Background

Problem 1. Deterministic Linear Programming Assignment Model

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Toolbox

Problem 2. Robust Model

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Toolbox

Problem 3. Stochastic Model

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Toolbox

 

Background

This case study considers the problem of optimal selection of  tests subject to several constraints on available resources (e.g. money, times, and people). There are no partial tests: each test is assumed to be either conducted or not conducted. If each resource estimate is assumed to be accurate, then the problem of optimal selection of  tests is formulated as Deterministic Linear Programming Assignment model with boolean decision variables. To take into account uncertainty in resource estimates two models are used: robust model and the stochastic model. The Robust model conservatively increases the need in each resource by 20% of its average consumption by 20% largest consumers. The Stochastic model is based on the assumption that resource consumption by each test is independent normally distributed random value. The Robust and Stochastic models provide more realistic solution of the problem of optimal selection of tests, than the Deterministic Linear Programming model. Moreover, the Stochastic model reduces many constraints to one constraint, and provides possibility of sensitivity analysis.

 

Problem 1

Deterministic Linear Programming Assignment Model.

 

Simplified Problem Statement

 

Maximize Linear (maximizing the value of selected tests)

 subject to

Linear ≤ Const1 (constraint on resources)

Box constraints (constraints on decision variables)

 

where

Box constraints = constraints on individual decision variables

 

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

Problem 2

Robust Model.

 

Simplified Problem Statement

 

Maximize Linear (maximizing the value of selected tests)

 subject to

Linear + Cvar_comp_pos ≤ Const2 (constraint on resources)

Box constraints (constraints on decision variables)

 

where

Cvar_comp_pos = Cvar Component Positive

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

Problem 3

Stochastic Model.

 

Simplified Problem Statement

 

Maximize Linear (maximizing the value of selected tests)

 subject to

Prmulti_pen_ni_g ≤ Const3 (constraints on Probability Exceeding Penalty for Gain Multiple Normal Independent)

Box constraints (constraints on decision variables)

 

where

Prmulti_pen_ni_g = Probability Exceeding Penalty for Gain Multiple Normal Independent

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS: