Method and system for encoding and fast-convergent solving general constrained systems

Abstract

The present invention provides a method and system that produces a near-optimum schedule in linear time by providing an optimal resource ordering scheme. The present invention is embodied in a scheduling computer program.

Description

TECHNICAL FIELD

[0002] The present invention relates to optimization methodologies and, in particular, to a method and system for encoding a class of constrained optimization problems, and then employing a generic, meta-level, iterative optimization technique to solve the encoded class of constrained optimization problems.

BACKGROUND OF THE INVENTION

[0003] It is well known that scheduling of scarce nonrenewable resources subjected to constraints is an NP-hard problem. Suppose that there is a set of tasks W and there is a set of N resources that can be assigned to tasks w e W. The problem that needs to be addressed is to schedule the N resources among the W tasks in an optimal or near optimal manner. Assume uwi(t) to be a piecewise constant function of the assignment of resources i to task w. Assume dw(t) to be a piecewise constant function of the demand for resources for task w. Then the optimization problem is as follows: 1$\underset{u_w^i\left(t\right),\dots ,u_w^N\left(t\right)}{\min }\sum _{w\in W}{\int }_0^T{c}_w\left(t\right)\&LeftBracketingBar;d^w\left(t\right)-\sum _{i=1}^N{u}_w^i\left(t\right)\&RightBracketingBar;dt$

[0004] where cw(t) is a time-varying cost of not satisfying demand for task w.

SUMMARY OF THE INVENTION

[0005] The present invention provides a method and system that produces a near-optimum schedule in linear time by providing an optimal resource ordering scheme.

DETAILED DESCRIPTION OF THE INVENTION

[0006] The present invention is embodied in a computer program that, using a state vector definition and a defined cost-go-go function, optimally orders resources for scheduling.

[0007] Define a state vector xwk(t) as follows:

x w k+1 (t)=xwk(t)+uwk(t),

x w 1 (t)=0,

x w 2 (t)=uw1(t),

wεW

[0008] The optimization problem described above then becomes:

φ(xwN(t))

[0009] subject to the above definition for xwk(t) where 2$\phi \left(x_w^N\left(t\right)\right)\text{:}=\sum _{w\in W}{\int }_0^T{c}_w\left(t\right)\&LeftBracketingBar;d^w\left(t\right)-x_w^N\left(t\right)\&RightBracketingBar;dt$

[0010] Define a cost-to-go function V(xwN(t),k)

V (xwN(t),k):={φ(xwN(t))}

x w N (t)=y(t)

[0011] Then by Bellman's principle of optimality,

V (y,k):={V(y(t)+uwk(t)),k+1}

x w N (t)=y(t)

V (xwN(t),N):=φ(xwN(t))

[0012] Optimal ordering of resources is based on the following weighting function:

ζ1ƒ1i+ζ2ƒ2i

[0013] where ζ1 and ζ2 are relative weight coefficients, ƒ1i is a variable that defines how a user values resource i, and ƒ2i is a variable that defines an actual cost of resource i.

[0014] The resources are arranged based on the respective values of the weighting function in such a way that the resources with the smallest values go first.

Claims

1. A method for scheduling scare, nonrenewable resources, the method comprising: defining a state vector; defining a cost-to-go function; and using Bellman's principle of optimality, optimizing the scheduling by optimally ordering the resources by a weighting function.