Integrating Inventory Management and Distribution Scheduling for Clinical Supplies in a Hospital Context

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Integrating Inventory Management and Distribution

Scheduling for Clinical Supplies in a Hospital Context

Maria Madalena Marques Romano Pinheiro de Lima

Thesis to obtain the Master of Science Degree in

Biomedical Engineering

Supervisor(s):

Prof. Inês Marques Proença

Examination Committee

Chairperson: Prof. Mónica Duarte Correia de Oliveira

Supervisor: Prof. Inês Marques Proença

Member of the Committee: Dr. Daniel Rebelo dos Santos

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Declarac¸ ˜ao

Declaro que o presente documento é um trabalho original da minha autoria e que cumpre todos os requisitos do C ódigo de Conduta e Boas Pr áticas da Universidade de Lisboa.

Declaration

I declare that this document is an original work of my own authorship and that it fulfills all the require-ments of the Code of Conduct and Good Practices of the Universidade de Lisboa.

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Acknowledgments

Começo por agradecer à minha orientadora, Professora In ês Marques, por me ter guiado antes e du-rante este longo semestre, e por de certeza continuar a faz ê-lo no futuro.

Agradeço tamb ém ao hospital que me proporcionou este trabalho, pela disponibilidade e ajuda ao longo da minha investigaç ão.

`

As minhas meninas de biom édica, um enorme obrigado. Foi t ão melhor fazer este percurso ao vosso lado. Constança, como j á disse antes, és a luz desta universidade. Joana, foste a minha companhia numa grande parte deste curso. Sofs, Ritinha, Paulininha, Sarah, C átuxa, n ão me esqueço de voc ês. Obrigada por todas as gargalhadas ao longo destes 5 anos e a muitas mais nos pr óximos.

Aos meus amigos mais antigos, Cata, Costa, Migas, Hortega, Fernandes e Ahmad, obrigada por todo o vosso apoio e por continuarem sempre ao meu lado.

Gabriel, foste o meu verdadeiro porto-seguro e constante ao longo deste anos. Nunca te vou conseguir agradecer o suficiente por todo o teu apoio.

Aos meus pais, tudo o que fizeram por mim trouxe-me aqui, e n ão podia estar mais agradecida. Aos meus irm ãos, os meus melhores e mais antigos amigos, obrigada por me tornarem quem sou hoje e por me fazerem acreditar que tudo é poss´ıvel.

Aos meus sobrinhos, a parte mais importante da minha vida, espero sempre ser um exemplo para v ´os e conseguir mostrar-vos que com trabalho, foco e empenho, tudo se alcanc¸a.

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Resumo

Os gastos com a sa úde t êm registado um aumento nos últimos anos, levando a um interesse na melho-ria das atividades log´ısticas em hospitais. Estes t êm, habitualmente, um armaz ém central que apoia os serviços, abastecendo os armaz éns perif éricos. S ão respons áveis por um grande n úmero de produtos, com diferentes caracter´ısticas e datas de validade. Cada serviço encomenda invent ário ao armaz ém central de acordo com as suas necessidades. A complexidade est á em decidir quando e quanto en-comendar, organizar e agendar as entregas, sem esquecer as restriç ões inerentes ao meio hospitalar. A gest ão de invent ário torna-se, ent ão, uma tarefa dif´ıcil, onde as encomendas e distribuiç ão t êm de ser coordenadas entre os serviços e o armaz ém central. Com o objetivo de reduzir custos e a variabilidade no tempo di ário de entregas, é proposto um modelo de otimizaç ão para o controlo de invent ário e agen-damento de entregas. Este trabalho é motivado pelo caso de um hospital portugu ês, focando-se no invent ário cl´ınico. S ão determinados um hor ário de entregas e pol´ıticas de invent ário para os produtos consumidos em cada serviço, considerando restriç ões de capacidade e recursos humanos, e tamb ém a localizaç ão dos serviços recorrendo a rotas. Para considerar a incerteza no consumo, é desenvolvido um modelo de otimizaç ão robusta, complementar à abordagem determin´ıstica. Os modelos constituem uma abordagem inovadora ao problema, integrando controlo de invent ário com agendamento de en-tregas. S ão testados com dados do hospital e as soluç ões revelam hor ários mais preenchidos e uma menor quantidade de invent ário nos serviços quando comparadas com a realidade no hospital.

Palavras-chave:

Gest ão de Serviços de Sa úde, Gest ão de Invent ário Hospitalar, Controlo de Invent ário, Agendamento da Distribuiç ão, Produtos cl´ınicos

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Abstract

Healthcare costs have been increasing over the past years, leading to an interest in improving logistic activities in hospitals. Hospitals usually have a central warehouse supporting the services and replen-ishing their warehouses. They are responsible for multiple items with different particularities, often with expiration dates. Each service is responsible for placing inventory orders to the central warehouse ac-cording to its needs. The complexity lies on deciding when and how much to order of each supply and scheduling the deliveries, while dealing with multiple constraints. Inventory management is therefore a challenging task where ordering, distribution and consumption must be coordinated between the dif-ferent services and the central warehouse, maintaining wastage levels to a minimum and preventing stock-outs. With the aim of reducing costs and decreasing the variability in the workload of deliver-ies, a model that allows the optimization of inventory control and delivery scheduling is proposed. This work is motivated by the case of a Portuguese hospital, focusing on clinical supplies. Inventory policies for the items in each service and a schedule of deliveries are determined, considering capacity and human-resource constraints and the services’ location considering routing. A robust optimization model is developed, complementary to the deterministic model, to address demand uncertainty. This work proposes an innovative approach to hospital inventory management, integrating inventory control with delivery scheduling. Both models are tested with hospital data and the obtained solutions show heavier schedules, with more deliveries, and less inventory at the services when compared with the hospital’s current situation.

Keywords:

Health Care Services Management, Hospital Inventory Management, Inventory Control, Distribution Scheduling, Clinical Products

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List of Tables

2.1 Activity of PPH in 2017 and 2018. . . 7

2.2 PPH services that receive clinical inventory from the CW. . . 7

2.3 Inventory Classification at PPH. . . 8

3.1 Summary of the literature on inventory policy and distribution scheduling. . . 25

3.2 Summary of the literature on inventory policy and distribution scheduling considering de-mand uncertainty. . . 28

4.1 Notation. . . 37

5.1 Generic characteristics of the toy instances. . . 50

5.2 Characteristics of the items in the toy instances. . . 50

5.3 Characteristics of the services of the first toy instance. . . 50

5.4 Characteristics of the services of the second toy instance. . . 51

5.5 Generic characteristics of PPH instances. . . 52

5.6 Characteristics of the 19 services of PPH. . . 52

5.7 Characteristics of the 10 PPH instances for testing the deterministic model. . . 53

5.8 Characteristics of the 12 PPH instances for testing the robust model . . . 54

5.9 Results for 10 instances with the deterministic model. . . 57

5.10 Dimensions of the robust optimization toy instance in common for all the tests made. . . . 61

5.11 Results of the robust optimization model with the second toy instance, for three different demand ranges and with different values of the parameters Γ. . . 61

5.12 Computational results with the robust optimization model for instances R1 to R12. . . 66

6.1 Comparison of the schedule in place at PPH with schedules determined for Instance D6 with α = 1 and Instance R5 with Γ = 0 and α = 1. . . 73

6.2 Comparison of the inventory holding costs for 8 services in PPH in 31st _{of December of} 2018 with the average holding costs determined by the models tested with the schedule in place in PPH and for the complete solutions. . . 74

6.3 Comparison of the inventory level at 31st _{of December of 2018, the average daily} con-sumption in 2018 and the average inventory level and daily concon-sumption in Instance R5, for the items considered in service 6. . . 75

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6.4 Comparison of the inventory level at 31st of December of 2018, the average daily con-sumption in 2018 and the average inventory level and daily concon-sumption in Instance R5, for the items considered in service 19. . . 75

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List of Figures

2.1 Total number of clinical supplies consumed in each service in 2018. . . 10

2.2 Number of different clinical supplies consumed in each service in 2018. . . 10

2.3 Average cost per clinical item in each service in 2018. . . 11

2.4 Cost with consumed clinical inventory in each service in 2018. . . 11

2.5 Multi-echelon system of the Inventory Management at PPH. . . 11

3.1 Illustration of hospital’s Supply Chain adapted from Volland et al. [2017]. . . 17

3.2 Common inventory policies, adapted from Volland et al. [2017]. . . 18

4.1 Inventory Management: Period review logic. . . 34

4.2 Inventory Management: reorder point (s) and order-up-to level (S) logic. . . 35

5.1 Demand data used in the deterministic and robust models. . . 53

5.2 Solution generated by the deterministic model with the second toy instance. . . 56

5.3 Solution obtained by the deterministic model for instance D6 with α = 1, including the inventory policy (order-up-to level S), schedule of deliveries from the CW to the services and inventory level, for three items (i = 36, i = 49 and i = 62). . . 59

5.4 Solution obtained by the deterministic model for instance D8 with α = 1, concerning the inventory policy (order-up-to level S) for three items (i = 36, i = 49 and i = 62) and schedule of deliveries. . . 60

5.5 Evolution of the objective value for different values of the parameter Γ, for the three differ-ent demand ranges used. . . 61

5.6 Solution obtained by the robust model with Γ = 0 for the second toy instance. . . 62

5.7 Solution obtained by the robust model with Γ = 0, 5 for the second toy instance. . . 63

5.8 Solution obtained by the robust model with Γ = 1 for the second toy instance. . . 63

5.9 Solution obtained by the robust model for Γ = 0 for instance R5 and α = 1 concerning the inventory policy (order-up-to level S) for three items (i = 6, i = 7 and i = 54), schedule of deliveries and routes. . . 68

5.10 Solution generated by the robust model for Γ = 0, 2 for instance R5 and α = 1 concerning the inventory policy (order-up-to level S) for three items (i = 6, i = 7 and i = 54), schedule of deliveries and routes . . . 69

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5.11 Solution obtained by the robust model for Γ = 0, 8 for instance R5 and α = 1 concerning the inventory policy (order-up-to level S) for three items (i = 6, i = 7 and i = 54), schedule of deliveries and routes. . . 69

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List of Acronyms

ADSE Instituto de Proteç ão e Assist ência na Doença

AMB Ambulatory Service

ANG Angiography

CARD Cardiology

CFO Chief Financial Officer

CP Central Pharmacy

CT Computed Tomography

CW Central Warehouse

DLM Deterministic Linear Model

DNLM Deterministic non-Linear Model

ER Emergency Room

GAST Gastroenterology

GPO Group Purchasing Organization

HU Head Unit

ICU Intensive Care Unit

IU4 Floor 4 Inpatient Unit

MAM Mammography

NHS National Health System

PHAR Pharmacy

PPH Portuguese Private Hospital

PRIV Private Consultation

RHA Regional Health Administrations

RLM Robust Linear Model

SAMS Serviços de Assist ência M édico-Social

SSU Small Surgical Unit

STE Sterilization Center

SU Surgical Unit

TSP Traveling Salesman Problem

URO Urology

VMI Vendor-Managed Inventory

VRP Vehicle Routing Problem

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Chapter 1

Introduction

The introductory chapter discusses the motivation and aim of this dissertation. A brief overview of the topic and the main contributions are addressed. Finally, an outline of the dissertation is made.

1.1 Motivation

Healthcare costs have been gradually increasing in developed countries over the past years. This leads to an emerging interest in improving logistic activities in hospitals. In fact, logistics activities constitute a large portion of hospital expenses, generally more than 30%, only surpassed by staff related costs (Volland et al. [2017]). Furthermore, there is a growing pressure in improving performance and deliver-ing healthcare more efficiently (Nicholson et al. [2004]), and, as a matter of fact, cost reductions from optimizing logistics activities do not affect patient care and safety. (Jarrett [1998]).

Although it is significantly studied in industrial environments, material logistics is less explored in the healthcare setting. Additionally, the demand of clinical supplies has an uncertain behavior. While some items can maintain regularly a high and relatively stable demand, this is not the case for all clinical materials that might show volatility and demand uncertainty. Moreover, hospitals are a complex setting and logistic activities tend to be performed by clinical staff without a background on logistics or inventory management, leading to experienced-based decisions as opposed to optimized solutions. Appropriate inventory policies and schedule of deliveries must be identified and applied, adapted to each service, allowing not only proper operation in each service but also the coordination between each service and the central warehouse, improving the functioning of the hospital as a whole.

The aim of this thesis is to optimize the schedule of deliveries and the inventory policy for items in each service of a hospital, satisfying the requirements and constraints faced by hospital inventory management but focusing on practical schedules. The schedule of deliveries must be adapted to the needs and specificities of each service while complying with the limitations of the hospital, minimizing the inventory holding costs and the variability in the time spent in deliveries throughout the week. A

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linear programming model that creates delivery schedules and inventory policies is proposed, and a robust optimization model is also developed, in order to take into account uncertainty in the clinical inventory demand. It is motivated by the case of a private hospital in Portugal (PPH) with the goal of determining inventory policies for the clinical items in each service, and a new schedule of deliveries from the hospital’s central warehouse to the services. The PPH is composed by multiple services that receive clinical supplies from the central warehouse. Although PPH handles multiple types of inventory (such as medicines and consignment inventory), the focus of the current work is solely on the clinical supplies (e.g. syringes and needles), and not on the pharmaceutical or in any other case, since the clinical inventory is considered to be the type of inventory with a more evident need of management changes and improvement. A schedule of clinical supply deliveries is already in place, but there are no safety stocks nor quantities to order defined. The current schedule of deliveries shows a considerable difference in the time spent in deliveries and in the number of services visited per day. The inventory quantities in multiple services are not in line with the average consumption, and excessive stock levels are commonly found. Due to obsolesce and expiration, a high wastage level is registered at PPH. Therefore, appropriate inventory policies and schedule of deliveries must be identified and put into practice.

1.2 Topic Overview and Contributions

Inventory logistics and management problems in healthcare can be divided in three hierarchical levels: strategic, tactical and operational levels, and the latter is further distinguished between offline and on-line decisions (Hans et al. [2012]). The strategic level addresses structural long-term decisions such as supply chain design and warehouse location. Tactical decisions concern the implementation of the strat-egy and organization of the execution, with a shorter planning horizon than strategic decisions but with less flexibility. An example is supplier selection. Operational planning is characterized by short-term decisions concerning the execution of processes. Decisions in advance (proactive) are offline opera-tional decisions (e.g. placing inventory orders), while online operaopera-tional planning are real-time decisions (reactive), such as emergency replenishment. In this work, two main decisions are addressed: to deter-mine a schedule for clinical inventory deliveries to the services in a hospital, and settle inventory policies for the items in each service. These decisions can correspond to either medium-term decisions falling into tactical planning, or offline operational planning. If the purpose of these decisions is to be adopted during a more extensive period (for example up to 1 year), this would correspond to a tactical plan. However, these decisions can be changed more frequently, for example weekly or monthly, leading to an operational decision. In this work, weekly and monthly approaches are addressed, and therefore this problem falls into the operational planning category.

In order to clarify the scope of this particular problem, this work is classified as a noitem multi-service inventory control and distribution problem. Inventory decisions within hospital inventory manage-ment can be divided in two particular approaches:(i) inventory oriented, regarding inventory policies and control, and (ii) schedule oriented, that are concerned with planning and scheduling of distribution

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ac-tivities.This work aims to coordinate both, in an integrated approach to all the services inside a hospital. By aiming to determine appropriate inventory policies for the clinical items consumed by each service this work falls within inventory control, also commonly referred to as inventory management and inven-tory policy. The distribution and scheduling stream approaches are commonly classified as inveninven-tory distribution problems, and by determining a schedule of deliveries to the services this work considers inventory distribution. It is worth noting that although this work addresses routing, it is not classified as an inventory routing problem since routing is not the focus. It is included to better estimate the time spent in deliveries and better perform the distribution, guaranteeing that the time available for deliveries per day is not exceeded. It is a part of the constraints and not a part of the objective function, and thus is not a crucial part of the problem. Accordingly, this work falls within a new approach in healthcare inventory management, the Inventory Control and Distribution problem.

The contributions of this study are fourfold. First, a new problem is addressed in this study and modelled integrating both inventory control decisions, by the definition of inventory policies, with distri-bution decisions, determining a schedule of deliveries from the warehouse to the services, adapted to the needs of each service but focusing on practical schedules. It is an innovative approach aiming to fill a gap in the hospital inventory management literature. Second, this work proposes a robust optimization model, to address the uncertainty in demand. This is the first work in the hospital inventory management field to consider a robust optimization approach. Third, routing is considered to better estimate the time spent in deliveries and take into account the distance between the services, which was not considered so far in this field. Finally, this work is applied to a real-case scenario of PPH, to address the current Inventory Control and Distribution problem, being tested with hospital data. A large number of items and services are considered, creating a complex setting and providing insights to hospital inventory manage-ment. Authentic recommendations and findings can be provided to the hospital and real improvements achieved.

1.3 Thesis Outline

This dissertation in composed by seven chapters.

First, the problem to be addressed is presented in the Introduction Chapter, including the main mo-tivation, scope and contributions of this work, as well as a brief introduction to the theme and the case study (PPH).

A description of the PPH is included in the second Chapter (Problem Contextualization). The in-troduction of the hospital within the National Health System, an outline of its activity and the current situation of inventory management problem are addressed.

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problem in the literature and provide an overview of the current state-of-the-art. First, hospital logistics, including inventory control problems and distribution problems are presented. Then, an overview of rout-ing and uncertainty approaches, emphasizrout-ing robust optimization, are included.

The model is introduced in Chapter Four (Methodology), starting by a description of the main fea-tures of inventory management, with a particular focus on PPH. The two developed models are then formulated: a deterministic and a robust optimization model. Extensions to the main models are also considered, so that other details can be considered.

The fifth Chapter (Results) is concerned with the application of the two proposed models to PPH. The considered instances are described, before the validation and testing of the models. Examples of solutions are presented and discussed.

The sixth Chapter (Validation) is concerned with the assessment of the solutions obtained by the models in contrast with the current situation at PPH. First, the schedule of deliveries in place and inven-tory levels registered at PPH are compared with the presented solutions. The validation of the solutions with PPH stakeholders is discussed, concluding with managerial insights gained from this work.

This dissertation concludes with some final remarks (Chapter seven, Conclusion), clarifying the limi-tations of this research. Suggestions for future work are included.

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Chapter 2

Problem Contextualization

Hospitals are organized in multiple services and usually have a central warehouse supporting their ac-tivity. Each service is responsible for placing inventory orders to the central warehouse according to its needs. The complexity lies on deciding when and how much to order of each medical supply, organizing and scheduling transportation and deliveries, as well as on dealing with capacity and human resource constraints, in an environment such as healthcare where stock-outs are usually not allowed. Inventory management is therefore a challenging task where ordering, distribution and consumption must be coor-dinated between the different services and the hospital’s central warehouse, maintaining wastage levels to a minimum. It is important to have proper inventory policies (i.e., inventory rules and guidelines) adapted to the activity, needs and particularities of each service but subject to the existing limitations (such as limited storage space). These should be supported by practical schedules with, for example, a balanced workload throughout the week.

This chapter describes the inventory management problem at the portuguese private hospital (PPH), presenting the current situation. Section 2.1 characterizes PPH, details the integration of the hospital within the National Health System as well as the hospital’s size, organization and activity. The current inventory management at PPH is summarized in Section 2.2. Section 2.3 defines the inventory man-agement problem tackled in this work, and Section 2.4 summarizes this chapter with concluding ideas on the current problem.

2.1 Portuguese Private Hospital

PPH is a private hospital unit that has been delivering health care services for many decades across multiple medical specialties.

2.1.1 PPH within the Portuguese Health System

The Portuguese Health System is composed by three different systems: the National Health Service (NHS), health subsystems and private voluntary health insurance (Sim ˜oes et al. [2017]). The health

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care is delivered by both public and private healthcare providers. Established in 1979, the NHS guar-antees universal coverage for all residents, nearly free at the point of use. It is financed mainly by taxes and consists of three levels: primary, secondary and tertiary health care. It has a gate-keeping structure, where the primary care physicians act as gatekeepers for access to secondary care. Health subsystems offer total or partial health care to certain professions or companies, and can be public or private. Examples include Instituto de Proteç ão e Assist ência na Doença, I.P. (ADSE, I.P.) and Serviços de Assist ência M édico-Social (SAMS). The main advantage of subsystems is that beneficiaries can go to a specialist appointment without referral from primary health centers, allowing the choice of provider (Barros [2017]). Private voluntary health insurances enable a faster access to healthcare, and consist of individual or group policies. In 2016 it was estimated that health subsystems covered 16% of the popu-lation and that 25,8% is covered by private health insurances (Sim ões et al. [2017]). The NHS, following the Law on the Fundamental Principles of Health (1990), is managed at a regional level by Regional Health Administrations (RHAs), whose directive body is appointed by the Minister of Health. RHAs are not only responsible for the strategic management of population health and primary care centers, the supervision of hospitals and the implementation of policies, but are as well responsible for contracting services with private providers. While the NHS more commonly provides primary, acute and special-ized hospital care, the private sector provides mainly diagnostic services, renal dialysis, rehabilitation and dental consultations, with some degree of public funding. The Ministry of Health is responsible for coordinating both public and private health care providers, as well as financing public ones. Public expenditure accounted in 2014 for 64.8% of total health expenditure, while private expenditure, which has been increasing in the recent years, represented 35,8% (Sim ões et al. [2017]). Public hospitals are funded through global budgets, and there is an increasing role of diagnosis-related groups. As for private insurers and health subsystems, these pay the providers for their services, but 76,3% of private health expenditure in 2014 corresponded to Out-of-pocket payments. In 2015 total healthcare expendi-ture consisted of 9% of the Portuguese gross domestic product, corresponding to a total of 16 132 190e and 1 557,5e per capita (PORDATA [2019]).

Similar to other private health care facilities, PPH has agreements with private health insurances and health subsystems (both private and public), enabling beneficiaries to have access to services provided by PPH. In addition, PPH has agreements with Administraç ão Regional de Sa úde for surgical procedures and medical exams. Concerning surgical procedures, arrangements can be also made directly with public hospitals.

2.1.2 Facilities, Activity and Yearly Results

Currently, PPH provides permanent health care, 24h a day, every day. The focus of the hospital remains on technology and innovation.

Regarding inpatient activity, PPH registered 4 079 admissions and 19 494 night stays in 2018. A total of 6 895 surgical procedures were performed, revealing an increase when compared to 2017. In

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addition, PPH provided 106 494 medical appointments and had 8 968 emergency room visits reported in 2018. An overview of the hospital’s activity in 2017 and 2018 is presented in Table 2.1.

2017 2018 Number of Patients Admitted (Ambulatory Unit) 4448 4079 Number of Inpatient Unit Stays 20043 19494 Surgical Procedures 5842 6895 Angiography Procedures 7432 4704 Medical Appointments 97961 106496 Medical Tests 469040 476638 Emergency Room Visits 8673 8968 Intensive Care Unit Stays 1539 1416 Pediatric Intensive Care Unit Stays 138 88

Table 2.1: Activity of PPH in 2017 and 2018.

As for infrastructure, the hospital has a Central Warehouse (CW), a Central Pharmacy (CP), and multiple services supported by warehouses or storage space. PPH is composed of 36 services. Of the 36 services, 24 correspond to services that consume clinical supplies, and the remaining 12 cor-respond to services such as help cabinets, administrative services, cleaning services, among others. Some services are outsourced, such as the Blood Service and Medical Laboratory, and therefore don’t commonly receive items from the CW. Among the 24 services, there are 19 services that have regu-lar clinical inventory deliveries from the hospital’s CW. The total stock in these services is evaluated in about 700 000e. The 19 services are shown in Table 2.2. This table also introduces the acronyms and number used in this work for the 19 services. To design routes for deliveries, the services are grouped into clusters, according to proximity. The CW is also considered as a cluster.

Main Building Separate Building Floor 0 Floor 1 Floor 3 Floor 5 Floor 7

Head Unit (18. HU) CT (1. CT) Mammography (2. MAM) Pharmacy (3. PHAR) X-Ray (4. XR) Urology (5. URO) Emergency Room (6. ER)

Small Surgical Unit (7. SSU) Gastroenterology

(8. GAST)

Surgical Unit (14. SU) Cardiology (9. CARD)

Inpatient Unit (13. IU5) Surgical Unit (14. SU)

Private Consultation

(17. PRIV)

Floor 4 Floor 6 Ambulatory Service(19. AMB) Inpatient Unit (10. IU4)

Angiography (11. ANG) Intensive Care Unit (12. ICU) Inpatient Unit (15. IU6) Sterilization Center (16. STE)

Table 2.2: PPH services that receive clinical inventory from the CW, with corresponding acronym and number. The two surgical units, in floor 3 and floor 5, function as one service in terms of inventory management and thus they share the same acronym and number in this work.

2.2 Inventory Management at PPH

There are several types of medical items handled by the hospital: Medicines, Clinical Supplies (such as syringes and catheters), and patient-specific surgical supplies, among others. Typically, each clinical item consumed by a patient is debited to the corresponding patient account at the hospital for billing purposes.

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In PPH the medical materials are classified into 5 categories (Table 2.3). Medicines (101) and Clin-ical Supplies (102) are used in every service that receives clinClin-ical supplies. There are three types of items only used in the Surgical Units and Angiography: 104, 105 and 808. Clinical Surgical Supplies (104) correspond to commonly used surgical materials. Materials type 105 are specific for a surgery (e.g. prosthesis) and are ordered purposely for a scheduled surgery for a specific patient. These mate-rials are often chosen by the physician performing the surgery, according to his preference. PPH also works with as well consignment materials (item type 808). Consignment items are stored at the Surgical Units’ warehouses, and are only charged to the hospital when the given package is open. In general, the supplier is the one responsible for the stock management. Furthermore, the hospital handles other types of materials such as administrative supplies, laundry and maintenance supplies. In total, there are more than 6000 different items handled by PPH.

Identifier Item Description 101 Medicines 102 Clinical Supplies 104 Clinical Surgical Supplies 105 Patient-Specific Surgical Supplies 808 Consignment items

Table 2.3: Inventory Classification at PPH.

Medicines (items type 101) are handled by the CP, which has the responsibility of ordering medicines from the suppliers. Orders are placed almost every day and, in regular situations - when there is not any problem at the supplier such as stock-outs - the normal lead time for medicines to be delivered to the CP is one or two days. The CP is then responsible for delivering every day to different services of the hospital: Ambulatory Service, Inpatient Unit (prepared at the CP specifically for each patient), among others. There are services within the hospital with automatic replenishment, where at the end of each day automatic orders reset the medicines to their initial level.

All clinical supplies (items type 102, 104, 105 and 808) are handled by the CW. There are currently stored about 180 000 to 200 000e of clinical supplies in the CW. The CW places orders for suppliers when it is needed, typically every day. There is a continuous review of the inventory levels, and these orders are made in relation to the material consumption at the services. After ordering to suppliers, the items are delivered to the hospital. Lead times depend on supplier, but without abnormal situations they are delivered in three or four days. Type 105, 808 and some 104 items are delivered directly to the Surgical and Angiography Units. Items of type 102 and remaining items of type 104 are delivered to the CW, where they stay until being delivered to each service.

The coordinator of each service is in charge of inventory management, being responsible for ordering material and medicines through the PPH computerized inventory system (named Gest ˜ao Hospitalar). If

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there is stock at the CW, it is delivered directly to the service. If not, an order is made to the respective supplier by the CW workforce. There is a schedule of deliveries from the CW to the services thus each service has fixed days for placing orders and receiving clinical inventory deliveries. The services can place orders until 12pm of the delivery day, to be delivered that same day. Therefore the expressions ”delivery day” and ”ordering day” have the same meaning in this work. The two Surgical rooms (3rd and 5th floor), can place orders and receive deliveries every day. In terms of inventory management they are considered as a single service. The remaining services have one, two or three days per week to place orders, depending on their activity and on the coordinator of the service. The current schedule is static meaning that every week it is cyclically repeated (see in Appendix A - Table A.1). This schedule defines the ordering/delivery days for the 19 services. Nevertheless, the schedule can be subject to changes, usually based on the feedback from the coordinators of each service.

The distribution is done in the afternoon period. From 12pm to 13pm the delivery is prepared by the warehouse workforce and then vehicle is loaded, in a process lasting from 15 to 30 minutes, depending on number of services visited and quantity to be delivered. If there is not enough capacity at the vehi-cle, the vehicle returns to the CW to load the remaining material. Arriving at each service, the delivery worker drops the material at a specified location, and the coordinator of the service is then responsi-ble for verifying the delivery. Emergency deliveries, if needed, can happen, either through the CW or from another service. The latter is more common and in this case, the stock levels at the lender unit are replenished afterwards. There is no definition of safety stocks, nor order-up-to or fixed quantities to order. Instead, at every ordering day, each service reviews its inventory levels, decides what to order and makes an estimate on the order quantity. This decision is based on the experience and intuition of the service coordinator. The inventory management is different from service to service, depending on the coordinator of the service.

PPH is a relatively small hospital, being located in confined facilities while providing a high number of services and numerous types of specialties and procedures. Consequently, it reveals space limitations. These limitations are reflected on the storage space. In some services this problem is not critical, which is the case, for example, of the X-Ray and Mammography. These are small services with a relatively stable activity using limited amount of items. However, in services that consume a larger number of items, with a more unpredictable demand and multiple procedures, the space available for storage of material is scarce. The surgical unit is the service with the more serious space limitations. In addition, while some services have physical warehouses to store inventory, such as the surgical and the inpatient units, others have storage spaces such as cabinets and drawers spread across the service, such as the Emergency Room. From here on out, the terms warehouse or service warehouse will be referred to the space allocated in each service for supplies, regardless of there being a physical warehouse destined for storage or only storage space used directly inside the services.

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Figure 2.1: Total number of clinical supplies con-sumed in each service in 2018.

Figure 2.2: Number of different clinical supplies con-sumed in each service in 2018.

services, such as the Emergency Room (ER), the level of activity is high, which leads to a larger ma-terial consumption. On the other hand, other services such as Private Consultation (PRIV) have less activity, and thus reveal a lower needs in terms of clinical material. Concerning inventory costs, there are services that generally use more expensive materials, which is the case of the Surgery Unit (SU), while in others the clinical supplies can be cheaper. Additionally, different services can have different levels of material diversity. Some may use a smaller amount of different clinical supplies, while others may use a more diverse set of inventory. This all translates into a lower or higher cost with the clinical ma-terial consumption per service. These four introduced metrics (total number of clinical supplies, number of different clinical supplies, average cost per item and cost with consumed inventory in each service) are quantified in Figures 2.1 - 2.4, regarding the clinical material consumption data in 2018 (items type 102 and 104) gathered from PPH. The Sterilization Unit (STE) is the service with a higher number of units consumed in 2018, followed by the Surgical Unit (SU) (see Figure 2.1). The Pharmacy, X-Ray and Urology services register lower material consumption. The SU is the service with the higher variability in items used (1094, a considerable high number compared to all the other services), seconded by the Angiography service (only consumed 386 different items) (see Figure 2.2). The Pharmacy and Urology are the services with smallest variability (20 and 21, respectively). Considering the average cost per item consumed in 2018 (Figure 2.3), the Surgical Unit is the service that consumes more expensive items, followed by the Ambulatory Service (AMB). The service with lower average cost per item con-sumed in 2018 is the small surgical room (SSU). Finally, in regard to the cost of the concon-sumed clinical inventory (Figure 2.4), the service with the higher cost is the Surgical Unit, once again with a consider-able difference to all the other services. The Pharmacy and the X-Ray are the services that registered a lower cost. It can be concluded that the SU is one of the most complex services in terms of inventory management. It registered the highest cost with consumed inventory in 2018 and average cost per item, revealing also a high number of clinical supplies consumed and the largest variability in terms of number of different items used. Services such as the Pharmacy, X-Ray and Urology reveal overall lower costs and less activity.

Services can be organized in two different ways with respect to inventory management and con-sumption. In most services, a track of the inventory levels in their warehouses is kept and, as previously mentioned, when supplies are used they are debited to the corresponding patient account. This act

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Figure 2.3: Average cost per item in each service in 2018.

Figure 2.4: Cost with consumed inventory in each service in 2018.

consists on the consumption of units of the given supplies from the service warehouse. However, some services only provide packages of procedures that are charged to the patient, already including supply consumption. Therefore, there is no registry of the consumption of clinical supplies in these services. This is the case of the CT service and in such situations, when there is an inventory delivery to these services it is automatically classified as consumed. Therefore, there isn’t any registry of stock nor debit of supplies to patients in these services. Although there is stock being kept, the computerized system (Gest ˜ao Hospitalar) does not keep track of the inventory levels. Another particularity of PPH inventory management is that, in some services, there are items grouped into families. When there is a consump-tion of these kinds of items, the debit is registered in the family, and not in the individual item.

Summarizing, the material management at PPH can be defined as a multi-echelon system with three levels: Suppliers, Central Warehouse and Services. The CW places clinical material orders to suppliers that then deliver the inventory to the CW. The central warehouse is responsible for making inventory deliveries to the services, according to the respective orders. Sometimes suppliers can deliver directly to the Surgical Unit, instead of delivering to the CW (particularly for items type 105, 808 and some items of type 104). A scheme is presented in Figure 2.5. The services have a periodic replenishment, with out-of-cycle orders if needed, while the central warehouse follows a continuous review policy.

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2.3 Inventory Management Problem at PPH

In order to successfully deliver the best clinical care possible, PPH is supported by a myriad of different activities such as procurement, logistics and inventory management, distribution of supplies through the different services, among others. These activities reflect the performance of the hospital, and so they must be properly organized and coordinated, allowing the adequate functioning of PPH.

Given that health expenditures have been increasing in developed countries over the past years, logistics activities account for the second biggest source of costs for hospitals, and more relevance is being placed on performance and on increasing service levels, there is a desire for improvements on the efficiency and effectiveness of logistics activities. In fact, improved material management can enable cost reductions (Moons et al. [2019]) that do not affect the quality of care (Jarrett [1998]). Furthermore, hospitals such as PPH deal with multiple constraints and restrictions, such as limitations on the storage capacity, with a small central warehouse and confined capacity at the services, and human resource constraints. Therefore, in order to decrease inventory related costs while improving care efficiency, thor-oughly reasoned delivery schedules and stock levels must be achieved, accounting also for the limited space to hold inventory and time allocated for replenishment activities, which includes not only the time spent in the central warehouse and at the services, but in the transportation time between them. The needs of each service must be satisfied, allowing staff to perform their daily activities without overload, avoiding stock-outs and minimizing emergency refills.

Currently, there is no inventory policy defined for the clinical items. Safety stocks are not determined, and no order quantities or order-up-to levels are defined. In some services, the quantities ordered reveal to be somewhat constant independently of the situation and time of the year. Conversely, there are ser-vices where these quantities are estimated based on the coordinators experience or the occupation level or types of procedures performed. Indeed, the consumption and order of clinical supplies in the Surgical Unit might vary depending on the procedures. Moreover, there is only one staff member performing deliveries from the CW to the multiple services. These deliveries are carried out from Monday to Friday in the afternoon period, and the number of services that receive clinical supplies visited per day varies between 5 and 11. The busiest days in terms of inventory deliveries are Tuesday and Wednesday, with Monday registering less activity (see Table A.1). In addition, a considerable amount of waste in clinical supplies is reported at PPH. In general, there is a very large stock of clinical items in services, due mainly to fear of stock-outs by the service coordinators. Practically every item has an expiration date and obsolescence is also a reality. Without taking into account the perishability of clinical supplies, and without an alignment of the consumption with order quantities and inventory levels, substantial obso-lesce and expiration levels are expected.

It can be also noted that the schedule currently in place has a considerable difference in terms of number of services visited per day and consequently on the time spent in deliveries, showing an

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unprac-tical schedule. It is important to create a schedule that is not only adapted to the needs and particularities of each service but that is also practical, with a balanced workload along the week.

Therefore, a proper management of inventory and deliveries is essential, with the aim of determining appropriate inventory policies and schedule of deliveries, leading to lower overall inventory levels and consequent lower holding cost at the services. The total amount of clinical inventory at the hospital is valued at about 900 000e with 700 000e (77,8%) at the services. The services are therefore a possible source for cost and stock level reductions, since the majority of stock is being kept there. Lower quantities can be stored at the services, taking advantage of the proximity to the central warehouse, which has the capability to properly support each service. This will not only decrease the inventory holding costs, but decrease wastage due to obsolesce and expiration. Suitable inventory policies and practical schedules of deliveries from the CW must be determined, adapted to each service but allowing the coordination between the central warehouse and the services, minimizing the inventory holding costs at the services and satisfying the capacity and human resource constraints faced by the hospital. Thus, the problem addressed in this study is an Inventory Control and Distribution problem, with the integration of inventory management policies and scheduling of deliveries at PPH. A model to determine the optimal policy for the items in each service and the schedule of deliveries from the CW to the services is developed.

2.4 Chapter Considerations

Hospitals must be supported by efficient inventory management since to provide proper care it is in-dispensable to have all the required supplies. In addition, logistics account for a considerable portion of hospital expenses, and a better and improved material management can lead to cost reductions. Moreover, an improvement on a hospitals performance and service levels is supported by a more effi-cient inventory management. However, there are limitations that must be considered such as capacity constraints, that directly influence inventory management policies, and human resource constrains, that include central warehouse and service warehouse activities as well as time spent in transportation.

This chapter presents an overview of PPH, detailing its activity and how it is organized. The man-agement of inventory at PPH is described, in order to better understand the current situation and the problem addressed in the present work.

The next chapter includes a review of the available literature related to inventory management, in-ventory control and scheduling of deliveries, with emphasis on healthcare and hospitals. Approaches on how to solve the proposed problems are presented.

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Chapter 3

Literature Review

The Inventory Control and Distribution problem addressed in this work aims at determining optimal in-ventory policies and schedule of deliveries for the clinical items in the services within a hospital. Routing is considered in order to better model the real setting of the hospital, with services distributed across different floors or separate locations. Demand uncertainty is taken into account since the clinical item demand is not known in advance and cannot be accurately estimated. This chapter is concerned with the published studies of interest to this work. First, a brief a review of logistics in healthcare is performed in Section 3.1. Section 3.2 focuses on inventory management and control (3.2.1) and distribution plan-ning and scheduling problems (3.2.2). In Section 3.3, two problems of interest to this work are analyzed: routing in subsection 3.3.1, and uncertainty in subsection 3.3.2 with focus on robust optimization, the approach chosen to solve the problem. Finally, Section 3.4 provides concluding remarks, connecting with the problem under study.

3.1 Inventory Logistics in Healthcare

Inventory management and logistics is an area where improvements leading to cost reductions can be made without jeopardizing patient care and safety (Jarrett [1998]). It is estimated that more than 30% of hospital costs arise from logistics, making logistics the second largest expenditure for hospitals af-ter personnel expenses (Volland et al. [2017]). Therefore, hospital logistics and inventory management have been increasingly studied in the literature.

Four areas can be identified within inventory logistics in hospitals (Volland et al. [2017]): (i) Supply and procurement, (ii) Inventory management, (iii) Scheduling and distribution, and (iv) Holistic supply chain management. The main goal of supply and procurement is to decrease purchasing costs and so this area includes the integration of hospitals and suppliers and the increase of purchasing power by the hospitals with for example the formation of group purchasing organizations (GPOs). The integration of hospitals and suppliers is supported by the recent practices in supply chain management of lean and out-sourcing activities, and examples include Vendor-Managed Inventory (VMI). Inventory management (ii)

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is concerned with location planning, inventory policies and inventory classification, whereas scheduling and distribution (iii) is focused on the transportation of the inventory itself, and can be divided between internal and external distribution. These two areas - (ii) and (iii), in particular inventory policies and in-ternal scheduling of deliveries - are the focus of the current work, and further discussed in Section 3.2 . Lastly, holistic supply chain management is mainly a qualitative approach for supply chain improvements.

In a brief overview of (i), Nicholson et al. [2004] compare the inventory costs of a multi-hospital three-echelon system with an outsourced two-three-echelon system, and conclude that outsourcing not only shows lower costs but does not affect the care provided. Within the application of outsourcing to real situations, Krichanchai and MacCarthy [2017] assess the benefits, challenges and limitations of adopting VMI in the pharmaceutical supply of a hospital, based on two real cases: unsuccessful situation on a private hospital and a successful public hospital case. The successful public hospital implementation had no costs and revealed lower risks when compared to the unsuccessful private case. Beaulieu et al. [2018] analyze the outsourcing of medical-supply distribution in a group of hospitals considering the opinion of multiple stakeholders, with the aim of making recommendations for the implementation of outsourcing. Six Sigma is a quality control method for continuous improvement of processes, eliminating unnecessary or inefficient steps and reducing defects that can be applied in supply chain management, and in par-ticular in healthcare (Conger [2015]). Al-Qatawneh et al. [2019] propose and analyze the effectiveness of a framework for the implementation of Six Sigma in a hospital. Regarding the increase of purchasing power, the creation of GPOs aims to obtain lower purchasing prices, reduce administrative costs and increase negotiation expertise (Nollet et al. [2018]). Indicators for assessing the performance of GPOs are proposed by Nollet et al. [2018]. Rego et al. [2014] elaborate on the advantages and disadvantages of GPOs, developing a model for decision support that, given a set of hospitals, recommends and eval-uates a GPO structure in terms of number, size and composition minimizing supply chain costs.

A wider look into the supply chain is taken in (iv), with the main goal of improving performance. Mul-tiple studies focus on providing benchmarks and measurements of performance for logistics activities in hospitals which is the case of Feibert et al. [2019], that propose performance indicators based on two hospital cases of bed and pharmaceutical logistics in the US and Denmark. Serrou and Abouabdellah [2016] focus on cost, quality and safety. Chen et al. [2013] are concerned with hospital-supplier integra-tion, technology integraintegra-tion, trust and knowledge exchange in SC performance.

Of the four identified areas within inventory logistics in hospitals, Inventory Management and Schedul-ing and distribution - (ii) and (iii) - are the focus of the current work. In particular, inventory policies and delivery scheduling are the main concern. Therefore, the following section provides a comprehensive review of the publications made within these two fields.

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3.2 Inventory Management and Distribution Scheduling

Typically, the material distribution in hospitals follows a multi-echelon network, with a central warehouse (CW) receiving items from suppliers and delivering them to each service. An illustration of a common supply chain is represented in Figure 3.1. Some studies put emphasis on the external supply chain, that includes manufacturers and suppliers (Kritchanchai et al. [2017]) while others are focused on the in-ternal supply chain, that comprises purchasing, inventory, distribution and consumption activities within the hospital grounds (Moons et al. [2019]). This work concerns the internal supply chain: integrating inventory and distribution activities.

Figure 3.1: Illustration of hospital’s Supply Chain adapted from Volland et al. [2017].

The literature on hospital inventory management addresses not only clinical use materials (the focus of this work) but a big emphasis is put on the pharmaceutical case (e.g. Guerrero et al. [2013], Kelle et al. [2012] and Maestre et al. [2018]). In addition, other types of goods are explored such as laundry (Banerjea-Brodeur et al. [1998]) and sterile items (Van de Klundert et al. [2008] and Tlahig et al. [2013]).

Within hospital inventory management, the most relevant and widely discussed topic is inventory policy. Regarding distribution planning and scheduling activities, internal distribution is concerned with scheduling the delivery of items within the hospital, and external distribution regards waste management and inter-hospital transportation. Within internal distribution, there is an enormous variability in the set-ting and setups of each study and of each case and therefore common and recurrent practices are rare (Volland et al. [2017]).

To the best of our knowledge, and based on the reviewed and researched literature, there aren’t any studies concerned with the problem currently addressed in this work, i.e., focusing on both inventory pol-icy of clinical items and distribution scheduling decisions for the services of a hospital. Multiple studies consider the optimization of inventory policies, while others concern distribution planning and scheduling activities. These two streams of literature in the healthcare setting are described, starting with optimiza-tion of inventory policies in subsecoptimiza-tion 3.2.1 followed by distribuoptimiza-tion planning and scheduling activities in

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subsection 3.2.2.

The presented and reviewed publications in the two following subsections (3.2.1 and 3.2.2) result from an exhaustive search of the studies found in the literature within the field. These are considered to have similarities and to be of interest to the problem addressed in this work.

3.2.1 Optimization of Inventory Policies

Hospitals are responsible for a significant number of items and inventory investments in healthcare are considerable. Therefore, inventory management and control have become more important and conse-quently the number of studies has been increasing over time (Volland et al. [2017] and Nicholson et al. [2004]). Generally there are two types of review cycles addressed, periodic and continuous. In contin-uous replenishment, an order is placed every time an item falls below the reorder point s. In periodic models an order can be placed in every review period R. The order can be made only if the stock level is below a reorder point s, or in every review period regardless of the stock level. Regarding the quantity to order, a fixed reorder quantity Q can be followed, or increasing the inventory to an order-up-to level S. An overview of inventory policies can be seen in Figure 3.2. Inventory decisions focus on determining the optimal policy: defining the inventory cycle and parameter setting s, R, Q and/or S.

Figure 3.2: Common inventory policies, adapted from Volland et al. [2017].

As previously mentioned, the hospital supply chain is composed by suppliers, a central warehouse, and multiple services storing inventory. Studies can be focused on individual services, they can consider the complete internal supply chain or even integrate the external part of the hospital supply chain. First, studies concerned with individual services are addressed, before discussing the integration of multiple services and central warehouse, and lastly three publications considering a wider network are reviewed.

Individual Warehouses. Several studies address the hospital inventory problem at individual

ser-vices or warehouses. In an early work, Dellaert and van de Poel [1996] employ a simple inventory rule to address the inventory problem at a hospital central warehouse, following a (R, s, c, S) policy, an ex-tension of the (R, s, S) policy. In this case, in every review period R, if the inventory level of an item is below the reorder point s, an order is placed, and, in addition, all other items from the same supplier with

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a level below the can-order level c are ordered as well, with the inventory reaching the order-up-to level S. The goal is total cost minimization, determining s, c and S levels for multiple items with a given review period R and service level. The demand is not deterministic (i.e., it is not known in advance). Instead, it is modeled as stochastic, where the demand is considered to follow a probabilistic distribution, in this case a normal distribution.

Little and Coughlan [2008] develop a model to optimize the inventory policy of a service in a hospital for sterile and bulk items considering space constraints, applied to the real situation of an intensive care unit in an Irish hospital. The service level, frequency of delivery and order-up-to level are determined for all items, within a prespecified range of values, following a periodic replenishment policy. Two different objective functions are considered: maximizing the minimum service level and maximizing the average service level. The demand is assumed to follow a normal distribution.

Bijvank and Vis [2012] propose two models for the management of disposable items in a service, one that considers capacity limitations, where the service level is maximized subject to a restriction on the capacity, and a second one that minimizes the capacity with a restriction on the service level. Multi-ple items are considered. Here, the demand is modeled as stochastic, following a Poisson distribution. A Markov model is employed (Markov models are addressed further ahead, in subsection 3.2.2). In addition, two periodic-review policies are compared: the (R, s, Q) policy, ordering a reorder quantity Q and the (R, s, S) policy, where the inventory position is raised to an order-up-to level S, for prespecified review periods R. The authors conclude that the latter uses capacity more efficiently (the fill rate - the fraction of demand satisfied directly from stock on hand - is higher), although with a high service level both policies perform well. Still, an inventory rule to find near optimal values for reorder level and order quantity is developed for the capacity model.

The different objectives in the pharmaceutical supply chain of a hospital, that can be conflicting among stakeholders, are explored by Kelle et al. [2012]. Regarding different levels of decision making, the differences between the operational, tactical, and strategic levels are analyzed, and for the oper-ational level, the inventory policy (reorder point s and order-up-to level S) is determined for a service, considering capacity constraints on a service warehouse. Three models are proposed for the (s, S) pol-icy with stochastic demand: two models that find the optimal space allocation for all items, minimizing the holding and ordering costs, and one model to minimize the total number of deliveries. The authors conclude that a reduction of inventory related pharmaceutical expenditures up to 70%–80% by imple-menting the models is possible.

Exploring a different approach by integrating the two different review cycles, Rosales et al. [2014] design a hybrid inventory policy for the management of a service combining a periodic-replenishment, which has lower costs, with a continuous replenishment in order to avoid stock-outs. A model for a single-item inventory system in a service is developed. Values for parameters (s, S) periodic policy and (R0,Q)

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for the out-of-cycle replenishment (i.e., extraordinary deliveries not following the review period) are de-termined for a prespecified review period, minimizing total costs. The parameter R0 corresponds to a critical inventory level below which an emergency out-of-cycle replenishment is triggered. A simulation-based optimization is developed and in 52 out of 96 scenarios tested, the result was a hybrid policy. The remaining scenarios suggest a period policy in 28 cases and a continuous policy in 16. Uncertainty in the demand is considered to follow a normal distribution but an approximation of the model is done for deterministic demand. The authors conclude that the hybrid policy can enable cost and inventory level reduction, especially when the review interval is large.

Internal supply chain. Hospitals’ internal supply chains are generally composed of multiple

ser-vices supported by a central warehouse, which is in turn responsible for ordering to suppliers. Baboli et al. [2011] propose two models for one warehouse, one pharmacy and multi-item inventory policies: a centralized model where the warehouse knows the real necessities of the pharmacy and decentralized model where the pharmacy places specific inventory orders to the warehouse. The distribution from the warehouse to the pharmacy is taken into account, with vehicles with different capacities, and the demand is assumed to be deterministic. The goal is to determine optimal R and quantities of replen-ishment, following the Economic Order Quantity model, and which vehicle to use (the replenishment is done by a single vehicle without dividing the order), minimizing total costs including transportation expenses. Both models are compared, showing that the centralization leads to lower costs.

In Guerrero et al. [2013], the problem of joint-optimization of inventory order-up-to level S policies is addressed, specifically for non-critical pharmaceutical products with long expiry-dates. A multi-item sys-tem with one central pharmacy and multiple services, considering emergency replenishment, is studied and the order-up-to level S is determined for each item in each service, with a prespecified review period R. The authors show that the optimal policy for services is obtained by setting the reorder point s one unit lower than the order-up-to level S, and that independently optimizing each department provides the same solution as the joint-optimization case. The objective is to minimize the stock-on-hand value at the supply chain while satisfying the minimal service level required by the hospital and dealing with capacity limitations. Stochastic demand is considered and a Markov Chain model is employed.

In Wang et al. [2015], a dynamic drum-buffer-rope (DDBR) inventory model for material management is proposed, considering multiple items, a central warehouse and multiple services. Drum-buffer-rope (DBR) is a planning and scheduling solution where the drum corresponds to the bottleneck of a system, the buffer is the material upstream of the bottleneck, and the rope is maintaining the system at the pace of the drum, in order to not disrupt the system, given the bottleneck. In this case the drum corresponds to the material consumption at each hospital service and the buffer to the materials. The system is dynamic, meaning that buffer size (stock level) is adjusted according to changes in the consumption of material. The optimal stock size and replenishment quantity are determined given the review period R, minimizing the total inventory quantity. It is then compared with conventional DBR and with the hospital’s

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reorder point original policy, achieving a better performance.

Complete network. For the case of a healthcare network with a central warehouse supporting

multi-ple hospitals each serving numerous services, Nicholson et al. [2004] develop two models to determine the service level and order-up-to-level S for the considered locations, for a single item in a periodic re-view cycle: one considering a three-echelon system and a second where a distributor would operate the central warehouse and deliver directly to the individual services (two-echelon system), with the objective of minimizing holding and back-ordering costs. The authors found that the latter always outperforms the former in terms of costs and provides equivalent service levels. The second model is close to a centralized distribution system, the case of a central warehouse serving several services within a single hospital.

Uthayakumar and Priyan [2013] consider the entire supply chain in the pharmaceutical setting. In-tegrating production and distribution in a two-echelon network with one pharmaceutical company and one hospital (with multiple pharmacies), the authors propose an optimization model for a continuous review policy (s, Q). Inventory lot size Q, lead time and number of deliveries in a production cycle to achieve the target service level are determined for multiple perishable items, minimizing total supply chain cost. The demand during lead time is modeled as stochastic following a normal distribution. In an extension to this model, Priyan and Uthayakumar [2014] address uncertainty in the hospital’s expiry rate and holding cost and in the pharmaceutical company’s production and screening rate and holding cost, with a fuzzy-stochastic environment (fuzzy approaches are explained and discussed in subsection 3.2.2), introduce uncertainty on the quantity received by the hospital following a stochastic approach with a normal distribution, and consider the lead time to be composed of m independent components.

The studies reviewed thus far concern inventory management and control, in particular the opti-mization of inventory policies. Some studies regard individual locations (services), others consider the hospital’s internal supply chain, integrating a central warehouse with one or more services, and there are approaches taking into account the external part of the supply chain as well. Nevertheless, the concern of the current work is not only inventory control, but also distribution planning and scheduling activities, which are addressed in the following subsection.

3.2.2 Optimization of Distribution Planning and Scheduling Activities

The management of materials is not only focused on inventory decisions such as reorder point, order-up-to-levels, but can instead turn its attention to distribution planning and scheduling decisions. These include how often and when to visit each care unit and when to order from suppliers. Inventory policies are not defined, but the order sizes might be determined complimentary to the distribution plan.

Integrating Inventory Management and Distribution Scheduling for Clinical Supplies in a Hospital Context