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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

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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

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Introduction I

Network design problems (NDPs) are essential for the development of modern societies Objective: Minimize flow cost and arc capacity modification cost

Figure : Internet cable network1 Figure : Road network2

1Source: http://jamesdudleyonline.com/wp-content/uploads/2011/03/connect-through-the-internet.jpg 2Source: http://static.panoramio.com/photos/large/8248689.jpg

S Shen, Z Chen INFORMS 2013 2/25

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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

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Introduction III

PNDP- cont- joint PNDP- bin- joint PNDP- bin-n PNDP- bin-c PNDP- bin-nc PNDP- cont-n PNDP- cont-c PNDP- cont-nc SNDP- cont-wp SNDP- cont- wop SNDP- bin-wop SNDP- bin-wp NDP SNDP PNDP PNDP- cont SNDP- cont PNDP- bin SNDP- bin SNDP or PNDP Binary or continuous With or without penalty SNDP or PNDP Binary or continuous Type of chance constraint

S Shen, Z Chen INFORMS 2013 4/25

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Notation I

Graph: G(N, A) Sets: W: Set of commodities Ow ⊆ N: Set of origins of commodity w ∈ W Dw ⊆ 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

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Notation II

Parameters: cij: Cost of allocating one unit of capacity at link (i, j) ∈ A qij: Fixed cost of adding link (i, j) ∈ A when capacity design variables are binary aijw: Unit cost of flowing commodity w ∈ W on link (i, j) ∈ A uij: Fixed capacity of link (i, j) ∈ A when capacity design variables are binary viw: Unit penalty cost of unmet demand of commodity w at destination i ∈ Dw

  • iw: Deterministic supply of commodity w at origin i ∈ Ow

diw: Random demand of commodity w at destination i ∈ Dw ξs

iw: Realization of random demand diw in scenario s ∈ Ω, ∀w ∈ W and i ∈ Dw

ps: Probability of scenario s ∈ Ω ǫ, ǫiw, ǫi, ǫw: Risk parameters associated with different forms of chance constraints

S Shen, Z Chen INFORMS 2013 6/25

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PNDP formulations I

PNDP-cont-joint: min

x,y

  • (i,j)∈A

cijxij +

  • w∈W
  • (i,j)∈A

aijwyijw s.t.

  • w∈W

yijw ≤ xij ∀(i, j) ∈ A (1)

  • j:(i,j)∈A

yijw −

  • j:(j,i)∈A

yjiw ≤ oiw ∀i ∈ Ow, w ∈ W (2)

  • j:(i,j)∈A

yijw −

  • j:(j,i)∈A

yjiw = 0 ∀i ∈ Ow ∪ Dw, w ∈ W (3) x ≥ 0, y ≥ 0 (4) P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw, ∀i ∈ Dw, w ∈ W   ≥ 1 − ǫ

S Shen, Z Chen INFORMS 2013 7/25

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PNDP formulations II

PNDP-cont-n: min

x,y

  • (i,j)∈A

cijxij +

  • w∈W
  • (i,j)∈A

aijwyijw s.t. (1)–(4) P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw, ∀w ∈ W   ≥ 1 − ǫi, ∀i ∈

  • w∈W

Dw PNDP-cont-c: min

x,y

  • (i,j)∈A

cijxij +

  • w∈W
  • (i,j)∈A

aijwyijw s.t. (1)–(4) P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw, ∀i ∈ Dw   ≥ 1 − ǫw, ∀w ∈ W

S Shen, Z Chen INFORMS 2013 8/25

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PNDP formulations III

PNDP-cont-nc: min

x,y

  • (i,j)∈A

cijxij +

  • w∈W
  • (i,j)∈A

aijwyijw s.t. (2);(3)

  • w∈W

yijw ≤ xij ∀(i, j) ∈ A x ≥ 0, y ≥ 0 P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw   ≥ 1 − ǫiw, ∀i ∈ Dw, w ∈ W

S Shen, Z Chen INFORMS 2013 9/25

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PNDP formulations IV

PNDP-bin-nc: min

β,y

  • (i,j)∈A

qijβij +

  • w∈W
  • (i,j)∈A

aijwyijw s.t. (2);(3)

  • w∈W

yijw ≤ uijβij ∀(i, j) ∈ A β ∈ {0, 1}|A|, y ≥ 0 P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw   ≥ 1 − ǫiw, ∀i ∈ Dw, w ∈ W

S Shen, Z Chen INFORMS 2013 10/25

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SNDP formulations I

SNDP-cont-wop: min

x,y

  • (i,j)∈A

cijxij +

  • s∈Ω

ps  

w∈W

  • (i,j)∈A

aijwys

ijw

  s.t.

  • w∈W

ys

ijw ≤ xij

∀(i, j) ∈ A, s ∈ Ω (5)

  • j:(i,j)∈A

ys

ijw −

  • j:(j,i)∈A

ys

jiw ≤ oiw

∀i ∈ Ow, w ∈ W, s ∈ Ω (6)

  • j:(i,j)∈A

ys

ijw −

  • j:(j,i)∈A

ys

jiw = 0

∀i ∈ Ow ∪ Dw, w ∈ W, s ∈ Ω (7) x ≥ 0, ys ≥ 0 ∀s ∈ Ω (8) −

  • j:(i,j)∈A

ys

ijw +

  • j:(j,i)∈A

ys

jiw ≥ ξs iw

∀i ∈ Dw, w ∈ W, s ∈ Ω

S Shen, Z Chen INFORMS 2013 11/25

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SNDP formulations II

SNDP-cont-wp: min

x,y

  • (i,j)∈A

cijxij +

  • s∈Ω

ps  

w∈W

  • (i,j)∈A

aijwys

ijw +

  • w∈W
  • i∈Dw

viwts

iw

  s.t. (5)–(8) −

  • j:(i,j)∈A

ys

ijw +

  • j:(j,i)∈A

ys

jiw + ts iw ≥ ξs iw

∀i ∈ Dw, w ∈ W, s ∈ Ω ts ≥ 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

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Big-M reformulation of chance constraints I

Approach: Add binary variable zs that takes value 1 if the chance constraint is violated by demand realization ξs 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

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Big-M reformulation of chance constraints II

Example: PNDP-cont-nc P  

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ diw   ≥ 1 − ǫiw, ∀i ∈ Dw, w ∈ W Create a new variable zs

iw such that

zs

iw =

  • 1

if

j:(j,i)∈A yjiw − j:(i,j)∈A yijw < ξs iw

if

j:(j,i)∈A yjiw − j:(i,j)∈A yijw ≥ ξs iw

, ∀s ∈ Ω, i ∈ Dw, w ∈ W

S Shen, Z Chen INFORMS 2013 14/25

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Big-M reformulation of chance constraints III

Chance constraint is equivalent to the following set of MIP constraints: −

  • j:(i,j)∈A

yijw +

  • j:(j,i)∈A

yjiw − ξs

iw + Miwzs iw ≥ 0

∀s ∈ Ω, i ∈ Dw, w ∈ W (9)

  • s∈Ω

pszs

iw ≤ ǫiw

∀i ∈ Dw, w ∈ W (10) ziw ∈ {0, 1}|Ω| ∀i ∈ Dw, w ∈ W (11) where M is an arbitrarily large number.

S Shen, Z Chen INFORMS 2013 15/25

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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

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ ξs

iw

for some realization ξs

iw, then

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ ξs′

iw

for any realization satisfying ξs′

iw < ξs iw.

S Shen, Z Chen INFORMS 2013 16/25

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Polynomial-time algorithm for PNDP-cont-nc II

ALGO1: for all w ∈ W, i ∈ Dw (i) Sort ξs

iw in ascending order and relabel the scenarios based on this order

(ii) Identify s′ ∈ {1, ..., |Ωiw|} such that

|Ωiw|

  • k=s

pk > ǫiw ≥

|Ωiw|

  • k=s′

pk (iii) Replace the (i, w)th chance constraint with

  • j:(j,i)∈A

yjiw −

  • j:(i,j)∈A

yijw ≥ ξs′

iw

(12) end for Solve PNDP-cont-nc as min

x,y

  

  • (i,j)∈A

cijxij +

  • w∈W
  • (i,j)∈A

aijwyijw : subject to (1)–(4); (12) ∀w ∈ W, i ∈ Dw   

S Shen, Z Chen INFORMS 2013 17/25

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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

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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

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Results for randomly generated networks II

1 2 3 4 5 6 7 8 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Ho−100−MIP Ho−200−MIP He−100−MIP He−200−MIP Ho−100−ALGO1 Ho−200−ALGO1 He−100−ALGO1 He−200−ALGO1

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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Results for randomly generated networks III

Summary of observations: Optimal objective values decrease as ǫ increases PNDP-cont-nc is less sensitive to changes in ǫ than PNDP-cont-joint ALGO1 is much more efficient than the MIP approach For MIP models, CPU time increases dramatically as ǫ is increased and as |Ω| is increased For ALGO1, CPU time increases is mostly unaffected by changes in ǫ and |Ω|, and the homogeneity of risk parameters

S Shen, Z Chen INFORMS 2013 21/25

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Results for real life network I

1 8 4 5 6 3 2 15 19 17 18 7 12 11 10 16 9 20 23 22 14 13 24 21 6 6 4 4 4 2 2 4 4 4 5 5 10 2 2 3 3 2 2 3 3 4 4 8 2 2 2 10 5 5 6 6 4 4 4 4 5 5 3 3 4 4 3 3 2 2 3 6 6 2 4 4 4 4 3 6 5 5 2 2 8 4 4 6 3 4 4 3 3 3 5 5 5 5 6 6

Figure : Sioux Falls road network

S Shen, Z Chen INFORMS 2013 22/25

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Results for real life network II

High inflow instance Higher mean demands for nodes closer to node 10 (center node) Compare sensitivity of optimal objective values to ǫ and v PNDP-cont-joint PNDP-cont-nc SNDP-cont-wp

S Shen, Z Chen INFORMS 2013 23/25

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Results for real life network III

0.05 0.1 0.15 85% 90% 95% 100% 20 25 30 35 40 85% 90% 95% 100% PNDP−joint PNDP−nc SNDP−wp

Figure : Percentage comparison of optimal objective values for PNDP-joint,

PNDP-nc and SNDP-cont-wp

S Shen, Z Chen INFORMS 2013 24/25

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Results for real life network IV

Summary of observations: Optimal values of PNDP-joint and PNDP-nc have a fairly linear relationship with ǫ Optimal values of SNDP-cont-wp are concave w.r.t. v (dominant term changes from real cost to penalty cost) Without first experimenting with several values of v, a suitable value for v may not be known PNDP models mitigate ambiguity in solution reliability

S Shen, Z Chen INFORMS 2013 25/25

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Conclusions and future research

Conclusions: Developed MIP formulations for PNDP models Developed polynomial-time algorithms for PNDP-cont-nc and special cases of PNDP-cont-n/c models that are far more efficient than MIP formulations Benchmarked PNDP models against SNDP to find that PNDP models are far less sensitive to small changes in parameters Future research: Risk parameters as variables, to seek and optimal combination of risk versus cost Special network topologies that may provide more effective algorithms

S Shen, Z Chen INFORMS 2013 26/25

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Thank you!

S Shen, Z Chen INFORMS 2013 27/25