APPLICATION OF OPTIMIZATION METHODS IN …...2 activities (in particular planning natural therapy treatments), but this was mainly due to the 3 progressive development of computers

ZESZYTY NAUKOWE POLITECHNIKI ŚLĄSKIEJ 2018

Seria: ORGANIZACJA I ZARZĄDZANIE z. 119

APPLICATION OF OPTIMIZATION METHODS IN PLANNING 1

OF PATIENTS ACCOMMODATION IN THE SPA COMPANIES 2

Adam R. SZROMEK 3

Politechnika Śląska, Zabrze; [email protected], Tel.: +48-322-777-336 4

Abstract: The aim of the article is to present the possibilities of using selected optimization 5

methods to planning accommodation in spa establishments. Therefore, two methods of 6

solving problems in the field of linear programming were used. The first is the north-west 7

angle method, and the second is the Vogel's approximation method. The methods used so far 8

in logistics in the field of planning a production program can be effectively used also in spa 9

services and improving the efficiency of business models of spa enterprises. 10

Keywords: spa, optimization, linear programming, key activities, business model. 11

1. Introduction 12

The improvement of the organization of spa services is an important activity that increases 13

the value for the client and improves relations with him within the framework of the business 14

model formulated. A. Osterwalder and Z. Pigneur (2010) specify that the business model 15

serves the description of the premises behind the way in which the organization creates value 16

and provides and derives profits from this generated value. According to this concept of the 17

business model, it can be described by means of customer segments, value propositions, 18

channels, customer relationships, revenue streams, key resources, key activities, key partners 19

and cost structure. 20

The essence of activities based on the improvement of the organization of spa services is 21

also confirmed by the experience of spa enterprises from the first decade of the 21st century, 22

as it significantly affects the building of competitive advantage. Nowadays, competitive 23

advantage is based on information that has become one of the key resources. However, the 24

multiplicity of information that an enterprise has to deal with can lead to the loss of 25

information relevant to information that is not of significant importance. 26

A competitive advantage based on information often involves two other types of 27

competitive advantage – a cost and quality advantage. Therefore, the analysis of data in the 28

possession of an enterprise may affect the amount of costs or the quality of products and 29

290 A.R. Szromek

services. This change may have a positive impact on the competitiveness of the company or, 1

on the contrary, plunge them into financial problems. This is no different in the case of spa 2

treatment plants, which at the end of the 1990s were placed before the fact of joining the 3

market processes and attempts to create a competitive advantage in the spa market. 4

A. Niezgoda and D.E. Jaremen (2009) remind that the competitiveness of a spa depends 5

on market conditions, which include: 6

the state of infrastructure: social, communication and technical-economic of general 7

purpose and technical special purpose, 8

tourist development, resources of production factors, including workforce, 9

unemployment rate and remuneration level, 10

research and development potential, 11

culture of the local community, its entrepreneurship and approach to innovation, 12

efficiency and flexibility of administrative activities, transparency of legal regulations, 13

transparency of the regulatory environment, concerning the economic, health and 14

tourism policy of the state and the region. 15

This is no different in the case of spa enterprises, in which case one of the key factors is 16

the optimization of the use of resources held by the spa. Logistic processes are of great 17

importance here. Optimizing even seemingly trivial logistic activities allows you to improve 18

the company's operations, and thus allows you to achieve a competitive advantage or an 19

attempt to match the competition. 20

The aim of the article is to present the possibilities of using selected methods of 21

accommodation optimization in spa establishments. To this end, two methods of solving 22

problems in the field of linear programming were used. These activities aim to improve the 23

organization of spa services, and thus aim to improve the business model. 24

2. Logistics of spa services 25

A contemporary look at the activities carried out by the company requires breaking the 26

limitations and unconventional solutions. However, their application requires skills aimed at 27

increasing efficiency, either by reducing operating costs or by increasing turnover. 28

Knowledge of the methods of optimization of processes taking place in an enterprise 29

providing spa services may increase the effectiveness of the activities of the manager and the 30

team coordinating particular areas of activity. The optimization of spa activities goes beyond 31

logistic activities, although it mainly concerns them. Although logistics is particularly 32

developed in production activities, its importance is also noticeable in services (Rzeczyński, 33

2001). 34

Application of optimization methods… 291

Computerization of spa enterprises partly led to the gradual optimization of some 1

activities (in particular planning natural therapy treatments), but this was mainly due to the 2

progressive development of computers and their capabilities, and was not caused by the 3

managers' awareness of optimization methods. Until the 90s of the twentieth century, mainly 4

sheets of paper were used, which were filled with the planning of treatments. At the turn of 5

the 20th and 21st centuries, some spa enterprises were already using software that would 6

enable coordination of individual treatments. However, the possibilities of the then programs 7

were limited and did not allow for taking into account many restrictive conditions. Nowadays, 8

the software used in spa plants allows for many possibilities, but it is expensive and at the 9

same time not very flexible. These defects usually result in a limited use of optimization 10

methods in the logistics of spa services or increasing the costs of business (Szromek, 2017). 11

Organizational activities in the enterprise providing services are different than those 12

observed in manufacturing enterprises. Thus, the logistics of the undertaken activities will 13

concern both different areas and different targeting of its objectives. Literature (Rzeczyński, 14

2001) indicates three basic areas of optimization in service logistics. Belong to them: 15

minimizing the time of waiting for a service or for the service itself, 16

service potential management, 17

service delivery through the distribution channel. 18

As part of these three goals, the optimization of many aspects of service and production-19

service activities that occur in a spa enterprise may be subject to optimization. You can list 20

here logistic problems like: 21

determining the optimal order of performed activities (services, orders) at specific 22

positions, 23

optimization of the location of service areas and individual service facilities in the spa, 24

optimization of the distribution of places (accommodation), 25

queues and waiting time for the service, 26

optimization of the size of raw materials stocks and their supplementary moments for 27

maintaining the continuity of service provision and minimizing storage costs, 28

selection of the price variant, production volume and provision of health services, 29

classification of services, etc. 30

Literature presents many interesting solutions. In the process of solving practical logistic 31

problems occurring in the organization of spa services, as well as in attempts to optimize the 32

organizational processes taking place, the scientific work of outstanding researchers of 33

optimization, forecasting and simulation methods and quantitative methods in logistics was 34

extremely useful and inspiring. These include S. Krawczyk (2001), who describes interesting 35

algorithms for determining the order in which orders are executed, which can also be used in 36

planning the activities of natural medicine facilities. Another representative is K. Kukuła 37

(Kukuła et al., 2000), who describes many optimization methods. The proposals of J.B. Gajda 38

292 A.R. Szromek

(2001), who points out many interesting solutions using simulation models. These are only 1

selected representatives of the researchers of data optimization methods. 2

3. Initial assumptions of the problem of optimizing the use of places in spa 3

establishments 4

The problem of accommodation of patients in spa facilities, and even in rooms of one 5

building requires taking into account many conditions and the purpose of running a business. 6

Optimization of accommodation can have two variants, resulting from the fact that the costs 7

of stay of the patients are different. It is possible to consider the situation in which the cost of 8

the stay of the patients is constant irrespective of the place of accommodation, the standard of 9

the room and the number of persons accommodated, and the situation in which the sanatorium 10

has rooms of various standards, of various sizes with different equipment. 11

To solve the problem of accommodation optimization, it was decided to use the most 12

commonly used methods for solving linear transport problems. It assumes knowledge of 13

transport costs and the amount of raw material that should be transported from one of the 14

warehouses to one of the recipients. Adequate to the formulated problem, this issue can be 15

used for accommodation of patients (from different regions, defined by branches of the 16

National Health Fund (NHF) with whom contracts have been signed) in sanatorium rooms 17

with different cost occupancy rates or in different sanatoriums of a sanatorium team. Thus, 18

when adapting the issue to the problem under consideration, it was assumed that the location 19

of m distribution centers, represented by NHF units (N1, N2, N3, ..., Nm) and the location of 20

n sanatoriums (S1, S2, S3, ..., Sn), is known, in which patients should be accommodated. 21

Instead of transport costs in the transformed issue there will be costs of stay, which is borne 22

by the plant before it receives the negotiated funds for the stay of the patient. This means that 23

to increase the profit of a spa facility, his manager is interested in minimizing these costs. 24

In addition, it is assumed that the facility contracted stays and patients (for i = 1, 2, 3, ..., 25

m) with Ni NHF units, and at the disposal of the patients is Sj sanatoria, where bj 26

accommodation facilities are located (for j = 1, 2 , 3, ..., n). The following assumptions were 27

made in the optimization problem: 28

in each of the sanatoriums you can accommodate a bather from any NHF unit and spa 29

visitors from each NHF unit can be accommodated in any sanatorium; 30

the total number of patients ai is equal to the total number of beds bj; 31

the accommodation costs of a patient in a given sanatorium are, depending on the 32

situation under consideration, equal or different; 33

the number of patients coming from the NHF Ni unit, accommodated in the 34

Sj sanatorium, is the decision variable xij. 35


The issue therefore seeks to accommodate all patients in such a way that the purpose 1

function, determined by the product of the number of accommodated patients and 2

accommodation costs, reaches its minimum. For a balanced problem, the linear program will 3

have the form: 4

a) Variables: 5

xij – the number of accommodated patients to i-th branch of NHF in j-th sanatoriums 6

(for i = 1, 2, 3, ..., m and j = 1, 2, 3, ..., n). 7

b) Goal function: 8

(1) 9

c) Limitations: 10

(2) 11

Assuming that the costs of the stay of the patients are constant, and the number of places 12

and contracted patients is the same, the issue is limited to the accommodation of the patients 13

in any way, because the purpose function in each solution will take the same value. However, 14

when costs vary and there is an imbalance between the numbers of patients and places, 15

finding optimal solutions requires the use of optimization methods. 16

The imbalance situation, in turn, raises two possible variants, one of which is only 17

theoretical. It is a situation in which the number of patients ai exceeds the number of bj places 18

in sanatoriums. As already mentioned, this is only a theoretical possibility, because 19

contracting more stays than places is pointless. 20

A more probable situation is the one in which the number of patients ai is smaller than the 21

number of places bj. Then an additional object of the origin of the patients Nm+1 should be 22

added to the task, supplementing the unbalanced number of patients with the value of am+1, 23

where: 24

(3) 25

and variables xm+1j (for j = 1, 2, 3, ..., n) and increase the total size of bj, because: 26

(4) 27

It is also necessary to take into account fictitious accommodation costs, cm+1, and patients 28

from the dummy branch Nm+1, which will be 0. Thus, the linear program will take the form: 29

a) Variables: 30

xij – the number of accommodated patients to i-th branch of NHF in j-th sanatoriums 31

(for i = 1, 2, 3, ..., m=1 and j = 1, 2, 3, ..., n). 32

294 A.R. Szromek

b) Goal function: 1

(5) 2

c) Limitations: 3

(6) 4

The solution of such a task requires using one of the methods of solving a linear program 5

or software. In further considerations, two methods of solving the acceptable linear program 6

will be cited, which often also give an optimal solution. 7

4. Application of the northwest angle method 8

The north-west angle method gives limited possibilities of linear program solution, 9

because it does not take into account the costs of stay cij, and therefore it can be used to search 10

for the optimal solution for fixed costs of stay. The patients are accommodated in individual 11

sanatoria in accordance with the principle based on the matrix, which is shown in Table 1. 12

Table 1. 13 Data table for a linear program 14

i\j NHF branch managing the patients stay’s Resources

(places) N1 N2 N3 … Nm

Sa

na

tori

um

s S1 x11 x12 x13 … x1m a1

S2 x21 x22 x23 … x2m a2

S3 x31 x32 x33 … x3m a3

… … … … … …

Sn xn1 xn2 xn3 … xnm am

Health resort’s

patients b1 b2 b3 bn

Source: author’s own work. 15

The northwest angle method, called NW for short, is based on the order of assigning 16

patients to beds at the highest possible altitude, starting from the first cell of the first column 17

of variables, that is from the northwest corner of the matrix. If a1>b1 (ie the number of places 18

exceeds the number of visitors), b1 patients stay in the S1 sanatorium (adjusting the number of 19

places available in the S1 sanatorium and the number of patients to accommodation). If, 20

however, b1>a1, in sanatorium S1, we provide accommodation for as many patients as 21

possible, and we can accommodate the remaining ones in subsequent sanatoriums, starting 22

from S2, until all the patients from the N1 branch are accommodated and we also make 23


corrections. When all the patients from the N1 branch are accommodated, we move right 1

(east) to the next cell and assign b2 patients to the sanatorium, where the last N1 patients are 2

accommodated, and if the number of places is insufficient, to the next sanatorium, etc. 3

Obtained solution though only acceptable, it will be optimal at the same time due to the fact 4

that there are the same costs of accommodation in all rooms. 5

This method has a certain weakness. It is the inability to take into account the varied costs 6

of stay. Thus, the method is only useful when these costs are the same for all accommodation, 7

but also when we are looking for an initial solution, being aware that it will be used in further 8

calculations to find the optimal solution. 9

According to the methodological principle, accepted in the northwest angle method 10

allocation of beds (being the stock) to patients (representing the party reporting the demand) 11

start from the upper left angle of the matrix, that is north-west angle, guided by the principle 12

that if the demand in the column is zero (that is, all the patients from the Ni branch have been 13

accommodated), then the new north-west angle is on the right side of the recently allocated 14

accommodation, but if the resources of the sanatorium Sj (sanatorium accommodation) were 15

exhausted earlier, the cell is a new northwest corner located under the recently assigned 16

accommodation. At the same time, if all the spa patients from the NHF unit are placed in one 17

or several sanatoriums, the amount of accommodation in other sanatoriums of patients from 18

this NHF unit is 0. Similarly, if all the places were allocated in the j-th sanatorium, by 19

allocating patients from the i-th branch NHF, the remaining allocations from other NHF units 20

to this sanatorium must amount to 0. 21

5. Application of the approximation method 22

If we want to take into account the varied costs of accommodation of patients, we can use 23

a different method of achieving an acceptable solution that is often optimal or close to such 24

a solution. This is the approximation method of W.R. Vogel (VAM), cited, among others by 25

S. Krawczyk (2001) in the context of its use in logistics. 26

Similarly to the northwest angle method, solving the problem should start with a matrix in 27

which not only the size of the demand (represented by patients and departments that delegate 28

them) are listed, as well as available resources (rooms in sanatoriums), but also the costs of 29

the stay of the patients in individual sanatoriums. This time, the costs incurred by the health 30

resort will be taken into account, so the aim of the issue will be to provide accommodation for 31

the patients, so that the cost is the smallest. 32

Of course, the task could also refer to the benefit of accommodation for a patient in 33

a specific room, and then it would be in the interest of the establishment to place places to 34

take the most expensive places first, ie the purpose function would maximize income. Such 35

296 A.R. Szromek

a problem can be successfully solved by this method, first by simply transforming the data 1

set, to include, for example, a complement to the maximum profit value, i.e. x’ij=xij-xijmax or 2

simply changing the character of each xij to the reverse and find the minimum function 3

purpose. 4

In the first stage, we find the two smallest values of the cost of stay for a specific branch 5

and subtract them from each other, i.e. ki=cis-cid, where cis≥cid and so in each column of the 6

matrix. Next, we find the two smallest cost values in each row in the matrix and calculate 7

their difference, i.e. kj=cjs-cjd, where cjs≥cjd,, i.e. the cost difference between NHF units for the 8

same sanatorium. In the same way, we follow each line. 9

From among the differences obtained, we select the largest and locate its location on the 10

matrix, realizing the accommodation in the maximum possible size. If the maximum 11

difference was related to the difference in costs within the same NHF unit, then in the 12

indicated sanatorium, we quarantine all patients from this NHF unit and from the next stages 13

eliminate the already accommodated patients (correcting the matrix). However, if the 14

maximum cost difference has been located in one of the sanatoriums, then to this sanatorium 15

we are accommodating as many patients as possible from the indicated NHF unit, and then 16

from further calculations we eliminate occupied accommodation. It should be remembered 17

that if all the patients of a certain NHF unit were assigned to one of the sanatoriums, one can 18

at the same time assume that in the other sanatoriums none of the patients of this NHF unit 19

will be accommodated (enter zero there). We repeat the whole procedure looking for further 20

differences between the costs and their maximum value, by placing further visitors until the 21

last of them is accommodated. Then the total product of the obtained values of 22

accommodation and costs assigned to individual rooms will indicate the value of the objective 23

function, which determines the lowest costs of accommodation of the patients or (if the value 24

of the matrix has been transformed) will indicate a solution maximizing the objective 25

function. 26

The described procedure is worth explaining exactly on the example. A spa company 27

faces the problem of optimizing the distribution of accommodation for contracted patients. 28

It is necessary to accommodate 800 patients in three sanatoria (S1, S2, S3). Their stays are 29

contracted with five NHF branches (N1, N2, N3, N4 and N5). Establish the accommodation 30

program in accordance with the data provided in Table 2, remembering that the daily costs of 31

stay (included in the table) depend on the standard of sanatoriums and contracts with NHF 32

branches. 33

Table 2. 34 The cost of accommodation for patients in sanatoriums [PLN] 35

Cost table

cij

Branch NHF Resources (places)

N1 N2 N3 N4 N5

Sanatoriums

S1 21 45 66 52 24 210

S2 24 22 33 34 45 310

S3 43 44 30 21 15 280

Health resort’s patients 100 120 110 220 250 800

Source: Author’s own study. 36


Table 2 shows the different costs of stays of the patients in the sanatorium. In order to 1

obtain a solution, we will use the approximation method. When analyzing the cost matrix, it 2

was noticed that the highest variation occurred in the costs of stays of patients from the N2 3

branch in individual sanatoria. The difference in the value of the lowest costs of stay (44-22) 4

is 22, and therefore the location of the lowest costs of S2N2 accommodation requires that 5

accommodation of patients from this matrix cell be started. Thanks to the fact that the 6

sanatorium at the moment has 310 places it is possible to accommodate all patients from the 7

N2 branch in this sanatorium. 8

Table 3. 9 The first stage of accommodation for patients in sanatoria by the approximation method 10

11

Source: Author's own study. 12

Another highest cost difference occurred in sanatoriums for patients from the N4 branch. 13

The lowest cost of accommodation occurred in the S3 sanatorium, which accommodated all 14

220 patients from this department. 15

Table 4. 16 The second stage of accommodation for patients in sanatoriums by the approximation method 17

18


This time the highest cost difference is noted in the costs of stays in the S3 sanatorium. 20

Due to the fact that the contract with the N5 branch allows for the lowest costs of stay, 60 21

patients from this branch were contracted in the S3 sanatorium. Unfortunately, it was not 22

possible to accommodate other people from N5, as all sanitary facilities were depleted in S3 23

sanatorium. 24

Table 5. 25 The second stage of accommodation for patients in sanatoriums by the approximation method 26

27


298 A.R. Szromek

Another maximum difference in the cost of stays was recorded in the case of patients from 1

branch N3. The most advantageous solution is to contract them in sanatorium S2. 2

Table 6. 3 The third stage of accommodation for patients in sanatoria by the approximation method 4

5


Of the remaining cost differences, the highest occurred in the case of non-registered 7

patients from the N5 unit and unoccupied places in the S2 sanatorium. Two first-rate solutions 8

were chosen from two equivalent solutions, which are most advantageously placed in the S1 9

sanatorium (because the costs there are lower than in S2, which also has vacant places). 10

Table 7. 11 The fourth stage of accommodation for patients in sanatoriums by the approximation method 12

13


The last comparison shows that the lowest costs of accommodation will be incurred by 15

placing patients from the N1 branch in sanatoriums S1 and S2. 16

Table 8. 17 The fifth stage of accommodation for patients in sanatoriums by the approximation method 18

19


In this way, all the patients were accommodated, obtaining the lowest costs of PLN 21

18.690. It is worth adding that the MS Excel SOLVER solution found the solution identified 22

as optimal. 23

24


Table 9. 1

The cost of accommodation for patients in sanatoria with the result of accommodation 2

Stay’s cost Branch of NHF

N1 N2 N3 N4 N5

Sanatoriums

S1 420 0 0 0 4560

S2 1920 2640 3630 0 0

S3 0 0 0 4620 900


The task is again a balanced issue, in other words, in which the reported demand is equal 4

to the resources held. In a situation where beds would be more than bathers, the matrix should 5

be transformed in such a way as to take into account an additional contracting entity, ie Nm+1, 6

supplementing the difference between the number of places in sanatoriums and the number of 7

lodgings. Then the cost of such a stay will be 0. 8

6. Summary 9

The analytical methods presented require some pre-emptive actions in the form of an 10

appropriate diagnosis of the problematic situation that is going to be solved. Appropriate 11

selection of analytical tools will also determine the effectiveness of solving the problem of 12

optimal use of places. 13

Modern spa facilities face the necessity to gain a competitive advantage and build better 14

relations with a direct customer than before. The optimization of the accommodation method 15

may allow an additional advantage over competitors by lowering the costs of running 16

a business and improving the organization of spa services. Thus, it will be a significant 17

activity improving the business model of a modern spa company. 18

Acknowledgments 19

This paper was published as part of the research project ‘A business model for health 20

resort enterprises’ No. 2017/25/B/HS4/00301, supervised and financed by the National 21

Science Center in Poland and as part of statutory research ROZ 1: BK-231/ROZ1/2018 22

(13/010/BK_18/0029). 23

24

300 A.R. Szromek

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APPLICATION OF OPTIMIZATION METHODS IN …...2 activities (in particular planning natural therapy treatments), but this was mainly due to the 3 progressive development of computers

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