Linear programming with interval right handsides - …gold/COST_slides/remli.pdf · Linear programming with interval right handsides ... Nabila Remli Linear programming with interval

Linear programming with interval right handsides

Linear programming with interval righthandsides

Nabila Remli

Lamsade, Université Paris Dauphine

Cost IC0602 International Doctoral SchoolSeptember, 17-21,2007, Han Sur Lesse, Belgium

Nabila Remli Linear programming with interval right handsides


Outline

1 Introduction

2 State of the artUncertainty on objective function coefficientsUncertainty in the LP coefficients

3 On going project: Uncertainty on right handsidesThe case of inequality constraintsThe case of equality constraints

4 Conclusion



Introduction

Outline

1 Introduction



4 Conclusion



Introduction

Motivation

Optimization problems with uncertain data. For example, supplychain problems with unknown demand, uncertain costs,...

Classical approaches:Sensitivity analysis.Stochastic optimization: parameters need probability laws.Robust optimization: take into account, in the optimization, thevariation of the parameters.



Introduction

Concepts and definitions

Uncertain parameter: a finite set of values or an interval number.A scenario is an assigned value to each parameter.The aim of robust optimization problem is to find a "relativelygood" solution for all scenarios.Classical criteria to mesure robustness: worst (best) casecriterion and maximum regret criterion.

ContextHere, we deal with linear programming with interval data.



State of the art

Outline

1 Introduction



4 Conclusion



State of the art

Uncertainty on objective function coefficients

Outline

1 Introduction



4 Conclusion



State of the art


Works: Ben-Tal, Inuiguchi and Sakawa (1995), Kouvelis and Yu(1997), Mausser and Laguna (1998), Ramadan (2000), Averbakh andLebedev (2005) ...ect.

Formulation:

Let (Pc) denote the following linear problem: (Pc)

{min cxs.t x ∈ X

with:

c ∈ [c, c];

X being a nonempty bounded polyhedron.

NotationGiven a linear optimization problem P. The value of the optimalsolution of P is denotes v(P).



State of the art


Classical criteria

1 Worst case criterionFor a solution x ∈ X , find the scenario that gives worst value:

fWOR(x) = maxc≤c≤c

cx

fWOR(x) is an upper bound for the value of x : absolute guarantee.

Then, the problem is to determine the solution xWOR ∈ X whichminimizes fWOR as follows:

fWOR(xWOR) = minx∈X

fWOR(x)

xWOR is the solution that gives the smallest upper bound.



State of the art


Classical criteria

ExampleLet (P1) be a LP:

(P1)

min c1x1 + c2x2s.t x1 + x2 ≥ 2

x2 ≤ 42x1 − x2 ≤ 10x1, x2 ≥ 0

Where c1 ∈ [2, 3] and c2 ∈ [1, 6].fWOR(x) = 30fWOR(xWOR) = 6with c = (3, 6).

..

x1

x2

x

Xx=(4,3)

=(2,0)worx 1



State of the art


Classical criteria

2 Best case criterionFor a solution x ∈ X , find the scenario that gives best value:

fBES(x) = minc≤c≤c

cx

Then, the problem is to determine the solution xBES ∈ X whichminimizes fBES as follows:

fBES(xBES) = minx∈X

fBES(x)

Less studied in decision theory, but provides an interestinginformation in robust optimization.



State of the art


Classical criteria


(P1)

min c1x1 + c2x2s.t x1 + x2 ≥ 2

x2 ≤ 42x1 − x2 ≤ 10x1, x2 ≥ 0

Where c1 ∈ [2, 3] and c2 ∈ [1, 6].fBES(x) = 11fBES(xBES) = 2with c = (2, 1).

..x1

x2

x

Xx=(4,3)

=(2,0)bes

x

1



State of the art


Classical criteria

3 Maximum regret criterionThe regret associated to a fixed cost vector c and a solution x ∈ X , isgiven by: r(x , c) = cx − v(Pc)The maximum regret value of the solution x is:

fREG(x) = maxc≤c≤c

r(x , c)

fREG(x): greatest distance from optimality: relative guarantee.

The optimal solution according to the maximum regret criterion will bedenoted xREG and checks:

fREG(xREG ) = minx∈X

fREG(x)

fREG(xREG ) the smallest relative guarantee.



State of the art


Classical criteria

Complexity results1 Worst case criterion: polynomial time problem (Averbakh and

Lebedev - 2005).2 Best case criterion: strongly NP-hard problem (Gabrel, Murat,

Remli - 2007).3 Maximum regret criterion: strongly NP-hard (Averbakh and

Lebedev - 2005). Moreover, the problem of computing fREG(x) fora specific x is also strongly NP-hard. Approximate algorithmshave been developed to solve this problem.



State of the art


Bertsimas and Sim approach (2003)

In the objectif function: cj ∈ [cj , cj + cj ] where cj is the nominalvalue and cj ≥ 0 the deviation from the nominal value cj .New parameter Γ0 "budget of uncertainty": maximum number ofcoefficients that can deviate from their nominal value.

Γ0 = 0: nominal problem.Γ0 = n: worst case criterion.

The same complexity of the nominal problem.



State of the art

Uncertainty in the LP coefficients

Outline

1 Introduction



4 Conclusion



State of the art



Bertsimas and Sim approach (2003)The same approach: for each constraint i , each coefficientδij ∈ [δij − δij , δij + δij ] where δij is the nominal value and δij ≥ 0the deviation from the nominal value δij .New parameter Γi : the maximum number of coefficients that candeviate from their nominal value in the constraint i .Same complexity as the nominal problem.



State of the art



Chinnek and Ramadan approach (2000)All coefficients are interval numbers.Find the best and the worst optimum. In minimization problem:

best optimum: the lowest minimum.worst optimum: the highest minimum.

It gives ”the range of variation of the objective function”.Different cases were separately studied depending of variablesigns (restricted or unrestricted in sign) and kind of constraints(equality or inequality constraints) but none complexity resultsnor general algorithm were given.



On going project: Uncertainty on right handsides

Outline

1 Introduction



4 Conclusion




Uncertainty on right handsides

MotivationIn real applications: stock management, flows problems,...the demand is often uncertain (not exactly known).In the corresponding LP modelization, this demandappears like RHS.

LP with uncertainty on RHS.





DifficultiesWhen uncertainty concerns RHS, the set of feasiblesolutions is not exactly known.A solution may not be feasible for all interval righthandsides. Especially, for problems with equalityconstraints.

How to do?Compute worst (best) optimum.Two cases: LP with inequality constraints and LP withequality constraints.






(P2)

min x1 + 3x2s.t x1 + x2 ≥ b1 ∈ [2, 4]

2x1 − x2 ≤ b2 ∈ [6, 10]x2 ≤ 4x1, x2 ≥ 0

=10

1x1

x2

1b2b2b1b =4 =6

x

=2

��

��




The case of inequality constraints

Outline

1 Introduction



4 Conclusion





The case of inequality constraintsDefinition:

(Pb≥)

{min cxs.t Ax ≥ b

for each constraint i , bi ∈ [bi , bi ].

For all b ∈ [b, b], we denote X b≥ the nonempty bounded

polyhedron defined by {x ∈ Rn : Ax ≥ b}.





The case of inequality constraints1 Best optimum solution:

Determine the minimum value of the optimal solution of (Pb≥)

when b ∈ [b, b]. Solve the problem :

(B≥)

{min v(Pb

≥)

s.t b ≤ b ≤ b

Theorem(B≥) can be solved in polynomial time.





Proof.

(B≥)

{min v(Pb

≥)

s.t b ≤ b ≤ b

is equivalent to:

(B≥)

{min

b≤b≤bmin cx

s.t Ax ≥ b

Linear problem: min cxs.t Ax ≥ b

b ≤ b ≤ b





The case of inequality constraints2 Worst optimum solution:

Find the maximum value of the optimal solution of (Pb≥) when

b ∈ [b, b]. The worst optimal solution problem is:

(W≥)

{max v(Pb

≥)

s.t b ≤ b ≤ b

Theorem(W≥) can be solved in polynomial time.






(P2)

min x1 + 3x2s.t x1 + x2 ≥ b1 ∈ [2, 4]

2x1 − x2 ≤ b2 ∈ [6, 10]x2 ≤ 4x1, x2 ≥ 0

x1x1

x2

1b2b2b1b

=2=4 =6 =10

XWOpt��

��

X BOpt. .




The case of equality constraints

Outline

1 Introduction



4 Conclusion





The case of equality constraintsDefinition:

(Pb=)

{min cxs.t Ax = b

bi ∈ [bi , bi ].

For all b ∈ [b, b], we denote X b= the nonempty bounded

polyhedron defined by {x ∈ Rn : Ax = b}





The case of equality constraints1 Best optimum solution:

The best optimal solution problem is

(B=)

{min v(Pb

=)

s.t b ≤ b ≤ b

Theorem(B=) can be solved in polynomial time.





The case of equality constraints2 Worst optimum solution: The problem of determining the

worst optimal solution can be formulated as follows

(W=)

{max v(Pb

=)

s.t b ≤ b ≤ b

TheoremUsing a reduction from the problem of computing the maximum regretvalue for a given solution, we prove that (W=) is strongly NP-hardeven if variables are restricted in sign.





Element of proof

(W=)

{max v(Pb

=)

s.t b ≤ b ≤ b

equivalent to:

(W=)

{max

b≤b≤bmin cx

s.t Ax = b

According to the strong duality theorem:

(W=)

max bty

s.t Aty = ct

b ≤ b ≤ b

Quadratic problem.



Conclusion

Outline

1 Introduction



4 Conclusion



Conclusion

Concluding remarks

When uncertainty is on RHS, Bertsimas and Sim approachis not relevant: one uncertain parameter per constraint.penalty model ↪→ Relation between worst (best) optimumand worst (best) case criterion.

Future worksWhat about maximum regret criterion?Find other criteria?What about discret set of value for RHS only?



Conclusion

Concluding remarks

When uncertainty is on RHS, Bertsimas and Sim approachis not relevant: one uncertain parameter per constraint.penalty model ↪→ Relation between worst (best) optimumand worst (best) case criterion.

Future worksWhat about maximum regret criterion?Find other criteria?What about discret set of value for RHS only?


Documents

Linear programming with interval right handsides - …gold/COST_slides/remli.pdf · Linear programming with interval right handsides ... Nabila Remli Linear programming with interval