## 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.

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## Notes

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).

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.

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

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

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.

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.

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

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 \(R^A({\hat{\mathbf{x}}}^j)\).

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## Acknowledgments

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

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\(\copyright \) 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.

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).

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) |

\(\displaystyle \sum _{{l}^{\prime }\in L_{lt}^\mathrm{d} ( \omega )} {y_{l{l}^{\prime }t} ( \omega )} \le \rho ^cb_{lt}^c ( \omega )+y_{lt}^o (\omega )\) | \(l\in L^c,t\in T,\omega \in \Omega _i^{m}\) | |

\(\displaystyle y_{l{l}^{\prime }t} ( \omega )\ge 0, y_{l^{\prime }t}^r ( \omega )\ge 0,y_{lt}^r ( \omega )\ge 0\) | \(l\in L^c,{l}^{\prime }\in L_l^d ,\,t\in T,\,\omega \in \Omega _i^m\) |

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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

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DOI: https://doi.org/10.1007/s00291-013-0327-6