A furniture company produces three types of desks: a...

A furniture company produces three types of desks: a children’s model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$\begin{array}{|l|l|l|l|} \hline \text {} &\text {Children’s Model} &\text {Office Model} &\text {Deluxe Model}\\ \hline {\text { Cutting }} & {2 \mathrm{hr}} & {3 \mathrm{hr}} & {2 \mathrm{hr}} \\ \hline{\text { Construction }} & {2 \mathrm{hr}} & {1 \mathrm{hr}} & {3 \mathrm{hr}} \\ \hline{\text { Finishing }} & {1 \mathrm{hr}} & {1 \mathrm{hr}} & {2 \mathrm{hr}}\\ \hline\end{array}$$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

A furniture company produces three types of desks: a children’s model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table.
$$\begin{array}{|l|l|l|l|} \hline \text {} &\text {Children’s Model} &\text {Office Model} &\text {Deluxe Model}\\ \hline {\text { Cutting }} & {2 \mathrm{hr}} & {3 \mathrm{hr}} & {2 \mathrm{hr}} \\ \hline{\text { Construction }} & {2 \mathrm{hr}} & {1 \mathrm{hr}} & {3 \mathrm{hr}} \\ \hline{\text { Finishing }} & {1 \mathrm{hr}} & {1 \mathrm{hr}} & {2 \mathrm{hr}}\\ \hline\end{array}$$
Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Key Concepts

Linear Programming

Linear programming is a mathematical method used for optimizing a linear objective function, subject to a set of linear equality or inequality constraints. It is commonly applied in resource allocation problems to determine the best possible outcome within given limited resources.

System of Linear Equations

A system of linear equations is a collection of equations that are linear in their variables. In production planning problems, such systems are used to represent the relationships between different processes and the resources they consume, ensuring that all resource constraints are met.

Resource Allocation

Resource allocation involves efficiently distributing limited resources among competing activities to achieve a specific goal. In manufacturing, it is crucial for ensuring each stage of production receives enough resources while meeting overall production requirements.

Constraint Satisfaction

Constraint satisfaction refers to the process of finding a solution that meets all the required limits or conditions imposed on a problem. This includes ensuring that the consumption of resources does not exceed the available amount in each category, a key aspect of production planning.

Manufacturing Process Optimization

Manufacturing process optimization focuses on improving the efficiency and productivity of production processes. It involves analyzing time or resource requirements at different stages, and adjusting production levels to fully utilize available resources while meeting demand.

Transcript

00:01 So for this question we have a table of different desk models and the work that you have to put into manufacture them.

00:09 Cutting, construction, finishing.

00:11 And we know that the company has a certain amount of hours for each type of stage.

00:18 And we want to work out how many of each desk we should be making.

00:22 So if we let our variables be the different models, say a is the children's model, b is the office model and c is the deluxe model.

00:35 And we will make some equations out of this.

00:37 So for cutting, we have 100 hours each week for cutting.

00:43 So our right hand side will be 100.

00:44 The children's model takes two hours, the office, three, and deluxe two.

00:52 And for construction, we have two for a, one for b and three for c.

01:01 And we have 100 hours for this and for finishing we have one a one for b and two for c and for this we have 65 hours yeah so now we have our three equations and when we solve this system we will get the number of different models that we're going to be producing now let's make an augmented matrix to solve this the difficult part here is just passing the question and working out how to translate the information into these equations.

01:37 From here on it's quite standard.

01:39 So i'm going to let this be my row one because it's got a one here.

01:43 This can be row two and this can be row three.

01:47 Okay so my first variable is going to be a.

01:51 We will have one, two, two...

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