Robust optimization of uncertain multistage inventory systems with inexact data in decision rules Frans de Ruiter (joint work with Aharon Ben-Tal, Ruud Brekelmans and Dick den Hertog) Tilburg School of Economics and Management Tilburg University July 13, 2015 Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 1 / 24
Overview Intro inexact data 1 Intro robust optimization 2 Robust optimization techniques 3 New methodology 4 Numerical example 5 Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 2 / 24
Intro inexact data Data uncertainty in practical applications Optimization problems are affected by uncertainty in their parameters due to: Measurement errors 1 physical experiments, weather observations, . . . Prediction errors 2 future demand, returns, . . . Implementation errors 3 optimal temperature, size, . . . System data errors 4 inventory records, miscodings, . . . Robust Optimization (RO) techniques find solutions that are robust against uncertainties in the parameters. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 3 / 24
Intro inexact data Evidence of poor data quality Despite developments in our Big Data era poor data quality is still a big issue. • Redman (1998): 1 − 5 % of data fields are erred. • DeHoratius and Raman (2008): Over 6 out of 10 inventory records are inaccurate. • Haug et al. (2011): Not even half of the companies is very confident in the quality of their data. . . . Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 4 / 24
Intro robust optimization Evolution of Robust Optimization Early 70s: First note on RO by Soyster. Late 90s: Research kicked off due to Ben-Tal, Nemirovski and El Ghaoui. 2004: Bertsimas and Sim’s budget uncertainty model. 2004: Adjustable Robust Optimization by Ben-Tal et al. 2009: Book Robust Optimization by Ben-Tal, Nemirovski and El Ghaoui. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 5 / 24
Intro robust optimization Robust Optimization Robust Optimization (RO): Decisions are here-and-now, to be made before data is revealed. 1 Decision maker is responsible for realisations in, and only in, the 2 uncertainty set. Constraints are “hard”, no violations allowed. 3 Advantages: • Only crude information (set of possible realisations) needed. • Computational tractability. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 6 / 24
Intro robust optimization Numerical example LP model (Ben-Tal et al. (2004)) 1 , 800 1 , 800 Nominal demand Random trajectory Minimize production costs over 24 1 , 600 1 , 600 Demand tube periods 1 , 400 1 , 400 subject to: Demand (units) 1 , 200 1 , 200 • Bounds on production 1 , 000 1 , 000 • Bounds on inventory levels ( V max and V min ) 800 800 • All uncertain demand is met 600 600 (production costs seasonal) 400 400 10 10 20 20 30 30 40 40 Time period (weeks) Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 7 / 24
Intro robust optimization Adjustable Robust Optimization Adjustabe Robust Optimization (ARO) is an extension of RO for multistage optimization problems where some decisions are wait-and-see. These adjustable decisions are functions of the revealed data from previous periods. Crucially, the wait-and-see decisions in ARO rely on exact revealed data. In practice, revealed data is also inexact which may lead to poor performance of ARO... Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 8 / 24
Intro robust optimization Numerical example - ARO assumption 1 , 800 1 , 800 Revealed data Demand tube 1 , 600 1 , 600 1 , 400 1 , 400 Demand (units) 1 , 200 1 , 200 1 , 000 1 , 000 “present” 800 800 600 600 400 400 10 10 20 20 30 30 40 40 Time period (weeks) Crucially, ARO relies on exact revealed data. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 9 / 24
Intro robust optimization What if revealed data is inexact? 1 , 800 1 , 800 Revealed data Demand tube 1 , 600 1 , 600 1 , 400 1 , 400 Demand (units) 1 , 200 1 , 200 1 , 000 1 , 000 “present” 800 800 600 600 400 400 10 10 20 20 30 30 40 40 Time period (weeks) Much evidence that revealed data is inexact! What are the consequences for ARO? Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 10 / 24
Intro robust optimization Contributions Reliance on data ‘as is’ may lead to poor performance of ARO if revealed 1 data is inexact. New method with decision rules based on inexact revealed data. 2 Uses convex analysis ( conjugates and support functions ). 1 Applicable to many types of convex problems and many different convex 2 uncertainty sets. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 11 / 24
Robust optimization techniques Robust counterparts Uncertain linear constraints of the form: ( a + A ζ ) ⊤ x + d ⊤ y ≤ 0 ∀ ζ ∈ Z x ∈ R n , y ∈ R m nonadjustable decision variables. a the nominal value of the the coefficient for x and A ∈ R n × L . ζ is the primitive uncertainty residing in a closed convex uncertainty set Z ⊂ R L . d ∈ R m is certain. How to derive equivalent tractable robust counterparts (RC) without ‘ ∀ ′ constraints? Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 12 / 24
Robust optimization techniques Tractable RC Introduce the indicator function δ : R L → { 0 , ∞} � 0 if ζ ∈ Z δ ( ζ |Z ) = ∞ if ζ �∈ Z and its support function: δ ∗ : R L → R � � δ ∗ ( v |Z ) = max ζ ⊤ v easy to compute for many U 0 ! ζ ∈Z δ ∗ ( v |Z ) Uncertainty set Z box { ζ : || ζ || ∞ ≤ θ } θ || v || 1 ball { ζ : || ζ || 2 ≤ θ } θ || v || 2 � b ⊤ z if B ⊤ z = v , z ≥ 0 polyhedral { ζ : b − B ζ ≥ 0 } min ∞ otherwise z Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 13 / 24
Robust optimization techniques Tractable RC Deriving the tractable RC: ( a + A ζ ) ⊤ x + d ⊤ y ≤ 0 ∀ ζ ∈ Z ⇔ � � ( a + A ζ ) ⊤ x + d ⊤ y ≤ 0 max ζ ∈Z ⇔ a ⊤ x + d ⊤ y + δ ∗ � � A ⊤ x |Z ≤ 0 . See also Ben-Tal, den Hertog and Vial (2014) Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 14 / 24
Robust optimization techniques Adjustable robust counterpart Uncertain linear constraints of the form: ( a + A ζ ) ⊤ x + d ⊤ y ( ζ ) ≤ 0 ∀ ζ ∈ Z x ∈ R n nonadjustable and y ( ζ ) ∈ R m adjustable. a the nominal value of the the coefficient for x and A ∈ R n × L . d ∈ R m is certain (fixed recourse). Linear decision rule based on exact revealed data y ( ζ ) = u + V ⊤ ζ with u ∈ R m and V ∈ R m × L . Tractable Affinely Adjustable Robust Counterpart (AARC): a ⊤ x + d ⊤ u + δ ∗ � � Ax + V ⊤ d |Z ≤ 0 Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 15 / 24
New methodology Inexact revealed data in decision rules Our new methodology deals with uncertain linear constraints of the form: ( a + A ζ ) ⊤ x + d ⊤ y ( � ∀ ζ, � ( � ζ − ζ ) ∈ � ζ ) ≤ 0 ζ ∈ Z , Z Affine decision rule based on inexact revealed data y ( � ζ ) = u + V � ζ with u ∈ R m and V ∈ R m × L . Estimation error ( � ζ − ζ ) resides in closed convex set � Z . Tractable AARC with decision rules based on inexact revealed data (ARCID): + δ ∗ � � a ⊤ x + d ⊤ u + δ ∗ � � + δ ∗ � � w | � A ⊤ x + w |Z V ⊤ d − w |Z Z ≤ 0 , with w ∈ R n an additional here-and-now decision variable. Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 16 / 24
New methodology Example with polyhedral uncertainty Consider the following constraint with decision rule y ( � ζ ) = u + V � ζ based on inexact revealed data: ( a + A ζ ) ⊤ x + d ⊤ y ( � ∀ ζ, � ( � ζ − ζ ) ∈ � ζ ) ≤ 0 ζ ∈ Z , Z , where Z = { ζ : b − B ζ ≥ 0 } and � Z = { ξ : r − R ξ ≥ 0 } are polyhedral uncertainty sets with given parameters B , R ∈ R l × p and b , r ∈ R p . Tractable AARCID: a ⊤ x + d ⊤ u + b ⊤ ( z 1 + z 2 ) + r ⊤ z 3 ≤ 0 B ⊤ z 1 = A ⊤ x + w B ⊤ z 2 = V ⊤ d − w R ⊤ z 3 = w z 1 , z 2 , z 3 ≥ 0 , where w , z 1 , z 2 , z 3 ∈ R n are additional here-and-now variables. Tractability: LP! Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 17 / 24
New methodology RC, ARC and the new ARCID ζ = � ζ ζ ≈ � ζ ζ ∈ Z Z Z RC ARCID ARC Red shaded region: Uncertainty set Z . Blue shaded region: Estimation uncertainty � Z . Large estimation uncertainty → ARCID boils down to RC (no extra value of inexact revealed data). Zero estimation uncertainty → ARCID ≡ ARC (revealed data is exact). In all other situations the new ARCID may outperform both RC and ARC! Frans de Ruiter (Tilburg University) RO with inexact data in decision rules ISMP July 13 18 / 24
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