Assignment Brief:

Please read the problem statement carefully and respond to questions $1$ and $2$ listed under Phase $1$ of the assignment.

Alloy Manufacturing Industry: A Case Study

A manufacturing company produces three types of metal alloys: Alloy X$,$ Alloy Y$,$ and Alloy Z$.$ The profit from selling each kilogram of Alloy X$,$ Alloy Y$,$ and Alloy Z is $$3,$ $$2,$ and $$3$ respectively. The production of these alloys requires manpower and two types of base metals: Metal A and Metal B$.$

Each kilogram of Alloy X requires $2$ hours of manpower, $3$ kg of Metal A$,$ and $1$ kg of Metal B$.$

Each kilogram of Alloy Y requires $1$ hour of manpower, $4$ kg of Metal A$,$ and $1$ kg of Metal B$.$

Each kilogram of Alloy Z requires $1$ hour of manpower, $2$ kg of Metal A$,$ and $1$ kg of Metal B$.$

The company has at most $2$ hours of manpower available. Additionally, the company must use at least $8$ kg of Metal A$.$

Respond to Questions $1$ and $2$ based on Phase $1$ case.

Question $1$

Formulate an LPP using the above details to maximize the total profit.

Question $2$

Using the M method, solve the LPP formulated in Question $4$ and find all the optimal solutions $($if they exist$).$

The responses will be based on the following parameters:

For Phase $1$ questions, you must find all the optimum solutions by formulating an LPP and solving it using the M method.

General instructions:

This case study$$based assignment is divided into two phases.

In Phase $1,$ you are required to solve the problem as a linear programming problem $($LPP$)$ based on the following objectives:

In the Phase $1$ problem, you are required to solve it as a linear programming problem. You will use optimization modeling and formulate linear programming problems.

Solving such problems requires bringing the system into standard form. You will analyze and apply algebraic solutions, need artificial variables, and dispose of artificial variables.

Identify special cases from the optimal table to enumerate other possible solutions to the problem. You will apply the basic concepts, non$-$basic optimal solutions, linear combinations, and so on$.$

Please do not use any AI chatbots as they are not able to answer this question.