Optimization Models for Differentiating Quality of Service Levels in - PowerPoint PPT Presentation

Optimization Models for Differentiating Quality of Service Levels in Probabilistic Network Capacity Design Problems Siqian Shen Zhihao Chen Department of Industrial and Operations Engineering, University of Michigan October 7th, 2013 S Shen, Z Chen INFORMS 2013 1/25

Introduction I Network design problems (NDPs) are essential for the development of modern societies Objective: Minimize flow cost and arc capacity modification cost Figure : Road network 2 Figure : Internet cable network 1 1 Source: http://jamesdudleyonline.com/wp-content/uploads/2011/03/connect-through-the-internet.jpg 2 Source: http://static.panoramio.com/photos/large/8248689.jpg S Shen, Z Chen INFORMS 2013 2/25

Introduction II NPDs under demand uncertainty and with multiple commodities Capacity design decisions are made before realization of demands Can be continuous or binary Probabilistic NDPs ( PNDPs ): Flow decisions made before realization of demands Flow decisions are made under probabilistic constraints Probabilistic constraints can be joint, or differentiated by node, commodity, or node and commodity Stochastic NDPs ( SNDPs ): Flow decisions made after realization of demands Expected flow costs Flow decisions may be penalized for unmet demand for greater flexibility in solution S Shen, Z Chen INFORMS 2013 3/25

Introduction III NDP SNDP or PNDP SNDP or PNDP SNDP PNDP Binary or Binary or continuous continuous SNDP- SNDP- PNDP- PNDP- bin cont bin cont With or without Type of chance penalty constraint SNDP- PNDP- PNDP- SNDP- SNDP- SNDP- PNDP- PNDP- PNDP- PNDP- PNDP- PNDP- cont- bin- cont- bin-wp bin-wop cont-wp bin-n bin-c bin-nc cont-n cont-c cont-nc wop joint joint S Shen, Z Chen INFORMS 2013 4/25

Notation I Graph: G ( N, A ) Sets: W : Set of commodities O w ⊆ N : Set of origins of commodity w ∈ W D w ⊆ N : Set of destinations of commodity w ∈ W Ω: Set of random scenarios where Ω = { 1 , . . . , | Ω |} S Shen, Z Chen INFORMS 2013 5/25

Notation II Parameters: c ij : Cost of allocating one unit of capacity at link ( i, j ) ∈ A q ij : Fixed cost of adding link ( i, j ) ∈ A when capacity design variables are binary a ijw : Unit cost of flowing commodity w ∈ W on link ( i, j ) ∈ A u ij : Fixed capacity of link ( i, j ) ∈ A when capacity design variables are binary v iw : Unit penalty cost of unmet demand of commodity w at destination i ∈ D w o iw : Deterministic supply of commodity w at origin i ∈ O w d iw : Random demand of commodity w at destination i ∈ D w ξ s iw : Realization of random demand d iw in scenario s ∈ Ω, ∀ w ∈ W and i ∈ D w p s : Probability of scenario s ∈ Ω ǫ, ǫ iw , ǫ i , ǫ w : Risk parameters associated with different forms of chance constraints S Shen, Z Chen INFORMS 2013 6/25

PNDP formulations I PNDP-cont-joint : � � � min c ij x ij + a ijw y ijw x , y w ∈ W ( i,j ) ∈ A ( i,j ) ∈ A � s.t. y ijw ≤ x ij ∀ ( i, j ) ∈ A (1) w ∈ W � � y ijw − y jiw ≤ o iw ∀ i ∈ O w , w ∈ W (2) j :( i,j ) ∈ A j :( j,i ) ∈ A � � y ijw − y jiw = 0 ∀ i �∈ O w ∪ D w , w ∈ W (3) j :( i,j ) ∈ A j :( j,i ) ∈ A x ≥ 0 , y ≥ 0 (4)   � �  ≥ 1 − ǫ y jiw − y ijw ≥ d iw , ∀ i ∈ D w , w ∈ W P  j :( j,i ) ∈ A j :( i,j ) ∈ A S Shen, Z Chen INFORMS 2013 7/25

PNDP formulations II PNDP-cont-n : � � � min c ij x ij + a ijw y ijw x , y ( i,j ) ∈ A w ∈ W ( i,j ) ∈ A s.t. (1)–(4)   � �  ≥ 1 − ǫ i , � P y jiw − y ijw ≥ d iw , ∀ w ∈ W ∀ i ∈ D w  j :( j,i ) ∈ A j :( i,j ) ∈ A w ∈ W PNDP-cont-c : � � � min c ij x ij + a ijw y ijw x , y ( i,j ) ∈ A w ∈ W ( i,j ) ∈ A s.t. (1)–(4)   � �  ≥ 1 − ǫ w , P y jiw − y ijw ≥ d iw , ∀ i ∈ D w ∀ w ∈ W  j :( j,i ) ∈ A j :( i,j ) ∈ A S Shen, Z Chen INFORMS 2013 8/25

PNDP formulations III PNDP-cont-nc : � � � min c ij x ij + a ijw y ijw x , y w ∈ W ( i,j ) ∈ A ( i,j ) ∈ A s.t. (2);(3) � y ijw ≤ x ij ∀ ( i, j ) ∈ A w ∈ W x ≥ 0 , y ≥ 0   � �  ≥ 1 − ǫ iw , P y jiw − y ijw ≥ d iw ∀ i ∈ D w , w ∈ W  j :( j,i ) ∈ A j :( i,j ) ∈ A S Shen, Z Chen INFORMS 2013 9/25

PNDP formulations IV PNDP-bin-nc : � � � min q ij β ij + a ijw y ijw β, y ( i,j ) ∈ A w ∈ W ( i,j ) ∈ A s.t. (2);(3) � y ijw ≤ u ij β ij ∀ ( i, j ) ∈ A w ∈ W β ∈ { 0 , 1 } | A | , y ≥ 0   � �  ≥ 1 − ǫ iw , y jiw − y ijw ≥ d iw ∀ i ∈ D w , w ∈ W P  j :( j,i ) ∈ A j :( i,j ) ∈ A S Shen, Z Chen INFORMS 2013 10/25

SNDP formulations I SNDP-cont-wop :   � �  � � p s a ijw y s min c ij x ij + ijw  x , y s ∈ Ω w ∈ W ( i,j ) ∈ A ( i,j ) ∈ A � y s s.t. ijw ≤ x ij ∀ ( i, j ) ∈ A, s ∈ Ω (5) w ∈ W � � y s y s ijw − jiw ≤ o iw ∀ i ∈ O w , w ∈ W, s ∈ Ω (6) j :( i,j ) ∈ A j :( j,i ) ∈ A � y s � y s ijw − jiw = 0 ∀ i �∈ O w ∪ D w , w ∈ W, s ∈ Ω j :( i,j ) ∈ A j :( j,i ) ∈ A (7) x ≥ 0 , y s ≥ 0 ∀ s ∈ Ω (8) � y s � y s jiw ≥ ξ s − ijw + ∀ i ∈ D w , w ∈ W, s ∈ Ω iw j :( i,j ) ∈ A j :( j,i ) ∈ A S Shen, Z Chen INFORMS 2013 11/25

SNDP formulations II SNDP-cont-wp :   � � p s  � � a ijw y s � � v iw t s min c ij x ij + ijw + iw  x , y ( i,j ) ∈ A s ∈ Ω w ∈ W ( i,j ) ∈ A w ∈ W i ∈ D w s.t. (5)–(8) � y s � y s jiw + t s iw ≥ ξ s − ijw + ∀ i ∈ D w , w ∈ W, s ∈ Ω iw j :( i,j ) ∈ A j :( j,i ) ∈ A t s ≥ 0 ∀ s ∈ Ω SNDPs can be solved as a two-stage problem using Benders’ decomposition SNDP-cont-wp is typically used to formulate cost-based NDPs A benchmark against which we compare our PNDP-cont reformulations S Shen, Z Chen INFORMS 2013 12/25

Big-M reformulation of chance constraints I Approach: Add binary variable z s that takes value 1 if the chance constraint is violated by demand realization ξ s and 0 otherwise The sum of probabilities of the realizations that violate the chance constraint must not exceed the tolerance level S Shen, Z Chen INFORMS 2013 13/25

Big-M reformulation of chance constraints II Example: PNDP-cont-nc   � �  ≥ 1 − ǫ iw , P y jiw − y ijw ≥ d iw ∀ i ∈ D w , w ∈ W  j :( j,i ) ∈ A j :( i,j ) ∈ A Create a new variable z s iw such that � j :( i,j ) ∈ A y ijw < ξ s 1 if � j :( j,i ) ∈ A y jiw − � z s iw iw = , ∀ s ∈ Ω , i ∈ D w , w ∈ W if � j :( j,i ) ∈ A y jiw − � j :( i,j ) ∈ A y ijw ≥ ξ s 0 iw S Shen, Z Chen INFORMS 2013 14/25

Big-M reformulation of chance constraints III Chance constraint is equivalent to the following set of MIP constraints: � � y jiw − ξ s iw + M iw z s − y ijw + iw ≥ 0 ∀ s ∈ Ω , i ∈ D w , w ∈ W (9) j :( i,j ) ∈ A j :( j,i ) ∈ A � p s z s iw ≤ ǫ iw ∀ i ∈ D w , w ∈ W (10) s ∈ Ω z iw ∈ { 0 , 1 } | Ω | ∀ i ∈ D w , w ∈ W (11) where M is an arbitrarily large number. S Shen, Z Chen INFORMS 2013 15/25

Polynomial-time algorithm for PNDP-cont-nc I An alternative method that does not require the use of binary variables Takes advantage of single-line chance constraints If � � y ijw ≥ ξ s y jiw − iw j :( j,i ) ∈ A j :( i,j ) ∈ A for some realization ξ s iw , then y ijw ≥ ξ s ′ � � y jiw − iw j :( j,i ) ∈ A j :( i,j ) ∈ A for any realization satisfying ξ s ′ iw < ξ s iw . S Shen, Z Chen INFORMS 2013 16/25

Polynomial-time algorithm for PNDP-cont-nc II ALGO1 : for all w ∈ W, i ∈ D w (i) Sort ξ s iw in ascending order and relabel the scenarios based on this order (ii) Identify s ′ ∈ { 1 , ..., | Ω iw |} such that | Ω iw | | Ω iw | p k > ǫ iw ≥ � � p k k = s k = s ′ (iii) Replace the ( i, w ) th chance constraint with � � y ijw ≥ ξ s ′ y jiw − (12) iw j :( j,i ) ∈ A j :( i,j ) ∈ A end for Solve PNDP-cont-nc as     � � � min c ij x ij + a ijw y ijw : subject to (1)–(4); (12) ∀ w ∈ W, i ∈ D w x , y  w ∈ W  ( i,j ) ∈ A ( i,j ) ∈ A S Shen, Z Chen INFORMS 2013 17/25

Polynomial-time algorithm for PNDP-cont-nc III Similar approaches can be used to develop polynomial-time algorithms for special cases of PNDP-cont-n/c PNDP-cont-n with each node having demand for no more than 1 type of commodity ⇒ single-line chance constraint PNDP-cont-c with each commodity having no more than 1 demand node ⇒ single-line chance constraint S Shen, Z Chen INFORMS 2013 18/25

Results for randomly generated networks I Compare computational times and optimal objective values for PNDP-cont-joint PNDP-cont-nc with homogenous (“-ho”) risk parameters PNDP-cont-nc with heterogenous (“-he”) risk parameters S Shen, Z Chen INFORMS 2013 19/25

Results for randomly generated networks II 100% 90% 80% 70% Ho−100−MIP 60% Ho−200−MIP He−100−MIP He−200−MIP 50% Ho−100−ALGO1 Ho−200−ALGO1 40% He−100−ALGO1 He−200−ALGO1 30% 20% 10% 0% 1 2 3 4 5 6 7 8 Figure : Percentage comparison of CPU time taken by ALGO1 and the MIP approach for PNDP-cont-nc instances (100% is the largest CPU time) S Shen, Z Chen INFORMS 2013 20/25