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
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