Mathematical formulation

The link between formulation & design is investigated. The establishment of variable definitions and the functional relationships of the variables needed to model the issue under study are the most important components of mathematical formulation. Creating an effective mathematical formulation is similar to other engineering design tasks. It necessitates a thorough understanding of the problem at hand and the capacity to devise new approaches to the problem. The following are the design considerations of mathematical formulation mentioned in this paper:

1. The mathematical formulation of the problem.
2. Definitions and relationships for a preliminary design.
3. Redesign to decrease the number of unknowns, variables, & constraints.
4. Refining of the redesign.
5. Generalisation to a multi-project scheduling issue.

Only algebraic manipulations are used, despite the complicated symbology.

A mathematical formulation of the problem

Let x and y represent the number of cabinets of kind 1 and 2 that he must produce. Non-negative constraints are non-negative limitations.

The company is allowed to invest 540 hours of labour and must build up to 50 cabinets. Hence,

15x + 9y <= 540

x + y <= 50

Linear constraints are defined as the above two equations.

Let Z be the profit he makes from producing x and y pieces of type 1 and 2 cabinets. Thus,

Z = 5000x + 3000

The goal here is to maximise Z. As a result, Z is referred to as the objective function. We employ graphs to obtain the answer to this question, which is referred to as the graphical approach of answering LPP. 

A mathematical formulation of a linear programming problem:

Linear programming, also known as linear optimisation, obtains the best possible solution to a mathematical model or maths formulation by taking into account specific linear relationships. LPP is another name for it. It encompasses issues such as maximising revenues, decreasing expenditures, and making the most efficient use of resources. The simplex approach or the graphical method can tackle these problems.

The phases in the technique for mathematically formulating a linear programming problem are as follows.

• Determine the variables that will be used to make a choice.
• Identify the goal function that must be maximised or minimised and describe it as a linear function of choice factors.
• Identify the collection of constraint conditions and describe them in terms of the decision variables as linear inequalities or equations.

A mathematical formulation of assignment problem

The assignment problem is a linear programming problem that entails allocating resources to individual tasks. It does so that the process’s cost or time is kept to a minimum while the profit or sale is maximised. Though similar problems can be handled using simplex or transportation methods, the assignment model provides a more straightforward solution.

The problem is formed on a one-to-one basis. A supervisor at a workplace might have six workers available & six tasks to fire. He will have to decide which work should be assigned to which job. This is a difficulty with an assignment.

Problem-solving strategy

Consider the minimisation type objective function. To solve this Assignment problem, you must take the following steps:

1. Beginning with the first row, find the smallest cost element in each row in the given cost table. This lowest element is now subtracted from each row’s element. As a result, each new table row will have at least one zero.

2. After constructing the table (like in step 1), take the table’s columns. Starting with the first column, find the cheapest element in each column. Subtract this smallest element from each column’s element. After completing steps 1 and 2, we should have at least one zero in each column of the reduced cost table.

3. The allocations for the smaller table are now made in the following manner:

(I) Rows are inspected one by one until the row with exactly one (zero) zero is discovered. This single zero is assigned by putting a square around it. All other zeros in the corresponding column are crossed out (x) because they will not be utilised to create additional assignments. For each row, a step is taken. 

(II) Now do Step 3 (I) on the columns as follows:

Columns are inspected one by one until one with exactly one zero is discovered. Now, by placing the square around this one zero, all other zeros in the associated rows are crossed out (x); a step is performed for each column.

(III) Steps 3 (I) and 3 (II) are repeated until all zeros have been crossed out or noted. The ideal solution has been found when the number of marked zeros or made assignments equals the number of rows or columns. There shall be exactly one assignment in each or no assignment in any of the columns. We’ll go to step 4 in this scenario.

4. At this point, draw the smallest number of lines (horizontal and vertical) required to cover all zeros in the matrices obtained in steps 3 and 4.

(I) Mark all of the rows that have no assignment.

(ii) Tick all the columns in the ticked rows with a zero value.

(iii) Tick all the rows that haven’t been marked yet and have an assignment in the designated columns.

(iv) Repeat steps (4(i), 4(ii), 4(iii) until there are no more rows or columns to mark.

(v) Now, draw straight lines that cross all unmarked rows and columns. It’s also worth noting that in a n x n matrix, if there is no solution among them, always less than ‘n’ lines will cover all the zeros.

5. If the number of lines produced in step 4 equals n or the number of rows, the solution is optimal; otherwise, proceed to step 6.

6. Among all the uncovered elements, choose the smallest element. This element is now deducted from all the elements that were previously uncovered & added to the intersection of two lines. For new assignments, this is the matrix.

7. Steps 3–7 should be repeated until the total number of assignments equals the total number of rows or columns.

Conclusion

A known function (maybe profit/loss or any output) is sometimes sought to optimise (maximise or minimise) under a set of linear restrictions. Linear programming problems (LPP) describe how to locate such an optimal function and the values that would optimise the needed function.

It’s one of the most crucial instruments in operations research. In almost all industries, it is widely used as a decision-making tool. LPP can be used in various domains, including economics, computer science, mathematics, and so on.