A paper mill sells rolls of paper to newspapers, which usually place orders for rolls of different widths. They have just received a large order for 1 million feet of 4-foot-wide paper, 3 million feet of 9-foot-wide paper, and 2 million feet of 12- foot-wide paper. The mill produces rolls of the following two sizes: (1) 2,000 feet long and 12 feet wide, at a cost of $400 per roll, and (2) 2,000 feet long and 18 feet wide, at a cost of $900 per roll. Large cutting machines are used to cut these rolls to rolls of desired widths. What should the mill do to satisfy this order at minimum cost? (a) Define decision variables. (b) Write a suitable LP problem formulation.
A paper mill sells rolls of paper to newspapers, which usually place orders for rolls of different widths. They have just received a large order for 1 million feet of 4-foot-wide paper, 3 million feet of 9-foot-wide paper, and 2 million feet of 12- foot-wide paper. The mill produces rolls of the following two sizes: (1) 2,000 feet long and 12 feet wide, at a cost of $400 per roll, and (2) 2,000 feet long and 18 feet wide, at a cost of $900 per roll. Large cutting machines are used to cut these rolls to rolls of desired widths. What should the mill do to satisfy this order at minimum cost? (a) Define decision variables. (b) Write a suitable LP problem formulation.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Question
A paper mill sells rolls of paper to newspapers, which usually place
orders for rolls of different widths. They have just received a large order for 1 million
feet of 4-foot-wide paper, 3 million feet of 9-foot-wide paper, and 2 million feet of 12-
foot-wide paper. The mill produces rolls of the following two sizes: (1) 2,000 feet long
and 12 feet wide, at a cost of $400 per roll, and (2) 2,000 feet long and 18 feet wide, at
a cost of $900 per roll. Large cutting machines are used to cut these rolls to rolls of
desired widths. What should the mill do to satisfy this order at minimum cost?
(a) Define decision variables.
(b) Write a suitable LP problem formulation.
Transcribed Image Text:**Problem Statement:**
A paper mill sells rolls of paper to newspapers, which usually place orders for rolls of different widths. They have just received a large order for 1 million feet of 4-foot-wide paper, 3 million feet of 9-foot-wide paper, and 2 million feet of 12-foot-wide paper. The mill produces rolls of the following two sizes:
1. 2,000 feet long and 12 feet wide, at a cost of $400 per roll
2. 2,000 feet long and 18 feet wide, at a cost of $900 per roll
Large cutting machines are used to cut these rolls to rolls of desired widths. What should the mill do to satisfy this order at minimum cost?
**(A) Define Decision Variables**
**(B) Write a Suitable LP Problem Formulation**
Transcribed Image Text:**Optimal Paper Roll Production Planning: A Cost-Effective Approach**
A paper mill specializes in supplying rolls of paper to newspapers, which customarily require rolls of various widths. Currently, the mill has received a substantial order for the following paper dimensions: 1 million feet of 4-foot-wide paper, 3 million feet of 9-foot-wide paper, and 2 million feet of 12-foot-wide paper.
The mill produces rolls available in two sizes:
1. **Size 1**: Rolls that are 2,000 feet long and 12 feet wide, priced at $400 per roll.
2. **Size 2**: Rolls that are 2,000 feet long and 18 feet wide, priced at $900 per roll.
To meet the specified dimensions, large cutting machines are employed to trim these rolls into the desired widths.
**Primary Objective**: The goal is to determine the optimal strategy for fulfilling this order at the lowest possible cost.
**Tasks**:
**(A) Define Decision Variables:**
- Identify the variables representing the number of rolls of each size to be produced and how they will be cut to satisfy the order requirements.
**(B) Write a Suitable LP Problem Formulation:**
- Establish a linear programming model to minimize production costs while meeting the demands for each type of paper width. This involves setting up constraints based on the roll cutting and assignment to different width requirements.
This approach ensures that the mill efficiently allocates resources and minimizes production costs while fulfilling large, varied orders from their clients.
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