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A manufacturing company makes two products (denoted as X and Y) through two ma- chines (denoted as A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and another 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and another 30 minutes processing time on machine B. At the beginning of this month, there are 49 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours, and on machine B is forecast to be 37 hours. In the current month, the demand for Xis forecast to be 85 units, and the demand for Yis forecast to be 95 units. Under the condition that the demand is met by the end of this month, the company's goal is to maximise the combined sum of the units of X and the units of Y in stock at the end of this month.

A manufacturing company makes two products (denoted as X and Y) through two ma- chines (denoted as A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and another 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and another 30 minutes processing time on machine B. At the beginning of this month, there are 49 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours, and on machine B is forecast to be 37 hours. In the current month, the demand for Xis forecast to be 85 units, and the demand for Yis forecast to be 95 units. Under the condition that the demand is met by the end of this month, the company's goal is to maximise the combined sum of the units of X and the units of Y in stock at the end of this month.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter10: Matrices
Section10.1: Solution Of Linear Systems
Problem 56E
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Question 5 A manufacturing company makes two products (denoted as X and Y) through two ma- chines (denoted as A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and another 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and another 30 minutes processing time on machine B. At the beginning of this month, there are 49 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours, and on machine B is forecast to be 37 hours. In the current month, the demand for Xis forecast to be 85 units, and the demand for Yis forecast to be 95 units. Under the condition that the demand is met by the end of this month, the company's goal is to maximise the combined sum of the units of X and the units of Y in stock at the end of this month. (1) Formulate the maximisation problem of deciding how many units of each product to make in the current month as a linear programming problem. You need to specify the target function to be maximised, and all constraints to be imposed on the target. (2) Present a hand-drawing graph to illustrate this maximisation problem. A ruler needs to be used when drawing straight lines, and the coordinates on the vertical axis and horizontal axis need to be marked. Note that you can take a photo of your graph and insert it into your Word document (or Latex document). (3) Find out all possible points, which are most likely to be the maximum point (that is, the point which maximises the target function). (4) Of all the points identified in (3), decide which point is the solution to this maximisa- tion problem.
Transcribed Image Text:Question 5 A manufacturing company makes two products (denoted as X and Y) through two ma- chines (denoted as A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and another 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and another 30 minutes processing time on machine B. At the beginning of this month, there are 49 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours, and on machine B is forecast to be 37 hours. In the current month, the demand for Xis forecast to be 85 units, and the demand for Yis forecast to be 95 units. Under the condition that the demand is met by the end of this month, the company's goal is to maximise the combined sum of the units of X and the units of Y in stock at the end of this month. (1) Formulate the maximisation problem of deciding how many units of each product to make in the current month as a linear programming problem. You need to specify the target function to be maximised, and all constraints to be imposed on the target. (2) Present a hand-drawing graph to illustrate this maximisation problem. A ruler needs to be used when drawing straight lines, and the coordinates on the vertical axis and horizontal axis need to be marked. Note that you can take a photo of your graph and insert it into your Word document (or Latex document). (3) Find out all possible points, which are most likely to be the maximum point (that is, the point which maximises the target function). (4) Of all the points identified in (3), decide which point is the solution to this maximisa- tion problem.
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