TriCity Manufacturing (TCM) makes Styrofoam cups, plates, and sandwich and meal
ID: 371239 • Letter: T
Question
TriCity Manufacturing (TCM) makes Styrofoam cups, plates, and sandwich and meal containers. Next week’s schedule calls for the production of 80,000 small sandwich containers, 80,000 large sandwich containers, and 65,000 meal containers.
To make these containers, Styrofoam sheets are melted and formed into final products using three machines: M1, M2, and M3. Machine M1 can process Styrofoam sheets with a maximum width of 12 inches. The width capacity of machine M2 is 16 inches, and the width capacity of machine M3 is 20 inches.
The small sandwich containers required 10-inch-wide Styrofoam sheets; thus, these containers can be produced on each of the three machines.
The large sandwich containers require 12-iinch-wide sheets; thus, these containers can also be produced on each of the three machines.
However, the meal containers require 16-inch-wide Styrofoam sheets, so the meal containers cannot be produced on machine M1.
Waste is incurred in the production of all three containers because Styrofoam is lost in the heating and forming process, as well as in the final trimming of the product. The amount of waste generated varies, depending upon the container produced and the machine used. The following table shows the waste I square inches for each machine and product combination. The waste material is recycled for future use.
Machine
Small Sandwich
Large Sandwich
Meal
M1
20
15
--
M2
24
28
18
M3
32
35
36
Production rates also depend upon the container produced and the machine used. The following table shows the production rates in units per minutes for each machine and product combination. Machine capacities are limited for the next week. Time available is 35 hours for machine M1, 35 hours for machine M2, and 40 hours for machine M3.
Machine
Small Sandwich
Large Sandwich
Meal
M1
30
25
--
M2
45
40
30
M3
60
52
44
a.Costs associated with reprocessing the waste material have been increasing. Thus, TCM would like to minimize the amount of waste generated in meeting next week’s production schedule. Formulate a linear programming model that can be used to determine the best production schedule.
b. Solve the linear program formulated in part (a) to determine the production schedule. How much waste is generated? Which machines, if any, have idle capacity?
Machine
Small Sandwich
Large Sandwich
Meal
M1
20
15
--
M2
24
28
18
M3
32
35
36
Explanation / Answer
Solution:
Minimize: 20S1+ 24S2+ 32S3+ 15L1+28L2+ 35L3+ 18M2+ 36M3
Limitation:
Small Sandwich---- 1S1+1S2+1S3 >80000
Large Sandwich---- 1L1+1L2+1L3 > 80000
Meal----- 1M2+1M3 > 65000
Machine 1---- 0.033S1+0.04L1 <2100
Machine 2----- 0.0222S2+ 0.025L2+0.033M2 < 2100
Machine 3---- 0.0166S3+ 0.0192L3+0.022M3 < 2400
Slack (Mach. 3) = 492min
Total waste: 5515887 objective functions
Schedule of the machines
X1
Machine1
0
X2
Machine2
0
X3
Machine3
80000
X4
Machine1
52500
X5
Machine2
0
X6
Machine3
27500
X7
Machine2
63006.3
X8
Machine3
1993.699
Waste from Every Machine
Mach. 1 = 0
Mach. 2 = 6,400,000,000
Mach. 3 = 3,594,273.18
TOTAL = 6,403,594,273
Schedule of the machines
X1
Machine1
0
X2
Machine2
0
X3
Machine3
80000
X4
Machine1
52500
X5
Machine2
0
X6
Machine3
27500
X7
Machine2
63006.3
X8
Machine3
1993.699
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