Keywords

1 Introduction

The implementation of networks of logistic hubs usually allows to decrease transportation costs and delivery delays in comparison with direct source/destination transportation [1]. Within logistic hubs, material flows coming from different origins are sorted, consolidated according to their destination then transported using unimodal or multimodal transport. Two families of hubs can be distinguished: pure logistic hubs, providing standard logistic services (warehousing, inventory management, packaging, labeling, orders preparation or cross-docking, sorting and transport/distribution) and combined logistic/industrial hubs, offering high added-value services on logistics (such as co-packing) and/or industrial functionalities allowing the final customization of the product.

Late customization of products (often called “postponement”) was considered for many years by some industries such as computers, printers, medical products and fertilizers [2]. Pushing this logic to its limits, multinational firms now attempt to customize their products within their distribution centers, like Hewlett-Packard producing DeskJet printers in its factory in Singapore and customizing them for the European and Asian markets within its European distribution center near Stuttgart. In that new context, this paper aims at defining the optimal design of a network of logistic hubs with postponement strategy integration, in a context of mass customization. The distribution network has four levels: production plants, regional logistic hubs, sub-regional logistic hubs, and urban/rural distribution centers. The network processes goods with possibly different packaging (raw materials/components and finished products). The main originality of this study is that the joint location of logistic hub and postponement units was rarely addressed in the literature on supply chain network design. This original concept challenges the classical production-distribution model where a clear distinction is made between production and distribution networks [3].

2 State of the Art

The hub network design problem, also known as the distribution network design problem, was intensively studied in the literature on global distribution networks [1, 4]. An analysis of this literature is summarized in Fig. 1.

Fig. 1.
figure 1

Logistic hub network review analysis

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Few authors consider a multi-level network [4], while no paper addresses the integration of industrial services. Origin and destination nodes might be either allocated to a unique hub or to multiple ones: [5] considers multiple allocation of clients to located hubs while [6] studies multiple allocation of plants and customers to intermediate hubs. Service level can be addressed through the definition of a maximum distance between distribution center and market zone [7] or through a delivery delay [8, 9]. Many papers assume that demand is deterministic, which is seldom true. Multiple commodity, allowing to consider products using different transportation means, is only considered by few researchers (cf. Fig. 1).

To our best knowledge, no paper considers a physical transformation of goods while transiting a hub (from bulk material to packs or pallets for instance). [10] states that one of the main characteristics of supply chain network design models is their multi-period nature. Many studies have analyzed the advantages and disadvantages of various postponement strategies [11,12,13]. However, quantitative models for postponement implementation decisions are scarce: [14] considers decisions on where to implement assembly and packaging functions in a distribution network while [15] addresses the problem of facility location-allocation (plants, warehouses) considering commonality and postponement strategies in the logistic network.

This quick analysis of the literature shows that no study gathers yet all the characteristics we have chosen to address in order to answer to present real problems, summarized by the “Proposal” column in Fig. 1.

3 Problem Formulation

The problem is to determine simultaneously the location of the logistic hubs and of the postponement services, while minimizing the total logistics costs. Postponement units will have as inputs raw materials and components coming from international plants, based on their specialization and logistic costs, and will provide bagged/assembled products in response to the requirements of the market zones. These requirements may differ in terms of packaging preferences and required response time. We assume that postponement units hold sufficient component inventories for meeting the customer deterministic demands. The capacities (processing and storage) will be determined a posteriori (through simulation) by considering the expected levels of service. Market zones are allocated to a unique hub based on logistic costs.

In this problem, the interdependence of the decisions makes it difficult to instantiate all the decision variables simultaneously: the location of postponement units will impact the management of the logistic flow, as they constitute decoupling points. On the other hand, their location depends on the location of the logistic hubs and of the allocated demand (volume, response time and product preferences). For addressing this problem, we have chosen as a first approach to decouple the initial problem in two sub problems: definition of the logistic network, then location of the postponements facilities, even if the network structure may in theory be set into question by the positioning of the postponement units. Each sub-problem will be modeled by a deterministic mixed integer linear programming models (cf. Sects. 3.1 and 3.2).

3.1 First Sub-problem: Logistic Hub Location Problem

Within this sub-problem we have to decide: (1) The location of hub h among potential locations H using the zh binary variable (2) The allocation of the origin and destination nodes to the located hubs, using transport mode m represented by the \( {\text{y}}_{{{\text{o}},{\text{d}}}}^{\text{m}} \) binary variable defined only if a modal link between node “o” and “d” exists (Link (o, d, m) = 1) and (3) The flow routing within the network \( {\text{x}}_{{{\text{o}},{\text{d}}}}^{{{\text{m}},{\text{k}},{\text{t}}}} \) i.e. the amount of final product k originated from plant p and transported from node “o” to node “d” using vehicle mode m and under packaging n at period t. The generated solution must provide the max benefit considering the initial investment to open hubs, the total transportation costs including customs, the external handling cost with seaport terminal or rail terminals and the internal handling cost within opened hubs (Eq. 1):

$$ \begin{aligned} Min(\text{Cos} t) = \sum\limits_{h \in H} {co_{h} \times Am \times z_{h}^{{}} } + \sum\limits_{m \in M,t \in T,o \in O,d \in D} {(ct_{m} + cd_{o,d}^{m} ) \times } NV_{o,d}^{m,t} \hfill \\ + \sum\limits_{\begin{subarray}{l} m \in M,n \in N,t \in T, \\ k \in K_{p} ,o \in O,h \in H, \\ d \in D|o \ne h \ne d \end{subarray} } {cm_{h}^{m} \times (NV_{o,h}^{m,t} + NV_{h,d}^{m,t} )} + cm^{n} \_\text{int}_{h} \times (x^{m,k,t,n}_{o,h} + x^{m,k,t,n}_{h,d} ) \times f_{{}}^{n} \hfill \\ \end{aligned} $$
(1)

Subject to:

$$ \sum\limits_{m \in M,h \in H} {y_{h,z}^{m} } = z_{h} \, \forall h \in H \, \wedge z \in Dz \wedge Link(h,z,m) = 1 \, $$
(2)
$$ \sum\limits_{{m \in M,h_{1} \in H}} {y_{{h_{1} ,h_{2} }}^{m} } = z_{{h_{2} }}^{{}} \, \forall h_{2} \in H \wedge Link(h_{1} ,h_{2} ,m) = 1 \, $$
(3)
$$ \sum\limits_{{m \in M,h_{1} \in H}} {y_{{h_{1} ,h_{2} }}^{m} } = z_{{h_{2} }}^{{}} \, \forall h_{2} \in H \wedge Link(h_{1} ,h_{2} ,m) = 1 \, $$
(4)
$$ y_{p,h}^{1} + y_{p,h}^{2} \le 2 \times z_{h}^{{}} \, \forall h \in H_{1} \wedge {\text{ p}} \in P \wedge m \in \{ 1,2\} \wedge \, Link(p,h,m) = 1 \, $$
(5)
$$ y_{{h_{1} ,h_{2} }}^{m} \le z_{h1}^{{}} \quad \forall m \in \{ 2,3\} \wedge h_{1} , {\text{h}}_{2} \in H \wedge {\text{Link( }}h_{1} ,{\text{h}}_{2} , {\text{m) = 1}} \wedge {\text{Dist(h}}_{1} , {\text{h}}_{2} , {\text{m)}} \le {\text{D(m) }} $$
(6)
$$ y_{{h_{1} ,h_{2} }}^{m} \le z_{h2}^{{}} \quad \forall m \in \{ 2,3\} \wedge h_{1} , {\text{h}}_{2} \in H \wedge {\text{Link( }}h_{1} ,{\text{h}}_{2} , {\text{m) = 1}} \wedge {\text{Dist(h}}_{1} , {\text{h}}_{2} , {\text{m)}} \le {\text{D(m) }} $$
(7)
$$ y_{h,z}^{3} \le z_{h}^{{}} \quad \, \forall h \in H \wedge z \in Dz \wedge {\text{Link( }}h,z , 3 ) { = 1} \wedge {\text{Dist(h,z,3)}} \le {\text{D(3)}} $$
(8)
$$ x_{{p,h_{1} }}^{1,k,n,t,p} \le BigM \times z_{h}^{{}} \quad \forall h_{{}} \in H1 \wedge {\text{ p}} \in P_{k} \wedge k \in Kp \wedge t \in T \, $$
(9)
$$ x_{{h_{1} ,h_{2} }}^{m,k,t,p} \le BigM \times z_{{h_{1} }}^{{}} \quad \forall t \in T \wedge m \in \{ 2,3\} \wedge k \in K_{p} \wedge h_{1} \in H \wedge Dist(h_{1} ,h_{2} ,m) \le D(m) $$
(10)
$$ x_{h,z}^{3,k,t,p} \le BigM \times z_{h}^{{}} \quad \, \forall t \in T \wedge {\text{h}} \in H \wedge z \in Dz \wedge k \in K_{p} \wedge Dist(h,z,3) \le D(3) $$
(11)
$$ \sum\limits_{{p \in P_{k} }} {x_{o,d}^{m,k,n,t,p} } \le BigM \times y_{o,d}^{m} \quad \forall {\text{m}} \in {\text{M}} \wedge {\text{n}} \in {\text{N}} \wedge {\text{t}} \in {\text{T}} \wedge {\text{ k}} \in {\text{K}}_{p} \wedge {\text{o}} \in O \wedge {\text{d}} \in D \wedge Link(o,d,m) = 1 \, $$
(12)
$$ y_{o,d}^{m} \, \le \sum\limits_{{p \in P_{k} }} {x_{o,d}^{m,k,n,t,p} } \, \quad \forall {\text{m}} \in {\text{M}} \wedge n \in N \wedge {\text{t}} \in {\text{T}} \wedge {\text{k}} \in {\text{K}}_{p} \wedge {\text{o}} \in O \wedge {\text{d}} \in D \wedge Link(o,d,m) = 1 \, $$
(13)
$$ \sum\limits_{\begin{subarray}{l} m \in M, \\ p \in P_{k} \end{subarray} } {x_{p,h}^{m,k,n,t,p} } = \sum\limits_{m \in M,z \in Dz,h' \in H} {x_{h,z}^{m,k,n,t + \Delta (p,h) + \theta ,p} } + x_{h,h'}^{m,k,n,t + \Delta (p,h) + \theta ,p} \quad \, \forall k \in K \wedge {\text{h}} \in H \wedge t \in T $$
(14)
$$ \sum\limits_{\begin{subarray}{l} h_{1} \in H, \\ m \in M \end{subarray} } {x_{{h_{1} ,h_{2} }}^{m,k,n,t,p} } = \sum\limits_{z \in Dz,m \in M} {x_{{h_{2} ,z}}^{{m,k,n,t + \Delta (h_{1} ,h_{2} ) + \theta ,p}} } \, \forall k \in K_{p} \wedge t \in T \wedge {\text{h}}_{2} \in H \, $$
(15)
$$ \sum\limits_{m \in M,h \in H} {x_{h,d}^{m,k,t} } = {\text{D}}_{z}^{k,t} \times {\text{y}}_{h,d}^{m} \, \quad \forall k \in K_{p} \wedge t \in T \wedge d \in D \, $$
(16)
$$ NV_{o,d}^{m,t} \ge \sum\limits_{{k \in K_{p} }} {\frac{{x_{o,d}^{m,k,t} }}{{CT_{m}^{{}} \times f_{{}}^{n} }}} \, \forall t \in T \wedge o \in O \wedge d \in D \wedge m \in M \wedge n \in N $$
(17)

Constraints (2, 3, 4) express the single allocation of nodes to hubs. Constraints (5, 6, 7, 8) control the allocation mode to physical links. Outgoing hub flows exist only if the hub is active (9, 10, 11) and require that this modal link should be already activated (12, 13). Constraints (14, 15) ensure flow conservation at each period of time where Δ (o, h) +θ is the sum of the transports to h and transit time within h. Outgoing flows toward distribution centers must be equal to their respective demand (16). Equation (17) computes the number of modal vehicles within the network where CTm is the capacity of a vehicle.

3.2 Second Sub-problem: Postponement Location Problem

Given a set of located hubs Hl \( \{ {\text{h}} \in {\text{H}}/{\check{\text{z}}}{\text{h}} = 1\} \cup {\text{Dz}} \) and a set of active links \( {\text{L}} = \{ \left( {{\text{o}},{\text{d}},{\text{m}}} \right);\,{\text{o}} \in {\text{O,}}\,{\text{d}} \in {\text{D,}}\,{\text{m}} \in {\text{M}}/{\hat{\text{y}}}^{\text{m}}_{{{\text{o}},{\text{d}}}} = 1\} \), we have to select the suitable location of postponement units in the designed distribution network. Location can be either on regional hubs, sub-regional ones or on local distribution centers, in order to minimize the total logistic costs (Eq. 18) where \( {\text{b}}_{\text{h}}^{\text{e}} \) is a binary variable equal to 1 if the postponement unit is located on hub h at echelon e, while Bin (H, e) is a Boolean value equal to 1 if hub h is located at level e.

$$ \begin{aligned} & Min(\text{Cos} \,t\_postp) = \sum\limits_{h \in Hl,e \in E} {co_{h} \times Am \times b_{h}^{e} } \\ &+ [\sum\limits_{m \in M,t \in T,o \in O,d \in D} (ct^{1}_{m} + cd_{o,d}^{m} + cm_{m}^{1} ) \times NBV_{o,d}^{m,t} + \sum\limits_{{m \in M,t \in T,k \in K_{b} ,o \in O,d \in D}} cm_{m}^{1} \_\text{int} \times x^{m,k,t}_{o,d} ] \\ & +[\sum\limits_{m \in M,t \in T,o \in O,d \in D} (ct^{2}_{m} + cd_{o,d}^{m} + cm_{m}^{2} ) \times NCV_{o,d}^{m,t} { + }\sum\limits_{\begin{subarray}{l} m \in M,t \in T,k \in K_{c} \\ ,o \in O,d \in D,p \in Pk \end{subarray} } cm_{m}^{2} \_\text{int} \times \bar{x}^{m,k,n,t}_{p,o,d} \, ] \\ \end{aligned} $$
(18)

Subject to:

$$ \sum\limits_{e \in E} {b_{h}^{e} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{z}_{h} } \quad \, \forall {\text{h}} \in H \wedge Bin(H,e) = 1 $$
(19)
$$ \sum\limits_{{m \in M,p \in P_{k} }} {{\text{x}}_{{h_{1} ,h_{3} }}^{m,k,t} } \le {\text{BigM}} \times \bar{y}_{{h_{1} ,h_{2} }}^{m} \times {\text{b}}_{{h_{3} }}^{3} \quad \forall t \in T \wedge k \in K_{b} \wedge h_{1} ,h_{3} \in Hl $$
(20)
$$ \sum\limits_{{m \in M,p \in P_{k} }} {{\text{x}}_{{h_{1} ,h_{2} }}^{m,k,t} } \le {\text{BigM}} \times \bar{y}_{{h_{1} ,h_{2} }}^{m} \times ({\text{b}}_{{h_{2} }}^{2} {\text{ + b}}_{{h_{3} }}^{3} )\quad \, \forall {\text{t}} \in {\text{T}} \wedge {\text{k}} \in {\text{K}}_{b} \wedge h_{1} ,h_{2} ,h_{3} \in Hl \, $$
(21)
$$ \begin{aligned} & \sum\limits_{m \in M} {x_{{h_{2} ,z}}^{m,k,t} } = \sum\limits_{\begin{subarray}{l} m \in M,k' \in K_{c} , \\ p \in P_{k} \end{subarray} } {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}_{{h_{2} ,z}}^{m} \times \bar{x}_{{p,h_{2} ,z}}^{{m,n,k',t + \Delta (h_{2} ,z) + \theta }} \times Conv(n) \times \, Cs(k,k') \times b_{z}^{3} } \\ & /\forall t \in T,h_{2} \in Hl,z \in Dz,k \in K_{b} \\ \end{aligned} $$
(22)
$$ \begin{aligned} & \sum\limits_{m \in M} {x_{{h_{1} ,h_{2} }}^{m,k,t} } = \sum\limits_{\begin{subarray}{l} m,m' \in M, \\ k' \in K_{c} , \\ p \in P_{k} ,z \in Dz \end{subarray} } {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}_{{h_{1} ,h_{2} }}^{m} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}_{{h_{2} ,z}}^{m'} \times \bar{x}_{{p,h_{2} ,z}}^{{m',n,k',t + \Delta (h_{1} ,h_{2} ) + \theta }} \times Conv(n) \times \, Cs(k,k') \times (1 - b_{{h_{1} }}^{1} )} \\ & /\forall t \in T,h_{1} ,h_{2} \in Hl,k \in K_{b} \\ \end{aligned} $$
(23)
$$ \begin{aligned} & \sum\limits_{m \in M} {x_{{h_{1} ,z}}^{m,k,t} } = \sum\limits_{\begin{subarray}{l} m \in M,k' \in K_{c} , \\ p \in P_{k} ,z \in Dz \end{subarray} } {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}_{{h_{1} ,z}}^{m} \times \bar{x}_{{p,h_{1} ,z}}^{{m,n,k',t + \Delta (h_{1} ,h_{2} ) + \theta }} \times Conv(n) \times \, Cs(k,k') \times (1 - b_{{h_{1} }}^{1} )} \\ & /\forall t \in T,h_{1} ,h_{2} \in Hl,k \in K_{b} \\ \end{aligned} $$
(24)
$$ NBV_{o,d}^{m,t} \ge \sum\limits_{{k \in K_{b} ,p \in P_{k} }} {\frac{{x_{p,o,d}^{m,k,t} }}{{CT_{m}^{1} }}} \times NV_{m} { ;}\quad \forall t \in T \wedge m \in M \wedge h \in Hl \wedge h_{3} \in Dz \, $$
(25)
$$ NCV_{{h,h_{3} }}^{m,t} \ge \sum\limits_{e \in E} {b_{h}^{e} } \times [\sum\limits_{{k \in K_{c} ,p \in P_{k} ,n \in N}} {\frac{{\bar{x}_{{p,h,h_{3} }}^{m,k,n,t} }}{{CT_{m}^{2} \times f^{n} }}]} /\forall t \in T \wedge m \in M \wedge h \in Hl \wedge h_{3} \in Dz \, $$
(26)
$$ NCV_{{h_{2} ,h_{3} }}^{m,t} \ge (b_{{h_{1} }}^{1} + b_{{h_{2} }}^{2} ) \times [\sum\limits_{\begin{subarray}{l} k \in K_{c} , \\ p \in P_{k,} n \in Pg \end{subarray} } {\frac{{\bar{x}_{{p,h_{2} ,h_{3} }}^{m,n,k,t} }}{{CT_{m}^{2} \times f^{n} }}]} /\forall t \in T \wedge m \in M \wedge h_{1} ,h_{2} \in Hl \wedge h_{3} \in {\text{Dz }} $$
(27)

Constraint (19) translates that a postponement activity can only be located on activated hubs, at only one level. “Continuous flows” \( {\text{x}}_{{{\text{o}},{\text{h}}}}^{{{\text{m}},{\text{k}},{\text{t}}}} \) exist only if postponement units are located before that hub if a modal link is activated (20, 21). (22, 23, 24) express a flow balance at each period and on each hub and computes the ingoing discrete flows to each hub depending on location of postponement units. Conv(n) is the conversion ratio from unit of packaging product under n commodity to a continuous unit (tons for example) and Cs(k, k’) is the amount of component k needed to produce a unit of product k’. (25, 26, 27) assess the number of bulk and container vehicles within the network.

4 Illustrative Study

This case study aims to illustrate the application of the proposed models. It concerns the location of blending units within East Africa for specific industries involving hybrid (discrete-continuous) flows, like the fertilizer industry. Three production zones located in Morocco, Ethiopia and Nigeria and considered. Five Regional hubs are defined: Kenya-Angola-Tanzania-Djibouti, so that ten sub-regional hubs: Nairobi-Kisumu-Dodoma-Arusha-Tabora-Kuito-Tete-Lichinga-Kigali. The considered data are summarized in Tables 1 and 2. The demand (aggregated on one year) varies for each zone. We assume that shipment is done every month and that all the market zones require the same service level. Distances and traveling times are extracted from Google Maps. The problem was solved using the Xpress-IVE solver tools. The results of models 1 and 2 are summarized in Tables 3 and 4.

Table 1. List of transport cost parameter
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Table 2. List of other cost parameters
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Table 3. Results of sub-model 1
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Table 4. Results of sub-model 2
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The location of the blending within the distribution network is the result of a comparison between investments and operational costs linked to postponement and to the transportation of bulk and packed materials. Especially, it requires to compare the profit generated by several small blending units, close to the final customers, and the one of larger units, located earlier in the network but requiring more time to adapt their production to the final demand. In real cases, good sense is not sufficient to figure out this decision problem. Our model helps the decision maker to assess quantitatively each possible solution.

5 Conclusion and Research Perspectives

In the context of mass customization, postponement activities may provide an answer to fulfill customized orders, increase customer responsiveness and increase service level. However, the literature combining design of logistic hub networks and implementation of postponement facilities is scarce and usually assumes that the location of the distribution centers is already known. In order to address the problem, we have developed and tested a two-phase deterministic mathematical programming model where, as a first step, we design incapacitated discrete logistic hub, then allocate postponement services on some hubs. In our future work, this model will be coupled with a discrete event simulation model in order to take into consideration uncertainties on the demand and on the availability of the resources. Simulation will also allow to assess postponement capacities and to refine logistic costs, these results being re-injected in the mathematical model as new constraints.