## 3 Statement of the Model  
We have considered a production planning problem in which we have chosen different types of machines for the production process, such as a milling machine, lathe, grinder, jig saw, drill press, and band saw. The main objective of every single production business is to make a profit to keep the company running smoothly. It is always advisable for a company to prepare a production plan based on scientific methods to get a clear direction as to how the production process should be carried out. The main objective of this study is to optimize the profit, product liability, quality, and workers' satisfaction. The following input information is essential before formulating the problem, i.e., information about the resources and the available facilities, including production equipment, hours of using the machine for manufacturing product, number of machines available for the production of the item, etc. The following assumptions and limitations which are essential for our industrial planning model are considered as:  
* The model has multi-objectives, where we maximize profit, product liability, quality, and workers' satisfaction in the industry. * The multi-item production model is to be considered. * One machine cannot perform more than one operation at a time. * There is no shortage of materials in production. * Demand should be only for final products. * Machine and storage capacity cannot exceed the maximum level in any case. The following notations which have been used during the formulation of the problem:  
**Nomenclature**  
**Indices:**  
\(k\)**--Index for multi-choices, \(k=1\), \(2\),..., \(K\)  
\(j\)**--Index for the manufactured product, \(\mathrm{j}=1\), \(2\),..., \(\mathrm{J}\)  
\(i\)**--Index for machines \(i=1\), \(2\),..., \(\mathrm{J}\)  
\(l\)**--Index for the level, \(l=1\),\(2\)  
\(m\)**--Index for the objectives, \(m=1\),\(2\),..., \(\mathrm{J}\)  
**Decision Variable:**  
\(x_{j}\)**--Manufactured items**  
**Parameters:**\(P_{j}\)--Profit related to the product  
\(L_{j}\)--Liability of the product  
\(Q_{j}\)--Quality of the product  
\(W_{j}\)--Workers' satisfaction  
\(m_{j}\)--Milling machine time on \(j\)th product  
\(M\)--The total available time of milling machine  
\(l_{j}\)--Lathe time on \(j\)th product  
\(L\)--The total available time of lathe  
\(g_{j}\)--Grinder time on \(j\)th product  
\(G\)--The total available time of grinder  
\(s_{j}\)--Jig saw time on \(j\)th product  
\(S\)--The total available time of jig saw  
\(d_{j}\)--Drill press time on \(j\)th product  
\(D\)--The total available time of drill press  
\(b_{j}\)--Band saw time on \(j\)th product  
\(B\)--The total available time of band saw  
Based on the above assumptions and notations, the mathematical model of the production planning problem is formulated:  
The objective function **I** is related to maximizing the profit:  
Every company wants to remain in business for a longer period, and this can be achieved by only making a profit. In other words, profit is the main oil which drives the wheels of a company, and in a mathematical form, it is represented as:  
\[Max\,Z_{1}\ =\ \sum_{j\ =\ 1}^{3}P_{j}x_{j}.\]  
Objective function **II** is related to maximizing product reliability:  
Every company wants to produce a product that must be reliable for their customers in terms of their longevity. The company's reputation in the market depends upon the reliability of the product produced by the company's during its production run, and in a mathematical form, it is represented as:  
\[Max\,Z_{2}\ =\ \sum_{j\ =\ 1}^{3}L_{j}x_{j}.\]  
Objective function **III** is related to maximizing the quality of the product:  
Quality has its importance when it comes to any product of a company. The quality of the product affects the company's reputation. It is very important to uphold the quality of the product and satisfy your customers while retaining their loyalty.