Linda L Zhang
email: l.zhang@ieseg.fr

Carman Lee

Shuzhu Zhang
email: shuzhu.zhang@connect.polyu.hk

An integrated model for strategic supply chain design: Formulation and ABCbased solution approach

This study develops a mixed integer nonlinear programming (MINLP) model to design supply chains. In view of the limitations of many available strategic supply chain design models, this model involves three major supply chain stages, including procurement, production, and distribution, and their interactions; it takes into account bill of materials constraints for modeling complex supply chain inter-relationships.

In addition, in accordance with the fact that companies nowadays develop product families, our model addresses multi-product supply chain design to respond to diverse customer requirements. Recognizing their importance, this study identifies and formulates constraints related to facility pairwise relationships and supplier priority along with the classical constraints from the available literature. To efficiently solve such a highly constrained, large scale MINLP model, we develop an approach based on an artificial bee colony (ABC) algorithm. Bicycle design and production is used to demonstrate the potential of the MINLP model for designing supply chains and the performance of the ABC-based solution approach in solving the model. The proposed model and solution approach can be considered as two fundamental components of an expert system in the broad sense. Thus, this study is expected to stimulate more future research on the development of practical expert systems for designing supply chains.

Introduction

A supply chain involves multiple facilities (i.e., supply chain members), such as raw material and component suppliers, final product producers, distribution centers. Based on customer requirements, supply chain members collaboratively design, produce, and deliver products while attempting to achieve the optimal performance of the cohort of the chain [START_REF] Safaei | Integrated multi-site production-distribution planning in supply chain by hybrid modeling[END_REF][START_REF] Zhang | Supply chain configuration with coordinated product, process and logistics decision: an approach based on Petri nets[END_REF]. As today's business competition is among supply chains, instead of individual firms, it is deemed important to design effective supply chains, which can help sustain competitive advantages for all chain members. This is well evidenced by the numerous articles reported for designing supply chains in the recent two decades [START_REF] Gebennini | An integrated productiondistribution model for the dynamic location and allocation problem with safety stock optimization[END_REF][START_REF] Ivanov | An adaptive framework for aligning (re)planning decisions on supply chain strategy, design, tactics, and operations[END_REF].

As pointed out in [START_REF] Gebennini | An integrated productiondistribution model for the dynamic location and allocation problem with safety stock optimization[END_REF][START_REF] Ivanov | An adaptive framework for aligning (re)planning decisions on supply chain strategy, design, tactics, and operations[END_REF][START_REF] Sabri | A multi-objective approach to simultaneous strategic and operational planning in supply chain design[END_REF][START_REF] Simchi-Levi | Managing the supply chain: the definitive guide for the business professional.N e wY o r k :M c G r a w -H i l l[END_REF][START_REF] Thanh | A dynamic model for facility location in the design of complex supply chains[END_REF], the supply chain design's central decisions include supplier selection, facility location and capacities, customer demand allocation, raw material, component and product flows, which are at the strategic level. To cope with these decisions, researchers have proposed a myriad of valuable strategic supply chain design models. Most of these models treat each stage (or at most two) of the chain as a separate system, e.g., only procurement or only production or the integration of production-distribution; a few studies have addressed an integrated supply chain design from material procurement to production to product delivery [START_REF] Sabri | A multi-objective approach to simultaneous strategic and operational planning in supply chain design[END_REF][START_REF] Li | A strategic capacity allocation model for a complex supply chain: formulation and solution approach comparison[END_REF]. However, as revealed in industrial projects [START_REF] Melo | Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning[END_REF], companies wish the simultaneous consideration of all the three important stages in their supply chain design. This indicates that designing a supply chain by considering multistage and their interactions yields realistic solutions. In a recent review article based on the analysis of 33 survey-based studies, [START_REF] Van Der Vaart | A critical review of survey-based research in supply chain integration[END_REF] point out that integrated supply chain design leads to higher supply chain performance. In view of the importance of considering multistage in supply chain design and the relative lack of models, in this study, we simultaneously consider procurement, production and distribution in the strategic supply chain design.

A bill of materials (BOM) is a very important product document and describes in detail product's constituent elements and their relationships. BOMs provide key information for coordinating activities between material procurement and production in a supply chain and are, thus, related to many complex supply chain inter-relationships [START_REF] Yan | A strategic model for supply chain design with logical constraints: formulation and solution[END_REF]. Researchers highlight that BOMs should be exploited to coordinate the behavior of suppliers with the production and distribution activities and should be considered as constraints in the strategic supply chain design [START_REF] Arntzen | Global supply chain management at Ditital Equipment corporation[END_REF][START_REF] Cohen | Resource deployment analysis of global manufacturing and distribution networks[END_REF][START_REF] Melo | Facility location and supply chain management: a review[END_REF]. However, due to the difficulties in formulating BOM-related constraints in a mathematical model, there is a lack of models with the inclusion of BOM constraints [START_REF] Vidal | Strategic production-distribution models: a critical review with emphasis on global supply chain models[END_REF][START_REF] Thanh | A dynamic model for facility location in the design of complex supply chains[END_REF][START_REF] Melo | Facility location and supply chain management: a review[END_REF]. In this study, we consider BOMs in the development of the integrated model for strategic supply chain design and formulate BOM-related constraints.

Among the available supply chain design models, some consider one product [START_REF] Li | A strategic capacity allocation model for a complex supply chain: formulation and solution approach comparison[END_REF][START_REF] Osman | A bilinear goal programming model and a modified Benders decomposition algorithm for supply chain reconfiguration and supplier selection[END_REF],t on a m e but three); some involve multiple products [START_REF] Das | Integrating effective flexibility measures into a strategic supply chain planning model[END_REF][START_REF] Safaei | Integrated multi-site production-distribution planning in supply chain by hybrid modeling[END_REF][START_REF] Thanh | A dynamic model for facility location in the design of complex supply chains[END_REF][START_REF] Yan | A strategic model for supply chain design with logical constraints: formulation and solution[END_REF]. Designing supply chains for multiple products is more practical. The reason for this is that in practice, companies develop families of related products (so called product families) to fulfill diverse individualized customer requirements. Due to the similarities among customized products in a family, a supply chain is normally utilized to design and produce one product family [START_REF] Huang | Towards integrated optimal configuration of platform products, manufacturing processes, and supply chains[END_REF]. In this study, we consider multiple products in developing the integrated, strategic supply chain design model.

To summarize, this study focuses on the strategic design of supply chains for multiple products by considering procurement, production, distribution and their interactions and BOM-related constraints. A mixed integer non-linear programming (MINLP) model is developed. In order to solve the MINLP model, swarm intelligence is employed in this study. Swarm intelligence is a new category of the meta-heuristics, which is inspired by the collective intelligence of insect colonies or animal societies in their operational behavior [START_REF] Bonabeau | Swarm intelligence: From natural to artificial systems[END_REF]. One of the typical examples of intelligent swarms is the bee colony, which demonstrates an amazing intelligence when foraging for food sources. In this study, we, thus, employ a bee colony inspired algorithm, named artificial bee colony (ABC) algorithm. The ABC algorithm is originally introduced by [START_REF] Karaboga | An idea based on honey bee swarm for numerical optimization[END_REF] and becomes a popular choice for solving constrained optimization problems in comparison to the evolutionary computation thanks to its simple yet robust framework and implementation simplicity. The ABC algorithm is efficient in both exploration and exploitation of the search space because of its unique design of multiple roles and phases [START_REF] Horng | Multilevel thresholding selection based on the artificial bee colony algorithm for image segmentation[END_REF][START_REF] Karaboga | A comprehensive survey: artificial bee colony (ABC) algorithm and applications[END_REF]. In view of the advantage of the ABC algorithm for solving optimization problems, we develop an approach based on the ABC algorithm to solve the proposed MINLP model. The effectiveness and efficiency of the ABC-based solution approach is demonstrated through a comparison with other approaches, including LINDO (a commercial solver) and a genetic algorithm (GA)-based approach.

The rest of the paper is organized as follows. We formulate the MINLP integrated, strategic supply chain design model in Section 2 by identifying diverse constraints. This is followed by the proposed ABC-based solving approach in Section 3. Bicycle design and production is used to demonstrate designing supply chains using the MINLP model in Section 4. Also provided in this section is the comparison between the proposed ABC-based solution approach and LINDO, the GA approach. We conclude this paper in Section 5 by pointing out the limitations and identifying potential avenues for future research.

Model formulation

Problem context and decisions

A set of products is to be designed, manufactured, and distributed by a set of facilities, including suppliers, production plants, and distribution centers (DCs), which will form a supply chain. A production plant is responsible for the overall design and production activities, whilst suppliers are expected to be involved in design and production of intermediate components. Each facility has limited capacity in fulfilling its tasks. Together, they complete the product design, production, and distribution according to customer requirements. Consistent with the current global manufacturing practice, while the set of production plants belong to one company, the suppliers and DCs may not belong to the same company (Chung, Lau, Choy, Ho, & Tse, 2010;[START_REF] Kim | Values of the balanced decision making between supply chain partners[END_REF]. In addition, the company makes the decision on the plants, suppliers, and DCs to be included in the supply chain. As indicated by [START_REF] Anussornnitisarn | Decentralized control of cooperative and autonomous agents for solving the distributed resource allocation problem[END_REF]Dominiquez and Lashkari, 2004;[START_REF] Moon | Adaptive genetic algorithm for advanced planning in manufacturing supply chain[END_REF], the above context where supply chains are formed is not uncommon in practice, especially when involving internationally operating companies.

In this study, the general decisions considered are related to the selection of supply chain facilities and the allocation of loading (e.g., component quantities) to these facilities. The specific decisions deal with: (i) the selection of specific suppliers, production plants, and DCs, (ii) the design tasks assigned to suppliers and plants, (iii) the amount of components and products to be produced and shipped among suppliers, plant, DCs, and customers, (iv) global capacity coordination for the involved suppliers and plants, and (v) distribution planning that determines optimal distribution channel and quantity. Taking into account these decision factors, we formulate the below MINLP model.

The MINLP model

Involving multiple products and multiple stages, the supply chain design problem requires a systematic approach to account for product-related characteristics, supply and demand matching, facility loading, and both inter-and intra-facility transactions. The supply chain designed should be cost-effective so that the total cost associated with design, production, holding inventory, and logistics can be reduced to a minimum level. In light of the above issues, the MINLP model attempts to minimize the total cost of a supply chain, subject to the capacities of suppliers, plants, and DCs, throughput constraints, and customer demand requirements as well. The model is formulated based on the following assumptions.

Assumptions:

(1) Each plant is able to produce any arbitrary product mix;

(2) Product fulfillment within a plant is divided into a finite set of sub-tasks, each being carried out by one or more than one supplier;

(3) If a supplier undertakes a task of designing a component, it is also responsible for producing the component; (4) Each plant purchases components from multiple suppliers and each supplier serves several plants; (5) Each DC is opened for any arbitrary product mix; (6) DCs deliver all received products to customers (Note: Like many available supply chain design models, drop shipping is not considered in this study. Thus, DCs receive products and deliver them to customers.); (7) According to geographical locations, customers are grouped into different zones, forming customer zones (CZs);

(8) The design and production capacities of suppliers and plants are known; (9) The throughputs of plants and DCs are known; (10) The transportation costs from facilities to facilities are known.

In accordance with the above decisions and assumptions, we define a list of notation, including indices, input parameters, and decision variables (see Appendix A). While most of the definitions, e.g., unit production cost, setup cost, are easily understandable, two input parameters: design capacity and service level are explained below. In line with the uncertainties of development capabilities of a facility, design capacity is the probability that a plant (or supplier) can undertake the design task [START_REF] Wang | A fuzzy hybrid decision-aid model for selecting partners in the design chain[END_REF]. It assumes a real value in (0, 1]. A service level is measured as the percentage of orders that can be delivered by a supply chain facility (i.e., a supplier, a plant, or a DC) to its downstream facility (Simchi-Levi, [START_REF] Simchi-Levi | Managing the supply chain: the definitive guide for the business professional.N e wY o r k :M c G r a w -H i l l[END_REF]. It also assumes a real value in (0, 1].

The objective function is to minimize the total cost, C,o ft h e entire supply chain during a time period ($/period) as follows:

Min C = rj U rj (η rj /δ rj ) + rj U rj f S rj + rj U rj p S rj Z rj + rjk U rj c SP rjk R rjk + ik V ik (η ik /δ ik ) + ik V ik f P ik + ik V ik p P ik X ik + ikl V ik c PD ikl Q ikl + il W il f D il + il W il p D il Y il + ilm W il c DZ ilm P ilm Subject to: k R rjk ≤ S rj , ∀r, j (1) k R rjk ≤ Z rj , ∀r, j (2) j R rjk • α j ≥ λ ri X ik , ∀r, i, k (3) l Q ikl ≤ P ik , ∀i, k (4) il Q ikl ≤ P k , ∀k (5)
i Q ikl • β k ≥ Y il , ∀k (6)
Y il = m P ilm , ∀i, l (7) l (P ilm γ l T ilm ) ≥ D im , ∀i, m (8) i Y il ≤ D lm , ∀l (9)
l T ilm ≥ 1, ∀i, m (10) X ik , Y il , Z rj , P ilm , Q ikl , R rjk ≥ 0, ∀i, j, k, l, m (11) U rj , V ik , W il , T ilm = 0 or 1, ∀i, k, l, m (12) j U rj Z rj -λ ri k V ik X ik ≥ 0, foralli (13) V ik -C ri (Urm + U rn +•••+U rp) ≤ 0 (14) W il -A i V ix + V iy +•••+V iz ≤ 0 (15) U rm ≥ U rn , for m, n ∈ J (16)
The total cost in the above objective function includes both fixed and variable costs. It contains cost factors pertaining to three stages in a supply chain: the procurement, production, and delivery stages. In the procurement stage, the cost consists of component design, production set-up, component purchasing, and transportation costs. Among them, design cost is derived from the supplier's capacity to design a specific component. In the production stage, the cost consists of product design, production set-up, and production ramp up costs, and transportation cost from plants to DCs as well. In the delivery stage, the cost consists of set-up and inventory holding costs of DCs and transportation cost from DCs to CZs.

The quantity of any component that can be shipped from one supplier to plants is limited to the supplier's production capacity, as indicated in Constraint 1. Constraint 2 ensures the material balance of a supplier, that is, the quantity of a component shipped from a supplier is equal to the total quantity provided by the supplier. Constraint 3 guarantees that the demand for component j of product i from plant k is always met. Constraint 4 is to reinforce the rule: the quantity of a product shipped from a plant cannot be larger than the plant's production capacity. Similarly, Constraint 5 is to ensure that the maximum quantity of mixed products provided by a plant does not exceed the plant's total capacity. To meet the demand of a product at a DC, Constraint 6 is formulated. A DC can always supply the right amount of a product to its CZs (i.e., matching supply to demand), as indicated in Constraint 7. Constraint 8 ensures that the demand of each CZ is satisfied. Constraint 9 limits the total quantity of products that a DC can receive to be no larger than its maximum throughput. Constraint 10 specifies that for any product, each CZ should be served by at least one DC. Constraint 11 ensures non-negativity of quantities shipped in different stages. Constraint 12 determines the binary variables. Component quantity requirement in accordance with products' BOMs are met in Constraint 13. Constraints 14 and 15 model the pairwise relationship among facilities. Constraint 16 ensures supplier priority.

To sum up, in the above constraint formulation, we consider the classical constraints (e.g., Constraints 1, 2, and 3) appearing in the available supply chain models. Complementing the existing studies, we identify and formulate new constraints associated with products' BOMs (Constraint 13), facility pairwise relationships (Constraints 14 and 15), and supplier priority (Constraint 16). These constraints along with their reasoning and importance are elaborated below.

BOM constraints

BOMs contain information about product components, such as raw material, parts and assemblies, their relationships, and their quantities for one unit of the final products. Each component is usually associated with multiple supply chain facilities, be they raw materials (or parts) suppliers, final product production plants, or DCs. BOMs, thus, provide a foundation for selecting suppliers, determining plants, and allocating production quantities in designing supply chains. They are commonly represented graphically (as shown in Fig. 1) or tabularly. Such representation formats do not allow the easy formulation of mathematical expressions. As a consequence, most reported supply chain models only consider the quantity relationships among product components while ignoring the interactive impact between final products and product components. In this study, we take into account not only the quantity relationships among components but also these between product components and final products, which give rise to interconnections among supply chain facilities, in particular component suppliers and final product producers.

Product P 1 in Fig. 1 is used to explain the quantity relationships between final products and product components. P 1 has three immediate child components, including c 1 (an assembly), c 2 (a part) and c 3 (a part). c 1 is formed by three parts: r 1 , r 2 ,a n dr 3 .T h ea ggregated quantities of these components are modeled as n r 1 →P 1 , n r 2 →P 1 , n r 3 →P 1 , n c 1 →P 1 , n c 2 →P 1 and n c 3 →P 1 .T a k en r 1 →P 1 as an example. It indicates that for each unit of P 1 to be produced in any plant, n r 1 →P 1 units of r 1 must be delivered by one or more suppliers. In the light of the above relationships, we formulate Constraint 13 to model the interconnections among supply chain facilities.

Facility pairwise constraints

In most available supply chain models, it is commonly assumed that all suppliers/plants serve all plants/DCs (e.g., [START_REF] Li | A strategic capacity allocation model for a complex supply chain: formulation and solution approach comparison[END_REF]. That is, each supply chain facility is connected with all immediate downstream (or upstream) facilities, indicating a full connectivity, as shown in Fig 2(a). (Note, the connection between two facilities at two adjacent levels of a supply chain is referred to as a facility pairwise relationship.) Although supply chain models formulated based on the full connectivity are easy to solve, they are not practical. This is because this assumption is often not realized in practice. In practice, due to the geographic and economical issues, a supply chain facility normally serves a part of, but not all, its downstream facilities, and receives products (or components) from a subset of its upstream facilities. In other words, each facility is connected with a subset of downstream (or upstream) facilities, indicating a partial connectivity in Fig. 2(b).

Let S, P,a n dD denote the sets of suppliers, plants, and DCs, respectively; p k : s j models that plant p k is served by supplier s j for one or more components. Accordingly, the facility pairwise relationship in Fig. 2(b) can be represented as six pairwise tuples: of the suppliers s m , s n ,...s p , m, n, p ∈ Jshould be opened; if DC d j is opened for one or more products, at least one of the plants p x , p y ,...p z , x, y, z ∈ K should be opened. In this study, we formulate Constraints 14 and 15 to capture these pairwise relationships by following the above logic. While Constraint 14 models the pairwise relationships between a plant and its suppliers, Constraint 15 captures these between a DC and the plants that can potentially serve it.

d 1 : p 1 , d 2 : p 1 , p 2 , d 3 : p 2 , p 3 , p 1 : s 1 , p 2 : s 1 ,

Supplier priority constraints

In practice, companies often take into account preference considerations when selecting suppliers. Preference considerations associate suppliers with different priorities. The implication is that given same or similar conditions (e.g., prices, delivery lead times), a plant may prefer supplier A, instead of supplier B. Suppose that both suppliers s 1 and s 2 can produce a same component for plant p 1 . In this regard, either s 1 ,o rs 2 , or both can be chosen as suppliers for meeting the demand of p 1 . Such general relationships among p 1 , s 1 ,a n ds 2 can be represented by three inequalities: -s 1 + p 1 ≥ 0, -s 2 + p 1 ≥ 0, and s 1 + s 2 -p 1 ≥ 0. Representing supplier selection in general, these inequalities, however, overlook the preference considerations (i.e., supplier priority).

Suppose p 1 prefers s 1 to s 2 . This preference can be modeled as s 1 -s 2 ≥ 0. This inequality ensures that s 1 always has a higher priority than s 2 . In line with this representation, we formulate Constraint 16 to model facilities' preference considerations when deciding on upstream facilities.

Incorporating such priority constraints in a model can ease problem solving as well. For example, suppose that six suppliers are able to provide same components at similar prices, delivery lead times, and quality. Three of them are required to fulfill a specific demand. Without considering supplier priority, there are C 3 6 = 60 possible combinations. If supplier priority (in this case, supplier 1 is preferred to supplier 3, and supplier 4 is preferred to supplier 2) is considered, only 10 possible combinations deserve further investigation, including suppliers 1, 4 and 5, suppliers 1, 4 and 6, suppliers 1, 5, and 6, suppliers 4, 5 and 6, suppliers 1, 3, and 4, suppliers 1, 3 and 5, suppliers 1, 3 and 6, suppliers 4, 2 and 1, suppliers 4, 2 and 5, and suppliers 4, 2 and 6.

ABC-based solution approach

The ABC algorithm is a typical example of the swarm intelligence. It simulates the intelligent foraging behavior of honey bees. In a bee hive, there are three types of bees: scout bees, employed bees, and onlooker bees. These bees interact with one another and the environment for finding the best food source around the bee hive. The process of the ABC algorithm consists of four distinct phases, including (1) initialization phase, (2) employed bee phase, (3) onlooker bee phase, and (4) scout bee phase, as shown in Fig. 3.

Involving different types of bees, each phase fulfills certain function. In the initialization phase, the scout bees are sent out to find the initial food sources around the bee hive at random. Subsequently, in the second phrase, the employed bees begin to exploit the food sources. During the course of exploiting the current food sources, the employed bees also try to explore better food sources which are nearby the current ones. Upon finding them, the employed bees replace the previous food sources with these new ones. This is termed as greedy selection in the second phase. During the third phase, the onlooker bees in the hive receive the food sources information from the employed bees and determine probabilistically to follow the employed bees to exploit and explore the food sources. Such determination is proportional to the richness of the food sources. Food sources are gradually consumed and finally exhausted. The exhausted food sources would be abandoned and replaced by other food sources found by the scout bees in the last phase.

Solution scheme

In the proposed MINLP model, the number of parameters, decision variables and constraints is very large. This is especially true when the supply chain network is complicated. When the network design problem has multiple stages and each echelon involves multiple parties, representing solutions using matrix scheme in the traditional way is not able to model the problem [START_REF] Yan | A strategic model for supply chain design with logical constraints: formulation and solution[END_REF]. Thus, in this study, a hybrid encoding system consisting of a binary-based encoding mechanism and a prioritybased encoding mechanism is proposed to represent the solution scheme. In accordance with the two encoding mechanisms, a solution, thus, has two sections. The binary-based encoding mechanism is employed to denote the open/close status of facilities in the supply chain network, while the priority-based encoding mechanism is employed to indicate the product transportation at each stage [START_REF] Altiparmak | A steady-state genetic algorithm for multi-product supply chain network design[END_REF]. Based on this representation, the correspondence between model elements and solution elements is established, as shown in Fig. 4. A supply chain corresponds to a solution, which can be divided into several fragments. Each fragment represents a supply chain stage, such as supply, production, and distribution. Each element of a fragment is termed as a cell. It consists of two properties: location and value. Representing the transportation of products among each pair of facilities in different echelons, the second section contains three fragments, each of which represents a stage. Three fragments correspond to the product transportation from supplier to plant, from plant to DC and from DC to customers, respectively. As an example, the priority-based encoding scheme for the third stage is shown in Fig. 6. In this example, there are three DCs and four customer zones. The location of the element at this stage indicates if it is a DC or a customer zone and the index of the corresponding facility. The value of the element at this stage is the priority of the corresponding facility. The combination of location and priority information guarantees the decoding of the solution scheme. The first and second stages are encoded in a same way. In accordance with the binary-based encoding for the first section and the prioritybased encoding for the second section, we encode the whole solution as shown in Fig. 7.

Node

Supplier

Plant DC 1 st stage 2 nd stage 3 rd stage 1 0 1 0 1 1 0 1 1 1 3 1 7 4 5 6 2 5 1 3 6 4 2 1 2 7 5 6 4 3 The decoding procedure begins from the third stage, through which the actual throughput for each DC is determined by referring to the open/close status of DCs in the first section of the solution. After that, the second stage is decoded considering the actual throughput of DCs acquired in last step. The first stage is decoded i nas a m ew a yw i t ht h ea c t u a lt h r o u g h p u to fD C s .

Constraint handling

The diverse constraints in the MINLP model originate from a variety of sources, such as demand, production capacity, and BOMs. These constraints are closely related to the set of possible solutions, from which an optimal one is to be found. Therefore, constraint handling suggests itself to be an important issue in solving the proposed model. In this study, we develop a two-step approach to deal with constraints. In the first step, constraints are classified into two types: capacity-related constraints and priority-related constraints. Capacity-related constraints are these constraints that are associated with entities themselves with respect to production capacity, set-up cost, DC throughput, and so on. Priority-related constraints relate to solution from a holistic view, thus modeling BOM constraints, supplier priorities and facility pairwise relationships. In other words, the capacity-related constraints have influence on entities themselves, whilst the priority-related constraints affect the structure of the supply chains.

In the second step, a constraint handling module is designed as a filter functioning at the beginning of the solving process and during the solving process. Initially, constraint rules are stored in a pool. Whenever a new solution is produced, the constraint handling module checks it against the available capacity and priority constraints in the pool. During the solving process, in order to facilitate a diversified evolution, it is necessary to include both feasible and infeasible solutions in the interactive search process. In this study, different penalty factors are assigned to the violations of capacity and priority related constraints at different degrees. Such a mechanism allows both the feasible and infeasible solutions to contribute to the diversified evolution.

Phase 1 -Random initialization

Using the ABC algorithm, in the first phase, a number of solutions are generated randomly as initial solutions. To produce feasible solutions and facilitate the convergent process, in the first section of the solution, the status settings of suppliers, plants, and DCs should consider the demand from customers. For the second section of the solution, the priority in each stage is randomly assigned considering the number of facilities in each stage.

Phase 2 -Employed bee phase

Once the initialization phase is done, employed bees are assigned with initially found food sources. When exploiting food sources, the employed bees also try to search around for better food sources. Such a procedure is implemented through a neighborhood search for each solution in the ABC algorithm. Fig. 8 illustrates how the neighborhood search is conducted. Given the current solution (S1 in the figure), another solution (S2 in the figure) is randomly chosen from the solution set. Then S1 interacts with S2 stage by stage proportionally. The new generated solution (New in the figure) is taken only if its fitness is over the current solution. This entails the greedy selection. Besides, a local search procedure is conducted for obtaining a new solution after the two available ones interact. In the local search, the value for each cell is regenerated considering its location and range. For instance, if it is located in the first section of the solution, the value could be only adopted as 1 or 0. Fig. 9 summarizes the local search in this phase.

Phase 3 -Onlooker bee phase

In this phase, the onlooker bees are assigned to explore and exploit the food sources by referring to the information that they have obtained from the employed bees. Their decision making is affected by the richness of food sources. The richer food source certainly could attract more onlooker bees, which facilitates the intensive search in the promising areas. In the ABC algorithm, such a

Supplier

Plant DC 1 st stage 2 nd stage 3 rd stage S1 1 0 1 0 1 1 0 1 1 1 3 1 7 4 5 6 2 5 1 3 6 4 2 1 2 7 5 6 4 3 1 0 0 1 0 0 S2 0 1 1 1 1 0 1 0 1 1 5 6 7 4 3 1 2 6 5 2 1 4 3 5 7 6 3 2 1 4

New 0 1 1 1 1 1 0 1 1 1 5 6 7 4 3 1 2 5 1 3 6 4 2 1 2 7 5 6 4 3 procedure is implemented using a roulette wheel selection mechanism considering the fitness of solutions. Once a decision is made, the onlooker bees become employed bees and conduct neighborhood search for better food sources.

Phase 4 -Scout bee phase

An abandonment criterion is necessary for escaping the local optima and for increasing the diversified search in the solutions space. In other words, if one solution can not be improved after a certain number of iterations, it is replaced by another solution, which is randomly found by scout bees.

An industrial example

Compared with other typical products (e.g., automobiles), bicycles are less complex, and do not involve too many product components. Moreover, the degree of product complexity and the number of product components suggest bicycles to be suitable industrial products for case studies [START_REF] Kotha | From mass production to mass customization: The case of the National Industrial Bicycle Company of Japan[END_REF][START_REF] Magill | Customizing bicycles through the integration of best practices[END_REF][START_REF] Ong | Web-based configuration design system for product customization[END_REF]. Thus, in this study, we use the bicycle to demonstrate the feasibility and potential of the proposed MINLP supply chain design model and the ABC-based solving approach. Fig. 10 shows a bicycle, the main components, and the general product structure. In general, a bicycle has three immediate child components, including a wheel (Wl) representing both the front and back wheels, mainframe (Mf), and paddle (Pa) denoting the paddle system including the chain. The mainframe is further formed by a saddle (Sa) and a frame (Fr). (Note, As Wl represents both the front and back wheels, each bicycle needs one Wl. In this regard, the unit per of each component in the figure is 1, thus being omitted.)

To produce and deliver bicycles, four layers are involved in the supply chain, including component suppliers, main plants, DCs, and CZs. While mainframes and bicycles are produced in the main plants, the other components are produced by suppliers. To validate the proposed model and ABC-based solution approach, we consider four scenarios, as shown in Table 1. These scenarios differ from each other in the supply chain structure with respect to the number of supply chain facilities. Along with the increase of the number of facilities in each layer, the problem complexity and size of these scenarios increases exponentially. To demonstrate the problem solving performance of the ABC-based approach by means of comparison, we adopt the GA and LINDO to solve the four scenarios. Both the ABC-based and GA-based approaches are implemented using Java.

In the first scenario, there are four suppliers (one for Wl, one for Pa, and two alternatives for Sa and Fr), three plants (for both Mf and Bc), three DCs and four CZs. The design capacities, production capacities, set-up costs, and demand of these facilities are given in Table 2. As shown in the table, customer demands (in K units) from the four CZs are 580, 430, 460, and 350, respectively. According to the definition in the proposed model, a supplier design

Table 3

Transportation costs between two facilities in scenario 1. S u p p l i e r / P l a n t# 1# 2# 3P l a n t / capacity indicates the probability that a supplier can undertake the task to design a component. For example, Sa/0.6 and Fr/0.85 represent that the probabilities for Supplier 2 to design Sa and Fr are 0.6 and 0.85, respectively. Similarly, the plant design capacities indicate the probabilities of the three plants to design mainframes and bicycles. Supplier/plant production capacities (in K units) represent the production capacities of the four suppliers and the three plants. Plant/DC set-up costs are the set-up costs (in K dollars) for plants/DCs to produce/deliver bicycles. DC maximum throughputs represent the maximum number (in K units) of bicycles that DCs can handle. In regard to supplier priority determination, supplier design capacities are used as reference. Higher priorities are assigned to suppliers with higher design capacities and lower priorities are assigned to suppliers with lower design capacities for the same components. For example, since Supplier #3 has higher design capacities for both Sa and Fr than Supplier #2, the priorities assigned to Supplier #3 for both components are higher than Supplier #2. In accordance with these priorities, Supplier #2 or Supplier #3 or both will be included in the possible supply chains that can be obtained. Since Supplier #1 and Supplier #4 are the only suppliers to produce Wl and Pa, respectively, they will be included in all possible supply chains.

D C# 1# 2# 3D C / C Z# 1# 2# 3# 4 #1 1 8 1 5 2 0 #1 8 6 9 #1 2 5 2 2 1 8 2 0 #2 1 2 1 4 1 6 #2 9 1 2 1 0 #2 1 7 2 1 2 4 2 2 #3 1 1 1 7 1 5 #3 7 1 0 1 3 #3 1 5 1 9 2 3 2 0 #4 8 1 3 1 1 S1 S2 S3 S4
The transportation costs (in K dollars) between two facilities at two adjacent layers are given in Table 3.

This scenario is first solved using LINDO. The optimal solution with respect to the facilities and their production/delivery quantities is given in Fig. 11. As shown, the supply chain involves all four suppliers, 2 plants (plants 1 and 2), 2 DCs (DCs 1 and 3), and all four CZs. In this solution, all customer requirements are met. Without exceeding its production capacity: 1820 K, Supplier #1 produces 900 K and 920 K wheels for plants 1 and 2, respectively. Supplier #2 delivers both 920 K saddles and 920 K frames to Plant #2; Supplier #3 transports 900 K saddles and 900 K frames to Plant #1. Supplier #4 produces 900 K and 920 K paddles for plants 1 and 2, respectively. While Supplier #1 fully utilizes its production capacity, suppliers 2, 3, and 4 do not fully utilize their capacities. Plants 1 and 2 deliver 900 K and 920 K bicycles to the two DCs. There is no production capacity left at Plant #1, whilst there is capacity left (i.e., 230 K) at Plant #2. DCs 1 and 3 deliver 460 K, 350 K, 580 K, and 430 K bicycles to the four CZs, respectively. The corresponding supply chain cost is 150260 unit of cost.

In using the proposed ABC-based approach to solve this scenario, the size of the colony is set as 80 including 40 employed bees and 40 onlooker bees. 40 initial solutions are generated during the process. Each operation is executed 10 times and the average value is used to denote the overall cost. The supply chain obtained consists of all four suppliers, two plants (plants 1 and 2), two DCs (DCs 1 and 2), and all four CZs, as shown in Fig. 12. Fig. 12 also shows the quantities shipped between two of these facilities. The cost for this supply chain is 151340 unit of cost. For this scenario, the GA-based approach acquires the same result as the ABCbased approach. Although the supply chain cost: 151340 obtained using the ABC-based approach is slightly higher than that obtained using LINDO: 150260, the increase percentage is only 0.7%. In this regard, the ABC-based solution approach is effective in finding near optimal solutions. (Note: As the ABC algorithm is meta-heuristics, the ABC-based solution approach is not able to find optimal solutions.) Fig. 13 shows the performance (with respect to the number of iterations and costs) of the ABC-based approach in solving all scenarios. The convergent curves fluctuate due partially to the newly introduced parameters, variables and constraints in scenarios 2, 3, and 4 and partially to the discreteness of decision variables. The ABC-based approach can find a satisfactory solution within 200 generations for Scenario 1. However, due to the increased problem complexity, it needs more generations to achieve a satisfactory convergence for Scenario 2, 3 and 4.

To compare the performance of LINDO, ABC-based and GAbased approaches in solving these four scenarios, the results in terms of the supply chain cost obtained are provided in Table 4. With the increase of the number of supply chain facilities, parameters, decision variables, and constraints in each scenario increase

Fig. 1 .

 1 Fig. 1. The BOM structure of product P 1 .

Fig. 2 .

 2 Fig. 2. Pairwise relationship among supply chain facilities.

Fig. 3 .

 3 Fig. 3. The process of the ABC algorithm.

Fig. 4 .Fig. 5 .

 45 Fig. 4. The correspondence between supply chains and solution elements.

Fig. 6 .

 6 Fig. 6. The priority-based encoding scheme for the third stage.

Fig. 7 .

 7 Fig. 7. The encoding scheme for the solution.

Fig. 8 .Fig. 10 .

 810 Fig. 8. The neighborhood search.

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. The supply chain obtained using LINDO.

Fig. 14 .

 14 Fig. 14. The comparison between the ABC-based and GA-based approaches.

Table 1

 1 The characteristics of the four scenarios.

		Number of	Number	Number	Number
	Scenario #	suppliers	of plants	of DCs	of CZs
	14	3	3	4
	28	5	6	8
	31 0	8	1 0	9
	41 5	1 0	1 2	1 0

Table 2

 2 Design/production capacities, set-up costs, and demand in scenario 1.

	Supply chain facility #	# 1	# 2	# 3	# 4
	Customer demand	580	430	460	350
	Supplier design capacity	Wl/0.9	Sa/0.6; Fr/0.85	Sa/0.8; Fr/0.9	Pa/0.75
	Supplier production capacity	1820	1900, 1900	2000, 2000	2100
	Plant design capacity	Mf/0.88; Bc/0.95	Mf/0.95; Bc/0.85	Mf/0.90; Bc/1
	Plant production capacity	900	1150	1380
	DC maximum throughput	1250	1080	1050
	Plant set-up cost	2500	2200	2800
	DC set-up cost	1200	1800	1000

substantially. This, in turn, increases problem size and complexity of the scenarios. LINDO cannot solve scenarios 3 and 4, which are of large problem size and high problem complexity. The ABCbased and GA-based approaches can solve all these scenarios with slightly increased solving time.

To further demonstrate the efficiency of the ABC algorithm, we compare the performance of the ABC-based and GA-based approaches in solving each scenario (in Fig. 14). As shown, the ABCbased approach exceeds the GA-based approach with better results and less fluctuation. There are several reasons for this. First, the ABC-based approach consists of other three phases besides the initialization phase and each phase can be treated as an individual process. Second, the greedy selection mechanism in the ABC-based approach facilitates the convergence of the solutions.

To conclude, the ABC-based approach can solve all the scenarios with good performance. For small size problems, LINDO may perform the best as it uses exact mechanism to search the whole solution space. However, for large scale problems, LINDO is not able to find the optimal solutions because of the large number of parameters, decision variables, and constraints. On the contrary, the ABC-based approach is more efficient in solving problems with many parameters, decision variables, and constraints. In addition, the unique structure and the inner mechanisms of the ABC-based approach guarantee the better performance against the GA-based approach.

Conclusions

This study developed an MINLP model and ABC-based solution approach for designing supply chains. In the broad sense, the model and solution approach form an expert system providing practical decision making support in supply chain design. First, unlike many available supply chain design models, the MINLP model takes into account three major supply chain stages: procurement, production, and distribution and their interactions and involves BOM-related constraints. Thanks to these unique model features, the expert system as a whole can help design realistic supply chains. Second, as today's companies develop product families, instead of single products, involving multiple products in the MINLP model also facilitates the design of practical supply chains. Third, thanks to the inclusion of constraints related to facility pairwise relationship and supplier priority, the expert system also contributes to designing practical supply chains. Last, as the ABC-based solution approach performs better than the commercial solver LINDO and GA-based approach, the system as a whole can help design more complex supply chains and reduce supply chain costs.

While this study has the above contributions and significance, like most of the reported studies, it has certain limitations. First, as an initial study investigating supply chain design by integrating all the three important stages, the MINLP model considers one period in the planning horizon, thus being deterministic by nature. Consequently, the model cannot deal with environmental dynamics especially with respect to demand variations, which can be observed in many industries nowadays. In this regard, the future efforts might be directed to develop a multi-period model which takes into account stochastic demand. Second, in formulat-ing the MINLP model, we set cost as a single criterion in the objective function. As in some situations, companies might pursue several performance measures at the same time, e.g., cost, profit. In this regard, the MINLP model is not applicable in the situations where companies consider other performance measures besides cost. Thus, the second potential avenue for future research is to develop a comprehensive model considering multi-objective, multi-period, and all the three stages. Third, there is also potential future research regarding the solution approach. Efforts might be made to integrate the ABC-based approach with other programming methods to further improve its performance. Fourth, an expert system in the narrow sense (i.e., a computer system), where the above comprehensive supply chain design model and a solution method are embedded as system components, might be developed. The system should be flexible enough such that it can be used to design supply chains in different situations. Average demand for product i at CZ m (units/period).

Appendix A. Notation used in the model

Appendix A.

Supplementary Materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.eswa.2015.12.035.


ARNAUD GUILLIN ♦ Patrick Cattiaux ♠

Feng-Yu Wang
email: wangfy@bnu.edu.cn

Liming Wu

26d10

Patrick Cattiaux
email: cattiaux@math.univ-toulouse.fr

Arnaud Guillin
email: aguillin@egim-mrs.fr

LYAPUNOV CONDITIONS FOR LOGARITHMIC SOBOLEV AND SUPER POINCAR É INEQUALITY

Keywords: Ergodic processes, Lyapunov functions, Poincaré inequalities, Super Poincaré inequalities, logarithmic Sobolev inequalities 47D07, 60G10, 60J60

. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).

Introduction.

During the last thirty years, a lot of attention has been devoted to the study of various functional inequalities and among them a lot of efforts were consecrated to the logarithmic Sobolev inequality. Our goal here will be to give a new and practical condition to prove logarithmic Sobolev inequality in a general setting. Our method being general, we will be able to get also conditions for Super-Poincaré, and by incidence to various inequalities as F -Sobolev or general Beckner inequalities. Our assumptions are based mainly on a Lyapunov type condition as well as a Nash inequality (for example valid in R d ). But let us make precise the objects and inequalities we are interested in. Let (X , F, µ) be a probability space and L a self adjoint operator on L 2 (µ), with domain D 2 (L), such that P t = e tL is a Markov semigroup. Consider then the Dirichlet form associated to L E(f, f ) := -Lf, f µ f ∈ D 2 (L)

Date: December 3, 2007.

with domain D(E). Throughout the paper, all test functions in an inequality will belong to D(L). It is well known that L possesses a spectral gap if and only if the following Poincaré inequality holds (for all nice f 's)

(1.1) Var µ (f ) := f 2 dµ - f dµ 2 ≤ C P E(f, f )
where C -1 P is the spectral gap. Note that such an inequality is also equivalent to the exponential decay in L 2 (µ) of P t . A defective logarithmic Sobolev inequality (say DLSI) is satisfied if for all nice f 's

(1.2) Ent µ (f 2 ) := f 2 log f 2 dµ -f 2 dµ log f 2 dµ ≤ C LS E(f, f ) + D LS f 2 dµ.
When D LS = 0 the inequality is said to be tight or we simply say that a logarithmic Sobolev inequality is verified (for short (LSI)). Dimension free gaussian concentration, hypercontractivity and exponential decay of entropy are directly deduced from such an inequality explaining the huge interest in it. Note that if a Poincaré inequality is valid, a defective DLSI, via Rothaus's lemma, can be transformed into a (tight) LSI. For all this we refer to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] or [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF]. Recently, Wang [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF] introduced so called Super-Poincaré inequality (say SPI) to study the essential spectrum: there exists a non-increasing β ∈ C(0, ∞), all nice f and all r > 0

(1.3) µ(f 2 ) ≤ r E(f, f ) + β(r) µ(|f |) 2 .
Wang moreover establishes a correspondence between this SPI and defective F -Sobolev inequality (F-Sob) for a proper choice of increasing F ∈ [0, ∞[ with lim ∞ F = ∞, i.e. for all nice f with µ(f 2 ) = 1

(1.4) µ(f 2 F (f 2 )) ≤ c 1 E(f, f ) + c 2 .
More precisely, if (1.4)holds for some increasing function F satisfying lim u→+∞ F (u) = +∞ and sup 0<u≤1 |uF (u)| < +∞, then (SPI) holds with β(u) = C 1 F -1 C 2 (1 + 1 u ) for some well chosen C 1 and C 2 . Conversely if a (SPI) (1.3) holds, defining

ξ(t) = sup u>0 1 u - β(u) ut ,
a (F-Sob) inequality holds with

F (u) = C 1 u u 0 ξ(t/2)dt -C 2
for some well chosen C 1 and C 2 . For details see [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF] Theorem 3.3.1 and Theorem 3.3.3. Note that these results are still available when µ is a non-negative possibly non-bounded measure.

In particular an inequality (DLSI) is equivalent to a (SPI) inequality with β(u) = ce c ′ /u . These inequalities and their consequences (concentration of measure, isoperimetry, rate of convergence to equilibrium) have been studied for diffusions and jump processes by various authors [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF][START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF][START_REF] Röckner | Supercontractivity and ultracontractivity for (non-symmetric) diffusion semi-groups on manifolds[END_REF][START_REF] Cattiaux | Trends to equilibrium in total variation distance[END_REF] under various conditions.

In this paper we shall use Lyapunov type conditions. These conditions are well known to furnish some results on the long time behavior of the laws of Markov processes (see e.g. [START_REF] Down | Exponential and uniform ergodicity of Markov processes[END_REF][START_REF] Fort | Subgeometric ergodicity of strong Markov processes[END_REF][START_REF] Douc | Subgeometric rates of convergence of f -ergodic strong Markov processes[END_REF]). The relationship between Lyapunov conditions and functional inequalities of Poincaré type (ordinary or weak Poincaré introduced in [START_REF] Röckner | Weak Poincaré inequalities and L 2 -convergence rates of Markov semigroups[END_REF]) is studied in details in the recent work [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF]. The present paper is thus a complement of [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF] for the study of stronger inequalities than Poincaré inequality. We will therefore suppose that (X , d) is a Polish space (actually a Riemannian manifold). Namely we will assume (L) there is a function W ≥ 1, a positive function φ > φ 0 > 0, b > 0 and r 0 such that

(1.5) LW W ≤ -φ + b 1 B(o,r 0 )
where B(o, r 0 ) is a ball, w.r.t. d, with center o and radius r 0 .

The main idea of the paper is the following one: in order to get some super Poincaré inequality for µ it is enough that µ satisfies some (SPI) locally and that there exists some Lyapunov function. In other words the Lyapunov function is useful to extend (SPI) on (say) balls to the whole space. General statements are given in section 2.

In particular on nice manifolds the riemanian measure satisfies locally some (SPI), so that an absolutely continuous probability measure will also satisfy a local (SPI) in most cases. The existence of a Lyapunov function allows us to get some (SPI) on the whole manifold. The aim of sections 3 and 4 is to show how one can build such Lyapunov functions, either as a function of the log-density or as a function of the riemanian distance. In the first case we improve upon previous results in [START_REF] Kusuoka | Some boundedness properties of certain stationary diffusion semigroups[END_REF][START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semi-groups[END_REF][START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF][START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] among others. In the second case we (partly) recover and extend some celebrated results: Bakry-Emery criterion for the log-Sobolev inequality, Wang's result on the converse Herbst argument. In particular we thus obtain similar results as Wang's one, but for measures satisfying sub-gaussian concentration phenomenon. This kind of new result can be compared to the recent [START_REF] Barthe | Mass transport and variants of the logarithmic Sobolev inequality[END_REF].

The main interest of this approach (despite the new results we obtain) is that it provides us with a drastically simple method of proof for many results. The price to pay is that the explicit constants we obtain are far to be optimal.

2. A general result.

Diffusion case.

To simplify we will deal here with the diffusion case: we assume that X = M is a d-dimensional connected complete Riemannian manifold, possibly with boundary ∂M . We denote by dx the Riemannian volume element and ρ(x) = ρ(x, o) the Riemannian distance function from a fixed point o. Let L = ∆ -∇V.∇ for some V ∈ W 1,2 loc such that Z = e -V dλ < ∞, and L is self adjoint in L 2 (µ) where dµ = Z -1 e -V dx. Note that in this case, we are in the symmetric diffusion case and the Dirichlet form is given by

E(f, f ) = |∇f | 2 dµ.
We shall obtain (SPI) by perturbing a known super Poincaré inequality.

Theorem 2.1. Suppose that the Lyapunov condition (L) is verified for some function φ such that φ(x) → ∞ as ρ(x, o) → ∞. Assume also that there exists T locally Lipschitz continuous on M such that dλ = exp(-T ) dx satisfies a (SPI) (1.3) with function β.

Then (SPI) holds for µ and some α : (0, ∞) → (0, ∞) in place of β.

More precisely, for a family of compact sets {A r ⊃ B(o, r 0 )} r≥0 such that A r ↑ M as r ↑ ∞, define for r > 0 :

Φ(r) := inf A c r φ, Φ -1 (r) := inf{s ≥ 0 : Φ(s) ≥ r}, g(r) := sup ρ(•,Ar)≤2 |V -T |, G(r) := sup ρ(•,Ar)≤2 |∇(V -T )| 2 , H(r) = Osc ρ(•,Ar)≤2 (V -T).
Then we may choose for s > 0, either (1) α(s) := inf ε∈(0,1)

5 2ε β εs 10 ∧ ε 16 ∧ 2(1 -ε) G • Φ -1 ( 4b ε ∨ 4 sε ) exp g • Φ -1 ( 4b ε ∨ 4 sε ) , or (2) α(s) := 2 exp 2H r 0 ∨ Φ -1 4 s ∨ bs 2 β s 8 e -H•Φ -1 ( 4 s ∨ bs 2 ) .
Proof. For r > r 0 it holds

f 2 dµ = A c r f 2 dµ + Ar f 2 dµ = A c r f 2 φ φ dµ + Ar f 2 dµ ≤ 1 Φ(r) f 2 φdµ + Ar f 2 dµ ≤ 1 Φ(r) f 2 -LW W dµ + b Φ(r) + 1
Ar f 2 dµ using our assumption (L).

The proof turns then to the estimation of the two terms in the right hand side of the latter inequality, a global term and a local one. For the first term remark, by our assumption on L that

f 2 -LW W dµ = ∇ f 2 W .∇W dµ = 2 f W ∇f.∇W dµ - f 2 |∇W | 2 W 2 dµ ≤ |∇f | 2 dµ - ∇f - f W ∇W 2 dµ which leads to (2.2) f 2 -LW W dµ ≤ |∇f | 2 dµ.
For the local term we will localize the (SPI) for the measure λ. To deduce a similar local inequality for µ we have two methods. For the first one we apply this inequality to f e -V /2+T /2 . It yields

Ar f 2 dµ = Ar f 2 e -V +T dλ ≤ 2s |∇f | 2 1I ρ(.,Ar)≤2 dµ + s 2 f 2 |∇(V -T )| 2 1I ρ(.,Ar)≤2 dµ +2s f 2 1I ρ(.,Ar)≤2 dµ + β(s)
|f | e (V -T )/2 1I ρ(.,Ar)≤2 dµ 2 so that if we choose s small enough so that sG(r) ≤ 2(1 -ε), we get

Ar f 2 dµ ≤ 2s |∇f | 2 dµ + (1 -ε) f 2 dµ + 2s f 2 dµ (2.4) + β(s) exp (g(r)) |f |dµ 2 .
Now combine (2.2) and (2.4). On the left hand side we get

1 - b Φ(r) + 1 (1 -ε + 2s) f 2 dµ .
For the coefficient to be larger than ε/2 it is enough that s ≤ ε/16 and Φ(r) ≥ 4b/ε. Assuming this in addition to sG(r) ≤ 2(1 -ε) we obtain that for such s > 0 and r,

µ(f 2 ) ≤ 2 ε 1 Φ(r) + 5s/2 µ(|∇f | 2 ) + 5 2ε β(s) exp (g(r)) µ(|f |) 2 .
If t is given, it remains to choose first

r = Φ -1 4b ε ∨ 4 εt ,
and then

s = εt 10 ∧ ε 16 ∧ 2(1 -ε) G(r) ,
to get the first α(t).

The second method is more naive but do not introduce any condition on the gradient of V .

Start with

f 2 1I Ar dµ = f 2 e -V +T 1I Ar dλ ≤ e -inf Ar (V -T ) f 2 1I Ar dλ ≤ e -inf Ar (V -T ) 2s |∇f | 2 1I ρ(.,Ar)≤2 dλ + +2s f 2 1I ρ(.,Ar)≤2 dλ + β(s) |f | 1I ρ(.,Ar)≤2 dλ 2 ≤ e -inf Ar (V -T ) e sup ρ(.,Ar )≤2 (V -T ) 2s |∇f | 2 1I ρ(.,Ar)≤2 dµ + +2s f 2 1I ρ(.,Ar)≤2 dµ + β(s)e sup ρ(.,Ar )≤2 (V -T ) |f | 1I ρ(.,Ar)≤2 dµ 2 ≤ e Osc ρ(.,Ar)≤2 (V -T ) 2s |∇f | 2 dµ + 2s f 2 dµ + +e 2Osc ρ(.,Ar)≤2 (V -T ) β(s) |f |dµ 2 .
If we combine the latter inequality with (2.2) and denote s ′ = 2s e Osc ρ(.,Ar)≤2 (V -T ) we obtain

1 - bs ′ Φ(r) f 2 dµ ≤ ≤ s ′ + 1 Φ(r) |∇f | 2 dµ + e 2Osc ρ(.,Ar )≤2 (V -T ) β(s ′ e -Osc ρ(.,Ar)≤2 (V -T ) /2) |f |dµ 2 .
Hence, if we choose , r = Φ -1 ( 4 s ∨ bs 2 ) and s ′ = s/4 we obtain the second possible α(s).

Remark 2.5.

(1) The previous proof extends immediately to the general case of a "diffusion" process with a "carré du champ" which is a derivation, i.e. if [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF] for more details on this framework).

E(f, f ) = Γ(f, f )dµ for a symmetric Γ such that Γ(f g, h) = f Γ(g, h) + gΓ(f, h) (see
(2) For a general diffusion process, say with a non constant diffusion term, as noted in the previous remark we have to modify the energy term so that it is no further difficulty and there are numerous examples where condition (L) is verified, i.e. consider L = a(x)∆ -x.∇ where a is uniformly elliptic and bounded (consider W = e a|x| 2 so φ(x) = c|x| 2 ). But our method as expressed here relies crucially on the explicit knowledge of V . Note however, that for the second approach, only an upper bound on the behavior of V over, say, balls is needed, which can be made explicit in some cases. ♦ Remark 2.6. We may for instance choose A r = Vr :

= {x ; |V -T |(x) < r} (i.e. a level set of |V -T |) provided |V -T |(x) → +∞ as ρ(o, x) → +∞.
However we have to look at an enlargement V r+2 = {x ; , ρ(x, Vr ) < 2} (not the level set of level r + 2).

If we want to replace V r+2 by the level set Vr+2 we have to modify the proof, choosing some ad-hoc function ψ which is no more 1-Lipschitz. It is not difficult to see that we have to modify (2.3) and what follows, replacing 1 (the 1 of 1-Lipschitz) by sup Vr+2 |∇(V -T )| 2 . So we have to modify the condition on s in (1) of the previous theorem, i.e.

(2.7)

2 inf ( Vr) c φ ≤ ε s ≤ 2 inf ( Vr) c φ + 2 (1 -ε) sup Vr+2 |∇(V -T )| 2 ,
i.e. we get the same result as (1) but with Φ(r) = inf ( Vr) c φ, g(r) = r + 2 and

G(r) = sup Vr+2 |∇(V -T )| 2 .
The second case (2) cannot (easily) be extended in this direction. ♦

Actually one can derive a lot of results following the lines of the proof, provided some "local" (SPI) is satisfied. Here is the more general result in this direction.

Theorem 2.8. In theorem 2.1 define λ Ar (f ) = λ(f 1 I Ar ) where A r is an increasing family of open sets such that r A r = M . Given two such families A r ⊆ B r , assume that for all r large enough the following local (SPI) holds,

(2.9) λ Ar (f 2 ) ≤ sλ Br (|∇f | 2 ) + β r (s) (λ Br (|f |)) 2 .
Then the conclusions of theorem 2.1 are still true if we replace ρ(., A r ) ≤ 2 by B r and β(s) by β r(s) (s) with r(s) = Φ -1 4b ε ∨ 4 εs for each given ε in case (1) and r(s

) = Φ -1 ( 4 s ∨ bs 2 ) in case (2).

General case.

We consider here the case of general Markov processes on a Manifold M , with a particular care to jump processes. Indeed, a crucial step in the previous proof is to prove (2.2) and it has been made directly taking profit of the gradient structure, but it can be proved in greater generality. However the second part relying on a perturbation approach seems more difficult. We therefore introduce a local Super-Poincaré inequality.

Theorem 2.10. Suppose that the Lyapunov condition (L) is verified for some function φ such that φ(x) → ∞ as ρ(x, o) → ∞. Assume also the following family of local Super Poincaré inequality holds for µ: for a family of compact sets {A r ⊃ B(o, r 0 )} r≥0 such that A r ↑ M as r ↑ ∞, there exists β(r, •) such that for all nice f and s > 0

(2.11) µ(f 2 1 Ar ) ≤ sE(f, f ) + β(r, s)µ(|f |) 2 .
Then, denoting

Φ(r) := inf A c r φ, Φ -1 (r) := inf{s ≥ 0 : Φ(s) ≥ r},
µ verifies a Super Poincaré inequality with function

α(s) = β(Φ -1 (2/s), s/2).
The proof relies on a simple optimization procedure between the weighted energy term and the local Super Poincaré inequality. We then only have to prove (2.2) which is done by the following large deviations argument.

Lemma 2.12. For every continuous function U ≥ 1 such that -LU/U is bounded from below,

(2.13) - LU U f 2 dµ ≤ E(f, f ), ∀f ∈ D(E).
Proof. Remark that

N t = U (X t ) exp - t 0 LU U (X s )ds is a P µ -local martingale. Indeed, let A t := exp - t 0 LU U (X s
)ds , we have by Ito's formula,

dN t = A t [dM t (U ) + LU (X t )dt] - LU U (X t )A t U (X t )dt = A t dM t (U ).
Now let β := (1 + U ) -1 dµ/Z (Z being the normalization constant). (N t ) is also local martingale, then a super-martingale w.r.t. P β . We so get

E β exp - t 0 LU U (X s )ds ≤ E β N t ≤ β(U ) < +∞.
Let u n := min{-LU/U, n}. The estimation above implies

F (u n ) := lim sup t→∞ 1 t log E ν exp - t 0 u n (X s )ds ≤ 0.
F (u n ) ≥ sup{ν(u n ) -I(ν|µ); ν ∈ M 1 (E)}.
Thus u n dν ≤ I(ν|µ), which yields to (by letting n → ∞ and monotone convergence)

(2.14) - LU U dν ≤ I(ν|µ), ∀ν ∈ M 1 (E).
That is equivalent to (2.13) by the fact that

E(|f |, |f |) ≤ E(f, f ) for all f ∈ D(E).
We will discuss examples on jump processes in future research, see however [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF]Th. 3.4.2] for results in this direction.

3.

Examples in R n .

We use the setting of the subsection 2.1 (or of the remark 2.5) but in the euclidean case M = R n for simplicity. Hence in this section λ is the Lebesgue measure, i.e we have T = 0. Recall that dµ = Z -1 e -V dx. It is well known that λ satisfies a (SPI) with

β(s) = c 1 + c 2 s -n/2
. However it is interesting to have some hints on the constants (in particular dimension dependence). It is also interesting (in view of Theorem 2.8) to prove (SPI) for subsets of R n . Hence we shall first discuss the (SPI) for λ and its restriction to subsets. Since we want to show that the Lyapunov method is also quite quick and simple in many cases, we shall also recall the quickest way to recover these (SPI) results.

P N t f ∞ ≤ C 1 t -n/4 f L 2 (A,dx) , (3.1.2) (provided n > 2) for all f ∈ C ∞ ( Ā), f 2 L 2n/n-2 (A,dx) ≤ C 2 A |∇f | 2 dx + A f 2 dx , (3.1.3) for all f ∈ C ∞ ( Ā), f 2+4/n L 2 (A,dx) ≤ C 3 A |∇f | 2 dx + A f 2 dx f 4/n L 1 (A,dx) , (3.1
.4) the (SPI) inequality A f 2 dx ≤ s A |∇f | 2 dx + β(s) A |f |dx 2 holds with β(s) = C 4 (s -n/2 + 1).
Furthermore any constant C i can be expressed in terms of any other C j and the dimension n.

These results are well known. they are due to Nash, Carlen-Kusuoka-Stroock ( [START_REF] Carlen | Upper bounds for symmetric Markov transition functions[END_REF]) and Davies, and can be found in [START_REF] Davies | Heat kernels and spectral theory[END_REF] section 2.4 or [START_REF] Saloff-Coste | Aspects of Sobolev type inequalities[END_REF]. Generalizations to other situations (including general forms of rate functions β) can be found in [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF] section 3.3.

If A = R n (3.1.1) holds (for all t) with C = (2π) -n/2 and α = n/2, yielding a (SPI) inequality with

(3.2) β(s) = c n s -n/2 = 1 4π n/2 s -n/2 ,
which is equivalent, after optimizing in s, to the Nash inequality

(3.3) f 2+4/n 2 ≤ C n |∇f | 2 dx f 4/n 1 , with C n = 2(1 + 2/n) (1 + n/2) 2/n (1/8π) n/4 .
For nice open bounded domains in R n , as we consider here, (3.1.2) is a well known consequence of the Sobolev inequality in R n (see e.g. [START_REF] Davies | Heat kernels and spectral theory[END_REF] Lemma 1.7.11 and note that the particular cases n = 1, 2 can be treated by extending the dimension (see [START_REF] Davies | Heat kernels and spectral theory[END_REF] theorem 2.4.4)). But we want here to get some information on the constants. In particular, when A is the level set Vr we would like to know how β r depends on r.

Remark 3.4. If n = 1, we have an explicit expression for p N t when A = (0, r), namely

p N t (x, y) = (2πt) -n/2 k≥0 exp - (x -y -2kr) 2 2t + exp - (x + y + 2kr) 2 2t .
It immediately follows that (3.5) sup

x,y∈(0,r)

p N t (x, y) ≤ (2πt) -n/2   2 + k≥1 exp - ((2k -1)r) 2 2t + exp - (2kr) 2 2t   ,
so that, using translation invariance, for any interval A of length r > r 0 and for 0 < t ≤ 1, sup x,y∈A p N t (x, y) ≤ c(r 0 ) (2πt) -n/2 . Hence (3.1.1) is satisfied, and a (SPI) inequality holds in A with the same function β r (s) = c B (s -n/2 + 1) independently on r > r 0 . By tensorization, the result extends to any cube or parallelepiped in R n with edges of length larger than r 0 . ♦ If we replace cubes by other domains, the situation is more intricate. However in some cases one can use some homogeneity property. For instance, for n > 2 we know that (3.1.2) holds for the unit ball with a constant C 2 (for n = 2 we may add a dimension and consider a cylinder B 2 (0, 1) ⊗ R as in [START_REF] Davies | Heat kernels and spectral theory[END_REF] theorem 2.4.4). But a change of variables yields

f 2 L 2n/n-2 (B(0,r),dx) ≤ C 2 B(0,r) |∇f | 2 dx + r -2 B(0,r) f 2 dx ,
so that for r ≥ 1 (3.1.2) holds in the ball of radius r with a constant C 2 independent of r.

The previous argument extends to A = Vr provided for r ≥ r 0 , Vr is star-shaped, in particular it holds if V is convex at infinity. This is a direct consequence of the coarea formula (see e.g [START_REF] Evans | Measure theory and fine properties of functions[END_REF] proposition 3 p.118). Indeed if f has his support in an annulus r 0 < r 1 < V (x) < r 2 the surface measure on the level sets Vr is an image of the surface measure on the unit sphere. This is immediate since the application x → (V (x), x |x| ) is a diffeomorphism in this annulus. Hence for such f 's the previous homogeneity property can be used. For a given r > r 0 large enough, it remains to cover Vr by such an annulus and a large ball (such that the ball contains Vr 0 and is included in Vr ) and to use a partition of unity related to this recovering. We thus get as before that for r large enough, C 2 can be chosen independent of r.

For general domains A, recall that (3.1.2) holds true if A satisfies the "extension property" of the boundary, i.e. the existence of a continuous extension operator

E : W 1,2 (A) → W 1,2 (R n ).
If this extension property is true, (3.1.2) is true in A with a constant C 2 depending only on n and the operator norm of E (see [START_REF] Davies | Heat kernels and spectral theory[END_REF] proposition 1.7.11). If A = Vr is bounded, as soon as ∇V does not vanish on ∂A, the implicit function theorem tells us that for all x ∈ ∂A one can find an open neighborhood v x of x, an index i x and a 2-Lipschitz function φ x defined on v x such that

v x ∩ A = v x ∩ {φ(y 1 , ..., y ix-1 , y ix+1 , ..., y n ) < y ix } .
To this end choose i x such that |∂ ix V |(x) ≥ |∂ j V |(x) for all j = 1, ..., n, so that, for y ∈ ∂A neighboring x, 2|∂ ix V |(y) ≥ |∂ j V |(y), and the partial derivative of the implicit function φ given by the ratio ∂ j V (y)/∂ ix V (y) is less than 2 in absolute value.

By compactness we may choose a finite number Q of points such that j=1,...,Q v x j ⊃ ∂A. Hence we are in the situation of [START_REF] Davies | Heat kernels and spectral theory[END_REF] proposition 1.7.9. This property implies the extension property but with some extension operator E whose norm depends on two quantities : first the maximal ε > 0 such that for all x ∈ ∂A, B(x, ε) ⊆ v x j for some j = 1, ..., Q; second, the maximal integer N such that any x ∈ ∂A belongs to at most N such v x j 's. This is shown in [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF] p.180-192. Actually an accurate study of Stein's proof (p. 190 and 191) shows that E ≤ C(n) (N/ε) (recall that we have chosen φ 2-Lipschitz). Now assume that

(3.6) there exist R > 0, v > 0 , k ∈ N such that for |x| ≥ R , |∇V (x)| ≥ v > 0 .
Then it is easy to check that for

A = Vr it holds ε ≤ ε 0 = c(v, R, n) θ -1 (r) with (3.7) θ(r) = sup x∈∂ Vr max i,j=1,...,n ∂ 2 V ∂x i ∂x j (x) .
But ε 0 being given, it is well known that one can find a covering of A by balls of radius ε 0 /2 such that each x ∈ Vr belongs to at most N = c n such balls for some universal c large enough. Hence N can be chosen as a constant depending on the dimension only. It follows that

β r (s) = C(n) θ n (r) (1 + s -n/2 ) .
For the computation of β r we used [START_REF] Davies | Heat kernels and spectral theory[END_REF]

= c(n) C n/4
2 . Finally the proof of [START_REF] Davies | Heat kernels and spectral theory[END_REF] theorem 2.4.6 gives β(s) = C 2 1 (1 + (s/2) -n/2 ).

Proposition 3.8 gives of course the worse result and in many cases one can expect a much better behavior of β r as a function of r. In particular in the homogeneous case we know the result with a constant independent of r.

Remark 3.9. Another possibility to get (SPI) in some domain A, is to directly prove the Nash inequality (3.1.3). One possible way to get such a Nash inequality is to prove some Poincaré-Sobolev inequality. The case of euclidean balls is well known.

According to [START_REF] Saloff-Coste | Aspects of Sobolev type inequalities[END_REF] theorem 1.5.2, for n > 2, with p = 2 and s = 2n/(n -2) = 2 * therein, for all r > 0 and all ball B r with radius r, if λ r is the Lebesgue measure on B r and fr = (1/V ol(B r )) Br f dx we have

(3.10) λ r |f -fr | 2n n-2 n-2 2n ≤ C n λ r |∇f | 2 1 2 ,
so that using first Minkowski, we have

(3.11) λ r |f | 2n n-2 n-2 2n ≤ C n λ r |∇f | 2 1 2 + 1 V ol(B r ) λ r (|f |) ,
and finally using Hölder inequality and Cauchy-Schwarz inequality we get the local Nash inequality

λ r (|f | 2 ) ≤ (λ r (|f |)) 4/(n+2) C n λ r |∇f | 2 1 2 + 1 V ol(B r ) λ r (|f |) 2n/(n+2) , (3.12) ≤ (λ r (|f |)) 4/(n+2) C n λ r |∇f | 2 1 2 + 1 V ol(B r ) λ r |f | 2 1 2 2n/(n+2)
.

Again, for r > r 0 we get a Nash inequality hence a (SPI) inequality independent of r with β r (s) = c n (1 + s -n/2 ). Notice that (3.10) is scale invariant, i.e. if it holds for some subset A, it holds for the homotetic rA (r > 0) with the same constants. That is why the constants do not depend on the radius for balls. If we replace a ball by a convex set, the classical method of proof using Riesz potentials (see e.g. [START_REF] Saloff-Coste | Aspects of Sobolev type inequalities[END_REF] or [START_REF] Davies | Heat kernels and spectral theory[END_REF] lemma 1.7.3) yields a similar results but with an additional constant, namely diam n (A)/V ol(A), so that if V is a convex function the constant we obtain with this method in (3.10) for Vr may depend on r.

Actually the Sobolev-Poincaré inequality (3.10) extends to any John domain with a constant C depending on the dimension n and on the John constant of the domain. This result is due to Bojarski [START_REF] Bojarski | Remarks on Sobolev imbedding inequalities[END_REF] (also see [START_REF] Hajlasz | Sobolev inequalities, truncation method, and John domains[END_REF] for another proof and [START_REF] Buckley | Sobolev-Poincaré implies John[END_REF] for a converse statement). Actually a John domain satisfies some chaining (by cubes or balls) condition which is the key for the result (see the quoted papers for the definition of a John domain and the chaining condition). But an explicit calculation of the John constant is not easy. ♦ 3.2. Typical Lyapunov functions and applications. We here specify classes of natural Lyapunov function: function of the potential or of the distance. As will be seen, it gives new practical conditions for super-Poincaré inequality and for logarithmic Sobolev inequality. First, since W ≥ 1 we may write W = e U so that condition (L) becomes (3.13) ∆U + |∇U | 2 -∇U.∇V + φ ≤ 0 "at infinity".

3.2.1.

Lyapunov function e aV . Test functions e aV for a < 1 are quite natural in that they are the limiting case for the spectral gap (see [START_REF] Bakry | Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré[END_REF]). Indeed, µ(e aV ) is finite if and only if a < 1 and a drift condition such that LW ≤ -λW + b1 C formally implies by integration by µ, that µ(W ) is finite. So in a sense, e aV are the "largest" possible Lyapunov functions.

Hence, if W = e aV , LW W = a ∆V -(1 -a)|∇V | 2 .
Introduce the following conditions (3.14.1) V (x) → +∞ as |x| → +∞, (3.14.2) there exist 0 < a 0 < 1, a non-decreasing function η with η(u) → +∞ as u → +∞ and a constant b 0 such that

(1 -a 0 )|∇V | 2 -∆V ≥ η(V ) + b 0 1I |x|<R , (3.14.3) lim sup |x|→+∞ η(V (x))/|∇V (x)| 2 < +∞.
Then for 0 < a < a 0 condition (L) is satisfied with

φ = a(a 0 -a)|∇V | 2 + aη(V ) . In addition inf ( Vr) c φ(V ) ≤ c sup Vr+2 |∇V | 2 .
Following remark 2.6 (we choose arbitrarily ε = 1/2 here) we obtain for some constant c

(3.15) f 2 dµ ≤ s |∇f | 2 dµ + c   1 + sup V2+η -1 (c/s) |∇V | 2   n/2 e η -1 (c/s) |f |dµ 2 .
We thus clearly see that to get an explicit (SPI) we need to control the gradient ∇V on the level sets of V .

If instead of using theorem 2.

f 2 dµ ≤ s |∇f | 2 dµ + C θ n (η -1 (c/s)) e 2η -1 (c/s) 1 + s -n/2 e nη -1 (c/s)/2 |f |dµ 2 .
We have obtained Theorem 3.17. Assume that (

3) are satisfied. Then µ will satisfy a (SPI) inequality with function β in one of the following cases (3.17.1)

for |x| large enough, |∇V |(x) ≤ γ(V (x)) and β(s) = C(1 + e η -1 (c/s) γ n (η -1 (c/s))), (3.17.2) for |x| large enough ∂ 2 V ∂x i ∂x j (x) ≤ θ(V (x)) and
β(s) = C 1 + θ n (η -1 (c/s)) s -n/2 e (n+4)η -1 (c/s)/2 .
Remark 3.18. If η(u) = u we thus obtain that µ satisfies a (defective) logarithmic Sobolev inequality provided either γ(u) ≤ e Ku or θ(u) ≤ e Ku . But (3.14.1) and (3.14.2) imply that µ satisfies a Poincaré inequality (see e.g. [2] corollary 4.1). Hence using Rothaus lemma we get that µ satisfies a (tight) logarithmic Sobolev inequality. Conditions (3.14.1) and (3.14.2), with η(u) = u, appear in [START_REF] Kusuoka | Some boundedness properties of certain stationary diffusion semigroups[END_REF] where the authors show that they imply the hypercontractivity of the associated symmetric semi-group, hence a logarithmic Sobolev inequality by using Gross theorem. In particular the additional assumptions on the first or the second derivatives do not seem to be useful. Another approach using Girsanov transformation was proposed in [START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semi-groups[END_REF] for a 0 = 1/2, again without the technical assumptions on the derivatives. This approach extends to more general processes with a "carré du champ".

Here we directly get the logarithmic Sobolev inequality without using Gross theorem, but with some conditions on V .

The advantage of theorem 3.17 is that it furnishes an unified approach of various inequalities of F -Sobolev type. In [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF] conditions (3.14.1) and (3.14.2) are used (for particular η's) to get the Orlicz-hypercontractivity of the semi-group hence a F -Sobolev inequality thanks to the Gross-Orlicz theorem proved therein. The use of this theorem requires some quite stringent conditions on η but covers the case η(u) = u α for 1 < α < 2, yielding a F -Sobolev inequality for F (u) = log α + (u) (more general F are also studied in [START_REF] Barthe | Isoperimetry between exponential and Gaussian[END_REF] section 7). Note that in theorem 3.17 we do no more need the restriction α < 2, but we need some control on the growth on γ or θ, namely we need again γ(u) ≤ e Ku (the same for θ). We also obtain a larger class of F -Sobolev inequalities thanks to the correspondence between F -Sobolev and (SPI) recalled in the introduction. The reader is referred to [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF] Proof. In all the proof D will be an arbitrary positive constant whose value may change from place to place. All the calculations are assuming that |x| is large enough. Consider first case 1. Choosing a small enough and using lemma 3.20, we see that φ Case 2 is similar but improving the lower bound for φ.

Indeed since D|x| ≥ V 1/b ′ (x), φ(x) ≥ D V b ′ +b-2 b ′ (
The first statement of the theorem is reminiscent to Wang's improvement of the Bakry-Emery criterion, namely if e (-ρ+ε)|x-y| 2 µ(dx) µ(dy) < +∞, µ satisfies a logarithmic Sobolev inequality. Our statement is weaker since we are assuming some uniform behavior. The third statement can thus be seen as an extension of Wang's result to the case of F -Sobolev inequalities interpolating between Poincaré inequality and log-Sobolev inequality. These inequalities are related to the Latala-Oleskiewicz interpolating inequalities [START_REF] Lata | Between Sobolev and Poincaré. in geometric aspects of Functional Analysis[END_REF], see [START_REF] Barthe | Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry[END_REF] for a complete description. It should be interesting to improve (3) in the spirit of Wang's concentration result. See [START_REF] Kolesnikov | Modified log-Sobolev inequalities and isoperimetry[END_REF][START_REF] Barthe | Mass transport and variants of the logarithmic Sobolev inequality[END_REF] for a tentative involving modified log-Sobolev inequalities introduced in [START_REF] Gentil | Modified logarithmic Sobolev inequalities and transportation inequalities[END_REF] and mass transport.

The general manifold case

In fact as one guesses, the main point is to get the additional Super Poincaré inequality, local as developed in Section 3.1, or global (and then using the localization technique already mentioned). It is of course a fundamental field of research which encompasses the scope of the present paper. We may however use our main results Theorem 2.1 and Theorem 2.8, with the same Lyapunov functionals as developed in Sections 3.2.1 and 3.2.2, replacing of course the euclidean distance by the Riemannian distance (w.r.t a fixed point), at least in two main cases. According to [START_REF] Croke | Some isoperimetric inequalities and eigenvalue estimates[END_REF], if the injectivity radius of M is positive then (1.3) holds for T = 0 and β(s) = c 1 + c 2 s -d/2 for some constants c 1 , c 2 > if in particular the injectivity is infinite, then one may take c 1 = 0, [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF] page 225.

Next, if the Ricci curvature of M is bounded below, then by [START_REF] Wang | Functional inequalities for empty essential spectrum[END_REF] Theorem 7.1, there exists c 1 , c 2 > 0 such that (1.3) holds for T = c 1 ρ and β(s) = c 2 s -d/2 . For simplicity, throughout this section we assume that (H) The injectivity radius of M is positive.

4.1.

Lyapunov condition e aV . In this context, one may readily generalizes the result of Theorem (3.17) for the first case (3.17.1), with the euclidean distance replaced by the Riemannian one, assuming (3.14.1), (3.14.2) and (3.14.3).

(s) = C(1 + e η -1 (c/s) γ n (η -1 (c/s))).
The second point of Theorem 3.17 is more delicate as it relies on finer conditions on the manifold and the potential, it should however be possible to give mild additional assumptions ensuring such a result (for instance the so called "rolling ball condition"). Remark that it extends to the manifold case Kusuoka-Stroock's result (giving life to Remark (2.49) in their paper). for some constant c > 0 provided ψ is positive for large ρ.

We may then extend Lemma 3.20 in the manifold context. Proof. For x ∈ M , let ξ : [0, ρ(x)] → M be the minimal geodesic from o to x. Let g(t) = t ∇ρ, ∇V (ξ t ), t ≥ 0. We have g ′ (t) = ∇ρ, ∇V (ξ t ) + tHess V (∇ρ, ∇ρ)(ξ t ) ≥ c 0 t + dV(ξ t ) dt .

This implies the desired assertion by integrating both sides on [0, ρ(x)].

We may thus state Proof. The proof follows exactly the same line than in the flat case so that case 1 and case 2 follows once it is noted that since Hess V ≥ 0 implies the convexity of the level sets Vr , we know that some Nash inequality holds on Vr according to the discussion in the previous subsection and the boundedness of these level sets ensured by our hypotheses on V .

Let us now consider Case 3. Since b = 2, our hypothesis implies that for 2a < ε, φ ≥ DV . But (3.19) and Hess ρ 2 ≥ 2 on Cartan-Hadamard manifolds imply that the level sets of x → H = V + c 0 ρ 2 /2 are convex. Since V ≥ Dρ 2 , one has cr ≤ V ≤ r on Hr . We may thus mimic case 1, just replacing Vr by Hr . Case 4 is similar to the previous one just improving the bound on φ as in case 2.

Remark 4.4. Remark that in full generality, according to [START_REF] Wang | Functional inequalities on arbitrary Riemannian manifolds[END_REF] Theorem 1.2 and the recent paper [START_REF] Chen | Construction of larger Riemannian metrics with bounded sectional curvatures and applications[END_REF], there always exists T ∈ C ∞ (M ) such that dλ := e -T (x) dx satisfies a logarithmic Sobolev inequality hence (SPI) with β(s) = e s -1 . Of course for practical purposes, this very general fact is not completely useful since T is unknown.

  On the other hand by the lower bound of large deviation in [31, Theorem B.1, Corollary B.11] and Varadhan's Laplace principle, defining I(ν|µ) = E( dν/dµ, dν/dµ)

Proposition 3 . 8 .

 38 If (3.6) is satisfied, the (SPI) (3.1.4) holds with A = Vr , θ defined by (3.7) and

3 . 2 . 2 .

 322 section 5.7 for related results in the ultracontractive case. ♦ Lyapunov function e a|x| b . If we try to use W = e a|x| b we are led to choose φ(x) = ab|x| b-2 ψ(x) with ψ(x) = x.∇V -n + (b -2) + ab|x| b provided the latter quantities are bounded from below by a positive constant for |x| large enough. Introduce now the standard curvature assumption (3.19) for all x, HessV (x) ≥ c 0 Idfor some ρ ∈ R. This assumption allows to get some control on x.∇V namely Lemma 3.20.If (3.19) holds, x.∇V (x) ≥ V (x) -V (0) + c 0 |x| 2 /2.Proof. Introduce the function g(t) = t x.∇V (tx) defined for t ∈ [0, 1]. (3.19) implies that g ′ (t) ≥ x.∇V (tx) + tc 0 |x| 2 and the result follows by integrating the latter inequality between 0 and 1. We may thus state Proposition 3.21. Assume that (3.19) is satisfied. Then one can find positive constants c, C such that µ satisfies some (SPI) with function β (given below for s small enough) in the following cases (3.21.1)c 0 ≥ 0, V (x) ≥ c ′ |x| b for |x| large enough some c ′ > 0 and b > 1, β(s) = C e c(1/s) b 2((b-1)∧1) . (3.21.2) c 0 ≥ 0, d ′ |x| b ′ ≥ V (x) ≥ c ′ |x| b for |x| large enough some d ′ , c ′ > 0 and b ′ ≥ b > 1, β(s) = C e c(1/s) b ′ b ′ +b-2 . (3.21.3) c 0 ≤ 0, for |x| large enough, V (x) ≥ (ε -c 0 /2)|x| 2 for some ε > 0, and β(s) = C e c(1/s) . (3.21.4) c 0 ≤ 0, for |x| large enough, d ′ |x| b ′ ≥ V (x) ≥ c ′ |x| b and β as in (3.21.2).

  (x) ≥ D |x| b-2 V (x). If b ≥ 2 we thus have φ(x) ≥ D V (x) while for b < 2, φ(x) ≥ D V 2(b-1) b (x) forlarge |x| according to the hypothesis. For φ to go to infinity at infinity, b > 1 is required. In particular on the level sets Vr we have either φ(x) ≥ D r or φ(x) ≥ D r 2(b-1)/b . Now since the level sets Vr are convex, we know that some Nash inequality holds on Vr according to the discussion in the previous subsection. We may thus use theorem 2.8 in the situation (2) of theorem 2.1. Choosing s = d/r or s = d/r 2(b-1) b for some well chosen d yields the result with an extra factor s -k for some k > 0. This extra term can be skipped just changing the constants in the exponential term.

( 2 )

 2 If (3.21.1) holds with b = 2, µ satisfies a logarithmic Sobolev inequality. In particular if ρ > 0, µ satisfies a logarithmic Sobolev inequality (Bakry-Emery criterion). (3) If (3.21.1) holds for some 1 < b < 2, µ satisfies a F -Sobolev inequality with F (u) = log 2(1-(1/b)) +

Theorem 4 . 1 .

 41 Assume (H) and that (3.14.1), (3.14.2), (3.14.3) are satisfied. Suppose moreover that for large ρ, |∇V |(x) ≤ γ(V (x)). Then µ will satisfy a (SPI) inequality with function β given by β

4. 2 .

 2 Lyapunov condition e aρ b . We suppose moreover here that M is a Cartan-Hadamard manifold with lower bounded Ricci curvature. If we try to use W = e aρ b for ρ ≥ 1, since ∆ρ is bounded above on {ρ ≥ 1} (see for example Th.0.4.10 in [29]), (L) holds for φ := abρ b-2 ψ with ψ := ∇ρ 2 , ∇V -c + abρ b

Lemma 4 . 2 .

 42 If (3.19) holds, then ρ ∇ρ, ∇V ≥ V -V (o) + c 0 ρ 2 /2.

Proposition 4 . 3 .

 43 Let M be a Cartan-Hadamard manifold with Ricci curvature bounded below. Let V satisfy(3.19). Then one can find positive constants c, C such that µ satisfies some (SPI) with function β (given below for s small enough) in the following cases(4.3.1) c 0 ≥ 0, V (x) ≥ c ′ ρ b for ρ large enough some c ′ > 0 and b > 1, β(s) = C e c(1/s) b 2((b-1)∧1) . (4.3.2) c 0 ≥ 0, d ′ ρ b ′ ≥ V (x) ≥ c ′ ρ b for ρ large enough some d ′ , c ′ > 0 and b ′ ≥ b > 1, β(s) = C e c(1/s) b ′ b ′ +b-2 .

( 4 .

 4 3.3) c 0 ≤ 0, for ρ large enough, V (x) ≥ (ε -c 0 /2)ρ 2 for some ε > 0, andβ(s) = C e c(1/s) . (4.3.4) c 0 ≤ 0, for ρ large enough, d ′ ρ b ′ ≥ V (x) ≥ c ′ ρ b and β as in (3.21.2).The first point of this proposition specialized tot the case c 0 > 0 enables us to recover [28, Th.1.3] which extends Bakry-Emery criterion to lower bounded Ricci curvature manifold. It then extends the result to various F -Sobolev.

  3.1. Nash inequalities for the Lebesgue measure. Let A be an open connected domain with a smooth boundary. For simplicity we assume that A = {ψ(x) ≤ 0} for some C 2 function ψ such that |∇ψ| 2 (x) ≥ a > 0 for x ∈ ∂A = {ψ = 0}. It is then known that one can build a Brownian motion reflected at ∂A, corresponding to the heat semi-group with Neumann condition. Let P N t denote this semi-group, and denote by p N t its kernel. Recall the following Proposition 3.1. The following statements are equivalent (3.1.1) for all 0 < t ≤ 1 and all f ∈ L 2 (A, dx),

  1.(1) we want to use theorem 2.1.(2) or more precisely theorem 2.8 we have to use proposition 3.8. Hence since (3.6) is satisfied we obtain for s small enough (3.16)

  x). It allows us to improve β.Let us now consider Case 3. Since b = 2, our hypothesis implies that for 2a < ε, φ ≥ DV . But the curvature assumption implies that the level sets of x → H(x) = V (x) + c 0 |x| 2 /2 are convex. Since V (x) ≥ D|x| 2 , one has cr ≤ V (x) ≤ r if x ∈ Hr . We may thus mimic case 1, just replacing Vr by Hr . Case 4 is similar to the previous one just improving the bound on φ as in case 2. If (3.21.3) holds, µ satisfies a logarithmic Sobolev inequality.

	Corollary 3.22.	(1)