Linear Programming to improve decision

Deterministic Model Using Mathematical Optimization

Optimization modeling for Decision Support System

Linear algebra is widely used for mathematical optimization and applications can be found almost in every industry operating under conflicting constraints. We will here work with a simple and quite common use case of cost optimization problem. The problem can be formulated as a standard linear optimization problem with the objective function is to minimize the transportation cost, subject to supply & demand with equality and inequality constraints. If we want to write a simple mathematical equation:

[d] = solver{O(), c1(), c2(), c3(), ….}

considering:

• d = decision (optimal input variables which constitute best decision
• solver = mathematical optimizer to calculate the best possible decision
• O() = objective function(minimum or maximum depends on use case)
• c1, c2, c3 = are constraints

So, we see here the whole idea is to optimize an objective function over a set of decision variables subject to set of constraints.

Let us create some synthetic data. For easy understanding and computational ease, the relevant information is in tabular format as shown below:

SPs are supply points here and DPs are demand points; we have 3 supply points and 4 demand points here. The table has the transport unit cost records from each SP to DP. The supply capacity column captures the value of maximum number of products can be supplied by each supply point and likewise demand row shows the value of maximum requirements. So, supply cannot be more than the total capacity and vice versa total demand cannot be less than actual requirements.

Here, we will try to formulate the problem in linear programming. Once formulated with equality or inequality constraints, then we can model the formula to come up with optimal solution.

Geometrically, the linear constraints define the feasible region, which is a convex polyhedron. A linear function is a convex…