Classifications and mathematical modelling

Categories and taxonomies of facility location models are found in Francis and White (1974), Bradndeau and Chiu (1989), Daskin (1995), Owen and Daskin (1998), ReVelle et al. (2008), Jia et al. (2007), Daskin (2008), Klose and Drexl (2005), and Eiselt and Marianov (2011).

In literature, categories and taxonomies of facility location models under different con- texts of OR, logistics, and SCM are overlapped. Riopel et al. (2010) and Langevin and

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Riopel (2010) discuss the physical facility network in the context of logistics decisions considering several inter-dependent key decisions, viz. the type and the number of dif- ferent facilities (warehouses, DCs and terminals), size and the location of each facility, the products and services provided from each facility and whether to use new or existing facilities or open a new one.

Riopel et al. (2010) categorise mathematical facility location models as followed: (i) the fixed charge facility location problem. Extensions of this model con-

sider: a) facility capabilities and single sourcing requirements, b) multi- ple echelons in the supply chain, and c) multiple products,

(ii) integrated location/routing models, (iii) integrated location /inventory models, (iv) planning under uncertainty,

(v) location models with facility failures.

Daskin et al. (2010) and Alizadeh-Shabdiz (2009) classify and review facility location models into three main groups. This research uses the above-mentioned classification system. Based on integrated location-routing models (ii) in this classification system, two-layer and three-layer models are proposed. This classification system is as follows.

 Classical/traditional models: This category consists of classical facility loca- tion problems which forms the basis of most location models. In this type of location problem a set of customer locations with known demand and a set of candidate facility locations are given. The problem is finding the location of facilities and the movement (shipment) pattern between the facilities and the customers in order to minimise the combination of facility location and movement (shipment) costs subject to the constraint that all customer demand be met (Daskin et al. 2010). Sub-models of this group of mod- els are:

i. fixed charge facility location problem,

ii. uncapacitated facility location model (with single sourcing), iii. capacitated facility location model,

iv. locating plants and distribution centres (with multiple commodity). Fixed charge facility location problem is discussed below.

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Fixed charge facility location problem: Fixed charge facility location problem is the basis of most of location models used in SC design. In this type of model, there is a set of data available: (a) a set of customer locations with known demand, (b) a set of candi- date facility locations, if the model maker decided to locate a facility in a known site then a fixed cost is implied and (c) a known unit delivery/transportation cost between each candidate site and each customer. The problem here is to find the optimum loca- tions of the facilities in order to minimise the costs of transportation and facilities, sub- ject to the requirement of meeting all customer demands. The mathematical formula- tions of the other three facility location models, viz. uncapacitated facility location model (with single sourcing), capacitated facility location model, and locating plants and distribution centres (with multiple commodity), are included in Appendix A.2.

 Integrated decision making models: A study of the literature reveals many studies, and a variety of models for the purpose of attempting to integrate any two of the important decisions regarding SC design (Daskin et al. 2005; Hassanzadeh et al. 2009). These efforts in integration and combining elements of SC design are categorised as:

i. Integrated location-routing models (LR) ii. Integrated inventory-routing models (IR) iii. Integrated location-inventory Models (LI)

LR is the main concern of this research. This group of combined facility location mod- els is discussed in more details in the following section.

 Other models: The following models do not fall into classical or integrated cat- egories and are classified as ‘other models’:

i. model with routing cost estimation,

ii. model with capacitated DCs (distribution centres), iii. model with multiple levels of capacity,

iv. model with service considerations,

v. profit maximising model with demand choice flexibility, vi. model with multiple commodities,

vii. model with unreliable supply, viii. model with facility failures, ix. planning under uncertainty.

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26 2.4. Integrated Location-Routing Problem

The introductory concept of a Location-Routing Problem (LRP) was first reported in the 1960s and early 1970s (Maranzana 1964; Webb 1968; Lawrence and Pengilly 1969; Higgins 1972; and Christofides and Eilon 1969). In the late 1970s and 1980s studies on LRP, as an integrated combined problem, began to appear separately (Jacobsen and Madsen 1978; Or and Pierskalla 1979; Laporte and Norbert 1981).

Integrated LRPs combine three important components of a SC design: facility location, customer allocation to facilities, and vehicle routing (Daskin et al. 2010). These prob- lems merge two different problems in nature, viz., facility location and vehicle routing. Literature discusses LRPs under both categories. Facility location decisions are strategic decisions by nature, while vehicle routing decisions are not.

In the early attempts LRP is mostly discussed as a part of Vehicle-Routing Problems (VPR). VRPs are complex themselves and LRPs are considered a component of the VRPs. Laporte et al. (1988), in one of the early reviews on different types of LRP mod- els and solution techniques, define VRP and LRP as follows:

‘The VRP is commonly defined as the problem of designing optimal delivery or collec- tion routes from one or several depots to a set of geographically scattered customers, under a variety of side conditions. LRPs are VPRs in which the optimal depot locations and route design must be decided simultaneously.’

Min et al. (1998) tried to offer a comprehensive categorisation of LRP studies by using a two way classification: (i) classifying LRP with regard to its problem perspective and (ii) classifying LRP problem with regard to its solution method. Table 2.1 presents these classifications.

Table 2.1 Classification of LRPs (Min et al. 1998) Classification of LRP with regard to

problem perspective

Classification of LRP with regard to solu- tion method

I. Hierarchical level a. Single stage b. Two stages

II. Nature of demand/supply a. Deterministic b. Stochastic III. Number of facilities

a. Single facility b. Multiple facility IV. Size of vehicle fleets

a. Single vehicle

I. Exact algorithm

a. Direct tree search / Branch & bound

b. Dynamic programming c. Integer programming d. Non-Linear programming II. Heuristic a. Location-allocation-first, route-second b. Route-first, location-allocation-second c. Savings / insertion d. Improvement / exchange

Chapter Two 27 b. Multiple vehicles V. Vehicle capacity a. Uncapacitated b. Capacitated VI. Facility capacity

a. Uncapacitated b. Capacitated VII. Facility layer

a. Primary b. Secondary VIII. Planning horizon

a. Single period (static) b. Multiple periods (dynamics) IX. Time window

a. Unspecified time with no deadline b. Soft time windows with loose dead-

lines

c. Hard time windows with strict dead- lines

X. Objective function a. Single objective b. Multiple objectives XI. Types of model data

a. Hypothetical b. Real-world

Then Min et al. (1998) states the LRP as: ‘In general, the combined location-routing

model solves the joint problem of determining the optimal number, capacity, and loca- tion of facilities (domiciles) serving more than one customer/supplier, and finding the optimal set of vehicle schedules and routes’.

There is a main difference between the LRP and the classical location-allocation. The classical location allocation problem ignores tours when locating facilities. This eventu- ally leads to more distribution costs (Salhi and Rand 1989; Min et al. 1998). Min et al. (1998) and Daskin (1995) consider two types of trips between facilities and customers, viz. direct trips and tour trips. Based on these two types of trips between facilities and customers, Hassanzadeh et al. (2009) in an effort to define LRP, describes the different types of serving customers (Figure 2.5):

Figure 2.5 Different types of servicing a customer (adopted form Hassanzadeh, 2009) Location Problem Selecting the location of new facilities

Customers visit the facility

Customers are being serviced in thier own location Direct trips (ambulances, fire engines) Tours (postmen, repairer)

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Hassanzadeh et al. (2009) define the LRP as: ‘a feasible set of potential facility sites

and locations and expected demands of each customer which are given. Each customer is to be assigned to a facility which will supply its demand. The shipments of customer demand are carried out by vehicles which are dispatched from the facilities, and oper- ate on routes that include multiple customers. The location of distribution facilities and the distribution of products from these facilities to customers are two key components of a distribution system.’

Nagy and Salhi (2007) look at the LRP as an approach for modelling and solving locational problems as the LRP is not a single well defined problem from their point of view. In a state-of-the-art survey on location-routing, Nagy and Salhi (2007) classify literature at the time based on a classification system that includes a number of criteria. They state that ‘classifying location-routing problems is at least as difficult a task as

that of classifying location problems, with added complexity provided by the variability in the underlying vehicle routing problems’.

Literature reports mathematical formulation of the integrated LRPs in two-layer (Berger 1997; Hassanzadeh et al. 2009) and three-layer (Perl 1983; Perl and Daskin, 1985) stages or echelons. In the literature an example of a four-layer LRP is also presented (Hamidi et al. 2012). The number of layers represents the main players on the demand side of the supply chain considered in these models. Figure 2.6 presents a general representation of a SC. It consists of the supply side, the demand side, and the connections between the supply and the demand sides with the focal company. Most SC network design models are focused on the demand side and physical distribution of products.

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Figure 2.6 A general representation of a SC (Bowersox et al. 2013)

The two-layer LRP considers two players on the demand side of the SC. These two- layers can be the plant(s) and DC(s), or plant(s) and customer(s) (Hassanzadeh et al. 2009). The mathematical formulations of a two-layer LRP developed by Berger (1997) and a three-layer LRP developed by Perl (1983) are presented in Appendix A.3.