The design of robust value-creating supply chain networks

Abstract

This paper provides a methodology for supply chain network (SCN) design under uncertainty. The problem is initially casted as a two-level organizational decision process: the design decisions must be made here and now, but the reengineered SCN can be used for daily operations only after an implementation period. The network structure can also be adapted during the planning horizon considered. When making the design decisions, the operational response and structural adaptation decisions taking place during the planning horizon must be anticipated. The methodology recognizes three event types to characterize the future SCN environment: random, hazardous and deep uncertainty events. At the design time, plausible futures are anticipated through a scenario planning approach. Several Monte Carlo scenario samples are generated and corresponding sample average approximation programs are solved in order to produce a set of alternative designs. A multi-criteria design evaluation approach is then applied to select the most effective and robust design among candidate solutions. An illustrative case, based on the location–transportation problem, is finally introduced to illustrate the approach, and computational experiments are performed to demonstrate its feasibility.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Notes

  1. 1.

    Order winners are value criteria enabling a firm to win orders in its product-markets, and thus to increase its market share and its revenues (Klibi et al. 2010b).

  2. 2.

    In some cases, random parameters, such as random process yield, may also be found in the technological matrix \(\mathrm{\mathbf{A}}_t\). This is however unusual and, to simplify the presentation, we assume in what follows that technological parameters are known and not affected by disruptions.

  3. 3.

    We assume that at the beginning of the planning horizon all depots are closed.

  4. 4.

    The next day delivery requirement is implemented through a 400-mile limit on the distance between depots and ship-to-points.

  5. 5.

    For larger sample sizes, the problem contains more than 3 million decision variables and constraints and cannot be solved to optimality within a time limit of 24 h.

  6. 6.

    There is a lot of variability in solution times from one instance to another depending on how CPLEX employs the incorporated heuristics in the resolution.

  7. 7.

    \(\mathrm{MSD}_P ({\hat{\mathbf{x}}}^j)=\frac{1}{M_P}\sum \nolimits _{\omega \in \Omega ^{M_P}} {\text{ max }\left[ {( {\mathbb{E }_{\Omega ^{M_P}\left| P \right. } ( {{\hat{\mathbf{x}}}^j})-\overline{\mathrm{VA}} ({\hat{\mathbf{x}}}^j,\omega )});0} \right] },P=A,S\).

  8. 8.

    The following weights were used: \(\varphi _A =\varphi _S =\psi =0\) in \(R^N({\hat{\mathbf{x}}}^j)\) and \(\varphi _A =\varphi _S =\psi =0.1\) in \(R^A({\hat{\mathbf{x}}}^j)\).

References

  1. Alonso-Ayuso A, Escudero LF, Ortuno MT (2007) On modeling planning under uncertainity in manufacturing. Stat Oper Res Trans 31:109–150

    Google Scholar 

  2. Artzner P, Delbaen F, Eber L, Health D (1999) Coherent measures of risk. Math Financ 9:203–228

    Article  Google Scholar 

  3. Banks E (2006) Catastrophic risk: analysis and management. Wiley Finance, New York

  4. Birge JR, Louveaux F (2011) Introduction to stochastic programming, 2nd edn. Springer, Berlin

  5. Carle MA, Martel A, Zufferey N (2012) The CAT metaheuristic for the solution of multi-period activity-based supply chain network design problems. Int J Prod Econ 139(2):664–677

    Article  Google Scholar 

  6. Cordeau J-F, Laporte G, Pasin F (2008) An iterated local search heuristic for the logistics network design problem with single assignment. Int J Prod Econ 113:626–640

    Article  Google Scholar 

  7. Ducapova J, Consigli G, Wallace SW (2000) Scenarios for multistage stochastic programs. Ann Oper Res 100:25–53

    Article  Google Scholar 

  8. Dong M, Chen FF (2007) Quantitative robustness index design for supply chain networks. In: Trends in supply chain design and management. Springer series in advanced manufacturing, 2007, Part II, Part 3. Springer, Berlin, pp 369–391

  9. Eppen G, Kipp Martin R, Schrage L (1989) A scenario approach to capacity planning. Oper Res 37:517–527

    Article  Google Scholar 

  10. Escudero LF, Galindo E, García G, Gomez E, Sabau V (1999) Schumann: a modeling framework for supply chain management under uncertainity. Eur J Oper Res 119:14–34

    Google Scholar 

  11. Figueira J, Greco S, Ehrgott M (2005) Multiple criteria decision analysis: state of the art surveys. Springer, New York

    Google Scholar 

  12. Gollmer R, Gotzes U, Schultz R (2011) A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse. Math Program Ser B 126:179–190

    Google Scholar 

  13. Gollmer R, Neise F, Schultz R (2008) Stochastic programs with first-order stochastic dominance constraints induced by mixed-integer linear recourse. SIAM J Optim 19:552–571

    Article  Google Scholar 

  14. Grossi P, Kunreuther H (2005) Catastrophe modeling: a new approach to managing risk. Springer, Berlin

  15. Gutierrez GJ, Kouvelis P, Kurawala A (1996) A robustness approach to uncapacitated network design problems. Eur J Oper Res 94:362–376

    Article  Google Scholar 

  16. Haimes YY (2004) Risk modeling, assessment, and management, 2nd edn. Wiley, New York

  17. Heitsch HR, Roemisch W (2009) Scenario tree modeling for multistage stochastic programs. Math Program Ser A 118:371–406

    Google Scholar 

  18. Hendricks KB, Singhal VR (2005) Association between supply chain glitches and operating performance. Manag Sci 51(5):695–711

    Article  Google Scholar 

  19. Klibi W, Lasalle F, Martel A, Ichoua S (2010a) The stochastic multi-period location–transportation problem. Transp Sci 44(2):221–237

    Article  Google Scholar 

  20. Klibi W, Martel A, Guitouni A (2010b) The design of robust value-creating supply chain networks: a critical review. Eur J Oper Res 203:283–293

    Article  Google Scholar 

  21. Klibi W, Martel A, Guitouni A (2010c) The impact of operations anticipations on the quality of supply network design models, CIRRELT research document CIRRELT-2010-45. Université Laval, Canada

  22. Klibi W, Martel A (2012a) Modeling approaches for the design of resilient supply networks under disruptions. Int J Prod Econ 135:882–898

    Article  Google Scholar 

  23. Klibi W, Martel A (2012b) Scenario-based supply chain network risk modeling. Eur J Oper Res 223(2012):644–658

    Google Scholar 

  24. Klose A, Drexl A (2005) Facility location models for distribution system design. Eur J Oper Res 162:4–29

    Article  Google Scholar 

  25. Kouvelis P, Yu G (1997) Robust discrete optimization and its applications. Kluwer, Dordrecht

  26. Lee H (2004) The Triple-A supply chain. In: Harvard business review, October. pp 102–112

  27. Lempert RJ, Groves DG, Popper SW, Bankes SC (2006) A general, analytic method for generating robust strategies and narrative scenarios. Manag Sci 52(4):514–528

    Article  Google Scholar 

  28. Lim M, Bassamboo A, Chopra S, Daskin M (2010) Flexibility and fragility: use of chaining strategies in the presence of disruption risks. University of Illinois, Urbana-Champaign, Working paper

  29. Martel A (2005) The design of production-distribution networks: a mathematical programming approach. In: Geunes J, Pardalos P (eds) Supply chain optimization. Springer, Berlin, pp 265–306

  30. Martel A, Benmoussa A, Chouinard M, Klibi W, Kettani O (2013) Designing Global supply networks for conflict or disaster support: the case of the Canadian Armed Forces. J Oper Res Soc 64:577–596

    Article  Google Scholar 

  31. M’Barek W, Martel A, D’Amours S (2010) Designing multinational value-creating supply chain networks for the process industry, Research Paper CIRRELT-2010-51. Université Laval, CIRRELT

  32. Pomper C (1976) International investment planning: an integrated approach. North-Holland, Amsterdam

  33. Qi L, Shen Z-JM, Snyder LV (2010) The effect of supply disruptions on supply chain design decisions. Transp Sci 44(2):274–289

    Article  Google Scholar 

  34. Revelle CS, Eiselt HA, Daskin MS (2008) A bibliography for some fundamental problem categories in discrete location science. Eur J Oper Res 184:817–848

    Google Scholar 

  35. Rockafellar RT, Uryasev S (2000) Optimization on conditional value-at-risk. J Risk 2:21–41

    Google Scholar 

  36. Rockafellar RT (2007) Coherent approaches to risk in optimization under uncertainty. In: Tutorials in operations research, INFORMS 2007. pp 38–61

  37. Roemisch W, Schultz R (2001) Multi-stage stochastic integer programs: an introduction. In: Groetschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems. Springer, Berlin, pp 581–600

  38. Santoso T, Ahmed S, Goetschalckx M, Shapiro A (2005) A stochastic programming approach for supply chain network design under uncertainty. Eur J Oper Res 167:96–115

    Article  Google Scholar 

  39. Schmitt AJ (2011) Strategies for customer service level protection under multi-echelon supply chain disruption risk. Available at SSRN: http://ssrn.com/abstract=1433512

  40. Schmitt AJ, Singh M (2009) A quantitative analysis of disruption risk in a multi-echelon supply chain. Available at SSRN: http://ssrn.com/abstract=1463417

  41. Schneeweiss C (2003) Distributed decision making, 2nd edn. Springer, Berlin

  42. Schultz R (2003) Stochastic programming with integer variables. Math Program Ser B 97:285–309

    Google Scholar 

  43. Schultz R, Tiedemann S (2006) Conditional value-at-risk in stochastic programs with mixed integer recourse. Math Program Ser B 105:365–386

    Google Scholar 

  44. Schutz P, Tomasgard A, Ahmed S (2009) Supply chain design under uncertainty using sample average approximation and dual decomposition. Eur J Oper Res 199:409–419

    Article  Google Scholar 

  45. Shapiro A (2003) Monte Carlo sampling methods, chapter 6. In: Ruszczynski A, Shapiro A (eds) Handbooks in OR. & MS., vol 10. Elsevier, Amsterdam

  46. Shapiro A (2008) Stochastic programming approach to optimization under uncertainty. Math Program Ser A 112(1):183–220

    Article  Google Scholar 

  47. Sheffi Y (2005) The resilient enterprise: overcoming vulnerability for competitive advantage. MIT Press, Cambridge

  48. Shen Z-J (2007) Integrated supply chain design models: a survey and future research directions. J Ind Manag Optim 3(1):1–27

    Google Scholar 

  49. Snyder L (2006) Facility location under uncertainty: a review. IIE Trans 38(7):537–554

    Article  Google Scholar 

  50. Snyder L, Atan Z, Peng P, Rong Y, Schmitt A, Sinsoysal B (2010) OR/MS models for supply chain disruptions: a review. Available at SSRN: http://ssrn.com/abstract=1689882

  51. Snyder L, Daskin M (2006) Stochastic p-Robust location problems. IIE Trans 38(11):971–985

    Article  Google Scholar 

  52. Snyder L, Scaparra M, Daskin M, Church R (2006) Planning for disruptions in supply chain networks. In: Tutorials in operations research, INFORMS 2006, chap 9

  53. Stadtler H, Kilger C (2005) Supply chain management and advanced planning: concepts, models, software and case studies, 3rd edn. Springer, Berlin

  54. Tang CS, Tomlin B (2008) The power of flexibility for mitigating supply chain risks. Int J Prod Econ 116:12–27

    Article  Google Scholar 

  55. Thanh P, Bostel N, Péton O (2008) A dynamic model for facility location in the design of complex supply chains. Int J Prod Econ 113:678–693

    Article  Google Scholar 

  56. Thanh P, Bostel N, Péton O (2012) A DC programming heuristic applied to the logistics network design problem. Int J Prod Econ 135:94–105

    Article  Google Scholar 

  57. Tomlin B (2006) On the value of mitigation and contingency strategies for managing supply chain disruption risks. Manag Sci 52(5):639–657

    Article  Google Scholar 

  58. Van der Heijden K (2005) Scenarios: the art of strategic conversation, 2nd edn. Wiley, New York

  59. Vila D, Martel A, Beauregard R (2007) Taking market forces into account in the design of production–distribution networks: a positioning by anticipation approach. J Ind Manag Optim 3(1):29–50

    Article  Google Scholar 

Download references

Acknowledgments

This research was supported in part by NSERC (Grant No DNDPJ 335078-05), Defence R&D Canada and Modellium Inc.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Walid Klibi.

Additional information

\(\copyright \) Copyright Walid Klibi and Alain Martel, 2011 (2013).

Appendices

Appendix A: LTP case data

For the problem instances solved, the following known data were used:

\(L_l^d {:}\,\)Subset of ship-to-points for which next-day-delivery is feasible from depot \(l\)

\(p_{lt} {:}\,\)Discounted unit price of products sold in period \(t\) to ship-to-point \(l (p_{l1} =\$ 23 \forall l)\)

\(c_{lt} {:}\,\)Discounted unit cost of products shipped in period \(t\) from depot \(l (c_{l1} =\$ 20.50 \forall l)\)

\(c_{lt}^o {:}\,\)Discounted additional unit cost incurred in period \(t\) when overtime is needed to ship a product from depot \(l\) \((c_{l1}^o =\$ 1 \forall l)\)

\(c_{ll^{\prime }t}^s {:}\,\)Discounted unit transportation cost between depot \(l\) and ship-to-point \(l^{\prime }\) in period \(t\) (derived from the estimated linear regression function \(0.1091+0.0057m_{ll^{\prime }},\,m_{ll^{\prime }} \) being the distance between depot \(l\) and ship-to-point \(l^{\prime }\)).

\(c_t^e {:}\,\)Discounted unit cost of products supplied from the external source in period \(t (c_1^e =\$ 24.50 \forall l)\)

\(a_{\mathrm{ln}}^+ {:}\,\)Discounted fixed cost of opening depot \(l\) at the beginning of cycle \(n \)(initial investment less discounted residual value at the end of the horizon). \(a_{\mathrm{l}1}^+ \) was generated in the range \([900\mathrm{K},1,000\mathrm{K}]\).

\(a_{\mathrm{ln}}^- {:}\,\)Discounted fixed cost of closing depot \(l\) at the beginning of cycle \(n \)(under the assumption that closed depots are sold only at the end of the planning horizon)—about 0.4 % of opening costs.

\(a_{\mathrm{ln}} {:}\,\)Discounted fixed cost of using depot \(l\) during cycle \(n \)—about 2.5 % of opening costs.

\(\rho ^c\) Monthly capacity correction factor to take depot congestion into account (80 %).

\(\rho ^d\) Maximum proportion of the monthly demand of a ship-to-point that can be supplied by a single depot (100 % for the recourse-avoiding resilience policy and 60 % for the multiple-sourcing policy).

All the prices/costs were discounted based on an annual weighted average cost of capital of 12.5 %.

We assumed that the demand for the ship-to-points \(l\in L^d \) follows a compound Poisson process characterized by exponential order inter-arrival times with expected time between orders \(\eta _l \), and by log-normal order quantities with mean \(\mu _l \) and standard deviation \(\sigma _l\). The SC network includes large, medium and small ship-to-points with the following proportions: 15, 65 and 20 %. Table 4 provides the probability distribution parameters used to generate orders for each customer type. Under a given evolutionary path \(k\), any increase/decrease in demand growth results from a change in inter-arrival times. Accordingly, the following linear function is applied to derive the expected time between orders in planning period \(t{:}\,\eta _{ltk} =(1-\varsigma _k t)\eta _l \), where the slope \(\varsigma _k \) is fixed at 0.15, 0.05 and 0.1 respectively, for \(k = 1, 2\) and 3. The probability of occurrence of the three evolutionary paths is 0.4, 0.4 and 0.2, respectively. The daily capacity of the depots under normal operations was selected in the interval [6,000, 7,500] cwt. The daily overtime capacity available for local recourse was fixed to 20 % of the regular capacity level.

Table 4 Demand process characteristics

The disruption model parameters were estimated as follows. The US states in the region covered by the network were used as hazard zones. The exposure levels and the multi-hazard arrival process for each zone (state) were estimated from historical data on major disasters provided by FEMA (http://www.fema.gov). We assumed that the time between two consecutive disasters (Table 5) follows an exponential distribution with an historical mean \(\bar{\lambda }_z \) varying for each zone \(z\in Z\). Under a given evolutionary path \(k\), the following linear function is used to derive the mean inter-arrival times in period \(t{:} \,\bar{\lambda }_{tk} =(1+\hat{\delta }_k t)\bar{\lambda }\), where the slope \(\hat{\delta }_k \) is fixed at \(-0.04, 0.03\) and 0.01 respectively for \(k= 1, 2\) and 3. Table 5 also gives the state exposure level \(g( z)\) estimated on a scale from 1 to 4 (i.e. low to high). We assumed that the impact intensity distribution is Uniform for all \(z\). For depots it is expressed in terms of capacity loss with lower and upper bounds [0,0.25), [0.25,0.5), [0.5,0.75), [0.75,1] for exposure levels \(g=1,2,3,4, \) respectively. For ship-to-points it is expressed in terms of demand surge/reduction with amplitude parameters [0,0.1), [0.1,0.2), [0.2,0.3), [0.3,0.4] for exposure levels \(g =1,2,3,4, \) respectively. The recovery functions, the impact-duration functions and the attenuation probabilities used are similar to the ones estimated in (Klibi and Martel 2012a, b).

Table 5 Multi-hazard exposure levels and mean inter-arrival times

In order to obtain a scenario \(\omega \), standard Monte Carlo techniques, based on the functions defined in the two previous paragraphs, are used to generate daily demands and capacities for all the working periods of the planning horizon. These demands are then aggregated by planning periods (months) to get:

\(b_{lt}^d ( \omega ){:}\,\)Ship-to-point \(l\in L^d\) demand in planning period \(t\in T\) under scenario \(\omega \).

\(b_{lt}^c ( \omega ){:}\,\)Depot \(l\in L^c\) capacity in planning period \(t\in T\) under scenario \(\omega \).

Appendix B: LTP case SAA program

SCN design model discussed in the illustrative case is detailed here. The following decision variables are required:

\(x_{\mathrm{ln}}^+ {:}\,\)Binary variable equal to 1 if depot \(l\) is opened at the beginning of reengineering cycle \(n\), and 0 otherwise

\(x_{\mathrm{ln}}^- {:}\,\)Binary variable equal to 1 if depot \( l\) is closed at the beginning of reengineering cycle \(n\), and 0 otherwise

\(x_{\mathrm{ln}}{:}\,\)Binary variable equal to 1 if depot \( l\) is in use during reengineering cycle \(n\), and 0 otherwise

\(x_{l{l}^{\prime }n} {:}\,\)Binary variable equal to 1 if ship-to point \({l}^{\prime }\) can be supplied by depot \(l \)during reengineering cycle \(n\), and 0 otherwise

\(y_{l{l}^{\prime }t} ( \omega ){:}\,\)Quantity of products supplied by depot \(l\) to ship-to point \({l}^{\prime }\) in period \(t\) under scenario \(\omega \)

\(y_{lt}^o ( \omega ){:}\,\)Quantity of products shipped during overtime at depot \(l\) in period \(t\) under scenario \(\omega \)

\(y_{{l}^{\prime }t}^r ( \omega ){:}\,\)Quantity of products supplied to ship-to-point \({l}^{\prime }\) by the external supply source in period \(t\) under scenario \(\omega \)

In the following design model, it is assumed that the decisions \({\hat{\mathbf{x}}}_n ,n>1,\) must be made at the beginning of the planning horizon. Thus, the variables \(x_{\mathrm{ln}},x_{\mathrm{ln}}^+,x_{\mathrm{ln}}^- \) and \(x_{l{l}^{\prime }n} ,\,n\in N\) are first stage variables and \(y_{l{l}^{\prime }t} ( \omega ), \,y_{lt}^o ( \omega ),\,y_{{l}^{\prime }t}^r ( \omega ),\,t\in T\) are second stage recourse variables depending on the prevailing scenario \(\omega \in \Omega _i^m \). This leads to the following SAA program:

\(\displaystyle R\!=\!{\text{ max }} \sum _{P=A,S} \frac{w_P^i }{m_P} \sum _{\omega \in \Omega _i^{m_P}}\left\{ \sum _{t\in T} \left[ \sum _{l\in L^c} \left( \sum _{l^{\prime }\in L_{l}^{d}} [p_{lt} -c_{lt} \!-\!c_{ll^{\prime }t}^{s}] y_{l{l}^{\prime }t} (\omega ) -c_{lt}^{o} y_{lt}^{o} (\omega )\right) \right. \right. \)  
      \(\displaystyle \!+\!\left. \left. \sum _{{l}^{\prime }\in L^{d}} {(p_{l^{\prime }t} \!-\!c_t^e )y_{l^{\prime }t}^r ( \omega )} \right] \right\} \!-\!\sum _{n\in N} {\sum _{l\in L^c} {(a_{\mathrm{ln}}^+ x_{\mathrm{ln}}^+ +a_{\mathrm{ln}}^- x_{\mathrm{ln}}^- \!+\!a_{\mathrm{ln}} x_{ln})}}\) (14)
subject to  
\(\displaystyle x_{\ln }-x_{\ln }^{+} +x_{\ln }^{-}-x_{l(n-1)} =0\) \(l\in L^c,\quad n\in N\) (15)
\(\displaystyle x_{\ln }^{+} +x_{\ln }^{-} \le 1\) \(l\in L^{c}, \quad n\in N\) (16)
\(\displaystyle x_{l{l}^{\prime }n} \le x_{\ln }\) \(l\in L^c, \quad {l}^{\prime }\in L_l^d, \;n\in N\)
\(\displaystyle x_{\mathrm{ln}} ,x_{l{l}^{\prime }n} \in \left\{ 0,1 \right\} \) \(l\in L^c,\quad {l}^{\prime }\in L_l^d ,\;n\in N\)
\(\displaystyle y_{l{l}^{\prime }t} ( \omega )\le \rho ^db_{l^{\prime }t}^d ( \omega )x_{l{l}^{\prime }n}\) \(l\in L^c,\,{l}^{\prime }\in L_l^d,\,t\in T_n ,n\in N,\omega \in \Omega _i^m\) (17)
\(\displaystyle \sum _{l\in L^c} {y_{ll^{\prime }t} ( \omega )} +y_{{l}^{\prime }t}^r ( \omega )=b_{l^{\prime }t}^d ( \omega )\) \({l}^{\prime }\in L^d,\,t\in T,\,\omega \in \Omega _i^m\) (18)
\(\displaystyle \sum _{{l}^{\prime }\in L_{lt}^\mathrm{d} ( \omega )} {y_{l{l}^{\prime }t} ( \omega )} \le \rho ^cb_{lt}^c ( \omega )+y_{lt}^o (\omega )\) \(l\in L^c,t\in T,\omega \in \Omega _i^{m}\)
\(\displaystyle y_{l{l}^{\prime }t} ( \omega )\ge 0, y_{l^{\prime }t}^r ( \omega )\ge 0,y_{lt}^r ( \omega )\ge 0\) \(l\in L^c,{l}^{\prime }\in L_l^d ,\,t\in T,\,\omega \in \Omega _i^m\)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Klibi, W., Martel, A. The design of robust value-creating supply chain networks. OR Spectrum 35, 867–903 (2013). https://doi.org/10.1007/s00291-013-0327-6

Download citation

Keywords

  • Supply chain network design
  • Uncertainty
  • Robustness
  • Anticipation
  • Scenario planning
  • Network disruptions
  • Multi-hazards
  • Stochastic programming