Deck 17: Appendix: Modeling Using Linear Programming

Question 1

A company has normal capacity for 2,500 units a month. If, x₁₁ and x₂₁ represent the production decisions variables, which of the following equations represents the occurrence of change?

A) x₁₁ + x₂₁ − 2500
B) x₁₁ − x₂₁ − 2500
C) x₁₂ − x₂₁ − 2500
D) x₁₂ + x₂₁ − 2500

Answer 1

A

Question 2

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

Answer 2

True

Question 3

If a company has a 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 equations represents 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

Answer 3

A

Question 4

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 5

The total project cost can be minimized by minimizing crash costs.

Question 6

The constraint x₁ and x₂ $\ge$ 0 refers to:

A) feasibility.
B) maximization.
C) binding.
D) nonnegativity.

Question 7

Any particular combination of decision variables is referred to as a(n) _____.

A) objective
B) solution
C) constraint
D) coefficient

Question 8

A company is considering a rate change, either an increase or a decrease in production, where 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 9

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

Question 10

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 11

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

Question 12

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

Question 13

The constraint that requires the ending inventory from previous month plus current production minus ending inventory of this month equals this month's demand is known as a demand requirement.

Question 14

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 are positive.
C) Both R_m and D_m are negative.
D) One variable can be positive and the other negative.

Question 15

If a single variable is used to represent the change in production level, only positive changes would be permitted, because of the nonnegativity requirement.

Question 16

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 17

Constant terms in the objective function are called objective function _____.

A) variables
B) decisions
C) coefficients
D) constraints

Question 18

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

Question 19

In production scheduling, (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/undertime
D) next month's inventory

Question 20

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 21

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

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 22

A cement company has three factories that they identify as Alpha, Beta, and Gamma. They supply cement to three warehouses referred to as 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:

a. How many decision variables are there in this problem?
b. What is (are) the constraint(s) corresponding to factory Alpha?
c. How many constraints are required for this problem?

Question 23

The precedence relationships in a network limit the activity _____.

A) nonnegativity time
B) overtime
C) crash time
D) inventory time

Question 24

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:
$\begin{array}{ll}\text { Ingredient }&\text { Cost/pound }\\\text { pork } & \$ 2.87 \\\text { hamburger } & \$ 1.63 \\\text { wheat filler } & \$ 0.22 \\\text { corn filler } & \$ 0.16\end{array}$
For 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 x₁ 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?

Question 25

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 matched capacities and requirements. Data on transportation costs (in dollars per unit), capacities, and requirements are given below:

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 26

Differentiate between a feasible solution and an optimal solution.

Question 27

Explain the essence of a transportation problem.

Question 28

A 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 television sets. 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 production capacity allows them to make only 975 sets in total including both types of television sets. 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 29

What characterizes a linear function?

Question 30

A clothing distributor has four warehouses that 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:
$\begin{array}{ccccc}\text { Warehouse } &\text { City E }&\text { City F }&\text { City G }&\text { City H }\\\text { A } & 0.53 & 0.21 & 0.52 & 0.41 \\\text { B } & 0.31 & 0.38 & 0.41 & 0.29 \\\text { C } & 0.56 & 0.32 & 0.54 & 0.33 \\\text { D } & 0.42 & 0.55 & 0.34 & 0.52 \\\text { City demand } & 2.000 & 3,000 & 3,500 & 5,500\end{array}$
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 31

The Alpha Beta Corporation makes laser and inkjet 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:
$\begin{array}{lcc}\text { Printer }&\text { Dept. A }&\text { Dept. B }\\\text { Laser } & 9 & 12 \\\text { Inkjet } & 6 & 8 \\\text { Total time per day } & 216 & 384\end{array}$
Sales commitments require at least 5 laser printers and 10 inkjet 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 optimal solution points for this problem?
d. What is the optimal value of the objective function?

Question 32

Explain the essence of a blending problem.

Question 33

ABC Products Inc. 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:
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 34

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:
$\begin{array}{ccccr}&&\text { Brazil }\\\text { Peru } & \text { A } & \text { B } & \text { C } & \text { D }\\1 & 2 & 6 & 17 & 5 \\2 & 12 & 8 & 12 & 9 \\\text { Capacities } & 60 & 40 & 75 & 55\end{array}$
Requirements for airfield 1 are 140 tons per year and 80 tons per year for airfield 2. The president of the airlines 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 35

Which of the following is generally TRUE of the transportation problem of linear programming?

A) A specified quantity of goods is needed at each demand location.
B) The objective is to maximize profit.
C) The shipping cost from each source to each destination is the same.
D) An exponential loading/unloading distribution is assumed.

Question 36

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

Question 37

The Northwest Flower Company owns a greenhouse, which supplies 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. A special fertilizer is required for the 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 38

Discuss the concept of optimization models with examples.

Question 39

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:
$\begin{array}{cccc}\text { Notebook } & \text { Dept. A } & \text { Dept. B } \\\text { Model } & \text { Manuf. Time } & \text { Manuf. Time }&\text { Assembly Time }\\410 & 5 & 14 & 25 \\540 & 10 & 7 & 40\end{array}$
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 40

A small lumber company produces two types of pine boards used in home construction: 2x4s and 2x6s (dimensions in inches). It is attempting to determine how many of each to produce so as to minimize its costs on a per-minute basis. It has sales commitments to produce four 2x4s and two 2x6s per minute, but the management thinks they should not produce any more than eight 2x6s because of market demand. The company is also trying to support the community by employing people. Thus, it wants to keep at least 12 men employed, but only needs 2 men to produce each 2x4 and 1 person to produce each 2x6 per minute. It costs the company $.50 to produce a 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 41

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 $\ge$ x_D + 10 - y_D or x_F $\ge$ x_E + 10 + y_E or y_D, y_E $\le$ 0 or x_F -x_D - x_E + 10 $\ge$ 3
b. Which of the following is correct? y_D $\le$ 3 or y_E $\le$ 3 or y_F = y_D + y_E or y_F $\le$ 3