2,000 \\\text { Z } & 4.12 & 3.16 & \text { 3.0 } & 3,000 \\ \text { Capacities } & 4,000 & 0,000 & 3,000 & \\\end{array}$$
a. How many variables are involved in the LP formulation?
b. How many constraints are there in this problem?
c. What is the constraint corresponding to plant 2?

Question 38

Given the partial project network below and the fact that F can be crashed three times if X_t = start time of Activity i, and Y_i = amount of crash time used for Activity i.
a. Which of the following is correct? X_F  X_D + 10  Y_D or X_F  X_E + 10 + Y_E or Y_D, Y_E  0 or X_F  X_D  X_E + 10  3
b. Which of the following is correct? y_D  3 or y_E  3 or y_F = y_D + y_E or y_F  3

Question 39

The Alpha Beta Corporation makes laser and ink jet printers for personal computers. Each laser printer yields $40.00 in profits and each ink jet printer provides $20.00. Each of the printers goes through two assembly areas. The following table provides processing times per unit in minutes) as well as total available processing times per department:
$$\begin{array} { l c c } \underline{\text { Printer} } & \underline{ \text { Dept. A }} & \underline{\text { Dept. B }} \\\text { Laser }& 9 & 12 \\\text { Ink jet } & 6 & 8 \\\text { Tatal time per day } & 216 & 384\end{array}$$
Sales commitments require at least 5 laser printers and 10 ink jet printers to be made per day. The company is interested in determining how many of each printer to produce so as to maximize its profit.
a. What is the objective function for this LP problem?
b. What are the constraints corresponding to Dept. A if X1 corresponds to laser printers?
c. What are the optimum solution points for this problem?
d. What is the optimal value of the objective function?

Question 40

A company has normal capacity for 2,500 units a month. If, for a given month, X = demand, which of the following correctly allows for either overtime or under-time in a given month?

A)X_t + O_t  U_t = 2,500
B)X_t  O_t + U_t = 2,500
C)X_t  O_t + U_t < 2,500
D)X_t + O_t  U_t > 2,500

Question 41

A cargo airline company in South America ferries materials from four different airfields in Brazil called (A, B, C, and D) to two different airfields in Peru numbered (1 and 2). Distances in hundreds of miles between the six different airfields are as shown below:
$\begin{array}{ccccc}&&\text { Brazil }\\ \underline{\text {Peru }}& \underline{\text {A}}& \underline{\text {B}}& \underline{\text {C}}& \underline{\text {D}}\\1 & 2 & 6 & 17 & 5 \\2 & 12 & 8 & 12 & 9 \\\text { Capacities } & 60 & 40 & 75 & 5\end{array}$
Requirements for airfield 1 are 140 tons per year and 80 tons per year for airfield 2. The president of the airline wants to determine how much material should be shipped from each airfield in Brazil to each airfield in Peru so as to minimize total travel distance.
a. How many decision variables are there in this problem?
b. What is the constraint corresponding to airfield 1?
c. What is the constraint corresponding to airfield B?

Question 42

A cement company has three factories that they identify as Alpha, Beta, and Gamma. They supply cement to three warehouses that they call X, Y, and Z. The company wants to determine how much cement should be shipped from each factory to each warehouse to minimize shipping costs. The cost to ship each 100-pound bag, along with warehouse requirements and factory capacities in bags are shown in the table below:
$\begin{array}{lrrrr}&&\text { Warehouses }\\ \underline{\text { Factories }} &\underline{X} &\underline{Y} & \underline{Z}& \underline{ \text { Capacities bags) }}\\\text { Alpha } & \$ .40 & \$ .60 & \$ .78 & 10,000 \\\text { Beta } & .65 & .78 & .59 & 9,000 \\\text { Gamma } & .56 & .61 & .71 & 7,000 \\\text { Requirements bags) } & 10,000 & 5,000 & 10,000 &\end{array}$
a. How many decision variables are there in this problem?
b. What is are) the constraints) corresponding to Factory Alpha?
c. How many constraints are required for this problem?

Question 43

A food processing company makes meatloaf to be sold in the frozen food section of supermarkets. Each week the recipe used changes based on the current cost of ingredients. Ingredients and current costs are as shown below:
$$\begin{array} { l c } \text { Inpredient } & \text { Costpound } \\\hline \text { pork } & \$ 2.87 \\\text { hamburger } & \$ 1.63 \\\text { wheat filler } & \$ 0.22 \\\text { corn filler } & \$ 0.16\end{array}$$ F
or each batch made, at least 300 pounds of pork and 100 pounds of hamburger are required. No more than 200 pounds of wheat filler can be used per batch and the amount of corn filler has to range between 50 and 150 pounds. Moreover, each batch must contain at least 500 pounds of meat and no more than 200 pounds of filler.
a. What is the objective function if the company seeks to minimize costs?
b. If Xl is the amount of pork used, what are the constraints associated with it?
c. If X3 is the amount of wheat filler used, what are the constraints associated with it?