
Concept explainers
(a)
Interpretation: A company is considering where to locate its cafeteria to service six buildings. The location of the buildings and the fraction of the company’s employees working at these locations are given below. The optimal location of the cafeteria to minimize the weighted rectilinear distance to all the buildings needs to be determined.
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Concept Introduction: When the differences between the
(a)

Answer to Problem 56AP
The optimal location of the cafeteria to minimize the weighted rectilinear distance to all the buildings has been determined as
Explanation of Solution
Given information: The locations of the buildings and the fraction of company’s employees working at these locations are as given below:
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
In order to find out the optimal location of the cafeteria to minimize the weighted rectilinear distance to all the buildings, the optimal coordinates a and b needs to be derived.
Optimal coordinate a:
The fraction of workforce needs to be multiplied by 12 for normalization.
Building | a | Weight | Cumulative Weight |
B | 1 | 1 | 1 |
A | 2 | 1 | 2 |
C | 3 | 2 | 4 |
E | 4 | 3 | 7 |
D | 5 | 3 | 10 |
F | 10 | 2 | 12 |
By dividing the total cumulative value by 12 and finding out the first location where the cumulative weight exceeds this value, following is calculated:
The first time where the cumulative weight exceeds the value
Optimal Coordinate b:
The fraction of workforce needs to be multiplied by 12 for normalization.
Building | a | Weight | Cumulative Weight |
B | 0 | 1 | 1 |
E | 2 | 3 | 4 |
C | 3 | 2 | 6 |
A | 6 | 1 | 7 |
F | 7 | 2 | 9 |
D | 9 | 3 | 12 |
By dividing the total cumulative value by 12 and finding out the first location where the cumulative weight exceeds this value, following is calculated:
The first time where the cumulative weight exceeds the value
Therefore the optimal location of the cafeteria to minimize the weighted rectilinear distance to all the buildings is
(b)
Interpretation: A company is considering where to locate its cafeteria to service six buildings. The locations of the buildings and the fraction of the company’s employees working at these locations are given below. The optimal location of the cafeteria to minimize the maximum rectilinear distance to all the buildings needs to be determined.
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Concept Introduction: When the differences between the
(b)

Answer to Problem 56AP
All the points on line connecting
Explanation of Solution
Given information: The locations of the buildings and the fraction of company’s employees working at these locations are as given below:
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Following is the calculation for finding out the optimal location of the cafeteria to minimize the maximum rectilinear distance to all the buildings:
Assuming
Recitlinear distance
A
2
6
8
B
1
0
1
C
3
3
6
D
5
9
14
E
4
2
6
F
10
7
17
Substituting the obtained values:
Thus
Hence all the points on line connecting
(c)
Interpretation: A company is considering where to locate its cafeteria to service six buildings. The locations of the buildings and the fraction of the company’s employees working at these locations are given below. The gravity solution needs to be determined.
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Concept Introduction: In a gravity modeling, locations are determined that minimizes the cost of transportation of raw materials from the supplier as well as manufactured goods to the market.
(c)

Answer to Problem 56AP
The optimal solution derived through gravity method is
Explanation of Solution
Given information: The locations of the buildings and the fraction of company’s employees working at these locations are as given below:
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
A company might be considering to provide a new facility in a region, for various reasons, like increasing the capacity of production of a particular existing product, introduction of a new product, or finding new markets in order to have business on their existing or new products. The smallest productive entity wherein a single commodity is manufactured is called a facility. In order to identify appropriate geographic locations in a region, gravity location models are used.
In the gravity solution method it is assumed that there is a linear growth in the transportation costs, with the quantity that is being shipped. In this case all the distances between two points are being calculated as the geometric distance on a plane.
Using gravity modeling, the optimal solution can be obtained by the following formula:
Following tabulation gives the values of coordinates multiplied by their respective weights:
a | b | Weight (w) | wa | wb |
2 | 1 | 1 | 2 | 6 |
1 | 1 | 1 | 1 | 0 |
3 | 2 | 2 | 6 | 6 |
5 | 3 | 3 | 15 | 27 |
4 | 3 | 3 | 12 | 6 |
10 | 2 | 2 | 20 | 14 |
Sum=12 | Sum=56 | Sum=59 |
Substituting the summation values of w, wa and wb in the formula:
Through gravity method the optimal solution arrived at is
(d)
Interpretation: A company is considering where to locate its cafeteria to service six buildings. The locations of the buildings and the fraction of the company’s employees working at these locations are given below. Supposing that the cafeteria must be located in one of the buildings; the particular building, minimizing the weighted rectilinear distance to all the buildings, needs to be identified.
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Concept Introduction: When the differences between the
(d)

Answer to Problem 56AP
The weighted rectilinear distance for C and E are minimum and hence the optimal solution will be either C or E.
Explanation of Solution
Given information: The locations of the buildings and the fraction of company’s employees working at these locations are as given below:
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
The tabulation showing the coordinates along with their weights for various buildings is as shown below:
Building | a | b | Weights |
A | 2 | 6 | 1 |
B | 1 | 0 | 1 |
C | 3 | 3 | 2 |
D | 5 | 9 | 3 |
E | 4 | 2 | 3 |
F | 10 | 7 | 2 |
Below mentioned formula gives the rectilinear distance of a building from other buildings:
Similarly for all the other buildings the weighted rectilinear distance is being calculated as follows:
Therefore weighted rectilinear distance of building B from other buildings
Weighted rectilinear distance of building C from other buildings
Weighted rectilinear distance of building D from other buildings
Weighted rectilinear distance of building E from other buildings
Weighted rectilinear distance of building F from other buildings
From the results the weighted rectilinear distance for C and E are minimum and hence the optimal solution will be either C or E.
(e)
Interpretation: A company is considering where to locate its cafeteria to service six buildings. The locations of the buildings and the fraction of the company’s employees working at these locations are given below. Supposing that the cafeteria must be located in one of the buildings; using Euclidean distance the particular building needs to be identified.
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Concept Introduction: The simple measurement of distance in a straight line between two facilities is called Euclidean Distance.
(e)

Answer to Problem 56AP
The optimal solution can be considered as building C, as the Euclidean distance is least for the same.
Explanation of Solution
Given information: The locations of the buildings and the fraction of company’s employees working at these locations are as given below:
Building | Fraction of Workforce | ||
A | 2 | 6 | |
B | 1 | 0 | |
C | 3 | 3 | |
D | 5 | 9 | |
E | 4 | 2 | |
F | 10 | 7 |
Following is the formula for weighted Euclidean distance:
For building A, the weighted Euclidean distance from other buildings can be calculated as follows:
For building B:
For building C:
For building D:
For building E:
For building F:
From the results, Euclidean distance is least for building C and building C is the optimal solution.
Want to see more full solutions like this?
Chapter 11 Solutions
EBK PRODUCTION AND OPERATIONS ANALYSIS
- Learning Activity 6: Canadian Business This Year Read the following webpage: The Business Development Bank of Canada. (2023). Canadian economic outlook for 2024: Shifting into neutral. https://www.bdc.ca/en/articles- tools/blog/canadian-economic-outlook-for-2024- shifting-into-neutral "Despite persistently high inflation and rising. interest rates, the news was generally better than expected for the Canadian economy in 2023" (BDC Blog 2024). Discussion Question: In your view, what are the most pressing problems for Canadian companies or consumers in 2024? Explain your answer using current examples of companies or consumer concerns.arrow_forwardTravelling and working internationally can lead to a life of adventure and unique career experiences. For businesses, selecting the right candidates to take on foreign assignments can propel, delay, or deny the success of the international ventures. As an international manager, identify key competencies you would look for in choosing expatriates. What might be some of their concerns in taking on overseas assignments? What are some best practices in supporting expats during and after their assignments?arrow_forwardResearch proposal: The Investigation of Career Development and Job satisfaction atEskom Rotek Industries 1. Introduction (250 words) 2. Research Context: Background (Research Context: Background (250 words) 3. The Research Problem (The Research Problem) 4. Aim of the study (Maximum 50 words) 5. Research Objectives (3 in total) 6. Research Questions (3 in total) 7. Justification or rationale of the study (150 words) 8. Literature Review (2-3 pages / 1000-1500 words) 9. Research Design and Methodology (1 - 2 page in total, from 9.1 to 9.6) 9.1 Research Philosophy 9.2 Research Methodology 9.3 Target Population and Sampling strategy 9.4 Data Collection 9.5 Data Analysis (100 words) 9.6 Pilot Study 10. Ethical considerations 11. Referencearrow_forward
- With your experience and research, discuss;(a) what type of “Leadership Style” is used in communications services (Digicel Group Limited)?(b) whether the style used is successful or not and why. Justify your position with relevant citation and references ( Rubic is attached to guide the essay ) NOTE: NO AI RESPONSES PLEASEarrow_forwardby the end of 2013, the IPL port was operational and the naphtha was flowing to the Himachal Fertilizer Corporation (HFC) plant in Central India after being granted the port contract in Summer 2012. The plant was located in an industrial area that had several pipelines in the general geographic area including a crude oil pipeline leading to a refinery, and a pipeline carrying pesticides to be processed and redistributed to local farms in the area. In July 2014, Ajay Patel, the logistics general manager received notification that there were some minor leaks along the pipeline that were being fixed. Workers in the area had experienced sudden headaches and lightheadedness and the company immediately moved to investigate the leak and address faulty seams that had eroded and weakened. The flow of naphtha was temporarily stopped for 72 hours so two portions of the pipeline could be removed, and the eroded portions replaced. Patel put enormous pressure on the repair team to get the repair…arrow_forwardin the MABE: learning to be multinational case report, what is the reccomended course of action?arrow_forward
- In the MABE case report what would be the reccomened course of action?arrow_forwardQuestion 6. An electrical engineering company is designing two types of solar panel systems: Standard Panels (S) and High-Efficiency Panels (H). The company has certain constraints regarding the hours of labor and material available for production each week. Each Standard Panel requires 4 hours of labor and 2 units of material and each High-Efficiency Panel requires 3 hours of labor and 5 units of material. The company has a maximum of 60 hours of labor and 40 units of material available per week. The profit from each Standard Panel is GH¢80, and the profit from each High-Efficiency Panel is GH¢100. The company wants to determine how many of each type of panel to produce in order to maximize profit. i) Solve this LPP by using graphical analysis ii) What will be the slack at the optimal solution point? Show calculation.arrow_forwardqusestion 6. An electrical engineering company is designing two types of solar panel systems: Standard Panels (S) and High-Efficiency Panels (H). The company has certain constraints regarding the hours of labor and material available for production each week. Each Standard Panel requires 4 hours of labor and 2 units of material and each High-Efficiency Panel requires 3 hours of labor and 5 units of material. The company has a maximum of 60 hours of labor and 40 units of material available per week. The profit from each Standard Panel is GH¢80, and the profit from each High-Efficiency Panel is GH¢100. The company wants to determine how many of each type of panel to produce in order to maximize profit. i. Formulate a linear programming model of the problem for the company. ii Convert the linear programming model formulated in (a) to a standard form.arrow_forward
- G ווח >>> Mind Tap Cengage Learning 1- CENGAGE MINDTAP Chapter 09 Excel Activity: Exponential Smoothing Question 1 3.33/10 e Submit 自 A ng.cengage.com C Excel Online Student Work G A retail store records customer demand during each sales period. 1. What is the f... Q Search this course ? ✓ Co Excel Online Tutorial Excel Online Activity: Exponential Smoothing A-Z A retail store records customer demand during each sales period. The data has been collected in the Microsoft Excel Online file below. Use the Microsoft Excel Online file below to develop the single exponential smoothing forecast and answer the following questions. Office Video X Open spreadsheet Questions 1. What is the forecast for the 13th period based on the single exponential smoothing? Round your answer to two decimal places. 25.10 2. What is the MSE for the single exponential smoothing forecast? Round your answer to two decimal places. 21.88 Activity Frame ? 3. Choose the correct graph for the single exponential…arrow_forwardNot use ai pleasearrow_forwardItems removed from the work area (5S) were taken to a storage area called ___________. Choose from: SORT, STORD, KNUJ, STUFF, FUDG SORT STORD KNUJ STUFF FUDGarrow_forward
- Practical Management ScienceOperations ManagementISBN:9781337406659Author:WINSTON, Wayne L.Publisher:Cengage,
