A new polynomial algorithm for nested resource allocation, speed optimization and other related problems Thibaut VIDAL 1 , 2 , Patrick JAILLET 1 , Nelson MACULAN 3 1 MIT – Massachusetts Institute of Technology, USA 2 PUC-Rio – Pontifical Catholic University of Rio de Janeiro, Brazil 3 COPPE – Federal University of Rio de Janeiro, Brazil ROUTE 2014 June 1-4 th , 2014

Contents 1 Research context Timing problems in vehicle routing Hierarchy of features Re-optimization 2 Problem statement Nested resource allocation problems ǫ -approximate solutions Existing algorithms A proximity theorem 3 Proposed Methodology A new decomposition algorithm Convergence and complexity 4 A remark on the expected number of active constraints 5 Computational experiments Research context Problem statement Methodology Remark Experiments Conclusions References 2 / 53 >

Contents 1 Research context Timing problems in vehicle routing Hierarchy of features Re-optimization 2 Problem statement Nested resource allocation problems ǫ -approximate solutions Existing algorithms A proximity theorem 3 Proposed Methodology A new decomposition algorithm Convergence and complexity 4 A remark on the expected number of active constraints 5 Computational experiments Research context Problem statement Methodology Remark Experiments Conclusions References 3 / 53 >

Timing problems in vehicle routing • General effort dedicated to better address rich vehicle routing problems involving many side constraints and attributes • Observation : many rich VRPs are hard because of their time features: (single, soft, or multiple) time windows, time-dependent, flexible or stochastic travel times, various time-dependent costs, break scheduling... • Timing subproblems: similar formulations in various domains: VRP, scheduling, PERT, resource allocation, isotone regression, telecommunications... • Cross-domain analysis of timing problems and algorithms: ◮ T. Vidal, T. G. Crainic, M. Gendreau, and C. Prins. A Unifying View on Timing Problems and Algorithms. Submitted & revised to Networks. Tech. Rep. CIRRELT 2011-43. Research context Problem statement Methodology Remark Experiments Conclusions References 4 / 53 >

Some examples • Four different applications • VRP with soft time windows . Optimizing arrival times for a given sequence of visits σ : | σ | | σ | � � min t ≥ 0 α max { e σ ( i ) − t σ ( i ) , 0 } + β max { t σ ( i ) − l σ ( i ) , 0 } (1.1) i =1 i =1 s.t. t σ ( i ) + δ σ ( i ) σ ( i +1) ≤ t σ ( i +1) 1 ≤ i < | σ | (1.2) Research context Problem statement Methodology Remark Experiments Conclusions References 5 / 53 >

Some examples • Four different applications • E/T scheduling . Optimizing processing dates for a given sequence of visits σ : | σ | | σ | � � min α i max { d σ ( i ) − t σ ( i ) , 0 } + β i max { t σ ( i ) − d σ ( i ) , 0 } (1.3) t ≥ 0 i =1 i =1 s.t. t σ ( i ) + p σ ( i ) ≤ t σ ( i +1) 1 ≤ i < | σ | (1.4) Research context Problem statement Methodology Remark Experiments Conclusions References 6 / 53 >

Some examples • Four different applications • Ship speed optimization . Optimizing leg speeds to visit a sequence of locations σ : | σ |− 1 d σ ( i ) σ ( i +1) � � � min d σ ( i ) σ ( i +1) ˆ c (1.5) t σ ( i +1) − t σ ( i ) t ≥ 0 i =1 t σ ( i ) + p σ ( i ) + d σ ( i ) σ ( i +1) s.t. ≤ t σ ( i +1) 1 ≤ i < | σ | (1.6) v max r σ ( i ) ≤ t σ ( i ) ≤ d σ ( i ) 1 ≤ i ≤ | σ | (1.7) Research context Problem statement Methodology Remark Experiments Conclusions References 7 / 53 >

Some examples • Four different applications • Isotonic Regression . Given a vector N = ( N 1 , . . . , N n ) of n real numbers, finding a vector of non-decreasing values t = ( t 1 , . . . , t n ) as close as possible to N according to a distance metric: min � t − N � (1.8) t =( t 1 ,..., t n ) s.t. t i ≤ t i +1 1 ≤ i < n (1.9) Research context Problem statement Methodology Remark Experiments Conclusions References 8 / 53 >

General Timing Problem • Timing problems: � � f x min α x y ( t ) (1.10) t ≥ 0 F x ∈F obj 1 ≤ y ≤ m x s . t . t i + p i ≤ t i +1 1 ≤ i < n (1.11) F x ∈ F cons , 1 ≤ y ≤ m x f x y ( t ) ≤ 0 (1.12) • Continuous variables t i following a total order . • Additional features characterized by functions f x y ( t ) for y ∈ { 1 , . . . , m x } , either in the objective or as constraints. • Many names in the literature: scheduling, timing, projection onto order simplexes, optimal service time problem... Research context Problem statement Methodology Remark Experiments Conclusions References 9 / 53 >

Features • Rich vehicle routing problems can involve various timing features Symbol Parameters Char. functions ξ Most frequent roles W Weights w i f i ( t ) = w i t i 1 Weighted execution dates f i ( t ) = ( t i − d i ) + D Deadlines d i 1 Deadline constraints, tardiness f i ( t ) = ( r i − t i ) + R Release dates r i 1 Release-date constraints, earliness. f i ( t ) = ( t i − d i ) + TW Time windows 1 Time-window constraints, +( r i − t i ) + TW i = [ r i , d i ] soft time windows. [( t i − d ik ) + MTW Multiple TW f i ( t ) = min 1 Multiple time-window constraints k MTW i = ∪ [ r ik , d ik ] +( r ik − t i ) + ] Σ c cvx Convex c cvx f i ( t ) = c cvx ( t i ) ( t i ) ( t i ) 1 Separable convex objectives i i i Σ c i ( t i ) General c i ( t ) f i ( t ) = c i ( t i ) 1 Separable objectives, time-dependent activity costs f ( t ) = ( t n − δ max − t 1 ) + DUR Total dur. δ max 2 Duration or overall idle time f i ( t ) = ( t i +1 − p i − t i ) + NWT No wait 2 No wait constraints, min idle time f i ( t ) = ( t i +1 − p i − ι i − t i ) + IDL Idle time ι i 2 Limited idle time by activity, min idle time excess f i ( t ) = ( t i + p i ( t i ) − t i +1 ) + P ( t ) Time-dependent 2 Processing-time constraints, min ac- proc. times p i ( t i ) tivities overlap f i ( t ) = ( t j − δ ij − t i ) + TL Time-lags δ ij 2 Min excess with respect to time-lags Σ c i (∆ t i ) General c i ( t ) f i ( t ) = c i ( t i +1 − t i ) 2 Separable functions of durations between successive activities, flex. processing times General c ij ( t , t ′ ) Σ c ij ( t i , t j ) f ij ( t )= c i ( t i , t j ) 2 Separable objectives or constraints by any pairs of variables Research context Problem statement Methodology Remark Experiments Conclusions References 10 / 53 >

Hierarchy of features • These features can be classified within a hierarchy (using many-one linear reduction relationships between the associated timing problems) • Features in the NP-hard area lead to NP-hard timing problems Research context Problem statement Methodology Remark Experiments Conclusions References 11 / 53 >

Re-optimization • Some particular features have been extensively studied in various fields. ◮ For example for the problem { Σ c cvx ( t i ) | ø } 30 algorithms from various i domains (routing, scheduling, PERT, isotonic regression) were inventoried, based on only three main concepts. • Key lines of research related to the resolution of series of similar timing problems within neighborhood searches, considering different sequences σ . � � f x min α x y ( t ) (1.13) t ≥ 0 F x ∈F obj 1 ≤ y ≤ m x s . t . t σ k ( i ) + p σ k ( i ) ,σ k ( i +1) ≤ t σ k ( i +1) 1 ≤ i < | σ | (1.14) F x ∈ F cons , 1 ≤ y ≤ m x f x y ( t ) ≤ 0 (1.15) Research context Problem statement Methodology Remark Experiments Conclusions References 12 / 53 >

Contents 1 Research context Timing problems in vehicle routing Hierarchy of features Re-optimization 2 Problem statement Nested resource allocation problems ǫ -approximate solutions Existing algorithms A proximity theorem 3 Proposed Methodology A new decomposition algorithm Convergence and complexity 4 A remark on the expected number of active constraints 5 Computational experiments Research context Problem statement Methodology Remark Experiments Conclusions References 13 / 53 >

One particular problem • Consider one particular timing problem with flexible travel times and deadlines: | σ |− 1 � min c i ( t σ ( i +1) − t σ ( i ) ) (2.1) t ≥ 0 i =1 t σ ( i ) + p σ ( i ) + d σ ( i ) σ ( i +1) s.t. ≤ t σ ( i +1) 1 ≤ i < | σ | (2.2) v max t σ ( i ) ≤ d σ ( i ) 1 ≤ i ≤ | σ | (2.3) t σ ( | σ | ) = B (2.4) • It is a vehicle speed optimization problem with convex – and possibly heterogeneous – cost/speed functions per leg. • Direct applications related to: ◮ Ship speed optimization (Norstad et al., 2011; Hvattum et al., 2013) ◮ Vehicle routing with flexible travel time or pollution routing (Hashimoto et al., 2006; Bektas and Laporte, 2011) Research context Problem statement Methodology Remark Experiments Conclusions References 14 / 53 >

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