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A paper mill manufactures three-meter wide rolls of paper. However, customers want to buy paper rolls of shorter width, and the mill has to cut such rolls from the three-meter wide rolls. Let us consider an order of 97 rolls of width 135 cm, 610 rolls of width 108 cm, 395 rolls of width 93 cm, and 211 rolls of width 42 cm. A standard three-meter wide roll could be cut into two rolls of width 93 cm, one roll of width 108 cm, and a rest of 6 cm which goes to waste. Another possibility could be to cut a standard three-meter wide roll into one roll of 135 cm, three rolls of width 42 cm, and a rest of 39 cm which goes to waste. List all possible ways of cutting a standard three-meter wide paper roll into a combination of rolls of the four widths requested by customers. Obviously, you will want to consider only possibilities for which the width of the wasted piece is less than 42 cm. Let P; denote the j-th cutting pattern on your list, and let x; be the number of standard three- meter wide rolls that ought to be cut according to cutting pattern P;. The mill owner wants to minimize the number E,x; of standard three-meter wide paper rolls that must be cut in order to satisfy the customers' orders. Cutting a fraction of the length of a roll according to one pattern before switching cutting pattern is technically possible, for the industrial process of cutting rolls requires that they be unwound. In other words, the decision variables x; are not required to take integer values. Write down the linear optimization problem the mill owner must solve.

A paper mill manufactures three-meter wide rolls of paper. However, customers want to buy paper rolls of shorter width, and the mill has to cut such rolls from the three-meter wide rolls. Let us consider an order of 97 rolls of width 135 cm, 610 rolls of width 108 cm, 395 rolls of width 93 cm, and 211 rolls of width 42 cm. A standard three-meter wide roll could be cut into two rolls of width 93 cm, one roll of width 108 cm, and a rest of 6 cm which goes to waste. Another possibility could be to cut a standard three-meter wide roll into one roll of 135 cm, three rolls of width 42 cm, and a rest of 39 cm which goes to waste. List all possible ways of cutting a standard three-meter wide paper roll into a combination of rolls of the four widths requested by customers. Obviously, you will want to consider only possibilities for which the width of the wasted piece is less than 42 cm. Let P; denote the j-th cutting pattern on your list, and let x; be the number of standard three- meter wide rolls that ought to be cut according to cutting pattern P;. The mill owner wants to minimize the number E,x; of standard three-meter wide paper rolls that must be cut in order to satisfy the customers' orders. Cutting a fraction of the length of a roll according to one pattern before switching cutting pattern is technically possible, for the industrial process of cutting rolls requires that they be unwound. In other words, the decision variables x; are not required to take integer values. Write down the linear optimization problem the mill owner must solve.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A paper mill manufactures three-meter wide rolls of paper. However, customers want to buy paper rolls of shorter width, and the mill has to cut such rolls from the three-meter wide rolls. Let us consider an order of 97 rolls of width 135 cm, 610 rolls of width 108 cm, 395 rolls of width 93 cm, and 211 rolls of width 42 cm. A standard three-meter wide roll could be cut into two rolls of width 93 cm, one roll of width 108 cm, and a rest of 6 cm which goes to waste. Another possibility could be to cut a standard three-meter wide roll into one roll of 135 cm, three rolls of width 42 cm, and a rest of 39 cm which goes to waste. List all possible ways of cutting a standard three-meter wide paper roll into a combination of rolls of the four widths requested by customers. Obviously, you will want to consider only possibilities for which the width of the wasted piece is less than 42 cm. Let P; denote the j-th cutting pattern on your list, and let x; be the number of standard three- meter wide rolls that ought to be cut according to cutting pattern P;. The mill owner wants to minimize the number E, x; of standard three-meter wide paper rolls that must be cut in order to satisfy the customers' orders. Cutting a fraction of the length of a roll according to one pattern before switching cutting pattern is technically possible, for the industrial process of cutting rolls requires that they be unwound. In other words, the decision variables x; are not required to take integer values. Write down the linear optimization problem the mill owner must solve.
Transcribed Image Text:A paper mill manufactures three-meter wide rolls of paper. However, customers want to buy paper rolls of shorter width, and the mill has to cut such rolls from the three-meter wide rolls. Let us consider an order of 97 rolls of width 135 cm, 610 rolls of width 108 cm, 395 rolls of width 93 cm, and 211 rolls of width 42 cm. A standard three-meter wide roll could be cut into two rolls of width 93 cm, one roll of width 108 cm, and a rest of 6 cm which goes to waste. Another possibility could be to cut a standard three-meter wide roll into one roll of 135 cm, three rolls of width 42 cm, and a rest of 39 cm which goes to waste. List all possible ways of cutting a standard three-meter wide paper roll into a combination of rolls of the four widths requested by customers. Obviously, you will want to consider only possibilities for which the width of the wasted piece is less than 42 cm. Let P; denote the j-th cutting pattern on your list, and let x; be the number of standard three- meter wide rolls that ought to be cut according to cutting pattern P;. The mill owner wants to minimize the number E, x; of standard three-meter wide paper rolls that must be cut in order to satisfy the customers' orders. Cutting a fraction of the length of a roll according to one pattern before switching cutting pattern is technically possible, for the industrial process of cutting rolls requires that they be unwound. In other words, the decision variables x; are not required to take integer values. Write down the linear optimization problem the mill owner must solve.
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