Planning Logistics in Healthcare Networks: Analysis and Resolution

º Tesis Doctoral Planning Logistics in Healthcare Networks: Analysis and Resolution Dep. Ingeniería de Organización y Gestión de Empresas I Escuela Técnica Superior de Ingeniería Universidad de Sevilla 2015 “Planificación de la Logística en Redes de Salud: Análisis y Resolución” Autor: José Luis Andrade Pineda Directores: Pedro Luis González Rodríguez José David Canca Ortíz ii Tesis Doctoral Organización Industrial y Gestión de Empresas Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) Autor: José Luis Andrade Pineda Directores: Pedro Luis González Rodríguez Profesor titular José David Canca Ortiz Profesor titular Dep. de Ingeniería de Organización y Gestión de Empresas I Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2015 iv Tesis doctoral: Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) Programa / Línea: Organización Industrial y Gestión de Empresas / Gestión de Procesos de Salud Autor: José Luis Andrade Pineda Tutores: Pedro Luis Gonz ález Rodríguez José David Canca Ortiz El tribunal nombrado para juzgar el Proyecto arriba indicado, compuesto por los siguientes miembros: Presidente: Vocales: Secretario: Acuerdan otorgarle la calificación de: Sevilla, 2015 El Secretario del Tribunal vi A mis padres, Juan y Bárbara, que me inculcaron el valor del esfuerzo y el tesón, y a quienes debo todo lo que soy. viii Lista de Papers Compendiados ·Paper #1 Andrade-Pineda, J. L., Gonzalez-R, P. L., & Framinan, J. M. (2013). A Decision-Making Tool for a Regional Network of Clinical Laboratories. Interfaces, 43(4), 360-372. http://dx.doi.org/10.1287/inte.2013.0688 ·Paper #2 Jose L. Andrade-Pineda, David Canca, Pedro L. Gonzalez-Rodriguez “A Generalized Benders Approach for solving a Capacitated Multi- Commodity Network Flow Problem with non-linear routing costs” Submitted for publication at Int.Transactions in Operational Research. ·Paper #3 Andrade-Pineda, J., Canca, D., & Gonzalez-R, P. (2015). On modelling non-linear quantity discounts in a supplier selection problem by mixed linear integer optimization. Annals of Operations Research, 1–46. http://doi.org/10.1007/s10479-015-1941-2 ·Paper #4 Jose L. Andrade-Pineda, David Canca, Pedro L. Gonzalez-Rodriguez “Approximate resolution of a multi-commodity network flow problem with non-convex routing costs”, VII ALIO/EURO Workshop on Applied Combinatorial Optimization Montevideo, Uruguay. 4-9 December 2014. xv for aligning operational models and organizational behavior around demand, capacity and network configuration. This suggests that although such integration projects are difficult, costly and complex, they seem worth undertaking. xvi Table of Contents Lista de Papers Compendiados viii Agradecimientos x Resumen xii Abstract xiv Table of Contents xvi List of Figures and Tables xx Notation xxii 1. Introduction 1 1.1 Background and Motivations 1 1.2 Aims and Objectives 2 1.3 Scope and Delimitations 3 1.4 Reading Guide 4 2. Healthcare Logistics Foundations 7 2.1. Healthcare Planning 8 2.2. Supply Chain Integration 10 2.2.1. Sourcing Integration 11 2.2.2. Transportation Decisions 12 2.2.3. Procurement Trends in Multi-site Organizations 13 2.2.4. Tactical Logistic Planning 15 2.3. Integrated Healthcare Logistics 16 2.3.1. Extensive Healthcare Networks 17 3. Network-based Modeling Approach 19 xvii 3.1. Problem Definition 20 3.1.1. Related Work 20 3.2. Assumptions and Notation 23 3.2.1. Piecewise Linear Cost Function 23 3.3. Problem Formulation 28 3.3.1. Model for AUD Shipping Costs 28 3.3.2. Model for MAUD Shipping Costs 30 4. Solution Strategies 37 4.1. Problem’s Structure: Bordered Block Diagonal 37 4.1.1. Variable Decomposition for AUD Problems 38 4.1.2. Generalized Bender Decomposition on a Bilinear Reformulation of AUD Problems 40 4.2. Addressing Tractability Issues for MAUD Problems 42 4.2.1. DCC Method to Piecewisely Linearize MAUD Shapes 43 4.2.2. LogDCC Method to Piecewisely Linearize MAUD Shapes 44 4.3. Heuristic for Approximate Resolution of MAUD Problems 47 4.3.1. Phase1: Reducing the Available Real-Shipping Connections 47 4.3.2. Phase2: Moving Traffic in Flat Towards Upper Bound 49 5. Case Study: The Andalusian Network of Clinical Laboratories (ANCL) 51 5.1. The New Business Perspective 52 5.2. ANCL Integrated Logistic Problem Definition 53 5.2.1. Background 53 5.3. The New Planning Procedure: Applying Our Method 55 5.3.1. Graph Representation 55 5.3.2. Evolving Shipping Costs and Solutions Approaches for the ANCL Planning Problem 56 5.4. Decision Support System for Supporting What-If Analysis 57 5.4.1. Requirements 57 5.4.2. The DSS Tool Developed 58 6. Papers Results and Discussion 61 6.1. The Road-map of ANCL Transformation 61 6.1.1. Paper #1: A DSS Tool around an Optimization Engine 61 6.1.2. The Process Changes 62 6.2. Tractability Improvements 63 6.2.1. Paper #2: A GBD Algorithm on the AUD Problem 63 xviii 6.2.2. Paper #3: Reformulation and Exact Approach to MAUD Problem 64 6.2.3. Paper #4: Approximate Heuristic Approach to MAUD Problem 65 6.3. The Contribution of This Thesis 65 7. Conclusions 67 7.1. Summing Up the Thesis 67 7.2. Further Research Lines 68 References 69 Other Publications in the Research Area 79 Appended Papers 81 xx LIST OF FIGURES AND TABLES Figure 1 – Factors affecting the negotiated discount ...................................................... 13 Figure 2 – The transportation costs: (i) The modified all-units discount (MAUD) shape, and, (ii) The simplified shape ignoring the possibility of over-declaring the quantity of freight committed (AUD) . ............................................................................... 23 Table 1 – Notation ................................................................................................................... 24 Table 2 – Numerical example of the increased number of ranges arisen from the MAUD shipping costs .......................................................................................................... 31 Figure 3 – The MAUD shape and the consideration of a reduced amount of intervals based on the used of flat-ramp gadgets ........................................................... 33 Table 3 – Numerical example of the gadget based reduction of the required binary indicator variables: from the 13 extended ranges in Table 2, to only 7 modified ranges ....................................................................................................................................... 33 Table 4 – Gray codes in use in LogDCC: the all-zero code reserved to the non-load case, which is helpful to accomplish a locally ideal formulation ................................ 45 Figure 4 – This directed graph represents the regional network of clinical laboratories. Biological samples collected at points of extraction (POEs) are shipped to demand transfer points (DTPs) or service centers (SCs) for analysis, or to an outsourcing center (OC) ........................................................................................................ 56 Figure 5 – The DSS provides a user-friendly interface based on the Google Maps API to allow a user explore a solution .............................................................................. 58 xxii Notation 3PL Third-Party Logistics partner AC Autonomous Communities AGA Average Geographical Accessibility indicator (a QoS measure in the ANCL’s delivery service) AHA Andalusian Health Administration, wider than SAS AMPL A Mathematical Programming Language ANCL Andalusian Network of Clinical Laboratories AUD All-units discount API Application Programming Interface BD Bender Decomposition CBD Classical Bender Decomposition CC-MCNFP Capacitated MCNFP with concave costs CORAL Applied Project for the effective integration among the public hospitals in the ANCL, around interoperability and tactical logistic planning. DCC Dissaggregated Convex Combination Method for Linearization of PWL functions DSS Decision Support System DTP Demand Transfer Point ESB Enterprise Service Bus GUI Graphical User Interface GBD Generalized Bender Decomposition xxiii ILS Information Laboratory System INSALUD “Instituto Nacional de Salud” HIP Healthcare Integration Platform IT InformationTecnologies LTL Less-than-truckload LogDCC Logarithmic Dissaggregated Convex Combination Method for Linearization of PWL functions LP Linear programming MAUD Modified all-units discount MC Multiple Choice Method for Linearization of PWL functions MFS Maximum Feasibility Subsystem MIBLP Mixed-integer bilinear programming problem MILP Mixed integer linear programming MINLP Mixed-integer non-linear programming problem NFP Network Flow Planning NP-hard Non solvable in Polinomial time hard problem PHP PHP: Hypertext Preprocessor POE Point of Extraction PWL Piecewise Linear function QoS Quality of Service RMP Relaxed Master Problem SAS “Servicio Andaluz de Salud” SC Serving Center SC Supply Chain SCM Supply Chain Management SOA Service Oriented Architecture SP Primal problem, sometimes just Subproblem 7 2. HEALTHCARE LOGISTICS FOUNDATIONS his chapter is devoted to identifying models and methods useful to support logistic planning problems in healthcare networks. Logistics has traditionally been defined as the process of planning, implementing, and controlling the efficient flow and storage of goods, services, and related information as they travel from origin to consumption points (Arshinder, Kanda, &Deshmukh, 2008). In order to overview the context of our target problems, we again recall that healthcare networks bring together a wide range of public and private actors (hospitals, specific service units, healthcare professionals and managers) at local or regional level that seek for management effectiveness and coordination efficiency. Planning is one of the management strategies to face the task of service delivery in such complex contexts. We give a glance of the planning processes needed within healthcare systems, the different lead times (planning horizon) associated with the decisions to be made outlining the body of work on planning healthcare according to strategic, tactical and operational levels. Once given the context, since the task of designing and operating healthcare networks belongs to the broad realm of supply chain management, we conduct a review of supply chain and logistics management models connected to our research. We confine our literature review to works explicitly dealing with logistic issues that apply mathematical optimization techniques. Finally, we draw the main lines of real-life healthcare logistics problems. To great extent, the challenges in healthcare supply chain management turn around hospitals as the key elements in healthcare delivery networks. They implement the activities and processes, and also are on the receiving end of a wide range of supplies that support the delivery of care. T 8 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) 2.1. Healthcare Planning At the strategic level, the Network Design Decisions are addressed from a wholesystem perspective to provide the network design that should allow for the better service provision, setting the layout and allocation of long-lasting resources to cover necessities in a long period –i.e. location of facilities, their capacity and the assignment of the territory to be served-. The emphasis is usually on objectives such as ease of access and/or speed of access, as well as the attainment of costefficient service delivery. However, many often the strategic configuration goes beyond network design and customer service considerations, and the major goal is long-term follow-up and disease monitoring to decrease the risk of complications and/or death (Charfeddine, Augusto, & Montreuil, 2010; Y. Zhang, Berman, Marcotte, & Verter, 2010). Charfeddine et al. (2010) address the districting problem in a Quebec province (Canada) to the redesign of chronic diseases services distribution so as to facilitate their accessibility to the assigned center (patient driving distance) for patients residing in distant zones while, at the same time, optimizing the opening of those services at the center. In Zhang et al. (2010) a network of mammography centers in Montreal is designed aiming at reducing the frequency of life-threatening illnesses by protection and early detection, proposing a non-linear optimization problem. For a review of location problems as a critical element of strategic planning we refer to (Daskin & Dean., 2005). Once the network is in place, the tactical level focuses on what to do in the midterm to help the organization achieve the long term objectives determined by strategic planning. As an example, the tactical resource planning referred to hospitals is often focused on elective patient admission planning and the intermediate-term allocation of resource capacities (Hulshof, Boucherie, Hans, & Hurink, 2013). In Hulshof et al. (2013) the main planning objectives are equitable access for patients, meeting production targets and/or serving the strategically agreed number of patients, while making an efficient use of resources. Another example in a wider scenario is reported in Govind et al. (2008), which copes with a network of hospitals in a predefined geographical area to determine the bed capacity that each hospital should devote to different disease classes in order to maximize speed of access to care. The presented model accounts for the spatio-temporal pattern of disease incidence in the area and the driving speeds on different types of roads in determining the speed of access. Similarly, Santibáñez, Bekiou, & Yip (2009) face the problem of the location of clinical services and allocation of bed capacity across Healthcare Logistics Foundations 9 the services at each hospital within a real network with 12 hospitals with multiple services. They develop a multi-period mathematical programming model to provide options for configuring the system based on population access, critical mass standards, and clinical adjacencies. Finally, the care delivery is achieved conditioned by the day-to-day operations, which are configured throughout the so-called operational level decisions. In a resource-constrained environment it is important to have rules to prioritize the most important works. Singularly, the staff management decisions play a key role in the service schedule planning for providing health care services in the real-life situations, since staff is generally required to react to unexpected events. In last decade, efforts aimed at redesigning the service delivery for better response to increased levels of unexpected demand. A typical scenario for the latter is the operating room (OR) theater (Guerriero & Guido, 2011)(Cardoen, Demeulemeester, & Beliën, 2010) in hospitals, which is a major cost driver –i.e. surgeons, managers, trustees, nurses… each with their conflicting interests – and the main generator of variance in the surgical activities. The operational patient-customer acuity is usually the driver of scheduling OR (also in emergency departments). However, this policy suffers from the risk of lengthy waiting times for patients with issues that were not life threatening. Increasingly, patients are turning to an emerging model: the retail clinic. Retail clinics have emerged as a low-cost and convenient alternative to the traditional model of ambulatory care, providing a discrete set of acute care and preventive services, on an as-needed basis (Landry & Beaulieu, 2013). The improvement of service scheduling and patient access (medical prescription, appointment, patient admission) are currently the main challenges for healthcare organizations. In general, a whole-system perspective is required to re-define processes in order to consider the needs of and value accrued by patients at each point in their interaction with the hospital (lean approach (Waring & Bishop, 2010)). To cope with the inherent variability and complex dynamics, this task is many often addressed using optimization models and/or simulation models attempting to mimic the behavior of the system so as to analyze the outcome of different scenarios. Particularly, much attention in the healthcare management literature has been paid on the patient flow and its modeling – refer to (Bhattacharjee & Ray, 2014) for a comprehensive review-. 10 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) 2.2. Supply Chain Integration A supply chain system (supply chain, SC for short) is a network that mediates the flow of entities involved in a product life cycle, typically passing through a number of facilities (contributing in the value addition of the product) before being delivered to the end customer (Tayfur& Benjamin, 2007). Such a network consists of nodes and arcs. Nodes represent suppliers, manufacturers, distributors, and vendors (e.g. retail stores), as well as their inventory facilities for storing products and transportation facilities for shipping them among nodes. Arc represent routes connecting the nodes along which goods are transported in a variety of modes (trucking, railways, airway…). According to (Chopra and Meindl 2007, p. 6), supply chain management (SCM) is “the management of flows between and among all stages of a supply chain to maximize total profitability”. Multitude of issues ranging from location, product, and marketing decisions to the management of information exchange and coordination across different stages of the SC are comprised in this definition. Therefore, SCM tackles with a wide range of problems, from strategic problems like logistics network design, to tactical problems like the coordination of inventory and transportation decisions, all the way through day-to-day operational problems like production scheduling, delivery mode selection, and vehicle routing. The SCM systems have been seen as fertile ground for cost reductions (Kumar, Ozdamar, & Ning Zhang, 2008), which owing to the inherent size and complexity of many of these problems makes optimization models to be essential for effective decision making (Ana Muriel & Simchi-Levi, 2003). One of the most interesting macro-level trends in logistics (both applied and theoretical) is SC integration (also, logistical integration) (Fabbe-Costes, Jahre, & Roussat, 2008). This integration takes many forms, from increasing information flows between different actors in the SC to the transfer of process ownership. According to (Arshinder, Kanda, & Deshmukh, 2008): “Generally, a SC consists of different functions: logistics, inventory, purchasing and procurement, production planning, intra- and interorganizational relationships and performance measures. The rise in papers on SC as well as the case studies in different areas in different industries motivates to study SC issues further. SCs are generally complex with numerous activities usually spread over multiple functions or Healthcare Logistics Foundations 11 organizations and sometimes over lengthy time horizons. Therefore, it is necessary to overlay a coordination system, which may include: an explicit definition of processes, responsibilities and structures aligned with overall objective of whole SC to bring together multiple functions and organizations.” In practice, the organization restructuring to be addressed must be accompanied by systems and optimization tools, and then the key aspect for SCM are information sharing and SC coordination as vehicle for redesigning decision rights, workflow, and resources between chain members to leverage better performance as a whole (Lee, 2000). 2.2.1. Sourcing Integration In the SCM literature, the sourcing decisions faced by buyers of goods attempts to minimize an objective function including the transportation cost and procurement cost, both mostly effected by discounts. The quantity discount policy arise as a classical mechanism to conciliate conflicting interests from buyers and suppliers (Arshinder et al., 2008): suppliers want buyers to commit themselves to purchasing large quantities in stable volumes with flexible delivery dates, and buyers prefer supply in small batches due to changing demand and their unwillingness to hold inventories. Classical price discounts falls into two categories depending on how they are granted: (a) they can be quantity discounts per product and/or order, (b) discounts based in a total purchase volume (whether expressed in total euros or in total number of units), once aggregated over several products and even over several purchasing departments within a company (Stadtler, 2007). Whether (a) or (b), the reduced price is obtained by applying one of the following two discounting approaches: (1) incremental (cumulative approach to price, different prices applied to the units belonging to different discounting levels) or (2) proportional or all-unit discount (once reached a discounting level, the reduced price is applied to all units starting from the first unit). Finally, a buyer may be given the chance of overdeclaring the quantity to take advantage of the next discounting level (the so called bumping clause). There is plenty of SCM literature where quantity discount policies are applied in 12 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) the inbound transportation or procurement decisions –i.e. the buyer of goods attempts to minimize an objective function including the transportation cost and procurement cost after discount-. 2.2.2. Transportation Decisions Transportation planning occupies a central place in SCM, as transport and storage of physical goods account for a significant share of the operational cost in a supply network. In many practical situations, discounts for larger quantities of freight may be applicable to transportation economies of scale, in terms of the number of unit loads delivered. Further, complicating factors as the delay in payments have to be considered in the pricing and ordering decisions. (Sheen & Tsao, 2007) discussed the channel coordination issue under trade credit and freight cost discounts. They assumed that the transportation cost includes quantity discounts due to economies of scale. Glickman and White (2008) addressed the optimal vendor selection problem in a multiproduct supply chain with truckload discount. The study by (Tsao & Lu, 2012) considers quantity discounts for transportation cost between central distribution centers and regional distribution centers in a national-wide network. The point of view of freight carriers has been studied in (Smith, Campbell, & Mundy, 2007), when addressing the pricing of the freight services to be expedited (namely, define the net rates once discounts are applied) in a the less-than- truckload (LTL) shipments company. According to the authors, the company sets base rates and then negotiates individual customer discounts which purportedly reflect the costs of providing service, competitive pressures, and the anticipated value of the customer relationship. The factors identified in the literature as affecting negotiated rates for LTL shipments, are shown in Figure 1. Healthcare Logistics Foundations 13 Figure 1 – Factors affecting the negotiated discount (reproduced from (Smi th et al., 2007)). The net freight rate paid by a shipper depends upon the published rate, the discount negotiated with the customer, and the blend of shipments (classification, weight and distance) that occurs. The discount is affected by the relative negotiating power of the two parties (carrier and shipper). It is further affected by the perceived risk (uncertainty) associated with contractual arrangements and by the extent to which the two parties enjoy a mutually constructive relationship that enables them to work synergistically. Contributing to the shipper’s buying power in negotiating discounts from published base rates are the volume of business (most generally measured by combination of weight and distance), attributes of cargo affecting the utilization of assets (e.g., density, access to loading facilities, or unbalanced directional flows), ease with which the cargo can be handle and degree of risk associated with the service (Shanahan, 2003). Contributing to the carrier’s competitive position are the basic rate structure (which determines the base rates on different shipping lanes), its available capacity on the routes used by the customer, its cost structure for local operation and line hauls, and the service levels (timing and reliability) offered. Contributing to the negotiating power of both parties are the alternative options available from competing carriers and the strength of existing relationships between the parties. 2.2.3. Procurement Trends in Multi-site Organizations In a wide sense, this thesis concerns to the procurement problem addressed by large multi-site organizations in order to select among multiple suppliers that 14 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) provide base prices and discounts. In the last decades these organizations have moved to a centralization of their procurement functions, what typically provides the ability to leverage the overall buying power to obtain significant supplier discounts (Balakrishnan & Natarajan, 2014). The business is then allocated across a diminished number of active suppliers, which often offer attractive quantity-based discounts in the expectation of benefitting from economies of scale in their production activities. Another common outcomes are the reduction of the administrative costs and the promotion of better relations with the contracted providers (Xu, Lu, & Glover, 2000). Precisely, the maintenance of long-term partnerships with suppliers has been identified as a competitive strategy pointing to fewer and more efficient providers (Ho, Xu, & Dey, 2010). Further, when the involved product/service has strategic impact on the buyer’s business, the supplier selection process should aim at creating a closer relationship with suppliers that make possible improving the results for both parties (Gonçalo&Alencar, 2014). At a more operative level the procurement settings include realistic issues of lotsizing: variability on demand, shortage, backorders, lead time, late delivery and also, the role of discounts as a purchase incentive offered by providers that conditions the tradeoffs among objectives in buyer decisions (Taleizadeh, Niaki, &Hoseini, 2009). Complex NP-hard problems arise from stochastic parameters which need to be solved using Meta heuristic algorithms (Taleizadeh, Niaki, & Wee, 2013; Taleizadeh, Wee, &Jolai, 2013). In a whole, the supplier selection process leads usually to complex decision making problems, which could include both quantitative (pricing and discount policy, limited availability), and qualitative (service quality and reliability) criteria, sometimes calibrating the uncertainty risks in the supply. We refer to (Gonçalo&Alencar, 2014), (Ho et al., 2010) and (Aissaoui, Haouari, &Hassini, 2007) for comprehensive reviews on the supplier selection literature. Aissaoui et al., (2007) claimed that very little attention had been paid to logistics costs in the supplier selection research, once addressed a comprehensive review of supplier selection and order lot-sizing modeling. However, this aspect has been considered in recent studies (Ayhan & Kilic, 2015; Hammami, Temponi, & Frein, 2014; Mansini, Savelsbergh, & Tocchella, 2012; Qin, Luo, Gao, & Lim, 2012). (Mansini et al., 2012; Qin et al., 2012) are bound up to the common practice to outsource the transportation activity to a carrier company on the basis of minimum quantity commitment. (Ayhan & Kilic, 2015; Hammami et al., 2014) include the Healthcare Logistics Foundations 15 realistic inclusion of the global marketplace from which selecting suppliers. In recent years, one of the research streams consists on modeling the complex buyer-supplier discount negotiation (Bichler, Schneider, Guler, & Sayal, 2011; Şen, Yaman, Güler, & Körpeoğlu, 2013; Yin, Nishi, & Grossmann, 2014). In Yin et al. (2014) complexity comes from the flexible purchase agreements guiding contract decisions among a manufacturer of electronic components and its suppliers, with the purpose of sharing risks of uncertain products’ demand in a short-life cycle market. In Şen et al. (2013) discount rates are tied to future random events that supplier and buyer can mutually verify, base prices and discounts offered in a bidding process. Bichler et al. (2011) propose a bidding language and an optimization model to express the complex discounts structures that arise in the context of combinatorial auctions. 2.2.4. Tactical Logistic Planning We can state that for large multi-site organizations, coordinated procurement entails deciding which suppliers to use to meet each site’s purchasing needs and sourcing preferences so as to minimize overall purchasing, logistics, and operational costs. Incorporating the interdependence of location, transportation, and inventory decisions leads to complex decision making problems, which could include both quantitative (pricing and discount policy, limited availability), and qualitative (service quality and reliability) criteria, sometimes calibrating the uncertainty risks in the supply. Relevant to our thesis, we concentrate next on studying papers that apply a tactical planning approach to such decisions. Supplier selection considering a single-period approach has been applied largely in the literature. Murthy, Soni, & Ghosh (2004) address the supplier selection problem for make-to-order items, when savings in the setup costs associated with producing a bundle of items allow suppliers to offer volume-based discounts for bundles. They make also an interesting review on optimization models for procurement planning. Goossens, Maas, Spieksma, & van de Klundert (2007) address the lotsizing problem with all-unit discount based on total purchase quantities, presenting a MILP model and a customized branch-and-bound technique, although it is outperformed by a general purpose mixed integer solver when 22 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) the cost of the entire load purchased. However, in Cohn et al. (2008) discounts on shipments are a function of the overall flow of the network; in our case study, we use discount factors attached to the flow in each connection. Since each connection can be potentially allocated to a different carrier, the aggregated flow over commodities (total freight) on each connection acts as the discount driver. More specifically, the aggregated flow along that connection is priced according to a piecewise linear function referred as all-unit discount (AUD) (Stadtler, 2007), structure that requires a mixed integer formulation. Further, since capacity constraints appear both on facilities production and transportation, the model that arises is a Capacitated MCNFP with concave costs (CMCNFP-CC). CMCNFP-CC is a NP-hard problem – see (Guisewite&Pardalos, 1990) for a complexity analysisfor which certain heuristic approaches have been proposed in the literature. Particularly, we refer to (Croxton, Gendron, &Magnanti, 2007) and (Muriel &Munshi, 2004) for structural insight intoMILP formulations of this problem. These authors propose linear programming (LP) heuristic approaches in order to approximate the concave cost function while obtaining good lower bounds. While Muriel &Munshi (2004) report a heuristic solution approach based upon the fact that the LP relaxation of the problem is as strong as the Lagrangian relaxation (applied for example in (Amiri&Pirkul, 1997)), Croxton et al. (2007) approximate the original cost function with its lower convex envelope in the space of commodities. Since for linear routing costs the CMCNFP-CC becomes easier to solve, another heuristic approach consists on approximating geometrically the original cost function by just a line in order to find near-optimal solutions (Hill &Galbreth, 2008). The aforesaid approaches are relevant to address quite general application problems in the assumption that the objective function is just written in terms of routing costs. However, our quantity discount and carrier selection real-life problems requires a generalized version of this problem considering not only concave routing costs but also operational costs due to product handling, overload and underutilization of facilities. Network-based Modeling Approach 23 3.2. Assumptions and Notation 3.2.1. Piecewise Linear Cost Function In presence of economies of scale in the transportation activity, the shipping costs shape considered in this thesis are as in Figure 2. The first one, referred to as modified all-unit discount (MAUD) cost structure, results from incorporating the bumping clause to the AUD one (Chan, Muriel, Shen, Simchi-Levi, & Teo, 2002). Figure 2 – The transportation costs: (i) The modified all-units discount (MAUD) shape, and, (ii) The simplified shape ignoring the possibility of over-declaring the quantity of freight committed (AUD). 24 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) Let ( , , )G V L K• be a directed graph representing the involved distribution network, with commodities set K (namely, the items routed over the network), vertices set V , links set L , demands and available capacities for each commodity in K at the vertices, and link capacities (resulting from whether physical or managerial limitations). Vertices are of different types: some of them have a supply of commodities, others a demand for them. In this model, the flow through a vertex incurs a consumption/processing cost (if the vertex was the destination facility) or a transshipment cost (if the vertex has to redirect the commodity as a next hop before reaching the destination facility). Our model also includes costs for flowing along links, whether they represent routing/shipping costs or outsourcing costs in case of an external purchase of test services. In next section, we present alternative CMCNFP-CC formulations of the concerned logistic problem with AUD and MAUD piecewise routing costs. Before, we introduce the following notations: Table 1. Notation Set of Indices K Set of commodities: 1|k k K k N • K . V Set of vertices in graph ( , )G V L• , with elements: , {0,.., 1} i v V i N • ! . L Set of links in graph ( , )G V L• , with elements: ( , ) i j v v L , ,|i j v v V {0,.., }, {1,.., 1}i N j N• • ! . V Set of vertices representing real-life center s, with elements: , {1,.., } i v V i N • . They are all the vertices in V except the demand insertion vertex 0 v and the outsourcing vertex 1N v! . Network-based Modeling Approach 25 L Set of links representing real shipping connections, with elements: ( , ) , , {1,.., }, {1,.., }, | i j i j v v L v v V i N j N i j• • • • • . j S Set of indices marking the successors of vertex j : { ( , ) }| j j m S m v v L• • • . j P Set of indices marking the predecessors of vertex j : { ( , ) }| j m j P m v v L• • • . G Set of package carriers companies considered for shipping services: 1|g g G g N• • K . They are g N different carriers. g R Set of discount ranges offered by carrier g : 1| g gr r R r N• • K . They are gr N different ranges for carrier g , so that the involved discounts depend on the traffic committed on each shipping connection ( , ) i j v v L• , if assigned to be operated by g . These ranges fully partition the feasible volume of total flow on each link: rg L and rg U are the lower and up per bounds on this flow for discount range r , and rg DF is the discount factor itself . l Index in which the alternative binary variables of the LogDCC formulation are defined, with 2 {1... log ( ) } g gr l N N ! • " # $ . Model Parameters ik a Demand of commodity k at vertex i v V• . ik C Capacity of commodity k at vertex i v V• . E ik c Execution cost of commodity k at vertex i v V• . S ij c Shipment cost for link ( , ) i j v v L• . 26 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) SOC ik c Outsourcing cost, standing for the price of type k commodity when processed in a external facility (due to the limited capacity at vertex i v V• ). T ik c Transshipment cost of commodity k at vertex i v V• , which is a fixed perunit based inventory cost . u Percentage to apply to ik C to obtain the threshold that triggers an overload penalty, due to excessive transshipments – i.e. over-transshipment -. w Per-unit penalization when over-transshipment. ij Y Upper bound for aggregated traffic for ( , ) i j v v L• . ijk S Upper bound for a type k traffic for ( , ) i j v v L• . ik W Minimal workload, non-zero only at vertex 0 | i ik v V C• . low w Per-unit penalization when the established minimal workload ik W is not committed –i.e. underutilization. rg L Lower bounds on flow for discount range r referred to carrier g . rg U Upper bounds on flow for discount range r referred to carrier g . rg DF Discount factor referred to carrier g G• to be applied if the total flow committed in a certain link is in [ rg L , rg U ] – i.e. in range g r R• -. rg Gadget MP Flat-ramp gadget midpoint: transition point from the zero slope segment to the constant slope one. rg F Constant bargain zones costs, to be applied if the total flow committed in a certain link is in [ rg Gadget MP , rg U ] – i.e. in the flat zone of range g r R• rg DFA Commodity-dependent costs, to be applied if the total flow committed in a certain link is in [ rg L , rg Gadget MP ] – i.e. in the ramp zone of range g r R• rg GRAY Gray code assigned to each range g r R• , where adjacent ranges only differ in one component . Network-based Modeling Approach 27 Variables ijk s Amount of type k commodities flowing through link ( , ) i j v v L• . ik e Amount of type k commodities consumptions at destination facility i v V• – i.e. the workload assignment -. ik t Amount of type k commodities transshipments in vertex i v V• . OT ik P Contribution to operative costs due to facility 0 | i ik v V C• , when its level of transshipments is above the threshold for the penalty trigger – namely, ik ik t u C ! , and consequently, the facility is in the overtransshipment range -. LW ik P Contribution to operative costs due to facility , 0| i ik ik v V C W• is not committing the specified minimal workload for type k commodities – i.e. ik ik e W" , and consequently, facility assigned a too low workload. rg ij z 1, if aggregated flow on link ( , ) i j v v L• is operated by carrier g with the discount for range g r R• ; 0 otherwise. rg ij y Traffic flowing on link ( , ) i j v v L• , shipped by carrier g and priced according to the discount factor for range g r R• . ij y Aggregated flow on link ( , ) i j v v L• . rg ij g Contribution to operative costs due to the MAUD shipping costs on link ( , ) i j v v L• . rg ij • Multiplier [ 0, 1] rg ij •• in use in order to express the commodity contribution to costs when the load lies into segment g r R• - i.e. the aggregated flow on link ( , ) i j v v L• lies in range g r R• -, as a linear combination of the costs at the extreme points of that segment. Variable in use in the DCC and LogDCC models. rg ij Multiplier [ 0, 1] rg ij •• in use in order to express the commodity contribution to costs when the load lies in segment g r R• -i.e. the 28 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) aggregated flow on link ( , ) i j v v L• lies in range g r R• -, as a linear combination of the costs at the extreme points of that segment. Variable in use in the DCC and LogDCC models. l ij b Binary variables in use in the LogDCC model, defined in the following indices: 2 0, 1 / ( , ) , {1... log ( ) } i j g gr i j N v v L l N N ! " # # $ • " • % & ' . ij MAXb Binary variable in use in the LogDCC model, equals to the maximum value among the components l ij b . Notice that for notational convenience we listed symbols specifying the transportation costs derived from the centralized management of carriers in terms of a set g r R• . This index r stands for one of the two sets of range in use throughout this section, as we explain before –see Figure 2-. 3.3. Problem Formulation 3.3.1. Model for AUD Shipping Costs Assume economies of scale in the transportation activity according to all-unit discount (AUD), a piecewise linear structure which requires a mixed integer formulation hence. A formulation of AUD problem of interest is presented in (Andrade-Pineda, Gonzalez-R, &Framinan, 2013), assuming routing costs as in Figure 2 (ii), that are linearized using a multiple choice model approach (Balakrishnan& Graves, 1989) – namely, using a binary indicator variable to make the selection of at most one of the g p P• intervals within a carrier offer-. Next we present a model that has been slightly improved as regards as the preliminary one. Network-based Modeling Approach 29 ( 1) ( ) 0ik / 0 / 0 (1 ) (1) , . . { 0 } (2) | | ik ik i g OT WL ik ik k K k K SOC E ik i N k ik ik v V k K k K vi pq S pg Tij ij ik ik g G v v k K vi i p P i ik i ik ik Min P P v V C v V C i i c s c e V c DF y c t L V s t s = a , k K, i v V a e C • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ! ! ! ! ! ! ! ! ! ! ! ! ! 0 0 0 0 / (3) 0 / { | 0} (4) 0 / (5) 0 , , / , ( , ) (6) , , / ,( , ) | 0 (7) j j j ik i jlk jk jk j ik i P i ijk jlk jk j i P l S i ijk ij i o i j k K s ij ij i o i j ij ijk i , k K, i v V s e t , k K, j v V C s s e , k K, j v V v s y i j v v v v L y Y i j v v v v L c s S • " • • " • • • • " • • • • # • • • • • • • # • • • • " • • # • " • • " • • ! ! ! ! , , / ,( , ) (8) { 0} (9) { , 0} (10) 1 , 0, 1 / ( , ) (11) 0 , , , /( , | | | | g jk i o i j OT ik ik ik i ik low low LW ik ik ik i ik ik pg ij i j g p P rg pg pg ij ij i , k K i j v v v v L w t P w u C , k K, i v V C w e P w W , k K, i v V C W z i j N v v L y L z , p g i j v • • • • " • • • • • • • • • • • • • • • • • • • • • " " • • • • • • • ! ! ) (12) 0 , , , /( , ) (13) , , /( , ) (14) , , , /( , ) (15) , , , 0 , 0, 0 (16) g j rg pg pg ij ij i j pg ij ij i j g p P pg pg ij ij i j OT LW pg pg ik ijk ik ij ik ik ij ij v L y U z , p g i j v v L y y i j v v L y z , p g i j v v L e ,s ,t y P P y z binary • • • • • • • # • • $ • • • $ $ $ ! ! 30 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) The non-convex objective function embeds shipping costs, transshipment costs, processing and outsourcing costs, and additional operational costs when laboratories receive too few or too many samples to be analyzed. Notice we have considered it to be a linear combination (1) with weights , , , , ,• • • • • • that are instance-dependent parameters. Demand is injected from 0 v ‘fictitious vertex’ – see constraints (2) - and then balance equations – see constraints (4 5)• - allow for the workload assignment to facilities as well as for the evaluation of transshipments in every real-life center in the distribution network. When a facility is set at a too-low workload, the underutilization penalty is imposed by constraints (10) . Similarly, when there is an excess of transshipments at a facility, the overload penalty is imposed by constraints (9) . Notice that the decision-maker can impose traffic bounds – see constraints (7 8)• - and capacity constraints for the processing levels – including the outsourcing vertex, see constraints (3) -. Dislike the model in Andrade-Pineda et al. (2013), we let the above model determining that a certain connection is not assigned a carrier –see (11) - and include a valid inequality –see (15) - to code that a null load implies a null indicator variable (Sridhar, Linderoth, & Luedtke, 2013), thereby benefiting the speed of computations. Besides, (9 10)• are Set Ordering Set 1 (SOS1) constraints linearizing the respective piecewise lineal penalty functions corresponding to overload and underutilization of facilities. 3.3.2. Model for MAUD Shipping Costs Now, let us assume economies of scale in the transportation activity according to the modified all-unit discount (MAUD) (Chan, Muriel, Shen, Simchi-Levi, & Teo, 2002) in Figure 2 part (i). This is the cost structure used in Papers #3 and #4. The most straightforward idea for the realistic assessment of MAUD transportation costs is applying a multiple choice approach as in the above shown model -see Table 2 for a numerical example-. Network-based Modeling Approach 31 Table 2. Numerical example of the increasing number of ranges arisen from the MAUD shipping costs. Discounts Offered by a Carrier with sending commitment Discounts Offered by a Carrier with sending commitment g PpÎ pg L0 pg U0 pg DF 0 1- g QqÎ qg L qg U qg DFA qg F p1 0 500 1.000 q1 0 465 1 0 p2 501 1000 0.93 q2 466 500 0 465.93 p3 1001 1850 0.880 q3 501 947 0.93 0 p4 1851 2500 0.800 q4 948 1000 0 880.88 p5 2501 4500 0.730 q5 1001 1682 0.88 0 p6 4501 9999 0.650 q6 1683 1850 0 1480.8 p7 10000 50000 0.456 à q7 1851 2282 0.80 0 q8 2283 2500 0 1825.73 q9 2501 4007 0.73 0 q10 4008 4500 0 2925.65 q11 4501 7015 0.65 0 q12 7016 9999 0 4560 q13 10000 50000 0.456 0 Unfortunately, proceeding like this results in a MILP model that does not behave well with the state-of-the-art solvers. Since the extended problem greatly increases the number of binary indicators variables (almost the double), the problem becomes intractable for many instances of realistic size. To address the deficiency of such a straightforward modeling approach, in (Andrade-Pineda, Canca, & Gonzalez-R, 2015) we have proposed a novel modeling technique based on the identification of flat-ramp gadgets. 38 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) , where the superscripts V and L denote whether a variable is concerning a vertex or a link. Notice also that every matrix and vector appears subscripted with the appropriate dimensions. Assume that: (i) vector 9 U [ encloses the variables capturing the level of activity at each facility v V• – e.g. production or inventory- , with different capacity level for each commodity, (ii) vector / V \ represents the commodities flows (both aggregated and disaggregated) over the links, constrained to certain capacity bounds, and (iii) vector / T \ represents an integer subset of variables necessary to incorporate carrier selection decisions in each link , according to a discount policy based on the volume of commodities. The transportation cost within the objective function (35) – denoted / / T T G \ - is a concave piecewiselinear function, hence. In addition, the objective function collects the contribution due to link flows and activity levels for each commodity at each facility in a linear form through constant unitary costs U K and V J , respectively. The feasibility region for the integer vector / T \ is first defined by the block of linear constraints (37) . Notice that all the variables except / T \ have been considered to be continuous because at the optimum the solution will be integer due to the underlying pure network flow structure. The constraints that provide this structure are embedded in the first block of linear constraints (36) , representing flow balances and bounds on variables. The rest of these constraints act as links1 among the integer variables / T \ and the continuous variables 9 U [ , / V \ . 4.1.1. Variable Decomposition for AUD Problems Assume the problem of interests formulated according to (1 16)! . We can split into continuous and integer parts and hence, restate it in the following compact form: 1 In our concerned problem, linking constraints accomplish the specific filling of the priced flows embedded in / T \ based on the continuous variables values 9 U [ , / V \ , thereby refining which integer realizations are feasible according to the whole structure. We refer to chapter 5 for the illustrative example of the ANCL planning problem. Solution Strategies 39 , [ ] (39) . . (40) , (41) , (42) MILP Min s t • • • • • • • • • • • • • • • R Z S T S S T T [ \ PS S PT T P S RS S R S T QT T Q T F [ G \ $ [ % \ E & [ F [ ( \ H \ , where the continuous variables are all embedded into a p-dimensional vector ( ],[ / V 9 US \[[ º which contributes to the objective function linearly with a constant vector ],[ VUS JKF º ) and the q-dimensional vector of complicating variables appears renamed as / TT \\ º . Similarly, for notational convenience we have renamed the all-unit discounted prices as / TT GG º . We devise a variable decomposition strategy, where the carrier-discount selections represent the main logistic decisions (attained now in terms of integer complicating variables), whereas flows and activity levels continuous variables S [ are settled in a lower level of decision. Indeed, it is a solution strategy that comprises an iterative coordination of these two levels of decision: · A lower level (”Which flows and activity levels?”) that, for given carrierdiscount choices, computes the associated optimal values for the “easy LP problem” on a vector of continuous variables S [ as well as pricing information (dual variables) and an upper bound for the original problem. · A higher level (”Which carrier-discount selections?”), consisting of a relaxed version of the original problem with the set of binary variables and its associated constraints (still a “difficult MILP problem”). At every iteration, this relaxed version is enriched using the outcome of the lower level to append a violated cut on the space of the complicating variables. The augmented relaxed problem is then solved to provide a new lower bound for the original problem, as well as a new tentative carrierdiscount choice to be used in the next iteration’s lower level problem. 40 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) 4.1.2. Generalized Bender Decomposition on a Bilinear Reformulation of AUD Problems Aiming at a more agile resolution procedure, we can restate the concerned CMCNFP-CC (1 16)• as a linearly constrained MIBLP model – see (Adams & Sherali, 1993) for an in-depth numerical study of general MIBLPs-, which is algebraically written in the following compact form: , ' ' [ ] (42) . . (43) , (44) , (45) {0,1} MIBLP Min s t ! ! ! ! ! " ! " # ! " # R S T S S T S T [ ] PS S PT T P S RS S R S T QT T Q T F[ G [ ] $ [ % ] E & [ F [ ( ] H ] Next, we explicitly present the non-traditional formulation equivalent to the MILP model (1 16)• on which we have applied our GBD iterative algorithm: ( 1) ( ) ( , ) / 0 / 0 (1 ) (46) , . . ik ik i g BP BP OT WL ik ik k K k K SOC E ik i N k ik ik v V k K k K vi pq T S pg ik ik ij ij ij k K g G v v v i i i p P F Min f Min P P v V C v V C i i c s c e V c t c DF y z V L s t • • • • • • ! " " # ! # # # ! # ! $ $ # # ! # # ! # # ! # # • # # % % % % % % % % % % % % % S T S S T S T [ ] F [ G [ ] . (2) (10) 0 ( , ) | 0 (47) 0 , , ( , ) | 0, 0 (48) (1 ) 0 , , ( , ) | 0 (49) g pg s ij ij i j ij g G r P pg pg pg s ij ij i j ij pg pg pg s ij ij ij ij i j ij Eq y M z , v v L c y L z , p g v v L c L y U z Y z , p g v v L c # • ! • # & $ # • ! # & ' $ $ & • # • # • & ' $ % % PS S PT T P $ [ % ] E Solution Strategies 41 1 1 2 2 1 2 1 2 1 2 1 , , , , 0, 1 / ( , ) (50) 1 0, 1 / ( , ) | 0 (51) g p g p g g g ij ij i j rg s ij i j ij g G r R z z p P p P p p i j N v v L z , i j N v v L c • • • • • • • • • • • • • • • • • • • ! " " OT T O ( ] H , , , 0 (52) 0 OT LW ik ijk ik ij ik ik pg ij e ,s ,t y P P z binary • • • • • • R Z S S T T [ ] Observe that here the only complicating variables are the binary variables pg ij z , and that the feasibility region implied by constraints (51) is such that each connection is operated by exactly one carrier. Again, we use valid inequalities in order to tighten relaxations – see constraints (27) -. Certain constraints link complicating variables [ ] pg q ij z ] and continuous variables S [ = [ , , , , ] OT LW ik ik ik ik ijk ij P P e ,t s y – see constraints (47 49)! -. Notice that constraints (47 50)! impose the required centralized management of package carrier services as follows. Constraints (48) imply that when the aggregated flow ij y is zero, then none of the binary variables pg ij z are activated. Constraints (47) together with constraints (48) ensure that the absence of selected carrier/range –i.e. 0 , pg ij z , p g " - implies null aggregated flow. Similarly, if the aggregated flow is ij y > 0 , constraints (47) warrantee that exactly one pair carrier/range is selected, according to the lower and upper limits for the discount range imposed by constraints (47 49)! . Finally, constraints (52) indicate that all the decision variables are continuous and positive, with the exception of the complicating binary variable disturbing the network flow structure. Notice that within the vector of easy variables S [ , there are components that are continuous by nature ( , OT LW ik ik P P ) whereas others do not need to be forced to integer ( , , ik ik ij ijk e ,t y s ) because the special network flow 42 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) structure of constraints (2 10)• ensures integrality at the optimum, provided that the values for capacities, demands and bounds ( , , pg pg ik ijk Y ,S L U ) are also integer. 4.2. Addressing Tractability Issues for MAUD Problems The relaxation of the sending commitment condition makes even more difficult solving the network problem concerned in this research. Constructing a valid MILP formulation following the basic seminal multiple choice approach (namely, written in extended set g q Q in Figure 1) is possible, but it reduces the effectiveness of commercial solvers significantly. Further, this basic modeling approach can lead to intractability when applied to large instances. The two drawbacks of the arising MILP are its large size – increased number of constraints and binary variables as consequence of the n 1• newly introduced discount intervals (see Figure 2)- and, the deep and unbalanced branch-and-bound tree that is obtained from the LP solver – consequence of branching on the binary variables (Geißler, Martin, Morsi, &Schewe, 2012)-. Firstly, one of the key issues that causes intractability is addressed by the multiple choice approach on the modified range set g r R , which significantly reduces the number of intervals where defining binary indicator variables. When the novel gadget is considered joined to a multiple choice approach linearizing the nonconvex MAUD transportation costs, an alternative formulation – see model (17 34)• in Chapter 3 - emerges which outperforms the seminal one. Comparing to the classical multiple choice method stated in range g q Q , our identification of flat-ramp gadgets significantly reduces the number of intervals where defining binary indicator variables and the constraints which are exclusive to them. To address the second deficiency, along this section we explore potential opportunities to improve the branch-and-bound mechanism of LP based solvers, once the novel gadget technique has been applied. A two-steps procedure is followed which produces two alternative methods to piecewisely linearize the MAUD function instead of MC: (i). The first one consists on the use of a Disaggregated Convex Combination (DCC) approach – see (Croxton, Gendron, & Magnanti, 2003) for a comparison with the multiple choice one- and then, Solution Strategies 43 (ii). The second one copes with the task of formulating the DCC model using fewer binary variables. As it has been pointed out recently (Keha, De Farias Jr., & Nemhauser, 2006; Li, Huang, & Fang, 2013), when considering n discount ranges, one can formulate the adjacency conditions required in a DCC model using just 2 log n • • • • binaries, instead of representing the discount range selection with n binaries. It is worth to remark that while (i) introduces certain adjacency constraints, (ii) applies a logarithmic disaggregated convex combination method similar to that devised in Vielma & Nemhauser (2011) in order to no longer use the auxiliary binary indicator variables but incorporate the adjacency constraints directly in the branching (Geißler et al., 2012). More precisely, we have applied the logarithmic method in Li et al. (2013), which has been further modified to get a new formulation (henceforth LogDCC), whose LP relaxation has extreme points that all satisfy the adjacency property. This is the so-called locally ideal property (Sridhar et al., 2013) and, according to our computational experience, makes our variant exhibiting enhanced performance. 4.2.1. DCC Method to Piecewisely Linearize MAUD Shapes The commodity-dependent contribution of flows can be incorporated in a different manner. Making use of the fact that the cost of a flow that lies in a certain segment is a convex combination of the cost of its two extremes, that contribution can be written as a function of multipliers , [ 0, 1] rg rg • • • –see (56) -. This is the disaggregated convex combination (DCC) linearization method, which is appropriate for addressing continuous and semi-continuous piecewise linear functions. In our motivating problem a lower semi-continuous shape stands for commodity-dependent costs –see (32) -. When applied to our challenging planning problem it results: 44 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) ( 1) ( ) / 0 / 0 (55) , . . (18) (26), (31) , 0, ik ik i g OT WL DCC ik ik k K k K SOC E ik i N k ik ik v V k K k K vi T S rg ik ik ij ij k K g G v v v i i i r R rg rg rg ij ij ij F Min P P v V C v V C i i c s c e V c t c g V L s t z i j N • • • • • • ! • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ! ! " " " " " " " " " " " " " 1 / ( , ) (56) [ ] , , /( , ) (55) [ ] , , , /( , ) (56) , , , /( , ) (57) , , g i j rg rg rg rg rg ij ij ij i j g r R rg rg rg rg rg rg ij ij ij i j rg rg rg rg rg ij ij ij i j O ik ijk ik ij ik v v L y DFA U L i j v v L g DFA U L , r g i j v v L U L z , r g i j v v L e ,s ,t y P ! ! ! • • • • • • • • # • • • • • • • • # • " " , 0, 0, , 0, 0 (58) T LW rg rg rg rg ik ij ij ij ij P y z binary !# # # # 4.2.2. LogDCC Method to Piecewisely Linearize MAUD Shapes The third method proposed to address the lower semi-continuous piecewise linear optimization problem, follows the ideas first devised in Vielma, Keha, & Nemhauser (2008) to reduce the number of binary variables and constraints of DCC by using a logarithmic representation of the MAUD shape. This method has been formerly applied for the case of discontinuous objective function in Vielma & Nemhauser (2011). Solution Strategies 45 Let us denote n to the g gr N N• definition intervals of the MAUD piecewise linear function. In order to employ new binary variables 2 log ( ) 1 2 , ,..., ,..., n l ij ij ij ij b b b b ! " # , the key idea is to use a Gray code to encode each interval, which we do by defining an injective function 2 log ( ) : {1... } {0,1} n GRAY n ! " # $$% with the additional property that for any number {1... }k n& the vectors ( )GRAY k and ( 1)GRAY k ' only differ in one component. Table 4 – Gray codes in use in LogDCC: the all-zero code reserved to the nonload case, which is helpful to accomplish a locally ideal formulation. Our third alternative formulation (LogDCC) for the challenging planning problem is stated as follows: 46 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) ( 1) ( ) / 0 / 0 (59) , . . (18) (26), (55), (56) ( ) ik ik i g OT WL LogDCC ik ik k K k K SOC E ik i N k ik ik v V k K k K vi T S rg ik ik ij ij k K g G v v v i i i r R rg l l ij F Min P P v V C v V C i i c s c e V c t c g V L s t b Bit GRAY • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2 [ ] , 0, 1 / ( , ) , {1... log ( ) } (60) [ ] , , /( , ) (61) [ ] , , , /( , ) (62) , /( , ) , {1... l g g rg rg ij ij i j g r R g gr rg rg ij ij ij i j g r R rg rg rg rg ij ij ij i j l ij ij i j i j N v v L l N N MAXb i j v v L g F , r g i j v v L b MAXb , i j v v L l ! ! ! • • • ! " " • • # $ ! • • % & • • ! • ' • • ! ! • ( ! • ! • 2 og ( ) } (63) , , , 0, 0, , 0, , 0 (64) g gr OT LW rg rg rg l ik ijk ik ij ik ik ij ij ij ij ij N N e ,s ,t y P P y b MAXb binaries ! # $ • % & ' ' ' ' Notice that now, the range selection constraint is modified to equal a new binary variable ij MAXb . When the latter is 1, the above model enforces the adjacency constraints in a similar way to Theorem 1 in Li et al. (2013) by means of considering equations (60 61)• . However, seeking for an enhanced computational efficiency in our large-scale problem, we have introduced a crucial modification: we have reserved the all-zero Gray code –i.e. avoid its use to encode intervals- and enforced the implications 0 0 0 rg rg rg ij ij ij ij MAXb g !• ) • • ) • with constraints (56), (60), (61) , (62) and (63) . Solution Strategies 47 For an example, consider that at the outset the binary variables are defined on the ranges listed formerly in Table 4. Hence, the seven definition intervals can be encoded as in Table 4, and the linear equations needed to incorporate the adjacency constraints on every connection ( , ) / 0, 1 i j v v L i j N• • • • are as follows: 1 4 4 5 5 6 6 7 7 2 2 2 3 3 4 4 5 5 3 1 1 2 2 5 5 6 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( r r r r r r r r ij ij ij ij ij ij ij ij ij r r r r r r r r ij ij ij ij ij ij ij ij ij r r r r r r r r ij ij ij ij ij ij ij ij ij ij a b b b c b d MAXb • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 1 2 2 3 3 4 4 5 5 6 6 7 7 ) ( ) ( ) ( ) ( ) ( ) ( ) r r r r r r r r ij ij ij ij ij ij ij ij r r r r r r ij ij ij ij ij ij • • • • • • • • • • • • • • • • • • • • • • • • • • • 4.3. Heuristic for Approximate Resolution of MAUD Problems After close examination of the models stated so-far, we have devised a computational procedure to get fairly good solutions to the hard MAUD problems in feasible times. It is a two-phase heuristics algorithm, built around the successive resolution of several auxiliary problems, all of them considering that the ranges which fully partition the feasible volume of total flow on each link are g p P• (namely, that in which carriers presented their offers). 4.3.1. Phase1: Reducing the Available Real-Shipping Connections First, we propose a way of choosing a subset of links ˆ L L! to be the only one where the discounts based on the volume are allowed. To this aim, we proceed as follows: · Step1: Solve the continuous linear program resulting from minimizing the objective function: 54 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) to hospitals within a city; they model the problem as a multiperiod travelling salesman problem. Osvald and Stirn (2008) formulate a problem of perishable food distribution as a vehicle routing problem time window with time-dependent travel times. Ambrosino and Sciomachen (2007) determine the distribution plan for two products (fresh food and frozen food) to minimize the total travelling costs by formulating a vehicle routing problem with split delivery. These three works use a directed graph structure, which is also appropriate to represent the ANCL logistical issues. Nevertheless, they are not applicable to the ANCL problem, because they are suitable only for small networks and do not consider production decisions. Henceforth, we take into account the above observations and concentrate in efficiently provide a tactical logistic plan for serving the forecasted demand. We model it as the distribution of some commodities (clinical samples) from the sources to the demand nodes, with transportation costs that exhibit the following complicating feature: each shipping connection will contribute to system-wide costs throughout a non-linear piecewise function of the flow along that connection –i.e. concave MCNFP-. Cohn, Davey, Schkade, Siegel, & Wong (2008) use a similar approach to deal with the cost reduction achieved by a package carrier firm when contracting with a cargo airline; they assume that the cargo airline presents a price offer to each client depending on the cost of the entire load purchased. However, in Cohn et al. (2008) discounts on shipments are a function of the overall flow of the network; in our case study, we use discount factors attached to the flow in each connection. Since each connection can be potentially allocated to a different carrier, the aggregated flow over commodities (total freight) on each connection acts as the discount driver. More specifically, the aggregated flow along that connection is priced according to a MAUD cost structure, which results from incorporating the bumping clause to the AUD scheme (Chan et al., 2002). Since capacity constraints appear both on facilities production and transportation, the model that arises is a Capacitated MCNFP with concave costs (CMCNFP-CC). Next, we present the models developed to provide the framework for efficient operations in the target networks. Specifically, this means deciding on where to best satisfy the demand for commodities (i.e., destination facilities) and how to best route them within the distribution network, with the primary objective of minimizing operational costs: shipping costs, transshipment costs, processing and outsourcing costs, and additional operational costs when facilities workloads are too above or too beyond a certain reference workload. As claimed in (Andrade- Case Study: The Andalusian Network of Clinical Laboratories (ANCL) 55 Pineda et al., 2013), the tradeoff between taking advantage of the volume-based price incentives in the transportation activity and minimizing the rest of operational costs arises as the key issue in addressing the ANCL annual logistics decisions. 5.3. The New Planning Procedure: Applying Our Method We create a CMCNFP-CC model explained in a directed graph ( , , )G V L K• representing the involved distribution network, with commodities set K (namely, the various clinical sample types travelling through the network), vertices set V , links set L , demands and installed capacities for each clinical test type at the vertices, and link capacities. In this model, the flow through a vertex incurs a processing cost (if the sample is analyzed) or a transshipment cost. Our model also includes costs for flowing along links, whether they represent shipping costs or outsourcing costs. 5.3.1. Graph Representation To provide a flexible representation of the network, we identify the following functional entities: · Serving center (SC): the clinical laboratory (vertex) that processes tests on samples; its main feature is its installed capacity. · Outsourcing center (OC): a single entity that models the service points located outside the ANCL boundaries. · Point of extraction (POE): the center that collects samples for analysis; it could be within ANCL (in an SC) or outside ANCL (in the OC). · Demand transfer point (DTP): the center at which samples coming from POEs are collected (a process that involves reception, stabilization, and conservation); the samples are shipped later—typically to an SC. We next represent ANCL by using a directed graph (see Figure 4), which contains N+2 vertices: N vertices represent the centers (i.e., SCs, POEs, DTPs), one artificial vertex represents the individual demand insertion point (i.e., the supersource 56 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) vertex that supplies the total demand for the planning horizon) that feeds the POEs, and another artificial vertex represents the sink vertex for the outsourcing flows. For sample stability, if the distances between vertices are excessive (> 250 km), we exclude shipping connections. Figure 4 – This directed graph represents the regional network of clinical laboratories. Biological samples collected at points of extraction (POEs) are shipped to demand transfer points (DTPs) or service centers (SCs) for analysis, or to an outsourcing center (OC). 5.3.2. Evolving Shipping Costs and Solutions Approaches for the ANCL Planning Problem At the outset, carriers requested sending commitment as a condition to offer increasing per-unit discount rates to be applied to price the whole freight Case Study: The Andalusian Network of Clinical Laboratories (ANCL) 57 committed in every connection. An AUD piecewise linear structure arises from the scale economies in the transportation activity, which requires a mixed integer formulation. Papers #1 and #2 consider the AUD costs problem. In recent years, the ongoing competition among carriers and the centralized management of their services have allowed ANCL planners to avoid the sending commitment. Hence, the shipping costs follow a MAUD shape. Papers #3 and #4 address this MAUD costs problem. 5.4. Decision Support System for Supporting What-If Analysis 5.4.1. Requirements ANCL authorities needed comprehensive quantitative analyses to make informed decisions about efficiency improvements. Using decision tools to assess the alternatives for enhanced coordination and logistics in this large and complex network would be necessary. Specifically, because ANCL had opened new facilities as part of a general plan to improve access to medical care (Rodríguez Diaz 2011), planners needed a tool to respond to events in the changing ANCL environment; for example, a planner must consider that in setting up new hospitals, a hospital may come online (i.e., be activated) although its laboratory is still part of ANCL. The DSS had to be an IT tool that would address three requirements. · It must be able to use input data from diverse sources to construct and resolve different scenarios, regardless of the amount of computer time needed. · It must allow planners to analyze the workload and the flow assignments by studying a resolved scenario (i.e., annual plan) in greater detail. · It must allow planners to compare scenarios. The tool must allow them to rank each potential plan based on the average expected satisfaction of patients, as if the potential plan had been in practice. Planners must be able to use the tool to make operating decisions on primary care, the ANCL service that drains the most resources. 58 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) 5.4.2. The DSS Tool Developed We coded the ANCL planning model in AMPL (Fourer, Gay, & Kernighan, 2003) to create a core computational application (i.e., optimization engine), which reads input data from a relational database that contains data collected from RNCL, computes the solution by resolving the large-scale MILP described in The ANCL Long-Term Planning section, and places the solution into the same database. We packaged the optimization engine as a component of a Web-based DSS, which uses state-of-the-art IT tools to allow planners to revise input data, change parameter settings, run the planning model, and perform detailed analyses of the plans generated. We embedded an interactive graphical user interface (GUI) to facilitate these steps and allow the user to generate the appropriate ANCL annual production and transportation plan. Figure 5 – The DSS provides a user-friendly interface based on the Google Maps API to allow a user explore a solution. We coded the GUI using the PHP server-side scripting language, which allows planners to run the optimization engine and shows them a graphical representation Case Study: The Andalusian Network of Clinical Laboratories (ANCL) 59 of the output solutions based on the Google Maps API. The GUI allows the user to revise or modify the plan and to project and analyze What-If scenarios by changing data from the relational database to create new scenarios. The DSS provides userfriendly tools to enable the user to study a resolved scenario in greater detail and save it for future study or comparison, making it possible to compare two archived scenarios. In addition to the Google Maps API standard functionalities (e.g., zoom, drag), our graphical representation of solutions is characterized by various icons (see Figure 5). It uses larger triangles (laboratories with higher values for the displayed decision variable) and thicker lines (links with heavy traffic for the displayed layer commodity k ) for easier identification of issues. In addition, interactive access to a resolved scenario is possible by clicking on a specific laboratory (i.e., displaying its workload variable values) or by clicking on a specific link (i.e., displaying its flow variable values), as Figure 5 illustrates. 61 6. PAPERS RESULTS AND DISCUSSION his chapter gives an overview of the main findings from the appended papers, describing the relation and connection between them and the research questions. A quick glance at the result of each publication and its interpretation, starting at Paper#1 (Andrade-Pineda et al., 2013), is presented next. 6.1. The Road-map of ANCL Transformation 6.1.1. Paper #1: A DSS Tool around an Optimization Engine The first result from (Andrade-Pineda et al., 2013) was a MILP model for addressing the integrated logistics annual decisions (an AUD formulation similar to that in chapter 3). Once the MILP is solved, the last step to fully specify the production and transportation plan is estimating the average geographical accessibility (AGA) indicator: ( 1) ( ) [ ] / (74) ,i ijk i N k ik k K v V k K v v v i i i AGA = s s a L V • • • • • • • • • • • • , which indicates the average number of transshipments each test needs to reach the laboratory that will process the sample and hence, estimates the expected quality of service (QoS) provided. The second result in this paper was a Web-based DSS tool that embeds the concerned optimization model and uses state-of-the-art IT tools to allow planners to revise input data, change parameter settings, run the logistic operational model, and perform detailed analyses of the plans generated, thereby fulfilling the quantitative analysis requirements dictated by ANCL authorities. T 62 Planning Logistics in Healthcare Networks: Analysis and Resolution (Planificación de la Logística en Redes de Salud: Análisis y Resolución) A sensitivity analysis based on design of experiments (DOE) (Montgomery, 2009) has been applied on the model arisen. As reported in (Andrade-Pineda et al., 2013), both DSS real-time usability and suitability to the ANCL plan are affected mainly by network size. Furthermore, the embedded model offers considerable flexibility to allow decision makers to implement a variety of policies because of the sensitivity of the expected QoS to the parameter settings. 6.1.2. The Process Changes To great extent, the changes in any organization have to overcome the resistance of affected people. As reported in the seminal work (Andrade-Pineda et al., 2013), in order to take into account the ANCL personnel perspective, a multidisciplinary committee has guided the practical implementation of the new planning procedure for addressing the ANCL annual logistics decisions. Since 2009 the DSS tool embedding our integrated logistics models has been used in the improvement of the primary healthcare service. Further, it has been unvaluable in guiding the transformation of this organization (for instance, in determining a schedule for the activation of the new facilities to be opened in planning the primary care service), as the generated production and transportation plans have performed progressively lower outsourcing and shipping costs and better access to testing services. Similarly to the DSS tool, the HIP adoption was carefully prepared through a pilot project, focused also in the testing service on primary health care. Hence, in 2010 the Laboratory Component was tested with the purpose of defining the procedures and validating the workflow scheme. By end 2011, there were a subset of 15 (out of 29) clinical laboratories that have fully adopted the new business perspective (HIP), serving 3.5 million people (corresponding to 464 primary care centers from a total of 1509). About 1440000 orders were derived in 2011 among them and electronically tracked within the HIP, the bill of outsourcing costs reduced in a 40% (from 1.45 M€ in 2010 to 900000€). Tracing the producing data, there were two main laboratories that have greatly increased their workload. They two are henceforth considered reference laboratories, providing the others with capabilities formerly purchased to external laboratories. Historical data by the end of 2011 indicate that the new operational Papers Results and Discussion 63 model has led to cost containment while promoting the best use of fully-equipped facilities. Productivity bonuses (wages) have been paid to the efficient producing labs. As regards to ordering laboratories, they have taken advantage of the easier workflow to derive and track test orders (no paper, all in the HIP) and to get early test results to offer to patients (0.5-1 days in comparison with orders not tracked). Further laboratories have followed in 2012 and 2013, to take advantage of progressively more attractive SLAs. In year 2013 all the members in the ANCL were in the HIP and on average a 9% of all orders were derived inside the ANCL. These changes have required the reorder of connections and dependencies among members in a dynamic pattern, to which planners have used the DSS tool. 6.2. Tractability Improvements Over the years, major efforts have been done to improve the optimization engine at the core of the DSS. We first addressed how to best exploit the block structure of AUD problem. Later, the challenging MAUD problem that got in use made the tractability problem in the ANCL case study even more issuing – refer to Paper #4 for tractability reports on the real-sized arisen instances-. Next we discuss our research results on improving the optimization engine in the core of the DSS tool. Some of them are contributions to general CMCNCF problems with concave costs, and other are case study contributions (e.g. enhanced responsiveness in coping with the ANCL annual planning problem in the DSS tool in use). 6.2.1. Paper #2: A GBD Algorithm on the AUD Problem In Paper #2 two versions of the CMCNFP-CC problem are addressed: (i) A MILP model from which we derived a CBD procedure, and (ii) A MIBLP model which lead to a GBD one. We tried out the effective specialization of both decomposition approaches on the ANCL case study, in order to overcome the difficulties regarding to efficiency and speed-up of computations. 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IIE Transactions, 42(12), 865–880. 79 OTHER PUBLICATIONS IN THE RESEARCH AREA ·Paper #5 Dios M.A., Molina-Pariente J.M., Fernández-Viagas V., Andrade-Pineda, J.L., Framinan, J.M. (2015), “A Decision Support System for Operating Room Scheduling”, Computers & Industrial Engineering, (88), 430-443. http://doi:10.1016/j.cie.2015.08.001