Transportation Research Part B 110 (2018) 60–78
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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb
Joint production–inventory–location problem with multi-variate normal demand Mehrdad Shahabi a, Amirmahdi Tafreshian a, Avinash Unnikrishnan b,∗, Stephen D. Boyles c a
Department of Civil and Environmental Engineering, University of Michigan, 2350 Hayward, 1036 GG Brown, Ann Arbor, MI 48109, USA Department of Civil and Environmental Engineering, Engineering Building, 301D, 1930 SW 4th Avenue, Portland State University, Portland, OR 97201, USA c Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, 301 E. Dean Keeton St. Stop C1761, Austin, TX 78712-1172, USA b
a r t i c l e
i n f o
Article history: Received 23 April 2017 Revised 21 December 2017 Accepted 2 February 2018 Available online 22 February 2018 Keywords: Joint production–inventory–location Supply chain network design Outer approximation
a b s t r a c t This paper develops a mixed integer nonlinear programming formulation for the production–inventory–location problem with correlated demands across the retailers. Several structural results for special cases of the problem are derived and studied. A solution method based on the outer approximation of the nonlinear terms is developed to solve the problem. The efficiency of the proposed model and solution approach is investigated through extensive numerical studies. Ignoring correlations can increase the total costs of a production–location–inventory system. Accounting for correlations may lead to changes in supply chain configuration. The effect of capacity on computational times was more pronounced in lower correlations than higher correlations. In addition, we show that the efficiency of the solution method increases significantly for two special cases – when all products have the same holding cost and when the number of orders for different products at each warehouse is constrained to be the same. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Effective supply chain operation and design decisions are critical to the success of many industries, given the increasingly uncertain nature of the operating environment (Friesz et al., 2011). Traditionally researchers have adopted a sequential approach in solving the strategic and tactical decisions associated with the supply chain. In the sequential approach, strategic or longer-term decisions such as where to locate facilities are taken separately from short-term tactical decisions involving inventory management. Over the last decade, several researchers have shown that such a sequential approach led to suboptimal supply chain design and management decisions and proposed the joint location-inventory problem (Daskin et al., 2002; Shen et al., 2003; Miranda and Garrido, 2004). In the joint location–inventory framework, a single optimization model is used to determine both strategic location and tactical inventory control decisions. Safety stock is often used as a risk pooling strategy in joint location–inventory problems to protect against retailer demand uncertainties. A majority of the works in the joint location–inventory literature assume independent retailer demand ∗
Corresponding author. E-mail addresses:
[email protected] (M. Shahabi),
[email protected] (A. Tafreshian),
[email protected] (A. Unnikrishnan),
[email protected] (S.D. Boyles). https://doi.org/10.1016/j.trb.2018.02.002 0191-2615/© 2018 Elsevier Ltd. All rights reserved.
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
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(Daskin et al., 2002; Shen et al., 2003; Miranda and Garrido, 2004; Vidyarthi et al., 2007; Park et al., 2010; Hua and Willems, 2016). Some studies show that demand can be highly correlated due to the recent developments in information sharing, the increase in online purchasing, and factors such as retailer competition, cross-selling, etc. (Liu et al., 20 0 0; Raghunathan, 2003; Zhang et al., 2011; Ganesh et al., 2014). Several studies have shown that accounting for demand correlation is critical for developing successful supply chain management strategies (Güllü, 1998; Raghunathan, 2003; So and Zheng, 2003; Helper et al., 2010; Zhang et al., 2011; Ganesha et al., 2014; Ahmadzadeh and Vahdani, 2017). Recently Shahabi et al. (2014) demonstrated how accounting for demand correlation in safety stock cost can change the location–allocation decisions of three-echelon supply chain. However, the impact of demand correlation on daily warehouse inventory planning and plant production decisions requires further investigation. For instance, demand correlation can impose restrictions on order quantity and plant production decisions since both warehouse and plants are operating based on a limited capacity. Therefore, ignoring the existence of demand correlation would lead to significant changes in key decisions of an integrated supply chain. The goal of this work is to examine the impact of demand correlation on production, inventory, location and allocations decisions of a multi-echelon supply chain simultaneously. In particular, we consider a three level supply chain comprising of plants, warehouses, and retailers. The retailers have uncertain and correlated demand for multiple products. A mixed integer nonlinear formulation is developed to determine: (i) locations of multiple capacitated plants and warehouses, (ii) allocation of plants to warehouses and warehouses to retailers, (iii) optimal inventory levels and safety stock of inventories of multiple products at warehouses, and (iv) total amount of production by each plant. An outer approximation algorithm (OA) is proposed and customized to handle the mixed integer nonlinear formulation and deliver globally optimal solutions. This article also identifies two specific cases where the optimal order quantity is obtained analytically, resulting in significant computational savings. 2. Literature review Daskin et al. (2002) were among the first to study the uncapacitated joint location–inventory problem. A nonlinear integer programming formulation was provided to minimize location, transportation, and inventory costs and solved using several Lagrangian relaxation-based heuristics. Shen et al. (2003) adopted a set covering formulation for Daskin et al. (2002) joint location–inventory framework and provided a column generation algorithm for two specific cases with demand variance either proportional to the mean or zero. Shen (2005) extended Daskin et al. (2002) to consider multiple commodities. Shu et al. (2005) further developed the Shen et al. (2003) column generation algorithm for general cases of demand variance. Snyder et al. (2007) studied a scenario based stochastic variant of Daskin et al. (2002). The authors focused on a specific case where the ratio of variance to the mean of demand at each retailer was the same for each scenario and solved the problem using a Lagrangian relaxation heuristic embedded in a branch and bound scheme. Shu and Sun (2006) adopted a column generation approach to solve the scenario based stochastic variant for the general case. Miranda and Garrido (2004), Miranda and Garrido (2006), and Ozsen et al. (2008) studied the capacitated variant of Daskin et al. (2002) and Shen et al. (2003) with subtle differences in the way capacity constraints were enforced. Miranda and Garrido (2004) considered throughput capacity only. Miranda and Garrido (2006) considered two types of capacity constraints – maximum order quantity and maximum inventory levels. Ozsen et al. (2008) had the most conservative capacity restriction enforcing the sum of order quantity, safety stock, and lead time demand to be less than the capacity of the warehouse. Ozsen et al. (2009) studied the multiple sourcing variant of Ozsen et al. (2008). All four papers – Miranda and Garrido (2004), Miranda and Garrido (2006), Ozsen et al. (2008), and Ozsen et al. (2009) – adopted a Lagrangian relaxation heuristic solution method. An improved Lagrangian relaxation heuristic was later introduced by Diabat et al. (2015) for handling larger size networks. Similarly, Puga and Tancrez (2017) developed a heuristic for solving large scale joint inventory location problems. All the works mentioned above assumed independent retailer demand. Atamtürk et al. (2012) used a mixed integer conic quadratic framework to solve the joint location–inventory problem with demand correlations efficiently. This paper focuses on a three level supply chain network whereas Atamtürk et al. (2012) studied a two level supply chain network. Other researchers have extended the two level joint location–inventory problem to a multiple echelon setting (Vidyarthi et al., 2007; Park et al., 2010; Shahabi et al., 2013). Tancrez et al. (2012) adopted a continuous approach and developed an approximation heuristic to solve the three level joint location–inventory problem. Vidyarthi et al. (2007) proposed a nonlinear integer formulation for the multiple product three level production–inventory–location problem with stochastic demands. This paper differentiates itself from Vidyarthi et al. (2007) in two aspects. First, we consider correlation in retailer demands. Second, Vidyarthi et al. (2007) used a piecewise linear approximation of the nonlinear safety stock costs and solved the resulting mixed integer linear approximation using Lagrangian relaxation whereas we adopt an outer approximation algorithm which guarantees global optimality. Park et al. (2010) studied a three level supply chain network with independent retailer demand and used a Lagrangian relaxation-based heuristic. Shahabi et al. (2014) extended the formulation of Park et al. (2010) by considering demand correlations and used an outer approximation algorithm to solve the model globally. The problem studied in this paper is a generalization of the model studied in Park et al. (2010) and Shahabi et al. (2014) in the following ways: •
•
Our formulation determines the optimal production at each plant making it a joint production–location–inventory problem. The proposed formulation considers multiple products with retailer demand correlations and capacities at plants.
62 •
•
•
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
Daily capacity constraints are considered which restrict the sum of order quantity, safety stock, and lead time demand to be less than the capacity of the warehouse, whereas Park et al. (2010) and Shahabi et al. (2014) only considered yearly throughput capacity for the warehouse. The capacity constraint considered in this work is more conservative and makes the problem harder to solve. The outer approximation (OA) framework used in this work is more general. The outer approximation in Shahabi et al. (2014) was proposed to only solve binary nonlinear program, whereas in this work we customize the algorithm to efficiently solve the mixed integer nonlinear programs. In the proposed solution framework, the OA nonlinear subproblems are efficiently solved by reformulating it as a conic quadratic program while in Shahabi et al. (2014) subproblems were solved through closed form equations. In Shahabi et al. (2014), the optimal order quantity is determined analytically. In this paper, we identify specific conditions where closed form equations exist for optimal order quantity to transform the nonlinear OA subproblem into a linear program.
All the works mentioned above use a variant of the continuous review (r, Q) inventory policy. In this policy, the inventory manager places an order of Q units when the inventory falls below a pre-specified level r. This study makes a similar assumption. Other researchers have adopted a periodic review (Berman et al., 2011), power of two (Uster et al., 2008; Keskin and Uster, 2012; Diabat et al., 2013), and an infinite horizon replenishment (Teo and Shu, 2004; Romeijn et al., 2007) based inventory policy in a two-echelon framework. While this paper focuses on retailer demand uncertainty and their correlations, we do not consider issues such as the impact of disruptions on supply chain performance or reliability (Masih-Tehrani et al., 2011; Friesz et al., 2011; Schmitt 2011). Readers are referred to Wang and Ouyang (2013), Chen et al. (2011), Peng et al. (2011), and Li and Ouyang (2010) for work on the impact of probabilistic disruptions on decision making in a joint location–inventory context. 3. Problem definition This section presents the mathematical programming formulation for the three level multi-product joint production– location–inventory problem, accounting for demand correlation among retailers. More precisely, we are looking for: •
• • •
The location of plants and warehouses of a multi-product three level supply chain comprising of plants, warehouses, and retailers. The allocation of retailers to located warehouses and located warehouses to located plants. The amount of production of each product in located plants. The optimal order quantity of each product in located warehouses.
The objective of this formulation is to minimize the facility location, transportation, and inventory costs while considering more than one product and demand correlation across retailers. This work makes the following assumptions: • •
• •
•
•
•
• • •
The model considers the production, transport, and storage of more than one product. Retailers’ demand for each product follows a multivariate normal distribution with a known mean vector and covariance matrix. There is no correlation between the demands of different products. Every plant and warehouse has a fixed and economically scaled setup cost for being opened. The production and transportation costs of one unit of each product from plants to warehouses and warehouses to retailers are proportional to the Euclidean distances. Each retailer is sourced only from a single warehouse. Single sourcing of retailers from the warehouses is an accepted assumption among supply chain practitioners. Due to capacity restriction on production of the plants, warehouses can be supplied from multiple plants. These assumptions are consistent with Vidyarthi et al. (2007). The inventory at each warehouse is managed using a continuous review (r, Q) inventory policy, i.e., each warehouse orders a fixed quantity Q from its supplier, once the inventory level at that warehouse falls below a reorder point r. Each warehouse holds a one-time safety stock over the planning horizon η to compensate for demand variations at retailer’s side. Each product at each warehouse has a lead time. Back order is not allowed in this model. Instead, we use a level of service to capture the probability of stocking out. Plants have production capacities over the planning period, and each warehouse has a daily capacity for holding inventories of different products. These assumptions are consistent with Vidyarthi et al. (2007).
The model indices, parameters, and variables used throughout the paper are specified below: Indices r, l index for retailers (1, ……, R) w index for warehouses (1, ……, W) p index for plants (1, ……, P) m index for products (1, ……, M) Parameters gp fixed setup cost for plant p ($)
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
fw tcwpm crwm Awm hwm CWw CPp RLwm Lwm
nα
fixed setup cost for warehouse w ($) per-unit production and transportation cost of product m between plant p and warehouse w ($) per-unit transportation cost of product m between warehouse w and retailer r ($) per-order fixed inventory ordering cost of product m at warehouse w ($) per-unit per year inventory holding cost of product m at warehouse w ($) daily capacity for warehouse w capacity for plant p over the planning period optimal reorder level of product m at warehouse w order lead time of product m at warehouse w (day) mean daily demand of product m at retailer r variance of daily demand of product m at retailer r correlation coefficient between daily demands of product m at retailer r and l number of working days per planning period α -percentile of the standard normal distribution
Variables Dwm Uwm ∗ Qwm SSwm
mean daily demand of product m at warehouse w covariance of daily demand of product m at warehouse w optimal order quantity of product m at warehouse w safety stock level of product m at warehouse w
μrm 2 σrm ρ rlm η
63
Decision variables for optimization formulation vp ∈ {0,1} takes value 1 if a plant is located at p zw ∈ {0,1} takes value 1 if a warehouse is located at w ywpm ≥ 0 amount of product m shipped from plant p to warehouse w xrw ∈ {0,1} takes value 1 if retailer r is assigned to warehouse w Qwm ≥ 0 inventory order quantity of product m at warehouse w The daily demand of product m at warehouse w follows a multivariate normal distribution with mean Dwm = r μrm xrw 2 2 and variance–covariance Uwm = r σrm xrm + σrm σlm xrw xlw . Considering the correlation coefficient ρ rlm = 1 for r=l l=r ρrlm l = r the covariance equation can be rewritten as Uwm = l r ρrlm σrm σlm xrw xlw . Therefore, the demand for product m at warehouse j during the lead time is also normally distributed with mean Dwm Lwm and covariance Uwm Lwm . Assuming that the probability of stock out p(Dwm Lwm ≤ RLwm ) = 1 − α is fixed for product m at warehouse w, the optimal reorder level (RLwm ) and the safety stock level (SSwm ) can be determined as (Park et al., 2010; Miranda and Garrido, 2004)
RLwm = Dwm Lwm + nα SSwm = nα
Uwm Lwm
(1)
Uwm Lwm
(2)
where α determines the service level, or equivalently the maximum probability of stock-out during the lead time, and nα is the α -percentile of the standard normal distribution. In this work, similar to Park et al. (2010) and Miranda and Garrido, (2004), we assume the service level α is fixed and the decisions are optimized for a given probability of stock out. Based on the above notation and equations, the three level multi-product joint production–location–inventory problem with retailer demand correlation can be formulated as follows: P1:
Min
g pv p +
p
+
w
m
+
f w zw +
w
m
m
hwm nα
Lwm
l
hwm
w
p
t cwpm ywpm + η
w
m
w
crwm μrm xrw
r
ρrlm σrm σlm xrw xlw
r
Qwm Awm η r μrm xrw + 2 Qwm
(3)
subject to:
xrw = 1,
∀r
(4)
w
m w
p
∀p
ywpm ≤ C Pp v p ,
ywpm = nα
Lwm
l
r
ρrlm σrm σlm xrw xlw + η
(5) r
μrm xrw
∀ m, w
(6)
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M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
Qwm +
m
nα
Lwm
m
+
m
l
ρrlm σrm σlm xrw xlw
r
μrm xrw Lwm ≤ CWw zw
∀w
(7)
r
Qwm, ywpm ≥ 0 , xrw , zw ,
∀ m, p, w, r
v p ∈ {0, 1},
(8)
The objective function corresponding to Eq. (3) minimizes the total cost of locating plants and warehouses, producing multiple products in plants, transportation of products from plants to warehouses and from warehouses to retailers, and holding safety stocks and inventories plus ordering costs in warehouses over the planning period η. Eq. (4) ensures that each retailer will be assigned to exactly one warehouse. Eq. (5) states that the amount of different products produced in each open plant is less than or equal to its capacity. Eq. (6) ensures that the amount of products produced in different plants and shipped to each warehouse for each product is equal to a one time safety stock plus the demand of that product at the associated warehouse over the planning period. Eq. (7) specifies that each warehouse (if it is opened) has the capacity of stocking order quantities, safety stocks, and the demand during the lead time for all products. Constraints (8) are standard non-negativity and integrality constraints. In several works in the joint location–inventory literature with continuous review (r, Q) inventory policy, researchers obtained analytical expressions for the optimal order quantity using the Karush–Kuhn–Tucker conditions. Depending on the assumptions made, the optimal order quantity is often shown to be equal to the economic order quantity (EOQ). The mathematical program can be marginally simplified to solve the remaining location and allocation decisions (Daskin et al., 20 02; Miranda and Garrido, 20 04; Shahabi et al., 2014). In this paper, the calculation of optimal order quantity is more complicated. ∗ Proposition 1. For each warehouse w , the general form of optimal order quantity Qw m for each product m is equal to:
2Aw m η r μrm xrw hw m + 2λw
Qw∗ m =
where λw ≥ 0 is the Lagrange multiplier corresponding to the capacity constraint of warehouse w (Eq. (7)). Note that this equation is equal to the associated EOQ if λw is set to zero. It is also worth mentioning that if the capacity constraint is binding then the optimal order quantity is always less than or equal to the corresponding EOQ. Proof. Given that Uw m = l r ρrlm σrm σlm xrw xlw for each warehouse w , we have the following set of equations for every warehouse:
min
Qw m Aw m η r μrm xrw hw m + 2 Qw m
m
subject to:
Qw m +
m
nα
Lw mUw m +
m
m
(9)
μrm xrw Lw m ≤ CWw zw
(10)
r
Qw m ≥ 0
(11)
Since the program is convex in Qw m , we use Karush–Kuhn–Tucker conditions for each product m as below:
Aw m η r μrm xrw hw m − + λw = 0, 2 Qw2 m
λw
Qw m +
m
Qw m ,
nα
Lw m Uw m +
m
m
μrm xrw Lw m − CWw zw = 0,
r
λw ≥ 0
If the capacity constraint is not binding, λw has to be equal to zero and the optimal order quantity is the corresponding EOQ. i.e.
Qw∗ m =
2Aw m η r μrm xrw hw m
(12)
If the capacity constraint is binding, λw can be either equal to zero or greater than zero. In this case, the optimal order quantity can be obtained as
Qw∗ m
=
2Aw m η r μrm xrw hw m + 2λw
(13)
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
65
Comparing Eqs. (12) and (13), we can conclude that if the capacity is binding the optimal order quantity is always smaller than EOQ. For certain specific cases, simplified analytical expressions can be determined for the optimal order quantity. For example, when the holding cost is independent of the product type, the optimal order quantity can be determined as shown below in Proposition 2. This can be the case when all the products managed by the supply chain have similar characteristics, for example, perishable or non-perishable. Proposition 2. For each warehouse w , if we assume that the holding cost is independent of product type, then the optimal order ∗ quantity for each product Qw m is equal to:
⎧ ⎨
min
2A
w m
⎩
η
r
μ x rm
rw
CWw zw −
,
hw m
m
nα
Lw mUw m −
m
m
r
⎫ μrm xrw Lw m ⎬
2Aw m η r μrm xrw 2Aw m η r μrm xrw
⎭
Proof. If the capacity constraint is not binding λw = 0 and the optimal order quantity is determined according to Eq. (12) presented in Proposition 1. If the capacity constraint is binding, the summation of optimal order quantities for each product m has to satisfy the capacity constraints for warehouse w . Since the holding costs are independent of the product type, for every warehouse w , we have: hw 1 = hw 2 = … = hw m . And, the following relation is also valid.
hw 1 + 2λw = hw 2 + 2λw = . . . = hw m + 2λw The above equalities are rewritten as below based on Eq. (13):
2Aw 1 η r μr1 xrw 2Aw 2 η r μr2 xrw 2Aw m η r μrm xrw = = ... = Qw2 1 Qw2 2 Qw2 m
Above equations can further be simplified as
2Aw 1 η
r
μr1 xrw
Qw 1
=
2Aw 2 η
r
μr2 xrw
= ... =
Qw 2
2Aw m η
r
μrm xrw
Qw m
Given the above relation and considering that the capacity constraint is binding, the following equation is derived:
Qw 1
2Aw 1 η r μr1 xrw 2Aw 2 η r μr2 xrw 2Aw m η r μrm xrw + Qw 1 + . . . + Qw 1 2Aw 1 η r μr1 xrw 2Aw 1 η r μr1 xrw 2Aw 1 η r μr1 xrw + nα Lw mUw m + μrm xrw Lw m = CWw zw m
m
r
The above equation is simplified by summing the terms containing Qw 1 , which yields the equation below:
Qw 1
m
2Aw m η r μrm xrw + nα Lw mUw m + μrm xrw Lw m = CWw zw 2Aw 1 η r μr1 xrw m m r
Therefore, the optimal order quantity for the first product is calculated as
Qw 1 =
CWw zw −
m
nα
Lw mUw m −
m
m
r
μrm xrw Lw m
2Aw m η r μrm xrw 2Aw 1 η r μr1 xrw
The general equation for the optimal order quantity for every product m is
Qw m =
CWw zw −
m nα
Lw mUw m −
m
m
r μrm xrw Lw m
2Aw m η r μrm xrw 2Aw m η r μrm xrw
Therefore, the optimal order quantity is calculated based on the minimum of the amount of two cases and the proof is complete. Similarly, if the number of orders that can be received by the warehouse for each product is constrained to be the same, then the optimal order quantity for each product can be analytically determined as shown in Proposition 3.
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Proposition 3. For each warehouse w , if we assume that the number of orders during a year for each product is the same, then ∗ the optimal order quantity for each product Qw m is equal to:
⎧ 2 ( m Aw m ) ⎪ ⎨ η μrm xrw , m hw m η r μrm xrw r
min
⎪ ⎩
CWw zw −
η
T Sw =
r
m
μr1 xrw
Qw 1
nα
√
Lw mUw m − m r μrm xrw Lw m η m r μrm xrw
=
η
r
μr2 xrw
Qw 2
= ... =
η
(η
r
r
μrm xrw )
μrm xrw
Qw m
i f xr w = 1
Proof. First, if the capacity is not binding, the optimal order quantity for each product m is calculated based on the following minimization problem:
Min
hw m
m
subject to:
TS
w
η
=
r
Qw m Aw m η r μrm xrw + 2 Qw m
μr1 xrw
Qw1
=
η
r
μr2 xrw
Qw 2
(14)
= ... =
η
r
μrm xrw
(15)
Qw m
Qw m ≥ 0
(16)
In this case, the above constrained minimization problem can be reduced to an unconstrained minimization problem presented below:
Qw 1 hw 1 2
Min f =
η r μr1 xrw Aw 1 η r μr1 xrw + η r μr1 xrw Qw 1
Qw 1 + hw 2 2
η r μr2 xrw Aw 2 η r μr2 xrw η r μr1 xrw + η r μr1 xrw Qw 1 η r μr2 xrw
Qw 1 η r μrm xrw Aw m η r μrm xrw η r μr1 xrw + . . . + hw m + 2 η r μr1 xrw Qw 1 η r μrm xrw
Qw 1 η r μr1 xrw Aw 1 η r μr1 xrw = hw 1 + 2 η r μr1 xrw Qw 1
Qw 1 η r μr2 xrw Aw 2 η r μr1 xrw + hw 2 + 2 η r μr1 xrw Qw 1
Qw 1 η r μrm xrw Aw m η r μr1 xrw + . . . + hw m + 2 η r μr1 xrw Qw 1
Q η r μr1 xrw w1 = hw 1 η μr1 xrw + . . . + hw m η μrm xrw + ( Aw 1 + . . . + Aw m ) 2η r μr1 xrw Qw 1 r r Based on the first order condition, the optimal value for Qw 1 is computed as
∂f
∂ Qw 1
( hw 1 η
μr1 xrw + . . . + hw m η r μrm xrw ) η r μr1 xrw − ( Aw 1 + . . . + Aw m ) 2η r μr1 xrw Qw2 1 (η r μr1 xrw )( m Aw m ) m hw m η r μrm xrw = − =0 2η r μr1 xrw Qw2 1 =
r
Since the functions having a general form of
(η
r
μr1 xrw )(
first product is calculated as below:
Qw 1 =
η
μr1 xrw
r
Q w 1
m Aw m )
is convex over Qw 1 , the optimal order quantity for the
2 ( m Aw m ) m hw m η r μrm xrw
The optimal order quantity for any product m can thus be derived as
Qw m =
η
r
μrm xrw
2 ( m Aw m ) m hw m η r μrm xrw
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
67
Second, if the capacity is binding, the capacity constraint has to be satisfied. Given that relation (15) holds between the optimal order quantities of different products, the following relation is derived:
Qw 1
η r μr1 xrw η r μr2 xrw η μrm xrw + Q + . . . + r Q η r μr1 xrw η r μr1 xrw w 1 η r μr1 xrw w 1 + zα Lw mUw m + μrm xrw Lw m = CWw zw m
m
r
The above equation is simplified as below:
η
Q w1 r
μr1 xrw
η μrm xrw + nα Lw mUw m + μrm xrw Lw m = CWw zw m
r
m
m
r
Thus, the optimal order quantity for the first product is calculated as:
CWw zw −
Qw 1 =
m
nα
Lw mUw m − m r μrm xrw Lw m (η r μr1 xrw ) η m r μrm xrw
The optimal order quantity for any product m can thus be derived as:
Qw m =
CWw zw −
m
nα
Lw mUw m − m r μrm xrw Lw m (η r μrm xrw ) η m r μrm xrw
Similar to Proposition 2, the optimal order quantity is calculated based on the minimum of the amount of two cases which concludes the proof. 4. Solution method The mathematical programming formulation (P1) due to the nonlinear terms in its structure belongs to the family of mixed integer nonlinear programs (MINLPs). Outer approximation (OA) is an important method for solving MINLPs (Duran and Grossman, 1986). OA belongs to the family of decomposition and cutting plane methods, separating the original MINLP into a nonlinear program (NLP) subproblem, and a mixed integer linear program (MILP) master problem with sub-gradient based generated cuts approximating the nonlinear functions. More specifically, the nonlinear subproblem is achieved by fixing the integer variables while the MILP master problem is constructed by substituting the nonlinear convex function by their first order linear or in some cases second order quadratic approximation. The solution of the master problem provides the algorithm with the lower bound whereas the upper bound is achieved by solving the subproblem. The algorithm iterates between the solution of the subproblem and the master problem until the gap (absolute/relative) between the upper and the lower bound meets the convergence criterion. The big advantage of the OA method is that given the convexity of the integer relaxed formulation of the initial MINLP, the OA algorithm will converge to the global optimal solution (Bonami et al., 2008). In other words, if the resulting formulation after relaxing the integer variables to the corresponding continuous variables is convex, OA will provide the optimal solution (Duran and Grossman, 1986 and Fletcher and Leyffer, 1994). Theorem 1. The original MINLP P1 is a convex MINLP. Proof. In order to prove that formulation P1 is a convex MINLP, we need to prove that the continuous relaxation of problem P1 is convex (Bonami et al., 2008). In order to prove the convexity of continuous relaxation of model P1, it suffices to prove that the holding safety stock cost and ordering cost terms in Eq. (3) are convex, since the other parts in this equation are the summation of linear terms which are obviously convex. First, we need to show that the safety stock cost r l ρrlm σrm σlm xrw xlw Lwm is convex. Since the covariance matrix is positive semi-definite, the term can be written as:
r
xrw vrlm vlrm xlw Lwm
l
where vrlm is the rl entity of the triangular matrix V for product m obtained from the Cholesky decomposition of the covariance matrix denoted as Q = VVt . The above term can then be rewritten as the Euclidian norm of a vector X:
r
l
xrw vrlm vlrm xlw Lwm = X V
where X = ( Lwm x1w , . . . . . . , Lwm xiw ). It is well known in the convex optimization literature that the second norm functions are convex (Agrali et al. 2012; Shahabi et al., 2014). For the ordering cost term, we can replace the binary (0, 1) variable xrw with the twice differentiable term x2rw which enables us to compute the corresponding Hessian matrix.
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M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
Therefore, it is sufficient to prove that the hessian matrix of f = be determined as:
⎡
⎤ ⎡ 2 ∂2 f 2xrw 3 ∂ Qmw ∂ xrw ⎥ = ⎢ Qmw ⎦ ⎣ 2 −2 xrw ∂ f 2 2 Qmw ∂ xrw
∂2 f 2 ⎢ ∂ Qmw H=⎣ 2 ∂ f ∂ xrw ∂ Qmw
x2rw Qmw
is positive semidefinite. The Hessian matrix of f can
⎤
−2xrw 2 ⎥ Qmw 2 ⎦ Qmw
A symmetric matrix is positive semidefinite if we have zt Hz ≥ 0 for any arbitrary nonzero real vector of z. Assuming that z = [a b]t and xrw , Qwm ≥ 0, we have:
⎡
2x2rw ⎢ 3 zt Hz = [a b]⎣ Qmw −2xrw 2 Qmw
⎤
−2xrw ax 2 2 4abxrw 2a2 x2rw 2b2 2 ⎥ rw t Qmw − + = − b ≥0 [a b] = ⎦ 3 2 2 Qmw Qmw Qmw Qmw Qmw Qmw
Therefore, H is a positive semidefinite matrix and the proof is complete.
The OA subproblem and master problem are described next. 4.1. Outer approximation subproblem The OA subproblem (OA-SP) as discussed earlier is achieved by fixing the integer variables in the original MINLP formulation. The goal of the subproblem is to optimize the continuous variables, which are then used to derive the linear approximations in the master problem. Therefore, assuming that x¯hrw and x¯hlw represent the assignment of the retailers to the h = h h warehouses, the demand variance for warehouse w and commodity m at iteration h, U¯ wm r l ρrlm σrm σlm x¯rw x¯ . Furlw
h, v ¯ hp as the location of warehouses and plants at iteration h, the OA-SP can be formulated as the following thermore, given z¯w nonlinear program: OA SP-P1:
F = Min
m
p
t cwpm ywpm +
w
m
w
Qwm Awm η r μrm x¯hrw hwm + + M1 b1 + M2 b2 2 Qwm
(17)
subject to:
m w
p
m
ywpm ≤ C Pp v¯ hp + b1 ,
ywpm = nα Qwm +
m
nα
h LwmU¯ wm +η
h LwmU¯ wm +
Qwm, ywpm , b1 , b2 ≥ 0
∀p r
(18)
μrm x¯hrw ,
m
r
∀ m, w
μrm x¯hrw Lwm ≤ CWw z¯wh + b2 ,
(19)
∀w
∀ m, p, w
(20) (21)
The goal of this NLP formulation is to find the optimal values for the continuous variables ywpm and Qwm . The optimal values for the continuous variables will then be passed to the master problem. Note that the NLP subproblem has to be solved to optimality which is sometimes computationally expensive. In addition, we consider two positive variables b1 and b2 to make sure that the subproblem always provides a solution. In the above formulation, M1 and M2 are large positive constants (Fletcher and Leyffer, 1994). With simple reformulations as explained below, OA SP-P1 can be reformulated as a conic quadratic program (QCP).
2 2 ∗ Qwm Twm Qwm Awm η r μrm x¯hrw 2 ∗ Awm η r μrm x¯hrw 2 + ≤ Twm ⇔ Qwm + ≤ 2 Qwm hwm hwm 2 ⇔ Qwm +
2 2 ∗ Qwm Twm 2 ∗ Awm η r μrm x¯hrw ≤ hwm hwm
⇔ (Qwm − Twm )2 +
2 2 ∗ Awm η r μrm x¯hrw ≤ (Twm )2 hwm
One advantage of such a reformulation is that the optimization solvers such as CPLEX and MOSEK are now offering efficient methods for solving QCP and MIQCP through interior point based algorithms. Solving the following program would yield the outer approximation upper bound (UB).
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
OA SP-P2:
Min
m
p
t cwpm ywpm +
w
m
69
hwm Twm + M1 b1 + M2 b2
(22)
w
subject to:
m w
p
m
∀p
ywpm ≤ C Pp v¯ hp + b1 ,
ywpm = nα Qwm +
m
nα
h LwmU¯ wm +η
(Qwm − Twm )2 +
h + LwmU¯ wm
2∗Awm η
r
r
(23)
μrm x¯hrw
m
μrm x¯hrw
∀w
(25)
∀ m, w
≤ (Twm )2
hwm
(24)
μrm x¯hrw Lwm ≤ CWw z¯wh + b2
r
2
∀ m, w
(26)
∀ m, p, w
Qwm, ywpm , b1 , b2 ≥ 0
(27)
The conic variant of the OA subproblem can be solved using optimization solvers in order to provide the optimal value for order quantity Qwm and amount of production ywpm at each iteration of the algorithm. But, there are specific cases as we already discussed in Propositions 2 and 3 where the optimal order quantity for every warehouse and product can be analytically calculated. Considering the special cases of Propositions 2 and 3 would reduce the nonlinear OA subproblem to a linear program where the optimal values for variable ywpm at each iteration of the subproblem are calculated. 4.2. Outer approximation master problem The outer approximation master problem (OA-MP) at each iteration is constructed by accumulating sub-gradient based h at iteration generated cuts to approximate the convex nonlinear functions. In particular, given the values for x¯hrw and Q¯ wm h, Theorem 2 provides details on how to derive the first order linear approximation of the nonlinear terms of the original master problem P1. h Theorem 2. Let Q¯ wm denotes the optimal solution for the nonlinear sub-problem and x¯hrw a feasible assignment of retailers to the
η
A
warehouses at iteration h. The nonlinear terms wm outer approximation cuts (28) and (29), respectively.
h 2 Awm rm x¯rw r +2 × h Q¯ wm h Awm rm x¯rw r h − Qwm − Q¯ wm 2 Q¯ hwm
Awm η
μ
η
r
+
μ
η
ρil σi σl Lwm xˆhrw
l
ρil σi σl Lwm xˆhlw
r
r
μrm x¯hrw
2
and
Qwm
xrw − x¯hrw
LwmUwm can be approximated by the following first order
≤ tˆ1hmw − (t1mw − tˆ1hmw )
xlw − xˆhlw
μ
h rm x¯ rw r h Q¯ wm
∀ m, w, h
(28)
xrw − xˆhrw
≤ 2tˆ2hmw (t2mw − tˆ2hmw )
∀ m, w, h
(29)
l
Proof. Consider F (xrw , t1mw ) =
Awm η
μrm x2rw
− t1mw as a function where its continuous relaxation is convex. Assume a feasible assignment point of (xˆhrw , tˆ1hmw ) at iteration h, Eq. (28) can be derived from sub-gradient inequality as below:
F xhrw , t1hmw + ∇ F xhrw , t1hmw
T
r
Qwm
xrw − xhrw ≤ F (xrw , t1mw ) ≤ 0 t1mw − t1hmw
(30)
Therefore, by taking the gradient according to the above Eq. (29) is obtained. function, Eq. (29) can be achieved by setting F (xrw , t2mw ) = r l Lwm ρrlm σrm σlm xrw xlw − t2mw , the linear approximation provides
r
l
Lwm ρrlm σrm σlm xrw xlw +
2
Lwm ρrlm σrm σlm xlw
r l r
l Lwm ρrlm σrm σlm xrw xlw
(xrw − xˆhrw )
Lwm ρrlm σrm σlm xrw + r l (xlw − xˆhlw ) − tˆ2hmw − (t2mw − tˆ2hmw ) ≤ 0 2 r l Lwm ρrlm σrm σlm xrw xlw
∀r, w, h
(31)
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M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
Furthermore, given the optimal solution of tˆ2hmw =
r
h h l Lwm ρrlm σrm σlm x¯rw x¯lw
at every iteration h, Eq. (31) can be
reduced to Eq. (32):
2
Lwm ρrlm σrm σlm xlw
r l
(xrw − xˆhrw ) Lwm ρrlm σrm σlm xrw xlw Lwm ρrlm σrm σlm xrw + r l (xlw − xˆhlw ) − (t2mw − tˆ2hmw ) ≤ 0 2 r l Lwm ρrlm σrm σlm xrw xlw r
l
Also, since tˆ2hmw =
r
the proof is complete.
r
+
ρil σi σl Lwm xˆhrw
h h l Lwm ρrlm σrm σlm x¯rw x¯lw
l
ρil σi σl Lwm xˆhlw
xlw − xˆhlw
∀r, w, h
(32)
and tˆ2hmw = 0, Eq. (33) is achieved by multiplying Eq. (32) by tˆ2hmw and
xrw − xˆhrw
≤ 2tˆ2hmw (t2mw − tˆ2hmw )
∀m, w, h
(33)
r
l
QED Finally, given the above linear approximations, the master problem for OA can be formulated as below: OA-MP:
LBh = Min
g pv p +
p
+
m
w
p
m
w
ywpm = nα t2mw + η Qwm +
m
hwm nα t2mw +
subject to:
m
nα t2mw +
r
μrm xrw ,
m
r
μ
η
l
+
ρil σi σl Lwm xˆhrw
r
ρil σi σl Lwm xˆhlw
r
μ
h rm x¯rw r h Q¯ wm
(34)
∀ m, w
xrw − x¯hrw
(35)
∀w
(36)
≤ tˆ1hmw − (t1mw − tˆ1hmw )
xlw − xˆhlw
t cwpm ywpm
w
μrm xrw Lwm ≤ CWw zw η
μ
p
Qwm hwm + t1mw 2 m w
h 2 Awm rm x¯rw r +2 × h Q¯ wm h Awm rm x¯rw r h − Qwm − Q¯ wm 2 h ¯ Q wm
Awm η
f w zw +
∀ m, w, h
(37)
xrw − xˆhrw
≤ 2tˆ2hmw (t2mw − tˆ2hmw )
∀ m, w, h
(38)
l
LBh ≤ UB
∀h ∈ H
t1mw , t2mw ≥ 0
∀ m, w
(39) (40)
Eqs. (4), (5) and (8) Note that Eq. (39) ensures that the generated lower bound at iteration h is less than the algorithm’s upper bound. The complete OA algorithm is outlined below in Fig. 1. 5. Computational experiments Two sets of numerical experiments are conducted in order to study the performance of the model and the solution methodology on varying size supply chain networks. The first set of experiments study the impact of neglecting demand correlation in a production–location–inventory problem setting. The second set of experiments investigates the computational efficiency of the OA algorithm. The OA algorithm was implemented in the GAMS/CPLEX platform. The computational tests are conducted on a 3.4 GHz Dell Optiplex 990 Pentium i7-2600 computer with the 64-bit version of the Windows 7 operating system with 8 GB RAM.
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
71
Fig. 1. Outer approximation steps. Table 1 Data generation. Mean daily demand of retailer r for product m Daily demand variance of retailer r for product m Warehouse capacity (daily capacity)
μrm = U[50, 300] 2 σrm = U[0, 100] μrmCW0 CWw = βw1
Plant capacity (capacity over the period η)
C Pp = ηβ p2
Fixed cost of locating plant ($) Fixed cost of locating warehouse ($) Per unit per year holding cost ($) Per unit per year ordering cost ($) Order lead time of product m at warehouse w (day)
gp = U[50 0, 10 0 0]+U[10 0 0, 20 0 0] C Pp √ gw = U[50 0, 10 0 0]+U[10 0 0, 20 0 0] CWw 1 hwm = γwm . U (250, 500) 2 . U (150, 300 ) Awm = γwm lmw = U[1, 5]
r,m
r,m
μrm
Table 2 Test networks. Problem instance
Total number of potential plants (P)
Total number of potential warehouses (W)
Total number of retailers (R)
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
5 5 5 7 7 7 10 10 10 10
5 5 10 10 15 15 15 15 20 20
15 20 20 20 20 40 20 40 40 60
5.1. Experiment setup In this paper, ten increasing size supply chain networks comprising of layers of plants, warehouses and retailers are selected as the test problems (see Table 2). The parameters of the supply chain are randomly generated similar to Vidyarthi et al. (2007). First, the candidate locations for plants, warehouses, and retailers are randomly selected from a square of (0, 1]. The production and transportation costs from plant to warehouse and from warehouse to retailer are assumed to be proportional to the Euclidean distance between the generated locations. The production and transportation costs (per unit distance and unit demand) from plants to warehouses and warehouses to retailers are assumed to be US$ 1 and β 2 are used to adjust the ratio of capacity to the demand. The parameter η is the 8.00. The random parameters βw p 1 and γ 2 number of working days per year and is equal to 250 days. γwm wm control the level of holding and ordering costs. 1 2 1 For the base case, γwm , γwm and βw are randomly generated between 0.7 and 1. β p2 is randomly generated between 2 and 10. CW0 denotes the base warehouse capacity, which is fixed at 8 units. Three products were considered across the entire numerical study. Tables 1 and 2 present additional information regarding the data generation and the test problems.
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M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78 Table 3 Percentage of feasible instances and savings for ρ = 0.1, ρ = 0.3. Test problem
ρ = 0.1
ρ = 0.3
Savings
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Savings
Ave
Max
Feasible instances (%)
Ave
Max
Feasible instances (%)
0.60 1.55 0.91 0.83 1.43 2.81 1.00 1.42 0.73 0.22
5.45 7.23 15.22 2.00 13.26 8.78 6.07 1.79 2.61 25.07
80 80 76 76 76 68 68 68 64 60
2.96 3.07 1.64 1.54 1.88 2.49 1.78 1.31 1.82 1.56
3.65 0.40 4.23 6.73 11.23 2.71 3.89 12.68 9.44 11.50
68 64 60 60 60 60 60 60 40 40
5.2. Impact of correlation on key decisions of a supply chain The first part of the numerical analysis section studies the impact of retailers’ demand correlation on the key decisions of a supply chain. In particular, our goal is to derive managerial insights regarding decisions such as the warehouse and plant locations as well as the amount of production and order quantity on a supply chain with correlated demand. Toward this end, we first focus on how neglecting demand correlation can increase the total costs of a production–location–inventory system and in some cases even results in a supply chain structure which is unable to handle demand uncertainty. Then, we focus on how order quantity and production at plants are subject to change with demand correlation. Moreover, by comparing the solution of a supply chain network with and without demand correlation, we show how the plants and warehouses locations can be different, and finally the impact of different levels of demand correlation on the layout of a supply chain is demonstrated. Across the tests conducted in this section, the OA convergence criterion (the relative difference between the achieved upper bound and lower bound) is set at 1%. The purpose of the first part of the numerical tests in this section is to demonstrate that ignoring demand correlation in a production–inventory–location problem may increase the total cost of the supply chain, or even lead to infeasible solutions. The test instances in this section are solved with and without considering demand correlation. In order to calculate the value of capturing the demand correlation within the structure of the supply chain, we evaluate the solution of the uncorrelated model within the model with demand correlation. Two scenarios are possible: (i) the solution of the uncorrelated model may be infeasible in the correlated model. This is due to the fact that the solutions of uncorrelated model may not satisfy the capacity restriction on plants or warehouses. (ii) The solution of the uncorrelated model is feasible in the correlated model. In this case, the costs of the correlated demand under the solution of the uncorrelated problem is calculated and the relative difference of these two costs shows the savings achieved as a result of considering demand correlation within supply chain network. Mathematically speaking, savings are calculated according to the following relation:
Savings(% ) := 100 ×
Ob jCorrelated (Uncorrelated Solution ) − Ob jCorrelated Ob jCorrelated (Uncorrelated Solution )
In this part, five different correlation coefficients (0.1, 0.3, 0.5, 0.7, and 0.9) are considered. It is also assumed that at each level of correlation any two retailers have the same correlation coefficient. Each supply chain problem instance is solved for twenty five randomly generated model parameters. Tables 3 and 4 show the percentage of feasible instances as well as their associated savings for each test problem. We report the average and maximum savings in costs as a result of following a correlated demand in the retailers and percentage of feasible instances of correlated model under the solution of the model without correlation. The results clearly show that with increase in correlation coefficients the percentage of the feasible instances across the test problems would decrease. This can be attributed to the fact that with increase in correlation coefficient, the safety stock cost increases which reduces the available capacity for inventories at each warehouse. Also, with increase in correlation coefficient, the savings tend to increase. This is mostly because of the fact that with increase in correlation factor, the share of safety stock costs in comparison to the other costs would increase. The results presented here clearly show that total cost of a supply chain is highly coupled to the underlying assumptions on demand uncertainty, as ignoring the correlation in retailers’ demand would result in escalations in costs. This part of the numerical section is designed to examine the impact of demand correlation on the plants’ production and the order quantity of the warehouses which are key decisions related to the operation of a supply chain. Our experiments are conducted on three test networks of P1, P5, and P10, and we show how the average production level per plant and order quantity per warehouse can change with an increase in correlation level. For this test, in addition to the uncorrelated case, we consider five correlation levels of 0.1, 0.3, 0.5, 0.7, and 0.9. The results are presented in Figs. 2 and 3. As the results show, both the order quantity and the production levels tend to change with an increase in demand correlation coefficient. In particular, the inventory level decreases and the production level increases with an increase in
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
73
Table 4 Percentage of feasible instances and savings for ρ = 0.5, ρ = 0.7 and ρ = 0.9. Test problem
ρ = 0.5
ρ = 0.7
Savings
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
ρ = 0.9
Savings
Savings
Ave
Max
Feasible instances (%)
Ave
Max
Feasible instances (%)
Ave
Max
Feasible instances (%)
3.76 1.21 1.48 0.09 2.13 1.47 1.03 2.33 4.10 3.24
0.08 7.15 2.89 9.37 8.29 11.84 9.44 5.46 11.36 8.16
44 44 44 44 40 32 28 28 24 20
0.59 2.56 1.06 1.94 3.09 1.55 3.14 3.17 3.57 6.74
2.49 8.52 11.65 10.43 14.51 5.61 0.82 2.44 6.43 1.39
36 36 36 32 32 32 28 24 24 20
1.38 3.57 3.08 3.32 5.18 3.74 3.94 2.83 3.93 4.36
8.95 4.59 6.41 5.52 6.50 4.22 4.05 8.09 10.03 0.17
32 32 32 28 28 24 24 24 12 12
P1
Order Quantity per Warehouse
3000
P5
P10
2500 2000 1500 1000 500 0
0
0.1
0.3
0.5
0.7
0.9
Correlation level Fig. 2. Impact of correlation on inventory level per warehouse.
P1
P5
P10
4000000
Production Level per Plant
3500000 3000000 2500000 2000000 1500000 1000000 500000 0
0
0.1
0.3
0.5
0.7
0.9
Correlation Level Fig. 3. Impact of demand correlation on production level per plant.
correlation level. The increase in production level per each plant is attributed to the fact that with an increase in demand correlation, we require keeping more safety stock at each warehouse which would increase production margins of plants. On the other hand, since more safety stock is stored, less quantity is ordered due to the limited daily capacity of a warehouse. The results also confirm that the difference between productions and order quantities of the uncorrelated and correlated case would increase with an increase in correlation coefficient which is expected. We can thus conclude that both inventory and order quantity which are highly critical for the operation of a supply chain can significantly be affected by retailers’ demand correlation and their optimal values in the correlated case are significantly different from the uncorrelated case. The next part looks specifically into the structure of a supply chain and shows how accounting for demand correlation leads to changes in the locations of plants and warehouses. Table 5 reports the location of plants and warehouses for an arbitrary case where the correlation coefficient ρ is equal to 0.5. For comparison purposes, we are also reporting the loca-
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M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78 Table 5 Location of plants and warehouse for ρ = 0.5 and ρ = 0.0. Test instance
Correlation coefficient ρ = 0.5
Correlation coefficient ρ = 0.0
Plant location
Warehouse location
Plant location
Warehouse location
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
P4,P5 P3,P4 P1,P4 P5,P7 P2,P7 P2,P5 P1,P4 P2,P5 P9,P10 P2,P4,P5
W1,W3,W5 W1,W2,W5 W1,W3,W4 W2,W4,W9 W8,W10,W13 W1,W8,W12,W14 W1,W4,W10 W3,W7,W8,W11 W1,W8,W12,W14 W8,W10,W11,W15,W16
P4,P5 P3,P4 P2,P4 P1,P7 P3,P7 P2,P4 P3,P4 P4,P5 P6,P10 P4,P9
W1,W3,W4 W2,W4,W5 W1,W4,W5 W2,W3,W6 W8,W12,W14 W8,W12,W14,W15 W5,W10,W11 W4,W7,W9,W11 W1,W8,W12,W14 W8,W10,W11,W14
Table 6 Solution time of OA algorithm for correlation coefficients ρ = 0.1 and ρ = 0.3. Test problem
ρ = 0.1
ρ = 0.3
Time (s)
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Iteration
Gap
Time (s)
Iteration
Gap
Min
Ave
Max
Min
Max
Ave(%)
Min
Ave
Max
Min
Max
Ave(%)
5.23 7.67 8.45 15.73 36.57 62.24 76.66 74.31 91.29 108.18
26.99 27.09 35.14 43.05 77.8 48.64 124.27 158.77 176.9 285.06
128.08 123.5 152.99 175.32 156.6 216.7 335.66 453.35 555.32 786.15
13 8 15 6 7 9 12 15 6 9
34 38 37 49 38 51 51 35 25 50
0.26 0.63 0.33 0.25 0.39 0.79 0.88 0.82 0.94 0.36
6.63 2.33 2.84 11.55 14.06 41.88 47.82 31.94 21.84 58.71
27.08 9.03 18.12 113.01 131.07 142.56 159.70 179.41 195.35 343.82
136.75 137.36 143.00 174.00 170.77 231.14 338.91 466.11 544.32 813.55
6 10 10 10 9 14 13 10 16 7
42 38 44 46 31 23 47 34 36 20
0.69 0.13 0.83 0.64 0.31 0.97 0.62 0.74 0.33 0.71
tion of the plants and warehouses for the uncorrelated case. As it can be concluded from the results, the location of the warehouses for each of the test problems tends to be different when demand correlation is considered. This is because the warehouses may require additional capacity under demand correlation and therefore, different locations would be selected. It is also observed that when the number of retailers is high (SC10), additional warehouses may be located to account for the effects of demand correlation. The plant locations also seem to change especially for the larger instances. In our largest case (SC10), more plants were located to deal with demand correlation compared with the uncorrelated case. Our analysis in this section reveals that both strategic (plant and warehouse locations) and operational (order quantity and production decisions) can highly be affected by demand correlation. As a consequence, neglecting demand correlation would not lead to the most efficient plan for the design and operation of a supply chain. 5.3. Computational analysis of the OA algorithm The second part of the numerical experiments specifically focuses on the analysis of the performance of the OA algorithm in handling capacitated supply chain network design proposed in this paper. In particular, we study the computational efficiency of the formulation with respect to correlation coefficients, different levels of capacity and the specific cases presented in Propositions 2 and 3. All the parameters are set according to Table 1 unless mentioned otherwise. 5.3.1. Impact of correlation on OA solution time The first section of this part investigates the impact of correlation levels on the OA computational times. We consider four levels of correlation coefficients. Similar to the previous tests, twenty-five instances of each example problem are randomly generated. We measure the efficiency of the OA algorithm in terms of the minimum, the maximum, and the average of the solution time (in seconds) as well as the minimum and the maximum of the iterations required by the algorithm to reach 1% of the optimality gap. We are also reporting the average of the optimality gaps in order to better show the performance of the methodology. The computational performance of the OA algorithm is illustrated in Tables 6–8. As we can observe from the results, solution time increases with increase in network size, which is an expected outcome. The average solution time marginally increases with increase in correlation coefficient. The number of retailers is a contributing factor to the increase in solution time of the algorithm. For instance, examples SC10 and SC9 share the same number of warehouses and plants. However, SC10 has 20 more retailers compared with SC9. The average and the maximum solution time of SC10 is significantly more
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75
Table 7 Solution time of OA algorithm for correlation coefficients ρ = 0.5. Test Problem
ρ = 0.5 Time (s)
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Iteration
Gap
Min
Ave
Max
Min
Max
Ave (%)
6.64 7.10 13.24 16.83 72.22 48.29 45.91 55.88 67.97 99.35
37.02 35.27 95.63 135.38 167.22 175.51 238.86 341.06 304.29 428.11
259.85 143.02 143.28 441.49 434.81 475.88 811.73 888.01 1088.01 1196.81
9 22 13 8 3 13 29 11 16 13
36 36 38 41 38 39 39 47 34 30
0.51 0.54 0.79 0.91 0.91 0.79 0.31 0.65 0.83 0.75
Table 8 Solution time of OA algorithm for correlation coefficients ρ = 0.7 and ρ = 0.9. Test problem
ρ = 0.7
ρ = 0.9
Time (s)
SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Iteration
Gap
Time (s)
Iteration
Gap
Min
Ave
Max
Min
Max
Ave(%)
Min
Ave
Max
Min
Max
Ave(%)
2.27 8.00 16.88 21.65 38.56 38.24 44.54 69.61 51.50 125.39
28.81 40.94 72.94 118.58 126.82 140.05 153.88 174.96 184.32 329.71
92.59 233.73 230.97 607.55 224.50 281.59 427.11 608.16 718.81 1053.67
9 17 4 8 7 13 20 11 9 13
33 39 42 37 38 32 42 46 35 32
0.45 0.62 0.60 0.55 0.53 0.74 0.74 0.87 0.79 0.99
10.1 14.75 19.11 25.2 92.95 99.77 157.29 172.64 219.55 206.31
45.83 70.11 65.42 81.37 125.3 182.26 216.01 360.02 284.87 459.73
252.15 200.38 317.42 347.31 261.03 469.68 765.13 919.18 1033.09 1204.81
15 8 8 14 10 7 7 16 11 13
36 44 40 39 41 36 44 31 34 69
0.63 0.45 0.39 0.43 0.68 0.93 0.96 0.74 0.98 0.93
than the average and the maximum solution time of SC9. Nevertheless, the effect of the number of retailers tends to be less significant in the small size problems with less number of plant and warehouses. 5.3.2. Impact of capacity on OA solution time The second section of this part studies the response of the OA algorithm to different levels of warehouse capacity. Warehouse capacity is a significant factor in affecting the algorithm’s solution time (Atamtürk et al., 20 0 0; Shahabi et al., 2014). In order to analyze the effects of capacity on the performance of the algorithm, we consider two other scenarios of CW0 = 5 and CW0 = 10 for the base capacity. The latter corresponds to the case of loose capacity and the former denotes the tight capacity case. Considering values less than 5 for the base capacity would produce infeasible instances. Therefore, with the current parameter set up, CW0 = 5 is the tightest possible value, which can be considered for capacity. In addition, we consider three levels of ρ = 0.1, 0.5 and 0.9 to capture the effect of demand correlation. Similar to the previous tests, twenty five instances of each test problem are randomly generated. We measure the efficiency of the OA algorithm in terms of the minimum, maximum, and the average of the solution time required by the algorithm to reach 1% of the optimality gap. The results presented in Tables 9 and 10 demonstrate the effects of capacity on the computational performance of the OA algorithm considering the two cases of tight and loose capacity. As expected, the computational time increases with tightness of capacity. On an average, the average computational times are 77% higher for the tight capacity case compared to the loose capacity case. The percentage of increase in computational time is higher for lower correlations (100%) when compared to higher correlation factor of 0.9 (58%). 5.3.3. Analysis of Propositions 2 and 3 The purpose of this section is to analyze the special cases presented in Propositions 2 and 3 where the optimal value for inventory control decision can be calculated analytically and OA subproblem would reduce to a linear program. The OA is solved to an optimality gap of 1%. Similar to the previous tests, 25 samples are randomly generated. To capture the effects of demand correlation on the structure of the problem three levels of 0.1, 0.5, and 0.9 are assumed for demand correlation. Tables 11 and 12 report the computational performance of OA method in handling the special cases of Propositions 2 and 3. OA solution time and the corresponding percentage of improvement for the minimum, the average and the maximum of the solution time compared with the general case have been presented. The results clearly confirm that the analytical solution of the inventory decisions and reducing the subproblem to a linear program would lead to significant reductions in
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Table 9 Computational time for the loose capacity (CW0 = 10). Computational time (s)
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
2.46 4.43 5.60 8.11 33.48 59.57 50.21 66.83 86.27 87.81
20.70 12.74 32.23 32.33 44.62 26.36 120.30 87.48 134.21 177.01
117.11 85.41 78.75 160.57 147.99 194.45 200.04 240.27 530.21 547.02
0.35 0.59 0.54 0.61 0.31 0.69 0.71 0.77 0.47 0.87
5.38 6.91 13.08 9.94 57.90 42.76 24.05 52.34 45.22 88.29
27.97 31.03 92.68 129.98 138.49 164.13 113.01 310.57 155.38 222.59
153.64 129.81 81.39 231.32 394.96 212.85 762.95 714.75 967.26 1073.35
0.39 0.90 0.10 0.77 0.36 0.78 0.34 0.58 0.69 0.55
8.39 9.60 14.77 20.07 69.79 86.16 135.96 159.24 203.25 172.05
41.06 63.81 62.69 79.26 94.35 156.71 98.62 305.12 262.33 273.78
213.04 180.95 285.55 242.67 252.22 435.15 692.49 724.45 600.36 897.09
0.71 0.76 0.96 0.25 0.81 0.88 0.75 0.88 0.72 0.82
Gap(%)
Table 10 Computational time for the tight capacity (CW0 = 5). Computational time (s)
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
7.68 11.33 12.52 17.38 40.64 70.79 106.07 107.90 113.98 130.93
38.07 32.74 40.75 52.03 91.39 78.54 172.03 239.15 224.45 341.68
195.38 170.65 164.99 230.84 230.57 280.24 361.63 547.29 690.65 720.69
0.18 0.95 0.16 0.29 0.90 0.13 0.25 0.30 0.84 0.64
7.81 9.14 14.42 21.62 82.25 60.41 47.11 61.39 81.58 136.32
39.30 40.07 115.27 150.78 254.46 268.02 318.77 378.53 406.37 470.78
106.08 144.28 175.38 495.83 622.14 478.52 886.82 1280.89 1200.68 1574.75
0.02 0.20 0.59 0.12 0.33 0.09 0.75 0.94 0.22 0.40
13.54 19.30 22.88 28.87 108.73 157.79 210.93 203.87 338.10 286.17
69.78 82.24 74.60 97.41 148.98 250.18 294.51 479.20 334.01 389.14
329.38 228.29 401.50 435.44 289.58 581.10 800.67 1293.47 1506.42 1506.90
0.99 0.66 0.61 0.98 0.56 0.44 0.52 0.73 0.62 0.87
Gap(%)
Table 11 Computational time for the special cases of Propositions 2. Computational time (s)-Proposition 2
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
2.56 6.57 7.11 7.68 24.96 45.27 58.94 69.77 74.64 95.54
21.64 13.22 17.40 33.94 38.04 24.00 93.78 77.62 87.97 108.98
87.67 81.60 84.69 85.66 120.39 106.90 191.58 222.54 498.49 451.03
0.97 0.77 0.30 0.73 0.70 0.30 0.89 0.52 0.88 0.48
5.79 5.68 6.48 15.57 36.35 46.59 25.36 53.93 58.00 79.99
35.10 26.75 55.17 66.09 84.27 124.59 125.25 183.64 301.84 212.98
139.67 126.60 135.38 234.88 313.20 333.31 487.97 433.52 534.23 584.23
0.27 0.98 0.18 0.59 0.37 0.20 0.65 0.90 0.88 0.92
8.50 12.84 10.09 12.32 48.23 49.45 91.09 149.57 107.67 107.56
22.48 44.90 40.73 56.59 110.90 144.05 204.14 315.49 216.96 205.82
123.95 127.55 155.13 185.56 186.44 428.17 505.64 465.55 511.16 600.42
0.27 0.98 0.18 0.59 0.37 0.20 0.65 0.90 0.88 0.92
Gap(%)
Improvement in computational time (s)-Proposition 2 (%)
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Gap(%)
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
51.14 14.32 15.84 51.18 31.74 27.26 23.12 6.11 18.24 11.68
19.84 51.18 50.48 21.15 51.11 50.67 24.54 51.11 50.27 49.32
31.55 33.93 44.64 51.14 23.12 50.61 42.93 50.91 10.23 34.27
0.27 0.98 0.18 0.59 0.37 0.20 0.65 0.90 0.88 0.92
12.83 20.04 51.06 7.46 49.67 3.52 44.77 3.49 14.67 19.49
5.18 24.16 42.31 51.18 49.61 29.01 47.56 46.15 0.80 50.25
46.25 11.48 5.51 46.80 27.97 29.96 39.88 51.18 50.90 51.18
0.97 0.77 0.30 0.73 0.70 0.30 0.99 1.00 0.88 0.38
15.80 12.97 47.23 51.10 48.11 50.44 42.09 13.37 50.96 47.86
50.95 35.96 37.74 30.45 11.49 20.96 5.49 12.37 23.84 42.78
50.84 36.34 51.13 46.57 28.58 8.84 33.92 49.35 50.52 40.25
0.27 0.98 0.18 0.59 0.37 0.20 0.65 0.90 0.88 0.92
M. Shahabi et al. / Transportation Research Part B 110 (2018) 60–78
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Table 12 Computational time for the special cases of Propositions 3. Computational time (s)-Proposition 3
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
2.94 6.44 8.11 10.01 28.17 47.09 69.93 70.58 82.14 104.10
25.46 14.97 22.67 38.61 42.39 24.09 98.87 84.70 92.10 115.39
91.56 84.46 89.29 87.78 123.13 109.95 195.89 224.00 502.66 499.22
0.04 0.44 0.21 0.07 0.41 0.87 0.87 0.34 0.64 0.57
6.03 6.71 9.38 15.70 39.58 41.08 45.60 47.36 59.25 81.82
36.99 27.17 65.22 73.03 89.46 127.24 134.35 191.90 300.56 419.68
148.62 131.24 140.46 258.59 342.74 335.17 501.28 448.48 539.80 634.41
0.84 0.20 0.65 0.80 0.54 0.45 0.91 0.79 0.19 0.24
8.06 13.37 15.58 16.03 49.62 55.87 92.63 158.00 110.21 116.35
25.55 51.06 43.20 64.49 122.07 155.74 215.08 316.69 227.38 216.90
130.75 133.46 169.65 195.20 190.35 457.99 519.00 467.54 532.98 606.86
0.68 0.31 0.68 0.96 0.20 0.25 0.58 0.23 0.12 0.15
Gap(%)
Improvement in computational yime (s)- Proposition 3 (%)
ρ = 0.1 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10
Gap(%)
ρ = 0.5
Gap(%)
ρ = 0.9
Min
Ave
Max
Ave
Min
Ave
Max
Ave
Min
Ave
Max
Ave
43.85 16.06 3.97 36.38 22.98 24.35 8.79 5.02 10.02 3.78
5.67 44.74 35.47 10.30 45.51 50.47 20.44 46.65 47.94 46.35
28.51 31.61 41.63 49.93 21.37 49.26 41.64 50.59 9.48 27.24
0.68 0.91 0.22 0.59 0.81 0.47 0.09 0.98 0.84 0.82
9.17 5.45 29.18 6.69 45.19 14.93 0.68 15.25 12.83 17.65
0.08 22.96 31.80 46.06 46.50 27.50 43.75 43.73 1.23 1.97
42.81 8.23 1.97 41.43 21.18 29.57 38.25 49.50 50.39 46.99
0.52 0.40 0.14 0.05 0.61 0.64 0.26 0.37 0.39 0.80
20.24 9.36 18.48 36.40 46.61 44.00 41.11 8.48 49.80 43.60
44.26 27.18 33.97 20.75 2.58 14.55 0.43 12.04 20.18 39.70
48.15 33.39 46.55 43.80 27.08 2.49 32.17 49.14 48.41 39.60
0.35 0.75 0.82 0.35 0.55 0.89 0.71 0.88 0.76 0.89
Gap(%)
computational time across all test problems. For instance, up to 50% of improvement in average computational time is observed for SC6 in the case of ρ = 0.1 for Propositions 2 and 3. Also, most improvement in computational time is observed for ρ = 0.5 for Proposition 2 across all the test problems. Overall, such reductions in computational time for Proposition 2 tend to be more than Proposition 3 for majority of the test instances.
6. Conclusion This paper provides a mixed integer nonlinear formulation for a joint production–location–inventory problem in a three level supply chain comprising of plants, warehouses, and retailers. The optimization formulation seeks to determine: (i) locations of multiple capacitated plants and warehouses, (ii) allocation of plants to warehouses and warehouses to retailers, (iii) optimal inventory levels and safety stock of inventories of multiple products at warehouses, and (iv) total amount of production by each plant over the planning period. It is shown that both strategic (plant and warehouse locations) and operational (order quantity and production) decisions can highly be affected by demand correlation. This research develops a solution approach based on outer approximation methodology, which guarantees the global optimality. The performance of the solution algorithm is demonstrated on test problems of varying sizes. Numerical experiments show that ignoring correlations can lead to infeasible solutions and increase the total costs of a production–location– inventory system. Accounting for correlations is found to lead to changes in supply chain configurations. The average solution time is found to marginally increase with increase in correlation coefficient. Capacity is found to have a significant impact on computational times. The impact of capacity on increase in computational times is found to be higher in lower demand correlations compared to higher demand correlations. In addition, two structural results are derived for two special cases of the problem – when all products have the same holding cost and when the number of orders for different products at each warehouse is constrained to be the same. It is shown that under these conditions, the optimal value of the order quantity can be determined analytically. The two special cases yield significant savings in computational time. The savings is found to be higher when all products are constrained to have the same holding costs compared to the special case where the number of orders for different products at each warehouse is constrained to be equal. There are various directions for extending the current work both in terms of solution methodology and modeling approach. One possible way is the integration of other inventory control decisions on the formulation. The performance of OA algorithm can also be enhanced through multi-cut generation strategy where various feasible integer points are generated at every iteration.
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Acknowledgments The authors would also like to acknowledge the National Science Foundation for funding this work through Award Nos. 1069141 and 1157294 “Collaborative Research: Stochastic and Dynamic Hyperpath Equilibrium Models” and Award Nos. 1562109 and 1562291 “Collaborative Research: Non-Additive Network Routing and Assignment Models”. Partial support was also provided by the Data-Supported Transportation Operations and Planning Tier 1 University Transportation Center. Any error, mistake, or omission related to this research paper is the sole responsibility of the authors. References Agrali, S., Geunes, J., Tas¸ kın, Z.C., 2012. A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements. J. Global Optim. 54 (3), 551–581. Ahmadzadeh, E., Vahdani, B., 2017. A location–inventory-pricing model in a closed loop supply chain network with correlated demands and shortages under a periodic review system. Comput. Chem. Eng. 101, 148–166. Atamtürk, A., Berenguer, G., Shen, Z.-J., 20 0 0. A Conic Integer Programming Approach to Stochastic Joint Location-Inventory Problems. Oper. Res. 123 (3), 490–503. Atamtürk, A., Berenguer, G., Shen, Z., 2012. A conic integer programming approach to stochastic joint location–inventory problems. Oper. Res. 60 (2), 366–381. Berman, O., Krass, D., Mahdi Tajbakhsh, M., 2011. On the benefits of risk pooling in inventory management. Product. Oper. Manag. 20 (1), 57–71. Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A., 2008. An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–204. Chen, Q., Li, X., Ouyang, Y., 2011. Joint inventory–location problem under the risk of probabilistic facility disruptions. Transp. Res. Part B 45 (7), 991–1003. Daskin, M.S., Coullard, C.R., Shen, Z.M., 2002. An inventory–location model: formulation, solution algorithm and computational results. Annal. Oper. Res. 110 (1), 83–106. Diabat, A., Battaïa, O., Nazzal, D., 2015. An improved Lagrangian relaxation-based heuristic for a joint location–inventory problem. Comput. Oper. Res. 61, 170–178. Diabat, A., Richard, J.P., Codrington, C.W., 2013. A Lagrangian relaxation approach to simultaneous strategic and tactical planning in supply chain design. Anna. Oper. Res. 203 (1), 55–80. Duran, M.A., Grossmann, I.E., 1986. An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339. Fletcher, R., Leyffer, S., 1994. Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66 (3), 327–349. Friesz, T.L., Lee, I., Lin, C.C., 2011. Competition and disruption in a dynamic urban supply chain. Transp. Res. Part B 45 (8), 1212–1231. Ganesh, M., Raghunathan, S., Rajendran, C., 2014. The value of information sharing in a multi-product, multi-level supply chain: impact of product substitution, demand correlation, and partial information sharing. Decis. Support Syst. 59, 79–94. Güllü, R., 1998. Base stock policies for production/inventory problems with uncertain capacity levels. Eur. J. Oper. Res. 105 (1), 43–51. Hua, N.G., Willems, S.P., 2016. Analytical insights into two-stage serial line supply chain safety stock. Int. J. Product. Econ. 181, 107–112. Helper, C.M., Davis, L.B., Wei, W., 2010. Impact of demand correlation and information sharing in a capacity constrained supply chain with multiple retailers. Comput. Indust. Eng. 59 (4), 552–560. Keskin, B.B., Üster, H., 2012. Production/distribution system design with inventory considerations. Naval Res. Logist. 59 (2), 172–195. Li, X., Ouyang, Y., 2010. A continuum approximation approach to reliable facility location design under correlated probabilistic disruptions. Transp. Res. Part B 44 (4), 535–548. Liu, L., Yuan, X.M., 20 0 0. Coordinated replenishments in inventory systems with correlated demands. Eur. J. Oper. Res. 123 (3), 490–503. Masih-Tehrani, B., Xu, S.H., Kumara, S., Li, H., 2011. A single-period analysis of a two-echelon inventory system with dependent supply uncertainty. Transp. Res. Part B Methodol. 45 (8), 1128–1151. Miranda, P.A., Garrido, R.A., 2004. Incorporating inventory control decisions into a strategic distribution network design model with stochastic demand. Transp. Res. Part E 40 (3), 183–207. Miranda, P.A., Garrido, R.A., 2006. A simultaneous inventory control and facility location model with stochastic capacity constraints. Netw. Spat. Econ. 6 (1), 39–53. Ozsen, L., Coullard, C.R., Daskin, M.S., 2008. Capacitated warehouse location model with risk pooling. Naval Res. Logist. 55 (4), 295–312. Ozsen, L., Daskin, M.S., Coullard, C.R., 2009. Facility location modeling and inventory management with multisourcing. Transp. Sci. 43 (4), 455–472. Park, S., Lee, T., Sung, C.S., 2010. A three-level supply chain network design model with risk-pooling and lead times. Transp. Res. Part E 46 (5), 563–581. Peng, P., Snyder, L.V., Lim, A., Liu, Z., 2011. Reliable logistics networks design with facility disruptions. Transp. Res. Part B Methodol. 45 (8), 1190–1211. Puga, M.S., Tancrez, J.S., 2017. A heuristic algorithm for solving large location–inventory problems with demand uncertainty. Eur. J. Oper. Res. 259 (2), 413–423. Raghunathan, S., 2003. Impact of demand correlation in the value of and incentives for information sharing in a supply chain. Eur. J. Oper. Res. 146, 634–649. Romeijn, H.E., Shu, J., Teo, C., 2007. Designing two-echelon supply networks. Eur. J. Oper. Res. 178 (2), 449–462. Schmitt, A.J., 2011. Strategies for customer service level protection under multi-echelon supply chain disruption risk. Transp. Res. Part B 45, 1266–1283. Shahabi, M., Akbarinasaji, S., Unnikrishnan, A., James, R., 2013. Integrated inventory control and facility location decisions in a multi-echelon supply chain network with hubs. Netw. Spat. Econ. 13 (4), 497–514. Shahabi, M., Unnikrishnan, A., Jafari-Shirazi, E., Boyles, S.D., 2014. A three level location–inventory problem with correlated demand. Transp. Res. Part B Methodol. 69, 1–18. Shen, Z.M., Coullard, C., Daskin, M.S., 2003. A joint location–inventory model. Transp. Sci. 37 (1), 40–55. Shen, Z.-J., 2005. A multi-commodity supply chain design problem. IIE Transactions 37 (8), 753–762. Snyder, L.V., Daskin, M.S., Teo, C., 2007. The stochastic location model with risk pooling. Eur. J. Oper. Res. 179 (3), 1221–1238. So, K.C., Zheng, X., 2003. Impact of supplier’s lead time and forecast demand updating on retailers order quantity variability in a two-level supply chain. Int. J. Product. Econ. 86, 169–179. Shu, J., Sun, J., 2006. Designing the distribution network for an integrated supply chain. J. Indust. Manag. Optim. 2 (3), 339–349. Shu, J., Teo, C., Shen, Z.M., 2005. Stochastic transportation–inventory network design problem. Oper. Res. 53 (1), 48–60. Tancrez, J., Lange, J., Semal, P., 2012. A location–inventory model for large three-level supply chains. Transp. Res. Part E 48 (2), 485–502. Teo, C., Shu, J., 2004. Warehouse-retailer network design problem. Oper. Res. 52 (3), 396–408. Üster, H., Keskin, B.B., Çetinkaya, S., 2008. Integrated warehouse location and inventory decisions in a three-tier distribution system. IIE Trans. 40 (8), 718–732. Vidyarthi, N., Çelebi, E., Elhedhli, S., Jewkes, E., 2007. Integrated production–inventory–distribution system design with risk pooling: model formulation and heuristic solution. Transp. Sci. 41 (3), 392–408. Wang, X., Ouyang, Y., 2013. A continuum approximation approach to competitive facility location design under facility disruption risks. Transp. Res. Part B 50 (1), 90–103. Zhang, R.Q., Kaku, I., Xiao, Y.Y., 2011. Deterministic EOQ with partial backordering and correlated demand caused by cross-selling. Eur. J. Oper. Res. 210 (3), 537–551.
