# Accelerated Benders’ Decomposition for Integrated Forward/Reverse Logistics Network Design under Uncertainty

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Problem Definition

#### 3.1. Model Description

- The periodic review policy is used for the distribution centers and manufacturers, in which the inventory levels are reviewed at certain intervals and the appropriate orders are placed after each review. The inventory level of raw material should meet a specific amount in each period. The production and shipment from the manufacturers to the distribution centers takes place, to raise the inventory level of distribution centers to the base-stock level (S) at the beginning of each period. This concept is referred to as the push strategy in the related literature. On the other hand, customer demands are met with the inventory kept by the distribution centers. The customers only place orders to the distribution centers. This system is known as a pull-based system.
- A hybrid concept for production plants is considered. Due to the fact that locating manufacture and remanufacture plants in the same potential place will reduce fixed costs, we are interested in locating hybrid plants.
- In distribution centers, a risk pooling strategy is considered, where both new and remanufactured products are held simultaneously. The “risk-pooling” strategy is an efficient way of managing demand uncertainty, for which inventory needs to be centralized at distribution centers (DC’s) arriving at a convenient service level. Each DC uses a base stock level inventory policy to satisfy demands from retailers, as well as safety stock to cope with the variability of customer demand at retailers, to achieve “risk-pooling” benefits.
- As mentioned above, the inventory level of a raw material should meet a specific amount in each period. To this aim, raw material is provided through wholesale contracts, spot markets and recycled materials. A wholesale contract is a long term agreement with suppliers to convey a certain proportion of raw materials in the beginning of each period. If the amount of provided raw material from a wholesale contract and recycled material do not meet the base stock level in each period, the shortage of raw material is compensated for by buying from spot markets, but at a higher price.

- A single-product, multi-stage, multi-period supply chain network is given.
- We assume a finite set of facilities (i.e., manufacturers and distribution centers) should be opened.
- There is no limitation on the capacity of the material flow through the network.
- We are faced with uncertainty for the demand of the customers to the distribution centers and return of used products to collection centers.
- Transportation costs are linearly dependent on the distance between stages.
- Distribution centers and raw material stock at manufactures incur inventory holding costs at the end of each period.
- All of the returned products must be collected, but a shortage is allowed, to satisfy the demands of second market customers.
- Customers’ locations are known and fixed.

#### 3.2. Model Formulation

Sets: | |

$I$ | Set of potential manufacturer locations $i,{i}^{\prime}\in I$ |

$J$ | Set of potential distribution center locations $j\in J$ |

$T$ | Set of periods in planning horizon $t,k\in T$ |

$C$ | Set of customers for new product $c\in C$ |

${C}^{\prime}$ | Set of customers for used product ${c}^{\prime}\in {C}^{\prime}$ |

$L$ | Set of potential collection center locations $l\in L$ |

$D$ | Set of disposal locations $d\in D$ |

$R$ | Set of seller products $r\in R$ |

$S$ | Set of scenarios $s\in S$ |

Parameters, constants, and coefficients: | |

Fixed costs: | |

${\mathrm{F}}_{\mathrm{i}}^{\mathrm{M}}$ | Fixed cost of locating manufacturer at location i |

${\mathrm{F}}_{\mathrm{i}}^{\mathrm{RM}}$ | Fixed cost of locating remanufacturer at location i |

${\mathrm{F}}_{\mathrm{j}}^{\mathrm{Dc}}$ | Fixed cost of locating distribution center for new product at location j |

${\mathrm{F}}_{\mathrm{j}}^{\mathrm{D}{\mathrm{c}}^{\prime}}$ | Fixed cost of locating distribution center for used product at location j |

${\mathrm{F}}_{\mathrm{l}}^{\mathrm{CL}}$ | Fixed cost of locating collection center at location l |

Capacity costs and saving costs: | |

${\mathrm{s}}_{\mathrm{i}}^{\mathrm{P}}$ | Saving cost of locating a hybrid manufacture/ remanufacture facility at location i |

${\mathrm{s}}_{\mathrm{j}}^{\mathrm{Dcs}}$ | Saving cost of locating a hybrid distribution center facility at location j |

${\mathrm{Vc}}_{\mathrm{i}}^{\mathrm{M}}$ | Cost for capacity of manufacturer i per unit of product |

${\mathrm{Vc}}_{\mathrm{i}}^{\mathrm{RM}}$ | Cost for capacity of remanufacturer i per unit of product |

${\mathrm{Vc}}_{\mathrm{j}}^{\mathrm{Dc}}$ | Cost for capacity of distribution center j per unit of new product |

${\mathrm{Vc}}_{\mathrm{j}}^{\mathrm{D}{\mathrm{c}}^{\prime}}$ | Cost for capacity of distribution center j per unit of used product |

${\mathrm{Vc}}_{\mathrm{l}}^{\mathrm{Cl}}$ | Cost for capacity of collection center l per unit of returned product |

Capacity of facilities: | |

${\mathrm{Cap}}_{\mathrm{i}}^{\mathrm{Max}-\mathrm{M}}$ | Maximum available capacity of manufacturing at location i |

${\mathrm{Cap}}_{\mathrm{i}}^{\mathrm{Max}-\mathrm{RM}}$ | Maximum available capacity of remanufacturing at location i |

${\mathrm{Cap}}_{\mathrm{j}}^{\mathrm{Max}-\mathrm{Dc}}$ | Maximum available capacity for new products at distribution center j |

${\mathrm{Cap}}_{\mathrm{j}}^{\mathrm{Max}-\mathrm{D}{\mathrm{c}}^{\prime}}$ | Maximum available capacity for second hand products at distribution center j |

${\mathrm{Cap}}_{\mathrm{l}}^{\mathrm{Max}-\mathrm{Cl}}$ | Maximum available capacity of collection center at location l |

${\mathrm{Cap}}_{\mathrm{i}}^{\mathrm{Max}-\mathrm{P}}$ | Maximum available capacity for production facilities at location i |

${\mathrm{Cap}}_{\mathrm{j}}^{\mathrm{Max}-\mathrm{Dcs}}$ | Maximum available capacity for distributing center facilities at location j |

Transportation costs: | |

${\mathrm{Tc}}_{\mathrm{ij}}^{\mathrm{M}-\mathrm{Dc}}$ | Cost of transporting, per unit of product, between manufacturer p and distribution center j |

${\mathrm{Tc}}_{\mathrm{jc}}^{\mathrm{Dc}-\mathrm{Cu}}$ | Cost of transporting, per unit of new product, between distribution center j and customer c |

${\mathrm{Tc}}_{\mathrm{j}{\mathrm{c}}^{\prime}}^{\mathrm{D}{\mathrm{c}}^{\prime}-\mathrm{C}{\mathrm{u}}^{\prime}}$ | Cost of transporting, per unit of used product, between distribution center j and customer $c{u}^{\prime}$ |

${\mathrm{Tc}}_{\mathrm{rl}}^{\mathrm{Sr}-\mathrm{Cl}}$ | Cost of transporting, per unit of product, between seller r and collection center l |

${\mathrm{Tc}}_{\mathrm{ld}}^{\mathrm{Cl}-\mathrm{Di}}$ | Cost of transporting, per unit of product, between collection center l and disposal d |

${\mathrm{Tc}}_{\mathrm{di}}^{\mathrm{Di}-\mathrm{M}}$ | Cost of transporting, per unit of recycled product, between disposal d and manufacturer i |

${\mathrm{Tc}}_{\mathrm{lj}}^{\mathrm{Cl}-\mathrm{D}{\mathrm{c}}^{\prime}}$ | Cost of transporting, per unit of product, between collection center l and distribution center j |

${\mathrm{Tc}}_{\mathrm{li}}^{\mathrm{Cl}-\mathrm{M}}$ | Cost of transporting, per unit of product, between collection center l and manufacturer i |

${\mathrm{Tc}}_{\mathrm{i}{\mathrm{i}}^{\prime}}^{\mathrm{M}-\mathrm{Rm}}$ | Cost of transporting, per unit of product, between manufacturer i and remanufacturer ${i}^{\prime}$ |

Inventory costs: | |

${\mathrm{Ic}}_{\mathrm{j}}^{\mathrm{Dc}}$ | Cost of holding, per unit of inventory, in distribution center j |

${\mathrm{Ic}}_{\mathrm{i}}^{\mathrm{M}}$ | Cost of holding, per unit of inventory, in manufacturer i |

Demand and return: | |

${\mathrm{D}}_{\mathrm{cst}}^{\mathrm{Cu}}$ | Product demand of customer c in scenario s at period t |

${\mathrm{Rs}}_{\mathrm{rts}}^{}$ | Product returns of seller r in scenario s at period t |

Other parameters: | |

${\mathrm{Pr}}_{s}$ | Probability of scenario s |

$\mathrm{BOM}$ | The quantity of raw material needed for one unit of a product |

${\mathrm{C}}_{\mathrm{sm}}$ | Cost of buying raw material from spot market |

Coefficients and ratios: | |

$\mathsf{\beta}$ | Rate of raw material shipping from disposal center to raw material stock |

$\mathsf{\lambda}$ | Rate of new product shipping from manufacture centers to distribution centers |

${\mathsf{\gamma}}_{1}$ | Rate of product shipping from collection centers to distribution centers |

${\mathsf{\gamma}}_{2}$ | Rate of product shipping from collection centers to disposal centers |

$\mathrm{M}$ | A large number |

${\mathrm{N}}_{\mathrm{t}}$ | Number of periods |

Decision variables: | |

Binary variables (relating to opening and locating facilities): | |

${x}_{i}^{M}$ | Binary variable equals 1 if a manufacturer is located at location i, 0 otherwise |

${x}_{i}^{RM}$ | Binary variable equals 1 if a remanufacturer is located at location i, 0 otherwise |

${y}_{j}^{Dc}$ | Binary variable equals 1 if a distribution center for a new product is located at location j, 0 otherwise |

${{y}^{\prime}}_{j}^{Dc}$ | Binary variable equals 1 if a distribution center for a used product is located at location j, 0 otherwise |

${x}_{{}_{i}i}^{p}$ | Binary variable equals 1 if a manufacturer and remanufacturer are located at location i, 0 otherwise |

${y}_{j}^{Dcs}$ | Binary variable equals 1 if a new product distribution center and used product distribution center are located at location j, 0 otherwise |

${z}_{l}^{Cl}$ | Binary variable equals 1 if a collection center is located at location l, 0 otherwise |

Continuous variables (relating to production and raw material acquisition): | |

${W}_{}^{C}$ | Quantity committed in wholesale contract |

${r}_{it}^{M}$ | Quantity committed in contract to manufacturer i at period t |

$s{m}_{ist}^{M}$ | Quantity bought from a spot market for manufacturer i in scenario s at period t |

$q{p}_{ist}^{M}$ | Quantity of production from manufacturer i in scenario s at period t |

Continuous variables (relating to capacity of facilities): | |

${c}_{i}^{M}$ | Capacity of manufacturer i |

${c}_{i}^{RM}$ | Capacity of remanufacturer i |

${c}_{j}^{Dc}$ | Capacity of distribution center j for new product |

${c}_{j}^{D{c}^{\prime}}$ | Capacity of distribution center j for used product |

${c}_{l}^{Cl}$ | Capacity of collection center l |

Continuous variables (relating to inventory decisions): | |

${b}_{j}^{Dc}$ | Base-stock level of distribution center j at the beginning of each period |

${b}_{i}^{M}$ | Base-stock level of manufacturer i at the beginning of each period |

$in{v}_{ist}^{M}$ | Inventory level of manufacturer i at the end of period t in scenario s |

$in{v}_{jst}^{Dc}$ | Inventory level of distribution center j for new products at the end of period t in scenario s |

$in{v}_{jst}^{D{c}^{\prime}}$ | Inventory level of distribution center j for second market products at the end of period t in scenario s |

Continuous variables (relating to flows on network): | |

${f}_{ijst}^{M-Dc}$ | Flow of production from manufacturer i transported to distribution center j at period t in scenario s |

${f}_{dist}^{Di-M}$ | Flow of material from disposal d transported to manufacturer i at period t in scenario s |

${f}_{ijst}^{RM-D{c}^{\prime}}$ | Flow of remanufactured product from remanufacturer i transported to distribution center j in scenario s at period t |

${f}_{i{i}^{\prime}st}^{M-Rm}$ | Flow of production from manufacturer i transported to remanufacturer ${i}^{\prime}$ in scenario s at period t |

${f}_{list}^{Cl-Rm}$ | Flow of returned product from collection center l transported to remanufacturer i in scenario s at period t |

${f}_{ljst}^{Cl-D{c}^{\prime}}$ | Flow of returned product from collection center l transported to distribution center j at period t in scenario s |

${f}_{ldst}^{Cl-Di}$ | Flow of returned product from collection center l transported to disposal d at period t in scenario s |

${f}_{jcst}^{Dc-Cu}$ | Flow of new product from distribution center j transported to customer c at period t in scenario s |

${f}_{j{c}^{\prime}st}^{Dc-C{u}^{\prime}}$ | Flow of used product from distribution center l transported to customer ${c}^{\prime}$ at period t in scenario s |

${f}_{rlst}^{Sr-Cl}$ | Flow of returned product from seller r transported to collection center l at period t in scenario s |

## 4. A Benders’ Decomposition-Based Solution Algorithm

Benders’ Decomposition Algorithm |

Step 0. Initialization |

i. ${Z}_{0}^{Upper}=+\infty .$ |

ii. ${Z}_{0}^{Lower}=-\infty .$ |

iii. k = 0. |

iv. Solve the initial master problem to obtain $\{{c}_{i}^{RM},{c}_{i}^{M},{c}_{i}^{Dc},{c}_{i}^{D{c}^{\prime}},{b}_{j}^{Dc},{b}_{i}^{M},{c}_{l}^{Cl},{w}^{c}\}.$ |

While (${Z}_{k}^{Upper}-{Z}_{k}^{Lower}>\epsilon $) |

Step 1. Solving the sub-problems |

For each s ∈ S |

Solve the sub-problems by determined |

$\{{\widehat{c}}_{i}^{RM},{\widehat{c}}_{i}^{M},{\widehat{c}}_{i}^{Dc},{\widehat{c}}_{i}^{D{c}^{\prime}},{\widehat{b}}_{j}^{Dc},{\widehat{b}}_{i}^{M},{\widehat{c}}_{l}^{Cl},{\widehat{w}}^{c}\}.$ |

End for |

Step 2. Updating the lower and upper bounds |

i. ${Z}_{k}^{Upper}={\displaystyle \sum _{s\in S}{\mathrm{Pr}}_{s}\left({Z}_{s,k}^{SP}\right)}+f+\gamma \left[\xi +\frac{1}{1-\alpha}\left({\displaystyle \sum _{s\in S}{\mathrm{Pr}}_{s}\times {\mu}_{s}}\right)\right]$ |

ii. ${Z}_{k}^{Lower}={\displaystyle \sum _{s\in S}{\mathrm{Pr}}_{s}{\theta}_{s}}+f+\gamma \left[\xi +\frac{1}{1-\alpha}\left({\displaystyle \sum _{s\in S}{\mathrm{Pr}}_{s}\times {\mu}_{s}}\right)\right]$ |

Step 3. Solving the master problem |

i. Add optimality cuts to the master problem for each scenario. |

$\begin{array}{l}{\theta}^{s}\ge {Z}_{s,k}^{SP}+{\displaystyle \sum _{t}{\pi}_{{}_{tsk}}^{{w}^{c}{}_{}}\times}\left({w}^{c}-{\widehat{w}}^{c}{}_{sck}\right)+{\displaystyle \sum _{i}{\pi}_{{}_{isk}}^{{c}_{i}^{RM}}\times \left({c}_{i}^{RM}-{\stackrel{\wedge}{c}}_{isk}^{RM}\right)}+{\displaystyle \sum _{i}{\pi}_{{}_{isk}}^{{c}_{i}^{M}}\times \left({c}_{i}^{M}-{\stackrel{\wedge}{c}}_{isk}^{M}\right)}\\ +{\displaystyle \sum _{j}{\pi}_{{}_{jsk}}^{{c}_{j}^{D{c}^{\prime}}}\times \left({c}_{j}^{D{c}^{\prime}}-{\stackrel{\wedge}{c}}_{jsk}^{D{c}^{\prime}}\right)}+{\displaystyle \sum _{j}{\pi}_{{}_{jsk}}^{{c}_{j}^{Dc}}\times \left({c}_{j}^{Dc}-{\stackrel{\wedge}{c}}_{jsk}^{Dc}\right)}+{\displaystyle \sum _{j}{\pi}_{{}_{jsk}}^{{b}_{j}^{Dc}}}\times \left({b}_{j}^{Dc}-{\stackrel{\wedge}{b}}_{jsk}^{Dc}\right)+\\ {\displaystyle \sum _{i}{\pi}_{{}_{isk}}^{{b}_{i}^{M}}}\times \left({b}_{i}^{M}-{\stackrel{\wedge}{b}}_{isk}^{M}\right)+{\displaystyle \sum _{l}{\pi}_{{}_{lsk}}^{{c}_{l}^{Cl}}\times \left({c}_{l}^{Cl}-{\stackrel{\wedge}{c}}_{lsk}^{Cl}\right)}\end{array}$ |

ii. k = k + 1. |

iii. Solve the master problem to obtain $\left\{{c}_{i}^{RM},{c}_{i}^{M},{c}_{j}^{Dc},{c}_{j}^{D{c}^{\prime}},{b}_{j}^{Dc},{b}_{i}^{M},{c}_{l}^{Cl},{w}^{c}\right\}$. |

End while |

#### Valid Inequalities

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

**Lemma**

**2.**

**Proof**

**of Lemma 2.**

**Lemma**

**3.**

**Proof**

**of Lemma 3.**

## 5. Computational Results

#### Data Generation for Parameters and Settings

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The proposed integrated forward/reverse logistic network model, consisting of suppliers, manufacturers, distribution centers, collection/inspection centers and disposal centers.

Category | Detail | Code | Category | Detail | Code |
---|---|---|---|---|---|

Model objectives | Cost minimization | CM | Features of model | Period | |

Profit maximization | PM | Single-period | S | ||

Responsiveness | R | Multi-period | M | ||

Quality | Q | Facility capacity | |||

Other | OT | Un-capacitated | U | ||

Features of model | Stochastic parameters | Capacitated | C | ||

Quantity of demand | D | Capacity expansion | CE | ||

Quantity of returns | R | Single sourcing | SS | ||

Quality of returns | RQ | Model | Mixed Integer Linear Programming | MILP | |

Recovery rate | RR | ||||

Recovery cost | RC | Mixed Integer Non-Linear Programming | MINLP | ||

Transportation cost | TC | ||||

Lead time | LT | Decision variables of model | Inventory decisions | I | |

Income | In | Facility capacity | Fc | ||

Other | OT | Demand satisfaction | D | ||

Product commodity | Transportation values | TV | |||

Single-commodity | S | Location/allocation | LA | ||

Multi-commodity | M | Transportation mode selection | TM | ||

Solution methodology | Technology selection | TS | |||

Exact solution method | EX | ||||

Heuristic solution method | HE |

Ref. | Model Obj. | Stoch. Param. | Product Com. | Period | Facility Cap. | Model | D.V. | Sol. Method | Solution Approach |
---|---|---|---|---|---|---|---|---|---|

[50] | PM | R | S | S | C | MILP | TV, LA | EX | B&C |

[51] | PM | D | M | M | C | MILP | TV, LA, Fc, TM | -- ^{a} | AIMMS |

[48] | PM | R, In | M | S | C | MILP | TV, LA | -- ^{a} | CPLEX |

[62] | PM | D, R | S | S | C | MILP | TV, LA, SS | EX | Integer L-Shape Method |

[49] | CM | TC, D, R | M | S | C | MILP | TV, LA, D | -- ^{a} | CPLEX |

[60] | PM | LT | S | S | C | MINLP | TV, LA, Fc, I | HE | Differential Evaluation (DE) |

[59] | CM | D, R | M | M | C | MILP | TV, LA | HE | SAA with SA |

[54] | CM, OT | TC, R, OT | M | S | C | MILP | TV, LA, TS | -- ^{a} | CPLEX10 |

[58] | CM | TC, D, R, RQ | S | S | C | MILP | TV, LA | -- ^{a} | LINGO |

[55] | PM | D, R | S | S | C | MILP | TV, LA | -- ^{a} | CPLEX |

[57] | CM | D, R | M | S | C | MILP | TV, LA | EX | SAA with CPLEX |

[53] | CM | RQ | S | S | MILP | LA | EX | SAA | |

[63] | CM, R, Q | D, R, RC, OT | M | S | C, SS | MILP | TV, LA, Fc | -- ^{a} | Commercial Solver |

[56] | CM | D, R | S | S | C | MILP | TV, LA | -- ^{a} | CPLEX |

[61] | PM | D, R | S | M | C | MILP | TV, LA, I | -- ^{a} | XpressSp |

[20] | PM | OT | M | S | C | MILP | TV, LA | -- ^{a} | CPLEX |

[64] | CM | D, R, RQ | S | S | C | MILP | TV, LA | -- ^{a} | CPLEX/GAMS |

[4] | PM | R, RQ | P | S | C | MILP | TV, LA | EX | SAA |

[65] | PM | D, R, TC | S | M | C | MILP | TV, LA | EX | Accelerated BD |

Our paper | CM | D, R | S | M | C | MILP | TV, LA, I | EX | Accelerated BD |

^{a}Commercial solver packages are used or the solution method is not reported.

**Table 3.**Nominal values of the model parameters. For most of the parameters, a uniform distribution is utilized. For demand and return, an autoregressive time series (AR) is used.

Parameter | Range | Parameter | Range |
---|---|---|---|

${\mathrm{F}}_{\mathrm{p}}^{\mathrm{M}}$ | ~Uniform (1,000,000, 4,000,000) | ${\mathrm{Tc}}_{\mathrm{p},{\mathrm{p}}^{\prime}}^{\mathrm{M}-\mathrm{Rm}}$ | ~ Uniform (10, 25) |

${\mathrm{F}}_{\mathrm{p}}^{\mathrm{RM}}$ | ~Uniform (500,000, 1,500,000) | ${\mathrm{Tc}}_{\mathrm{i},\mathrm{p}}^{\mathrm{Cl}-\mathrm{M}}$ | ~ Uniform (10, 20) |

${\mathrm{F}}_{\mathrm{dc}}^{\mathrm{Dc}}$ | ~Uniform (500,000, 2,500,000) | ${\mathrm{Ic}}_{\mathrm{dc}}^{\mathrm{Dc}}$ | ~ Uniform (20, 25) |

${\mathrm{F}}_{\mathrm{dc}}^{{\mathrm{Dc}}^{\prime}}$ | ~Uniform (400,000, 600,000) | ${\mathrm{Ic}}_{\mathrm{p}}^{\mathrm{M}}$ | ~ Uniform (30, 40) |

${\mathrm{F}}_{\mathrm{i}}^{\mathrm{Cl}}$ | ~Uniform (300,000, 900,000) | ${\mathrm{D}}_{\mathrm{cu},\mathrm{t},\mathrm{sc}}^{\mathrm{Cu}}$ | $AR(1):{\mathrm{D}}_{\mathrm{cu},\mathrm{t},\mathrm{sc}}^{\mathrm{Cu}}=\alpha +{\beta}_{1}{\mathrm{D}}_{\mathrm{cu},\mathrm{t}-1,\mathrm{sc}}^{\mathrm{Cu}}+{\epsilon}_{cu,t,sc}$ |

${\mathrm{Vc}}_{\mathrm{p}}^{\mathrm{M}}$ | ~Uniform (1000, 1800) | α ~ Uniform (20, 40) | |

${\mathrm{Vc}}_{\mathrm{p}}^{\mathrm{RM}}$ | ~Uniform(2000,2800) | ${\beta}_{i}$ ~ Uniform (0.15, 0.2) | |

${\mathrm{Vc}}_{\mathrm{dc}}^{\mathrm{Dc}}$ | ~Uniform (1500, 3000) | ${\epsilon}_{cu,t,sc}$ ~ N(0, Uniform (20, 35)) | |

${\mathrm{Vc}}_{\mathrm{dc}}^{{\mathrm{Dc}}^{\prime}}$ | ~Uniform (900, 1500) | ${\mathrm{D}}_{\mathrm{cu},\mathrm{t}-1,\mathrm{sc}}^{\mathrm{Cu}}$ ~ Uniform (30, 50) | |

${\mathrm{Cap}}_{\mathrm{dc}}^{\mathrm{Max}-\mathrm{Dc}}$ | ~Uniform (7000, 15,000) | ${\mathrm{Rs}}_{\mathrm{sr},\mathrm{t},\mathrm{sc}}^{}$ | $AR(1):{\mathrm{Rs}}_{\mathrm{sr},\mathrm{t},\mathrm{sc}}^{}=\alpha +{\beta}_{1}{\mathrm{Rs}}_{\mathrm{sr},\mathrm{t}-1,\mathrm{sc}}^{}+{\epsilon}_{cu,t,sc}$ |

${\mathrm{Cap}}_{\mathrm{dc}}^{\mathrm{Max}-{\mathrm{Dc}}^{\prime}}$ | ~Uniform (1000, 2000) | α ~ Uniform (10, 20) | |

${\mathrm{Cap}}_{\mathrm{i}}^{\mathrm{Max}-\mathrm{Cl}}$ | ~Uniform (1000, 5000) | ${\beta}_{i}$ ~ Uniform (0.15, 0.2) | |

${\mathrm{Tc}}_{\mathrm{p},\mathrm{dc}}^{\mathrm{M}-\mathrm{Dc}}$ | ~Uniform (10, 30) | ${\epsilon}_{cu,t,sc}$ ~ N(0, Uniform (10, 25)) | |

${\mathrm{Tc}}_{\mathrm{dc},\mathrm{cu}}^{\mathrm{Dc}-\mathrm{Cu}}$ | ~Uniform (15, 30) | ${\mathrm{Rs}}_{\mathrm{sr},\mathrm{t}-1,\mathrm{sc}}^{}$ ~ Uniform (20, 30) | |

${\mathrm{Tc}}_{\mathrm{dc},{\mathrm{cu}}^{\prime}}^{{\mathrm{Dc}}^{\prime}-{\mathrm{Cu}}^{\prime}}$ | ~Uniform (10, 30) | $\mathrm{M}$ | 60 |

${\mathrm{Tc}}_{\mathrm{i},\mathrm{di}}^{\mathrm{Cl}-\mathrm{Di}}$ | ~Uniform (20, 35) | $\mathsf{\beta}$ | 0.7 |

${\mathrm{Tc}}_{\mathrm{di},\mathrm{p}}^{\mathrm{Di}-\mathrm{M}}$ | ~Uniform (10, 30) | $\mathsf{\lambda}$ | 0.95 |

${\mathrm{Tc}}_{\mathrm{sr},\mathrm{i}}^{\mathrm{Sr}-\mathrm{Cl}}$ | ~Uniform (15, 30) | ${\mathsf{\gamma}}_{1}$ | 0.4 |

${\mathrm{Tc}}_{\mathrm{i},\mathrm{dc}}^{\mathrm{Cl}-{\mathrm{Dc}}^{\prime}}$ | ~Uniform (10, 20) | ${\mathsf{\gamma}}_{2}$ | 0.4 |

**Table 4.**Characteristics of test problems. Four test cases are generated for each small, medium, and large test problems. Each test case has a specific distinction to the other cases.

Size of Test Problems | ID | $\mathit{i}$ | $\mathit{j}$ | $\mathit{l}$ | $\mathit{C}$ | ${\mathit{C}}^{\prime}$ | $\mathit{r}$ | $\mathit{d}$ | $\mathit{S}$ | $\mathit{T}$ |
---|---|---|---|---|---|---|---|---|---|---|

Small | 1 | 4 | 8 | 8 | 10 | 15 | 10 | 2 | 20 | 12 |

2 | 4 | 8 | 8 | 10 | 15 | 10 | 2 | 40 | 12 | |

3 | 5 | 10 | 10 | 12 | 15 | 12 | 2 | 20 | 12 | |

4 | 5 | 10 | 10 | 12 | 15 | 12 | 2 | 40 | 12 | |

Medium | 5 | 8 | 18 | 12 | 18 | 15 | 15 | 2 | 20 | 12 |

6 | 8 | 18 | 12 | 18 | 15 | 15 | 2 | 40 | 12 | |

7 | 10 | 20 | 12 | 20 | 15 | 15 | 2 | 20 | 12 | |

8 | 10 | 20 | 12 | 20 | 15 | 15 | 2 | 40 | 12 | |

Large | 9 | 15 | 40 | 30 | 40 | 15 | 20 | 2 | 20 | 12 |

10 | 15 | 40 | 30 | 40 | 15 | 20 | 2 | 40 | 12 | |

11 | 20 | 60 | 40 | 60 | 15 | 20 | 2 | 20 | 12 | |

12 | 20 | 60 | 40 | 60 | 15 | 20 | 2 | 40 | 12 |

ID | Number of Variables | No. of Constraints | No. of Scenarios | |
---|---|---|---|---|

Binary | Continuous | |||

1 | 44 | 117,213 | 35,116 | 20 |

2 | 44 | 234,333 | 70,156 | 40 |

3 | 55 | 169,316 | 43,532 | 20 |

4 | 55 | 338,516 | 86,972 | 40 |

5 | 90 | 358,747 | 67,586 | 20 |

6 | 90 | 717,307 | 135,026 | 40 |

7 | 102 | 433,183 | 75,682 | 20 |

8 | 102 | 866,143 | 151,364 | 40 |

9 | 195 | 1,439,176 | 143,584 | 20 |

10 | 195 | 2,877,976 | 287,167 | 40 |

11 | 280 | 2,750,921 | 202,516 | 20 |

12 | 280 | 5,501,321 | 405,032 | 40 |

**Table 6.**A comparison of the proposed accelerated Benders’ Decomposition (BD) to classic BD and CPLEX, for small, medium, and large size test problems.

CPLEX | Classic BD | Accelerated BD | ||||
---|---|---|---|---|---|---|

ID | Optimality Gap (%) | CPU (s) | Optimality Gap (%) | CPU (s) | Optimality Gap (%) | CPU (s) |

1 | 0 | 210 | 4.231 | 330.12 | 0.8197 | 320.64 |

2 | 0 | 721.18 | 7.3141 | 645.56 | 0.4826 | 642.61 |

3 | 0 | 400.5 | 11.8911 | 400.5 | 0.5528 | 393.76 |

4 | -- ^{b} | >3 h | 15.0164 | 779.74 | 0.8998 | 780.02 |

5 | 0 | 2751.16 | 11.4512 | 1312.51 | 1.3446 | 1268.44 |

6 | -- ^{b} | >5 h | 14.7121 | 2669.98 | 1.5875 | 2618.37 |

7 | -- ^{b} | >5 h | 15.1241 | 1591.56 | 2.6123 | 1540.67 |

8 | -- ^{b} | >5 h | 16.0195 | 3090.12 | 3.4303 | 3089.33 |

9 | -- ^{b} | >10 h | 15.9184 | 5093.42 | 4.9106 | 5009.21 |

10 | -- ^{b} | >10 h | 17.412 | 10,274.84 | 7.2837 | 10,121.71 |

11 | -- ^{b} | >10 h | 18.1027 | 7421.12 | 6.2287 | 7021.13 |

12 | -- ^{b} | >10 h | 19.8193 | 14,573.69 | 8.585 | 14,011.87 |

^{b}The dashes mean that admissible time to solve the problem with CPLEX has reached without reaching to optimality.

**Table 7.**Effectiveness of valid inequality cuts in terms of lower bound, optimality gap, and CPU time.

ABD-I | ABD-1 | ABD-2 | |||||||
---|---|---|---|---|---|---|---|---|---|

ID | Lower Bound ${\mathit{Z}}_{\mathit{l}\mathit{b}}$ | Gap (%) | CPU (s) | Lower Bound ${\mathit{Z}}_{\mathit{l}\mathit{b}}$ | Gap (%) | CPU (s) | Lower bound ${\mathit{Z}}_{\mathit{l}\mathit{b}}$ | Gap (%) | CPU (s) |

1 | 127,007,212 | 0.81 | 320.64 | 126,982,020 | 0.83 | 327.21 | 126,994,615 | 0.82 | 325.16 |

2 | 129,448,338 | 0.48 | 642.61 | 129,422,577 | 0.50 | 643.28 | 129,448,338 | 0.48 | 643.12 |

3 | 151,583,859 | 0.55 | 393.76 | 151,312,985 | 0.73 | 396.30 | 151,508,519 | 0.6 | 396.74 |

4 | 137,039,534 | 0.89 | 780.02 | 136,998,797 | 0.92 | 780.53 | 137,012,373 | 0.91 | 781.46 |

5 | 210,742,541 | 1.34 | 1268.44 | 210,368,884 | 1.52 | 1296.98 | 210,700,958 | 1.36 | 1270.14 |

6 | 215,373,195 | 1.58 | 2618.37 | 215,140,222 | 1.69 | 2660.02 | 215,288,419 | 1.62 | 2622.63 |

7 | 230,643,712 | 2.61 | 1540.67 | 229,859,667 | 2.96 | 1573.89 | 229,881,994 | 2.95 | 1576.68 |

8 | 215,260,805 | 3.43 | 3089.33 | 213,485,714 | 4.29 | 3090.01 | 215,219,189 | 3.45 | 3090.81 |

9 | 418,327,768 | 4.91 | 5009.21 | 416,027,739 | 5.49 | 5064.46 | 418,287,897 | 4.92 | 5060.58 |

10 | 444,456,828 | 7.28 | 10,121.71 | 440,514,861 | 8.24 | 10,245.74 | 442,888,060 | 7.66 | 10,199.11 |

11 | 617,187,359 | 6.22 | 7021.13 | 615,680,328 | 6.48 | 7142.71 | 616,085,342 | 6.41 | 7130.65 |

12 | 644,450,944 | 8.58 | 14,011.87 | 636,073,843 | 10.01 | 14,315.41 | 642,026,640 | 8.99 | 14,149.29 |

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**MDPI and ACS Style**

Vahdat, V.; Vahdatzad, M.A. Accelerated Benders’ Decomposition for Integrated Forward/Reverse Logistics Network Design under Uncertainty. *Logistics* **2017**, *1*, 11.
https://doi.org/10.3390/logistics1020011

**AMA Style**

Vahdat V, Vahdatzad MA. Accelerated Benders’ Decomposition for Integrated Forward/Reverse Logistics Network Design under Uncertainty. *Logistics*. 2017; 1(2):11.
https://doi.org/10.3390/logistics1020011

**Chicago/Turabian Style**

Vahdat, Vahab, and Mohammad Ali Vahdatzad. 2017. "Accelerated Benders’ Decomposition for Integrated Forward/Reverse Logistics Network Design under Uncertainty" *Logistics* 1, no. 2: 11.
https://doi.org/10.3390/logistics1020011