Nowadays, the number of medical treatment and medicines increases which allows a spectacular growth of the health care sector. Despite this development, the sector suffers from inefficient management and ineffective planning . Managing patients, nurses and physicians is a difficult problem that needs to be solved. Hospital bed planning is a central problem that affects hospital capacity, health care quality and also management of nurses and physicians. During the last decades, hospitals are a non profit organization where the demand is not a primary concern for the manager of these hospitals. Today, many private hospitals are acting with a primary objective to satisfy the demand and to provide outstanding services to compete with other private hospitals . The hospital is not just a medical care unit but also is providing hotel and transportation services. To insure competiveness of hospitals we need to improve the quality of services and to satisfy as much as we can the demand. Therefore, hospitals need to look for their supply chain and how to manage it.
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In this report, we focus on the supply chain management of hospitals in Dubai. Dubai’s health services are internationally recognized and due to their high standard and their modern facilities equipment, are comparable to other developed countries. The location of hospitals in Dubai is strategic to ensure accessibility for patients. There are approximately 20 clinics and hospitals distributed across the Emirate. The ratio of clinics/hospitals to patients is 1:78,000. One of the more impressive practices of medical professionals in Dubai is the post-clinic, private medical call. These are considered as part of their responsibilities. Medical attention is provided, regardless of residency or nationality. In general, Dubai aims to improve the over-all wellbeing of its people. Its strategy is to provide patient-specific care. The most popular medical services provided by healthcare providers in Dubai include immunizations and vaccinations, psychiatric treatments, medical fitness examinations, community services (such as marriage and family counseling), adult and infant yoga therapy, rehabilitation, and education on health and nutrition.
We focus on this report on Rashid private hospital in Dubai, UAE. We mainly present a multiple objective stochastic programming for the bed capacity planning taking into account the quality of the service and the stochastic demand in that hospital.
In the next chapter, we present a general overview of the hospital supply chain in general before we present in chapter 3 some of the Rashid hospital operations. In chapter 4, we focus on hospital bed capacity planning in order to introduce to the multiple objective stochastic program that we are going to propose for Rashid hospital bed capacity planning. The obtained model is transformed in chapter 6 into its certainty equivalent and solved in chapter 7 using data from Rashid hospital.
Hospital supply chain
Health is defined as ââ‚¬Å“a state of complete physical, mental and social well beingââ‚¬. The health care sector is an important sector as regards to the welfare of people. Health services require the synchronization of various resources, such as Human resources, medicines and medical equipment.
In any organization, a supply chain must be designed in accordance with its mission. The mission of all hospitals includes the maximization of the level of patient care. The size of a hospital, geographical location, diversification, and the various specializations all affect the nature of care provided in a hospital and, therefore, the goals of its supply chain.
The hospital chain may have some of the following goals :
To secure the availability of product ,
To Reduce the storage space and to maximize the patient care,
To reduce time and cost of handling the medical team (nurse, pharmacist, physician)
Minimize the stocks of inventory
The main functions of hospital supply chain are defined as follows :
To allocate the main resources (technical platforms, beds, physicians, nurses …) and their location in the hospital.
To plan for extra resource needed (medical staff, medical equipment), and to schedule the care activities.
To organize transportation of patients and equipment.
Generally, the hospital supply chain may be split into two parts (see Figure1): the external chain and the internal chain .
Fig 1: Hospital supply chain 
2.2. The external supply chain
The external chain begins with companies specialized in the creation of the raw material (patent, drug, machinery, etc). The raw material can be materialized (machine, drug, etc.) or immaterialized (know how to cure). The manufacturer may itself be the creator or a company that works in relationship with him. In this case, the company is responsible for the duplication (making molecules on a large scale and add excipients or drug) for the test and for the control. Once the product is ready to be used and receives the necessary certifications, the role of the distributor is to place the product on the market. The market is generally formed by a central purchasing (WHO, national distributors, NGOs, etc) or individual (hospital, pharmacy, etc). Each health facility may maintain direct relations with manufacturers so that products pass through certain distributors.
2.3. The internal supply chain
The health establishment is the last link in a supply chain consisting of manufacturers and distributors from various industries (medical supplies, pharmaceuticals, food, laundry, maintenance, etc…).
The supply chain within the hospital is complex. The size of the hospital, the geographical location, the diversification, various specializations, the high cost and perishable goods, all affect its supply chain. The first characteristic of the appropriate health care supply chain is its diversity in distribution channels. Inside the hospital, the hospital product is made up of items at low prices or high-prices and durable and perishable goods that are consumed in large or small quantities.
A health institution is composed of five main activities that manage different types of flows to offer many services or products to patients. These activities are defined as follows:
Intralogistics activities which are the fact that the hospital acquire, receives and distribute different supplies used in the service.
The demand management that is the planning and the coordination between the different necessary resources.
Operations and services given to the patient within the hospital from admission to discharge.
External logistic represented by the medical follow-up for the patient.
Services to the patient which are all auxiliary activities that are not linked to medical activities offered to the patient (gift shop, religious programs, etc).
The supply chain within the hospital can therefore be presented as follows:
The hospital supply chain must be developed for a specific product based on its unit cost, demand variability and the physical size. We can say that integration of the supply chain in the health care sector requires the synchronization of internal and external supply chains to each individual service. A good supply chain management within a hospital is necessary and must be performed efficiently
Operations in Rashid hospital
Rashid Hospital is a 454-bed general medical/surgical hospital in Dubai, the United Arab Emirates, and is a part of the Dubai Government Dubai Health Authority. Rashid Hospital is considered in Dubai as one of the first medical facilities for trauma, emergency, ambulatory care and critical care which provide a high-quality of services to all patients within the community. The Rashid hospital provides also leadership in the training and education of health care professionals. In the emergency, Rashid Hospital is considered as one of the most reputable and prominent medical centers in the Gulf region. It receives the majority of complicated case other hospitals are destined to Rashid hospital which coordinates also closely with the Dubai Civil Defense and Police for the training of emergency medical staff inside the airports
In Rashid hospital, two types of admissions are used: the outpatient admission and the admission through emergency department
3.2. Outpatient registration
This type of admission or registration is present in all hospitals and it can be defined as follows: “An outpatient admission is presented when a patient is admitted to the hospital, surgical center or ambulatory center for a surgical or nonsurgical operation, therapeutic procedure or diagnostic procedure, that does not require an overnight hospital stay. The preparation for outpatient admission varies with each procedure .
In Rashid hospital, the responsible physician, the treating physician and the admitting physician are responsible of the admission procedure of the outpatient. The registration of the outpatient is done after the patient gets a discharge from the emergency department or the inpatient unit. This must is done by the physician who gives the patient an outpatient appointment for follow up with the required specialty. After that the patient will be transferred to the required specialty.
The next step is the direct admission which must be done during the same day. The admitting physician/clinic nurse informs the case manager and the admission office that the patient requires admission, and then the admission of the concerned patient is linked with the availability of a bed. Next, the account department or the admission office informs the patient about charges for treatment as per the hospital payment policy. The clinic nurse will inform the patient about the admission conditions and about provisional diagnosis. But if the hospital can’t find an available bed, the treating physician will give to the patient another appointment or ask for a transfer of the patient to another healthcare unit (if the case is urgent).
In Rashid hospital, urgent case admission is directed to the emergency department. The admission in this department is different from the outpatient admission. In the next section, we are going to overview admissions procedures in the emergency department.
3.3. Admission through the emergency department
This type of admission is different from the outpatient admission because patient must access directly to health due to the urgency of his/her case of illness. It can be defined as housing the patient in the hospital to provide special interventional procedure(s) or definitive treatment. We can distinguish three types of patients in this admission. First, the unstable patients who will suffer irreversible damage or loss of life if not admitted immediately. Second, the stable patients who are the patients that requires urgent treatment or interventional procedures(s) that cannot be accomplished on an outpatient basis. Third, patients are not suffering loss life or serious damage if not admitted .
In the emergency department, the emergency physician has to observe and to investigate to know if the patient needs admission and to refer the patient to the on call physician. The emergency physician and on call physician will decide about the required screening and diagnostic tests after examining/before admitting the patient.
The emergency department must inform the case management about the admission, provisional diagnosis and level of care needed and check for the availability of bed.
If there is no available bed in the selected department, the case manager can admit the patient temporarily in another department where bed is available (with adequate equipment). But if there are no available beds throughout the hospital, the case manager has to refer patient to another hospital.
The patient flow in Rashid hospital can be presented as the following figure :
Bed flow in hospital
3.4. Rashid hospital departments
At Rashid Hospital it exists many specialized medical and paramedical departments all equipped to receive all kinds of patients and also patients from neighboring hospitals.
The existing specialities in this hospital are:
Rashid Hospital aims to provide an outstanding service to all outpatients and patients that are admitted through the emergency department. This aim cannot be achieved if the hospital has not the adequate capacity in terms of hospital bed and human resources (physicians and nurses). At the same time the hospital must run in profit to ensure the future of its activity.
In this study we will try to answer this important question of hospital capacity planning in order to determine both the level of beds and the number of resources that Rashid hospital needs to satisfy the random demand.
Hospital capacity planning
The capacity is defined as the quantity of service that the health care institution must provide to satisfy patients need. Capacity management is related to the control of the impact of demand variability on the management of the health care institution. It concerns the good coordination of resources through the management of medical equipments, human resources and bed occupancy. Hospital capacity has long been an indicator of the importance of the hospital structure and for budget allocation .
The capacity planning is a component of the internal hospital supply chain. This planning is usually used to help hospitals, to do well their objectives which are:
Trying to avoid an underestimating of the number of beds,
planning for the future
maintain a good service quality,
optimize resource use,
satisfy the requirements of internal and external security.
4.2. Bed capacity management
In hospitals, capacity planning usually focuses on the total capacity of beds, the capacity of the surgical system, the allocation of beds for different services, equipment capacity, the ability of auxiliary services, and the number of staff and their competence .
Before we plan capacity in a hospital, the following issues must be clarified :
The length of the planning horizon (operational, tactical and strategic)
The level of the provided care (primary, secondary)
The type of care (provided to inpatient and / or outpatient)
The quality, cost and types of available resources (physicians, nurses, technicians, rooms, beds, medical equipments and all what constitute an input for health)
The hospital capacity depends not only on the number of beds, but also how these beds are used.
The hospital capacity can be influenced by several factors:
The geographic distribution of patients: each locality has its own hospital.
The type of resources currently in use: a patient who wants to have a particular diagnosis by the nearest hospital must visit the hospital where it exist the necessary equipment.
availability of nurses, physicians, and support equipment in the hospital
Hospital bed management may affect cost, quality and accessibility of care. The daily management of beds is closely related to the management of the hospital. To properly determine the capacity of beds, we need to track the activities of hospital patients (admission, assignment, stay and leave) . The essential role of the hospital bed manager is to ensure balance between supply and demand for hospital beds.
Bed management has a long-term component, which is the choice of the overall number of beds as well as sharing among different departments, and a short-term component for the daily bed allocation to patients.
We conclude that hospital beds are important measure to determine the hospital capacity. The bed management does not only affect the overall capacity but it also impacts on cost, quality and accessibility of care .
4.3. Models for hospital bed capacity management
Many models were elaborated to determine the optimal number of beds inside a hospital. The simple and the most used models to evaluate the adequate capacity of a hospital department are based on the following index:
N = (length of stay * number of patient)/number of days
= number of patient per day / number of days
The transfer between departments and the randomness of some of the index parameters are not considered in the above model. To overcome this shortness in the index model more elaborate stochastic models can be used. These models can be used for the short term (daily problem), the long term (monthly problem) or even for the case of a disaster.
The Queuing models are short term models that are usually related to the operational level of the hospital capacity planning. These models characterize the relationship between the number of beds, the average occupancy levels and the number of patients transferred from one department to another based on the arrival time of patients, the nature of patients transferred from one unit to another and the period of use of each type of bed by the patients. .
The simulation models have the ability to consider the results of a decision on an item without carrying out the experiment on the actual item [9, 19]. They represent an artificial reproduction of what will happen when random parameters change their values. Sally C. Brailsford  proposed a simulation model to plan for the capacity of an intensive care in hospital using software called SIMUL8.
Nowadays, the health sector, an increasingly privatized sector, seeks to find an effective planning of his resources for the long term. Taking into account the benefit t and also the quality of offered service. The medical ethics and money profit are two conflicting criteria. Multiple objective programming is a model that can deal with several criteria. Chu and Chu  proposed a goal programming model for hospital beds allocation in Hong Kong. The model takes into account the constraints of location, the demand constraint and constraints related to manpower.
Black and Carter modeled the problem of allocating physicians to hospital department using a linear goal programming model . The model focuses on the number of cases handled by a physician taking into account that the hospital must be able to generate enough revenue to cover fixed costs and variable production.
The models developed for the hospital bed capacity planning problem are mostly categorized as stochastic models. These models are suitable for short and medium term. In this study, we are more concerned with the long term. This is way we focus on multiple objective programming models to plan for the bed capacity in Rashid hospital.
In this document, we follow Ben Abdelaziz and Masmoudi model to determine the optimal bed capacity in Rashid hospital . The model was first developed for bed capacity planning in all public Tunisian hospital to evaluate of missing beds.
l: specialty in a hospital department, . We have two kinds of specialties. Those called primary health specialties for which we cannot transfer the patient to another hospital and secondary healthcare specialties that in case of no hospital bed available can be transferred to another hospital.
: A subset of primary healthcare specialties that can be served by the same hospital bed (for which we are using the same equipment), .
: A subset of secondary healthcare specialties that can be served by the same hospital bed, .
: the set of specialties that may be served by the same type of nurses ,
: the set of specialties that may be served by the same type of physician ,
5.2. The parameters
: Existing beds in specialty in the hospital, .
: the number of beds that can be added in the specialty in the hospital, .
: ratio of nurses per bed, i.e. the number of nurses needed to serve one patient in the specialty l, .
: ratio of physicians per bed: The number of physicians needed to serve one patient in the specialty l, .
: the stochastic yearly demand for the specialty in the hospital where express the random demand.
5.3. Decision variables
: number of beds in the specialty in the hospital.
5.4. Constraints of the model
Maximum and minimum number of beds in the hospital
The demand for the set of specialties in the hospital must be satisfied
The demand for the set of specialties must be satisfied otherwise transferred to another hospital
where express the number of vacant beds in the set of specialties and the number of missing beds in the set of specialties .
5.5. Objective functions
The first objective function is to minimize the cost of adding and managing new beds
where is the daily cost of creating and managing an additional bed of the specialty in the hospital during the period of investment.
The stochastic constraint (1) is related to the satisfaction of the demand in secondary health care specialties. This transfer generates an additional cost (transfer cost). We have to use a recourse approach to get certainty equivalent constraint. In a recourse approach a penalty in the objective function is generated when the solution does not satisfy the random constraint. Here the penalty is the transfer cost.
The expected transfer cost is
where is the expected transfer cost.
The third group of objective functions is to minimize the number of nurses in the groups of specialities in the hospital
The fourth group of objective functions is to minimize the number of physicians in the groups of specialities in the hospital
5.6. The final model
The final model is expressed as the following multiple objective stochastic program
To solve the above multiple objective program, we need to transform it into an equivalent mathematical program. This transformation must be done following the problem hypotheses. In the next chapter, we will review these hypotheses and we will provide a suitable transformation of the program (P) into its certainty equivalent program.
The certainty equivalent program
The program (P) is a stochastic program as it presents two stochastic constraints (P.5) and (P.6) and a multiple objective program as it has several objective functions to minimize. To solve a multiple objective stochastic program, we need to transform it into its certainty equivalent program, under predefined approaches. In the next sections and using a chance constrained approach for the constraint (P.5), a discretization technique for the constraint (P.6) and a goal programming approach to deal with the two objective functions (P.3), and (P.4), we are going to build such a certainty equivalent program to the program (P).
6.2. Chance constrained approach
The chance constrained approach transforms the random constraint into a deterministic constraint by considering as feasible solution those satisfying the uncertain constraints with a predefined level of probability . Therefore, under a chance constrained approach, the following stochastic linear constraint
where , and are random variables, will be transformed into the following deterministic constraint
where is fixed level of probability. It means that a feasible solution must satisfy the uncertain constraints for all scenarios with a probability of occurrence higher than .
The constraint (P.5) expresses the satisfaction of the demand on primary health care specialties (the demand on these specialties cannot be transferred to another hospital). It is difficult and not justified to satisfy the demand for all scenarios and especially scenarios with a small probability of occurrence. In the following, we propose a chance constrained approach to deal with the constraint (P.5). Therefore, the demand on the primary health care specialties Ar must be satisfied with a given fixed probability level as follows
The constraint (3) is a chance constraint.
Using the model hypotheses, the random daily demands are normally distributed with a mean of and standard deviation of . Note that,
Then, we can rewrite the chance constraint (3) as follows
6.3. Discretization approach
We must satisfy almost surely the constraint (P.6). In stochastic programming, the normal distribution is approximated by a discrete distribution and then the constraint (P.6) can be rewritten as follows:
The total recourse cost and the monthly transfer cost for secondary health care specialities are transformed using the discretization of the normal distribution of demands as follows:
6.4. Goal programming approach
Charnes and Cooper  are the first to introduce the goal programming approach which is essentially used to transform multiple objective linear program into a linear program. This transformation consists on these steps:
First, to fix a target values for some or all objectives (called also goals)
Second, to transform the objective functions to constraints
and third minimizing the difference between objective functions value and these goals.
Using a goal programming approach, the following objective functions
can be transformed to constraints as follows
where and are the negative and the positive difference, respectively, between the fixed goals and the achievement , and the new objective function to optimize is expressed as follows
where and are weights of the negative and the positive deviation, respectively.
The objective functions (P.3) and (P.4) minimize the number of nurses and physicians in each hospital. As the actual number of nurses and physicians can not be reduced, a goal programming approach is used to deal with objectives (P.3) and (P.4) where goals must be equal to the number of nurses and physicians already working in hospitals.
Let us denote by and the number of nurses and physicians, respectively, who already work on the specialty in the hospital. We denote by and the goals for the objective functions (P.3) and (P.4), respectively, and are expressed as follows
where is the number of nurses in shortage in the group of specialties in the hospital, is the number of nurses in excess in the group of specialties in the hospital, is the number of physicians in shortage in the group of specialties in the hospital and is the number of physicians in excess in the group of in the hospital. From these goal constraints the additional cost that gives monthly salary of new nurses and physicians is as follows:
where is the nurse salary per month in the group of specialty in the hospital and is the physician salary per month in the group of specialty in the hospital. The monthly salary of nurses and physicians who work in hospitals is fixed.
Now, as all objective functions represent yearly expenses, we propose to combine all cost objectives which are the yearly transfer cost, the yearly cost of creating and managing new beds and the yearly salary of new nurses and new physicians, into a single objective function expressed as follows:
6.5. The certainty equivalent
Finally, under a chance constrained approach and a goal programming approach, the certainty equivalent program to the multiple objective stochastic program (P) is expressed as follows:
The chance constrained and the goal programming approaches are used to generate the certainty equivalent program. Their use is motivated by the problem hypotheses. In the next chapter, we are going to test the model using real data from Rashid hospital.
The experimental study
In this chapter, we discuss the results obtained by the previously presented model for hospital bed capacity planning using data from Rashid hospital. The data was obtained from the administration of the hospital and is related to a recent period (2009-2011). The quality of results here is highly linked to the quality of the input data. We are going in the following to report some of the data given to us as well as the model output.
7.1. Model parameters
From the Rashid hospital we collected data related to the following parameters:
Number of patients / specialty
New admissions/ day
Discharges / day
Stay of every patient
Number of Physicians / specialty
Number of physicians / team
Number of teams / specialty
Number of hours worked by each physician
Number of patients assigned to each team / day
Number of nurses / specialty
Number of beds / specialty
A description of the system of operation of each specialty.
In this document we cannot disclose the information that was given to us. We refer the reader to the manuals that the hospital published yearly and that are related to his yearly activity.
7.2. Lingo 12.0
To solve the linear programming (CE), we used the commercial software Lingo 12.0. Recently Lingo was ranked by INFORMS (www.informs.org) as one of the most valuable package for linear and nonlinear mathematical programming problems. For the mixed integer linear program (CE), Lingo uses a modified Branch and Bound algorithm .
7.3. Hospital beds
The Rashid hospital must have 467 beds in the total. It means that 15 supplementary beds must be added to the hospital. The number of optimal beds in each speciality is presented in the following table:
Current number of beds
Table 1: number of optimal beds
Only two specialities require additional beds. These specialities are the Neurosurgery where 5 beds must be added and the trauma speciality which requires 10 additional beds. This difference between the optimal number of beds and the current beds is also represented with the following histogram:
The Rashid hospital needs to hire 3 additional nurses to the hospital to cover the demand. The optimal number of nurses per specialty is represented in the following table:
Current number of Nurses
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