Two Integer Linear Programming Approaches for Solving the Car - PowerPoint PPT Presentation

Two Integer Linear Programming Approaches for Solving the Car Sequencing Problem Matthias Prandtstetter G¨ unther R. Raidl Institute for Computer Graphics and Algorithms Vienna University of Technology Vienna, Austria www.ads.tuwien.ac.at 10 th Aussois Workshop on Combinatorial Optimization Aussois, France, January 2006. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Searching for an optimal arrangement of commissioned cars along a production line in automobile industry. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Searching for an optimal arrangement of commissioned cars along a production line in automobile industry. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Searching for an optimal arrangement of commissioned cars along a production line in automobile industry. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Searching for an optimal arrangement of commissioned cars along a production line in automobile industry. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition “At most s consecutive cars are allowed to be painted with the same color.” Color changes have to be minimized. Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition “Only l c of m c cars are allowed to require component c .” Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Given n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . number of cars C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of components F ⊆ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . set of colors K = { k : k ⊆ C , | k ∩ F | = 1 } . . . . . . . . . . . . . .set of configurations cost c . . . . . . . . costs for violating a constraint w.r.t. component c l c , m c . . . . . . . . . . . . . . . . . . . . . . . . ratio constraints for component c � � 1 if c ∈ k 1 if c required by car at pos i a ck = e ci = 0 otherwise 0 otherwise Solution sequence X = � x 1 , . . . , x n � : { 1 , . . . , n } → K Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition Minimize n n � � � obj ( X ) = change ( i ) + viol ( i , c ) i = 1 c ∈ C \ F i = 1 change ( i ) i − 2 i − 1 i i + 1 · · · · · · 0 0 1 0 viol ( i , c ) ( l c = 2 , m c = 5) i − 5 i − 4 i − 3 i − 2 i − 1 i i + 1 . . . – – – . . . Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Car Sequencing - Definition change ( i )  � ( a fx i − a fx i − 1 ) · cost f � if i ≥ 2 max  f ∈ F change ( i ) = � � ( a f 1 − e f 1 ) · cost f max otherwise  f ∈ F viol ( i , c )  � � i  � max 0 , a cx i − l c · cost c if i ≥ m c     j = i − m c + 1 viol ( i , c ) = � � i m c − i  � � e cj − l c · cost c max 0 , a cx j + otherwise     j = 1 j = 1 Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Work Done So Far Proof of NP-hardness Decision Problem (without colors): Kis (2004), Gent (1998) Optimization Problem (including colors): Hu (2004) Greedy Heuristics (Gottlieb et. al. 2003) Local Search (Jaszkiewicz 2004, Puchta 2002) Large Neighborhood Search (Perron 2004) Ant Colony Optimization (Gravel et. al. 2005, Gottlieb et. al. 2003) Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Work Done So Far (continued) ROADEF Challenge 2005 Exact Approaches (Gravel et. al. 2005, Hu 2004) Basic Ideas (Gravel et. al. 2005) Grouping identical cars into classes Only counting the number of positions violating a constraint No colors included Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program C-ILP Basic Ideas Placing components along the production line Making sure only commissioned cars are produced Counting the number of violations of assembly shop constraints Including colors Making sure no paint shop constraints are violated Counting the number of color changes Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program Placing Components variables � 1 c at pos i b ci = ∀ c ∈ C , ∀ i ∈ { 1 , . . . , n } 0 otherwise constraints � b fi = 1 ∀ i ∈ { 1 , . . . , n } (1) f ∈ F n � b ci = d c ∀ c ∈ C (2) i = 1 Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program Only Commissioned Cars variables � 1 config k at pos i ∀ k ∈ K , ∀ i ∈ { 1 , . . . , n } p ki = 0 otherwise constraints p ki ≤ a ck · b ci + ( 1 − a ck ) · ( 1 − b ci ) ∀ k ∈ K , ∀ c ∈ C , ∀ i ∈ { 1 , . . . , n } (3) n X p ki = δ k ∀ k ∈ K i = 1 (4) Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program Number of Violations of Assembly Shop Constraints variables g ci ∈ { 0 , . . . , m c − l c } ∀ c ∈ C \ F , ∀ i ∈ { 1 , . . . , n } constraints m c − i i � � g ci ≥ b cj + e cj − l c ∀ c ∈ C \ F , ∀ i ∈ { 1 , . . . , m c − 1 } (5) j = 1 j = 1 i � g ci ≥ b cj − l c ∀ c ∈ C \ F , ∀ i ∈ { m c , . . . , n } (6) j = i − m c + 1 Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program Paint Shop Constraints constraints i s − i + 1 � � e fj ≤ s ∀ f ∈ F , ∀ i ∈ { 1 , . . . , s } b fj + (7) j = 1 j = 1 i � b fj ≤ s ∀ f ∈ F , ∀ i ∈ { s + 1 , . . . , n } (8) j = i − s Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Component Integer Linear Program Counting Color Changes variables � 1 if the color changed w fi = ∀ f ∈ F , ∀ i ∈ { 1 , . . . , n } 0 otherwise constraints w fi ≥ 0 ∀ f ∈ F , ∀ i ∈ { 1 , . . . , n } (9) w f 1 ≥ b f 1 − e f 1 ∀ f ∈ F (10) w fi ≥ b fi − b f ( i − 1 ) ∀ f ∈ F , ∀ i ∈ { 2 , . . . , n } (11) Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Configuration Integer Linear Program K-ILP Basic Ideas Placing configurations along the production line Avoiding symmetries with identical cars Counting the number of violations of assembly shop constraints Including colors Making sure no paint shop constraints are violated Counting the number of color Changes Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Configuration Integer Linear Program Placing Configurations variables � 1 if config k at pos i p ki = ∀ k ∈ K , ∀ i ∈ { 1 , . . . , n } 0 otherwise constraints � p ki = 1 ∀ i ∈ { 1 , . . . , n } (12) k ∈ K n � p ki = δ k ∀ k ∈ K (13) i = 1 Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Configuration Integer Linear Program Number of Violations of Assembly Shop Constraints variables g ci ∈ { 0 , . . . , m c − l c } ∀ c ∈ C \ F , ∀ i ∈ { 1 , . . . , n } constraints i m c − i � � � g ci ≥ a ck · p kj + e cj − l c ∀ c ∈ C \ F , ∀ i ∈ { 1 , . . . , m c − 1 } j = 1 k ∈ K j = 1 (14) i � � g ci ≥ a ck · p kj − l c ∀ c ∈ C \ F , ∀ i ∈ { m c , . . . , n } j = i − m c + 1 k ∈ K (15) Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Configuration Integer Linear Program Paint Shop Constraints constraints i s − i + 1 � � � a fk · p kj + e fj ≤ s ∀ f ∈ F , ∀ i ∈ { 1 , . . . , s } j = 1 k ∈ K j = 1 (16) i � � a fk · p kj ≤ s ∀ f ∈ F , ∀ i ∈ { s + 1 , . . . , n } j = i − s k ∈ K (17) Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Configuration Integer Linear Program Counting Color Changes variables � 1 if the color changed w fi = ∀ i ∈ { 1 , . . . , n } , ∀ f ∈ F 0 otherwise constraints w fi ≥ 0 ∀ i ∈ { 1 , . . . , n } , ∀ f ∈ F (18) � w f 1 ≥ a fk · p k 1 − e f 1 ∀ f ∈ F (19) k ∈ K � � w fi ≥ a fk · p ki − a fk · p k ( i − 1 ) ∀ i ∈ { 2 , . . . , n } , ∀ f ∈ F (20) k ∈ K k ∈ K Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP

Comparison C-ILP vs. K-ILP C-ILP # of vars: n · ( 2 | C | + | K | ) � � # of constraints: n · 1 + 2 | F | + | C | · ( 1 + | K | ) + | C | + | K | K-ILP # of vars: n · ( | C | + | K | ) # of constraints: n · ( 1 + | C | + 2 | F | ) + | K | C-ILP < - > K-ILP both formulations are equivalent, but performance using CPLEX 9.0 is different Matthias Prandtstetter, G¨ unther R. Raidl Two ILP Approaches for Solving the CarSP