** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

**PROBLEM 1: problem_CS_Project_Selection_FXCHG**

Maximize Linear (maximizing net present value)

subject to

Fxchg ≤ Const (constraint on available initial capital)

Box constraints (bounds on positions)

——————————————————————–——–——–

Fxchg = Fixed Charge

Box constraints = constraints on individual decision variables

——————————————————————–——–——–

The same problem is solved with two different solvers: CAR and CARGRB (which needs GUROBI).

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 7 | 1 | 610 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 2: problem_CS_Project_Selection_linear**

Maximize Linear (maximizing net present value)

subject to

Linear ≤ Const (budget constraint)

Box constraints (binary position variables, x = Boolean)

——————————————————————–——–——–

Box constraints = constraints on individual decision variables

——————————————————————–——–——–

The same problem is solved with two different solvers: VAN and VANGRB (which needs GUROBI).

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 7 | 1 | 610 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**CASE STUDY SUMMARY**

This case study demonstrates an optimization setup and relevant graphs for a project selection problem. A similar problem is described in Luenberger (1998), p. 104. The model allows selecting among several different projects. Each project, if chosen, requires an initial capital outlay. Projects are selected to maximize the net present value of the investment subject to constraint on the initial available capital. This problem belongs to the class of “knapsack optimization” problems. We present two equivalent problem formulations having different presentation of the knapsack constraint: the first formulation uses a linear function with Boolean decision variables, and the second one uses PSG “fxchg_pos” function.

**References**

• Luenberger, D.G. (1998): *Investment Science*, Oxford University Press.