1. Producing widgets involves fourteen production tasks, lettered a through n, below. For each task, the time required to accomplish it is given, and the immediately preceding task.

The manager expects to produce 300 widgets per eight-hour shift. The workers are allowed two 10-minute coffee breaks per shift, and 40 minutes is allocated to cleanup and maintenance tasks.

1. Is it possible to accomplish the objective of 300 widgets per shift with one workstation?
2. What is the maximum possible output with 14 workstations?
3. Draw the precedence diagram for this process.
4. What is the minimum number of workstations which will be required to achieve an output of 300 pieces per shift?

2. Refer to the data in Problem 1.

1. Assign the tasks to workstations according to the heuristic rule of the largest number of following tasks.
2. How many workstations are required under the rule of the largest number of following tasks?
3. What is the percent of idle time (or the balance delay) for this layout?

3. Refer to the data in Problem 1.

1. Assign the tasks to workstations according to the heuristic rule of the positional weight.
2. How many workstations are required under the rule of the greatest positional weight?
3. What is the percent of idle time (or the balance delay) for this layout?

4. The XYZ Co. manufactures a variety of specialized products according to the orders of the customers. The plant consists of 5 departments, numbered 1 through 5 below. While some variation in the work flow is experienced, depending on the composition of the orders received, the following matrix is considered to be representative of the traffic between the departments per five-day workweek, measured in the number of forklift loads or the number of pull truck loads.

The floor plan of the factory is displayed in the diagram below. (See next page). The plant has an L shape, and the lettered areas represent spaces that are to be occupied by the five departments.

1. In how many ways can these five departments be arranged in the factory?
2. Obtain the distance matrix by measuring from the center of one space to the center of the other space. For example, the distance from area B to area E is 18 ft. + 24 ft. = 42 ft. The distances are the same, both ways.
3. Assign departments to the areas. Try to keep departments with heavy traffic close to each other.
4. Obtain the load-distance index for your layout. (The load-distance index is the sum of the products of the traffic between two departments and the distance between the two departments.)
5. Have you obtained the minimum value of the load-distance index?

5. (One step beyond; based on matrix algebra) The textbook mentions some software packages which perform the lengthy computations involved in obtaining the load-distance index and which also test several layouts in search of a low value of the index. If you do not have access to any of these packages, you can obtain the load-distance index on your computer by multiplying the traffic matrix and the distance matrix. Let us symbolize the traffic matrix as L(i,j) and the distance matrix as D(i,j). Then the product matrix is: P(i,j) = L(i,j)*D(i,j).

1. Show how to construct the traffic matrix and the distance matrix so as to perform this multiplication for the layout in 4-c. Hint: consider the traffic matrix to be fixed, and rearrange the distance matrix.
2. Show that the load-distance index is the trace of the product matrix.

6. Here is the Muther diagram for the departments in Problem 4.

A = absolutely necessary to locate the departments next to each other.
U = undesirable to locate the departments next to each other.
X = departments must not be located next to each other.

1. Assign the departments to the areas so as to satisfy, as far as possible, the conditions of the Muther diagram.
2. What is the value of the load-distance index of this layout?

Solutions

1. a. The operating time (OT) is (8)(60) - 20 - 40 = 420 minutes per shift. One workstation would require 9.1 minutes to produce one widget. In one shift it could produce 420/9.1= 46 widgets, and the objective cannot be achieved.
b. With 14 workstations, the output is limited by the station with the longest time, which is station k with 1.2 minutes. In eight hours, fourteen stations could produce 420/1.2 = 350 widgets, which is more than adequate.
c.
d. The minimum number of workstations = D*S t/OT = 300(9.1)/420 = 6.5 workstations.

2. a. Count the tasks which follow a particular task on a path through the precedence diagram which was obtained in #1c.

The cycle time (CT) = OT/D = 420/300 = 1.4 minutes. Each workstation can spend a maximum of 1.4 minutes accomplishing a set of tasks.

Here is a set of feasible assignments. (It may not be the only one.) The idle time per station is the maximum total time (1.4 minutes) - the total station time.

b. 9 workstations are required.
c. The percent of idle time = (Idle Time per Cycle)x100/(number of stations)×(maximum station time) = 3.5(100)/[(9)(1.4)] = 27.78%.

3. a. The positional weight of a task is the sum of its own time and the times of all following tasks on the precedence diagram (#1 - c).

The assignment of tasks to workstations will be identical with Problem #2a. Because there must be a high correlation between the number of following tasks and the sum of the task times, these two rules must produce very similar layouts.

b. 9 workstations will be required.
c. The same as #3c.

4. a. 5! = 5(4)(3)(2)(1) = 120 possible layouts.
b. The distance matrix.

c. One layout is shown in the diagram below. This one starts with departments 3 and 4, since they have the heaviest traffic.
d. Find the products of the loads and distances for every pair of departments.

e. It is not known whether this is the minimum possible value among all of the 120 values of the load-distance index.

5. a. In the traffic matrix, the rows represented "from" and the columns represented "to." In the distance matrix, the rows will represent "to" and the columns will represent "from." Use the diagram in #4c to align the elements of the distance matrix with the elements of the traffic matrix. Thus, column 1 of the distance matrix must be "from E," since department 1 is in area E. Row 3 of the distance matrix must be "to D," since department 3 is in area D. Here is the rearranged distance matrix:

b. Each element on the principal diagonal of the product matrix will be the sum of the products of the traffic from each department and the distances from each department to all of the other departments.

6. a. One assignment of departments to areas is shown in the diagram.
b. The Load-Distance Index = 36,000. Compare this value with the value in #4d. The constraints on the locations of the departments have caused the value to increase.