Mathematical Programming Mathematical models Operations Research and Management (Decision) Science

Mathematical models A mathematical model is a description of a system or problem using mathematical concepts, tools and language. Mathematical model is a function, an equation, inequations, or system of equations or inequations, which represents certain aspects of the physical system or problem modelled. Ideally, by the application of the appropriate techniques the solution obtained from the model should also be the solution to the system problem.

Optimization model Problem III Suppose that a farmer has 35 acres of farm land to be planted with either cauliflower or kohlrabi. The farmer wants to have at least 8 acres of cauliflower. Farmer supposes increasing variable costs of cauliflower according to the function C1 = 0,25 x12-3x1, and costs of kohlrabi according to the function C2 = x22-4x2 (x1,2 – areas of plants).

Terminology Variables Decision variables Slack variables Artificial variables Constraints also called conditions or restrictions Capacities or Capacity constraints Requirements or Requirement constraints Balance constraints Definitional constraints Objective function also called criteria function

Terminology Feasibility region Search space Choice set Set of candidate solutions or Set of feasible solutions Objective function also called criteria function, cost function, energy function, or energy functional

Terminology Feasible solution Basic solution Infeasible solution Optimal solution Alternative solution Suboptimal solution

Solution of problem III x1 - area of cauliflower (ar) x2 - area of kohlrabi (ar) Minimize: f(x) = 0,25x12 - 3x1 + x22 – 4x2  min (minimize the costs) Subject to: x1 + x2  35 (limit on total area) x1  8 (limit on area of cauliflower ) x1 ≥ 0, x2 ≥ 0 (nonnegative area).

Mathematical programming Optimization model min f(x)  qi(x)  0 , i = 1, ..., m , xT=(x1, x2, ..., xn)T  Rn  f(x) and qi (x) - real function of many variables and x – vector of variables from vector space Rn.

General optimality problems Feasibility problem The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. Minimum and maximum value of a function The problem of finding extrema of function without regard to some constraints.

General optimality problems Mathematical optimization (alternatively, optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. One constraints in the form of line (contour line, curve) More then one constraints (subset of space of decision variables)

Classification of optimization models Number of criteria Single optimization Multiple optimization Type of criteria Minimization Maximization Goal problem Type of functions Linear model Nonlinear model Convex model Nonconvex model

Nonconvex or nonconcave function

Nonconvex feasibility region

Historical notes Fermat and Lagrange found calculus-based formulas for identifying optima. Newton and Gauss proposed iterative methods for moving towards an optimum. Historically, the first term for optimization was "linear programming", which was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939. Dantzig published the Simplex algorithm in 1947 John von Neumann developed the theory of duality in the same year.