Interval-Parameter Conditional Value-at-Risk Two-Stage Stochastic Programming Model for Management of End-of-Life Vehicles

Abstract

The management of end-of-life vehicles conserves natural resources, provides economic benefits, and reduces water, air, and soil pollution. Sound management of end-of-life vehicles is vitally important worldwide thus requiring sophisticated decision-making tools for optimizing its efficiency and reducing system risk. This paper proposes an interval-parameter conditional value-at-risk two-stage stochastic programming model for management of end-of-life vehicles. A case study is conducted in order to demonstrate the usefulness of the developed model. The model is able to provide the trade-offs between the expected profit and system risk. It can effectively control risk at extremely disadvantageous availability levels of end-of-life vehicles. The formulated model can produce optimal solutions under predetermined decision-making risk preferences and confidence levels. It can simultaneously determine the optimal long-term allocation targets of end-of-life vehicles and reusable parts as well as capital investment, production planning, and logistics management decisions within a multi-period planning horizon. The proposed model can efficiently handle uncertainties expressed as interval values and probability distributions. It is able to provide valuable insights into the effects of uncertainties. Compared to the available models, the resulting solutions are far more robust.

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Acknowledgements

The author is grateful to the advisory editor and the anonymous reviewers for their insightful comments and suggestions.

Funding

This work was partially supported by Ministry of Education, Science and Technological Development of the Republic of Serbia through the project TR 36006 for the period 2011–2019.

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Appendix. Notation

Appendix. Notation

Indices and sets

t—index of time period, t ∈ {1, …, T}

c—index of collection center, c ∈ {1, …, | C| }

a—index of authorized treatment facility, a ∈ {1, …, | A| }

x—index of specialized recovery facility, x ∈ {1, …, | X| }

d—index of dealer of reusable parts, d ∈ {1, …, | D| }

i—index of import center, i ∈ {1, …, | I| }

v—index of vehicle recycling factory, v ∈ {1, …, | V| }

s—index of ELV availability level, s ∈ {1, …, | S| }

C—set of collection centers located in the region

A—set of authorized treatment facilities located in the region

X—set of specialized recovery facilities located in the region

D—set of dealers of reusable parts (i.e., used parts market) located in the region

I—set of import centers located in the region

V—set of considered vehicle recycling factories located in the region

St—set of ELV availability levels in period t

\( \mathrm{S}=\underset{t\in \left\{1,...,\mathrm{T}\right\}}{\cup }{\mathrm{S}}_t \)—set of ELV availability levels

Parameters

T—number of analyzed time periods

f±—expected profit to ELV management system over the multi-period planning horizon

α—confidence level

λ—weighting factor specified by decision makers to trade-off expected profit with risk

\( pro{b}_{st},s\in {\mathrm{S}}_t\left( pro{b}_{st}>0,\forall s,t;\sum \limits_{s\in {\mathrm{S}}_t} pro{b}_{st}=1,\forall t\right) \)—probability of ELV availability level s in period t

\( C{Q}_{sct}^{\pm },c\in \mathrm{C},s\in {\mathrm{S}}_t \)—available quantity of ELVs in collection center c with probability level probst in period t

\( R{E}_{vt}^{\pm },v\in \mathrm{V} \)—revenue to ELV management system per weight unit of depolluted, dismantled, and flattened ELVs allocated to vehicle recycling factory v in period t (first stage parameter)

\( P{E}_{vt}^{\pm },v\in \mathrm{V}\left(P{E}_{vt}^{\pm }>R{E}_{vt}^{\pm },\forall \mathrm{v},t\right) \)—penalty to ELV management system per weight unit of depolluted, dismantled, and flattened ELVs not delivered to vehicle recycling factory v in period t (second stage parameter)

\( R{D}_{dt}^{\pm },d\in \mathrm{D} \)—revenue to ELV management system per weight unit of reusable parts and components allocated to dealer d in period t (first stage parameter)

\( P{D}_{dt}^{\pm },d\in \mathrm{D}\left(P{D}_{vt}^{\pm }>R{D}_{dt}^{\pm },\forall d,t\right) \)—penalty to ELV management system per weight unit of reusable parts and components not delivered to dealer d in period t (second stage parameter)

Δt—duration of planning period t in time units

ϒat—total working time of authorized treatment facility a in period t

\( ZM{E}_{vt\;\min}^{\pm },v\in \mathrm{V} \)—the minimum quantity of depolluted, dismantled, and flattened ELVs that must be allocated to vehicle recycling factory v in period t to avoid cessation

\( {\rho}_t^{\pm } \)—share of hazardous substances that are required to remove from ELVs during depolluting process in period t

\( {\tau}_t^{\pm } \)—share of reusable parts and components with market value in depolluted ELVs in period t

\( I{O}_c^{\pm },c\in \mathrm{C} \)—initial inventory weight of ELVs piled up in collection center c

\( I{\varXi}_a^{\pm },a\in \mathrm{A} \)—initial inventory weight of hazardous substances in storage tanks of authorized treatment facility a

\( I{\varOmega}_a^{\pm },a\in \mathrm{A} \)—initial inventory weight of reusable parts and components dismantled from depolluted ELVs and stored in authorized treatment facility a

\( I{H}_a^{\pm },a\in \mathrm{A} \)—initial inventory weight of depolluted, dismantled, and flattened ELVs piled up in storage area of authorized treatment facility a

\( I{\varPsi}_i^{\pm },i\in \mathrm{I} \)—initial inventory weight of depolluted, dismantled, and flattened ELVs piled up in import center i

\( C{T}_{cat}^{\pm },c\in \mathrm{C},a\in \mathrm{A} \)—transportation cost per weight unit of ELVs from collection center c to authorized treatment facility a in period t

\( D{T}_{adt}^{\pm },a\in \mathrm{A},d\in \mathrm{D} \)—transportation cost per weight unit of dismantled reusable parts and components from authorized treatment facility a to dealer d in period t

\( S{T}_{axt}^{\pm },a\in \mathrm{A},x\in \mathrm{X} \)—transportation cost per weight unit of hazardous substances from authorized treatment facility a to specialized recovery facility x in period t

\( A{T}_{avt}^{\pm },a\in \mathrm{A},v\in \mathrm{V} \)—transportation cost per weight unit of depolluted, dismantled, and flattened ELVs from authorized treatment facility a to vehicle recycling factory v in period t

\( I{T}_{ivt}^{\pm },i\in \mathrm{I},v\in \mathrm{V} \)—transportation cost per weight unit of depolluted, dismantled, and flattened ELVs from import center i to vehicle recycling factory v in period t

\( DP{C}_{at}^{\pm },a\in \mathrm{A} \)—depolluting cost per weight unit of ELVs in authorized treatment facility a in period t

\( DS{C}_{at}^{\pm },a\in \mathrm{A} \)—dismantling cost per weight unit of depolluted ELVs in authorized treatment facility a in period t

\( FL{C}_{at}^{\pm },a\in \mathrm{A} \)—flattening cost per weight unit of depolluted and dismantled ELVs in authorized treatment facility a in period t

\( TR{C}_{xt}^{\pm },x\in \mathrm{X} \)—treatment cost per weight unit of hazardous substances in specialized recovery facility x in period t

\( {U}_{it}^{\pm },i\in \mathrm{I} \)—importing cost per weight of depolluted, dismantled, and flattened ELVs ordered at the international secondary metal market and delivered to import center i in period t

\( U{B}_{t\;\max}^{\pm } \)—the maximum allowed expenses for ordering depolluted, dismantled, and flattened ELVs at the international secondary metal market in period t

\( SC{O}_{ct}^{\pm },c\in \mathrm{C} \)—inventory holding cost per weight unit and time unit for ELVs piled up in collection center c in period t

\( SC{\varXi}_{at}^{\pm },a\in \mathrm{A} \)—inventory holding cost per weight unit and time unit for hazardous substances in storage tanks of authorized treatment facility a in period t

\( SC{\varOmega}_{at}^{\pm },a\in \mathrm{A} \)—inventory holding cost per weight unit and time unit for reusable parts and components dismantled from depolluted ELVs stored in authorized treatment facility a in period t

\( SC{H}_{at}^{\pm },a\in \mathrm{A} \)—inventory holding cost per weight unit and time unit for depolluted, dismantled, and flattened ELVs piled up in storage area of authorized treatment facility a in period t

\( SC{\varPsi}_{it}^{\pm },i\in \mathrm{I} \)—inventory holding cost per weight unit and time unit for imported depolluted, dismantled, and flattened ELVs piled up in import center i in period t

\( O{C}_{at}^{\pm },a\in \mathrm{A} \)—operational capacity of authorized treatment facility a in period t

\( CP{\varXi}_a^{\pm },a\in \mathrm{A} \)—capacity of tanks for hazardous substances in authorized treatment facility a

\( CP{\varOmega}_a^{\pm },a\in \mathrm{A} \)—inventory storage capacity for reusable parts and components dismantled from depolluted ELVs in authorized treatment facility a

\( CP{H}_a^{\pm },a\in \mathrm{A} \)—capacity of storage area for depolluted, dismantled, and flattened ELVs in authorized treatment facility a

\( CP{\varPsi}_i^{\pm },i\in \mathrm{I} \)—capacity of storage area for depolluted, dismantled, and flattened ELVs in import center i

Variables

\( {\zeta}_{st}^{\pm } \)—auxiliary variable to compute the CVaR

\( {\eta}_t^{\pm } \)VaR in period t when the confidence level is α

\( Z{E}_{vt}^{\pm },v\in \mathrm{V} \)—allocation target of depolluted, dismantled, and flattened ELVs for vehicle recycling factory v in period t (the first stage decision variable)

γvt, v ∈ V(γvt ∈ [0, 1], ∀v, t)—the first stage decision variable that is used for identifying \( Z{E}_{vt}^{\pm }=Z{E}_{vt}^{-}+\varDelta Z{E}_{vt}\cdot {\gamma}_{vt} \) (where \( \varDelta Z{E}_{vt}=Z{E}_{vt}^{+}-Z{E}_{vt}^{-} \)), which is an optimal allocation target of depolluted, dismantled, and flattened ELVs for vehicle recycling factory v in period t

\( M{E}_{svt}^{\pm },v\in \mathrm{V},s\in {\mathrm{S}}_t \)—quantity by which allocation target of depolluted, dismantled, and flattened ELVs for vehicle recycling factory v is not met under ELV availability level s in period t (the second stage decision variable)

\( Z{D}_{dt}^{\pm },d\in \mathrm{D} \)—allocation target of dismantled reusable parts and components for dealer d in period t (the first stage decision variable)

μdt, d ∈ D (μdt∈[0, 1], ∀d,t)—the first stage decision variable that is used for identifying \( Z{D}_{dt}^{\pm }=Z{D}_{dt}^{-}+\varDelta Z{D}_{dt}\cdot {\mu}_{dt} \) (where \( \varDelta Z{D}_{dt}=Z{D}_{dt}^{+}-Z{D}_{dt}^{-} \)), which is an optimal allocation target of reusable parts and components dismantled from ELVs for dealer d in period t

\( M{D}_{sdt}^{\pm },d\in \mathrm{D},s\in {\mathrm{S}}_t \)—quantity by which allocation target of dismantled reusable parts and components for dealer d is not met under ELV availability level s in period t (the second stage decision variable)

\( A{Q}_{scat}^{\pm },c\in \mathrm{C},a\in \mathrm{A},s\in {\mathrm{S}}_t \)—quantity of ELVs allocated between collection center c and authorized treatment facility a under ELV availability level s in period t

\( HS{Q}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—quantity of hazardous substances removed from ELVs in authorized treatment facility a under ELV availability level s in period t

\( AS{Q}_{saxt}^{\pm },a\in \mathrm{A},x\in \mathrm{X},s\in {\mathrm{S}}_t \)—quantity of hazardous substances allocated between authorized treatment facility a and specialized recovery facility x under ELV availability level s in period t

\( D{Q}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—quantity of depolluted ELVs in authorized treatment facility a under ELV availability level s in period t

\( DP{Q}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—quantity of reusable parts and components dismantled from depolluted ELVs in authorized treatment facility a under ELV availability level s in period t

\( AP{Q}_{sadt}^{\pm },a\in \mathrm{A},d\in \mathrm{D},s\in {\mathrm{S}}_t \)—quantity of dismantled reusable parts and components allocated between authorized treatment facility a and dealer d under ELV availability level s in period t

\( B{Q}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—quantity of depolluted, dismantled, and flattened ELVs in authorized treatment facility a under ELV availability level s in period t

\( V{Q}_{savt}^{\pm },a\in \mathrm{A},v\in \mathrm{V},s\in {\mathrm{S}}_t \)—quantity of depolluted, dismantled, and flattened ELVs allocated between authorized treatment facility a and vehicle recycling factory v under ELV availability level s in period t

\( U{Q}_{sit}^{\pm },i\in \mathrm{I},s\in {\mathrm{S}}_t \)—quantity of depolluted, dismantled, and flattened ELVs imported from the international secondary metal market and delivered to import center i under ELV availability level s in period t

\( UI{Q}_{sivt}^{\pm },i\in \mathrm{I},v\in \mathrm{V},s\in {\mathrm{S}}_t \)—quantity of imported depolluted, dismantled, and flattened ELVs allocated between import center i and vehicle recycling factory v under ELV availability level s in period t

\( {O}_{sct}^{\pm },\mathrm{c}\in \mathrm{C},s\in {\mathrm{S}}_t \)—weight of ELVs piled up in collection center c at the end of period t under ELV availability level s

\( {\varXi}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—weight of hazardous substances in storage tanks of authorized treatment facility a at the end of period t under ELV availability level s

\( {\varOmega}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—weight of reusable parts and components dismantled from depolluted ELVs and stored in authorized treatment facility a at the end of period t under ELV availability level s

\( {H}_{sat}^{\pm },a\in \mathrm{A},s\in {\mathrm{S}}_t \)—weight of depolluted, dismantled, and flattened ELVs piled up in storage area of authorized treatment facility a at the end of period t under ELV availability level s

\( {\varPsi}_{sit}^{\pm },i\in \mathrm{I},s\in {\mathrm{S}}_t \)—weight of depolluted, dismantled, and flattened ELVs piled up in import center i at the end of period t under ELV availability level s

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Simic, V. Interval-Parameter Conditional Value-at-Risk Two-Stage Stochastic Programming Model for Management of End-of-Life Vehicles. Environ Model Assess 24, 547–567 (2019). https://doi.org/10.1007/s10666-018-9648-9

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Keywords

  • End-of-life vehicle
  • Risk control
  • Conditional value-at-risk
  • Two-stage stochastic programming
  • Interval-parameter programming
  • Uncertainty