Mathematics

Problem 1 (20 points)
All products in a smaller manufacturing shop have been sorted in eight part families, named A through H. All
parts are made on eight machines, numbered 1 through 8, which are organized in a job shop layout (process
layout). Process routing sheets were analyzed and the summary of part routings is
given in the following table. Your goal is to try to reorganize the job shop in a cellular layout for two
reasons. Because this is a small manufacturing system, the primary goal is to improve efficiency through
better scheduling. The next step would be gradual automation of the processes. For the data given:
(a) [5 points] Develop the part-machine incidence matrix. (b) [5 points] Identify manufacturing cells.
(d) [5 points] Determine the most logical sequence of machines in each manufacturing cell.
(e) [5 points] How to implement transformation to cellular manufacturing in this particular case? Can it be
implemented in stages, or not? Provide justification. Take in consideration the size of this manufacturing
system.

Part    Weekly quantity    Machine routing        Part    Weekly quantity    Machine routing
A    50    3 ? 5        E    45    8 ? 4
B    20    6 ? 3 ?5        F    80    5 ? 6
C    15    4 ? 8        G    70    2 ? 7
D    30    1 ? 7 ? 2        H    85    7 ? 1 ? 2

Problem 2 (20 points)
Now, one of the cells from previous problem is to be automated. The breakeven point is to be determined for
two production methods, one is existing manual method and the other is the newly proposed automated cell.
Both methods work one shift, 40 hours per week, 50 weeks per year. The manual method requires two workers at
$18.00/hr each. On average, material cost $30/peace. Together, they produce at a rate of 1.5 units/hour
(what cell is that?). Existing machines have paid off themselves and their maintenance costs $500 per year.
Their combined operation cost is $50/hour, which includes the average cost of tools, coolants, and energy.
The automated method has an initial cost of $165,000, a 4-year service life, no salvage value, and 25% rate
of return. The annual maintenance cost is $3000. The cost to run this cell alone is $75/hour, which includes
the average cost of tools, coolants, and energy. One worker services the cell, but his/her duties are split
between two other cells, so the cost of labor is $6/hour. The production rate for the automated machine is 3
units/hour. The cell will work with reduced scrap rate and smaller allowances, so the material cost is
reduced by 25%.
a) [8 points] Determine the breakeven point for the two methods in parts/year. b) [2 points] Can the

proposed method achieve the breakeven point?
c) [8 points] The automated cell will work 3 shifts, or 6,000 hour/year. Due to night shifts, the average

cost of labor is now $9/hour. Everything else stays the same. Calculate the new breakeven point. Can the

proposed method make that many parts?
d) [2 points] Draw a conclusion what is needed to justify the change of existing to automated manufacturing

method.

Problem 3 (10 points)
a)  [7 pts] Apply the rank order clustering technique to the part-machine incidence matrix in the following

table to identify logical part families and machine groups. Parts are identified by letters and machines are

identified numerically.
b)  [3 pts] Is there any overlapping machines? Describe briefly how you would solve this problem.

Parts
A    B    C    D    E    F    G    H    I
Weekly quantity    50    20    15    30    45    80    70    85    20

Machines    1            1    1    1
2    1    1                    1    1    1
3                        1    1    1
4    1        1        1
5            1        1
6        1                        1    1
7    1        1    1
8        1                1        1    1

Question 1 [2 points]
The part reference zero (X = 0, Y = 0, Z = 0) is at the top surface of the workpiece, in the lower left
corner of the top surface. What is wrong in this sequence of G-code?

… T01 M06
G00 X2. Y1.5
G43 H01 Z0.1
G01 Z-0.1 F5. X1.12 Y.52 F10.
…

Question 2 [2 points]
The part reference zero (X = 0, Y = 0, Z = 0) is at the top surface of the workpiece, in the lower left
corner of the top surface. What is wrong in this sequence of G-code? Hint: this is related to exercise in

Lab 4.
… T01 M06
G00 X2. Y1.5 S1500 M03
G43 H03 Z0.1
G01 Z-0.1 F5. X1.12 Y.52 F10.
…

Question 3 [2 points]
Describe what enables running the same G-code on different machines with different setups, provided that

machines use the same G-code setting (the same syntax for G, M, S, and other commands).

Question 4 [2 points]
List some applications where CNC type of motion control is used. In general, when is application of numeric
control appropriate?

Question 5 [2 points]
Describe briefly the meaning of these word addresses (commands): N, G, X, Y, Z, H, D, M, S, T, F. What are
they used for? What information they convey to the machine tool?

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