Show that this can be expressed as a nonlinear optimization

Show that this can be expressed as a non-linear optimization problem with real variables.

Solution

I am taking an example here to explain:

model : how to find the optimal path to transport materials.

The objective is to find the route that minimizes the probability of having an accident, or equivalently, the route that maximizes the probability of not having an accident.

Creating the mathematical model to represent the problem:

Each decision variable indicates whether or not a particular route is taken (they are known as binary variables). We will define these variables in following way: Xij = 1 , if the route from node i to node j is selected, and Xij = 0 otherwise. Let Pij be the probability of having an accident while travelling from node i to node j (1- Pij is the probability of not having an accident).

Objective function: Minimize the probability of having an accident or equivalently, maximize the probability of not having an accident. Note that this objective function is nonlinear

Maximize f(X₁₂, X₁₃,….) = (1-P₁₂*X₁₂) (1-P₁₃*X₁₃) (1 – P₁₄*X₁₄) (1 – P₂₄*X₂₄) ………. (1 - P_9.10*X)

we have following set of constraints as

Node 1: + X57 + X67 – X78 – X7,10 = 0

Node 2: + X48 + X68 + X78 – X8,10 = 0

Node 3: + X79 – X9,10 = 0

Node 4: + X7,10 + X8,10 + X 9,10 = 1   and so on

thus we can represent binary models as a non-linear optimization problem with real variables.