**3. Results**

#### **3.1. Construction of a model for estimation of task time devoted to patients for whom nurse is responsible**

Our 'model for estimation of task time devoted to patients for whom nurse is responsible' is a regression analysis model using 'care time devoted to a given patient for whom a given nurse is responsible' as the dependent variable. When formulating a plan for the creation of a virtual environment, we entered the factor 'nursing intensity' as an independent variable on the assumption that task time devoted to patients for whom a nurse is responsible would be affected by patient severity. Bearing in mind that changes in shift time and in the number of nurses actually on duty on a shift would also have a great effect on care time devoted to patients, we included the factor 'shift' as well.

The subject of analysis consisted of records, extracted from the data set created by integrat‐ ing time study data and patient condition information, which revealed responsibility rela‐ tionships between nurses and patients. The data extracted related to a total of 425 patients (a total of 57 nurses on 57 shifts).

### *3.1.1. Multilevel analysis*

Fig. 3 shows that there was a strong negative correlation between time spent on 'tasks relat‐ ing to patients for whom the nurse is responsible' and time spent on 'tasks relating to other patients' and 'other tasks,' and that the correlation of 'rest time' with other task items was low. It appears at first glance that the correlation coefficient between 'tasks relating to other patients' and 'other tasks' is high at 0.727, but the partial correlation coefficient of the two is 0.019 and almost no direct correlation was observed. We were therefore able to judge that this was a spurious correlation influenced by 'tasks relating to patients for whom the nurse is responsible.' In other words, what this shows is that the relationship between the two is not such that when one increases the other decreases, but such that when time spent on 'pa‐

In accordance with the plan formulated under 2.1, a virtual ward environment was created

**1.** Construction of a model for estimation of task time devoted to patients for whom the

**2.** Calculation of number of patients for whom one nurse is responsible and of care time

spent by that nurse on patients for whom she is responsible.

sponsible,' 'other tasks,' and 'rest time,' all of which are correlated.

**1.** Estimation of number of patients on the ward by nursing intensity

**3.** Effect of increase or decrease in number of nurses on each task time

Construct a model to estimate the kinds of factors that influence care time devoted to a

Determine which patients a given nurse is responsible for and find the total task time

On the basis of the total task time devoted to 'patients for whom the nurse is responsi‐ ble' calculated under 2., estimate time spent on 'patients for whom the nurse is not re‐

In the virtual environment we had created, we carried out the following test experiments

**3.1. Construction of a model for estimation of task time devoted to patients for whom**

Our 'model for estimation of task time devoted to patients for whom nurse is responsible' is a regression analysis model using 'care time devoted to a given patient for whom a given

tients for whom the nurse is responsible' increases, the two decrease together.

**2.4. Construction of a virtual ward environment**

according to the following procedure:

**3.** Estimation of task time by purpose

**2.5. Test experiments in virtual ward environment**

and investigated the difference from actual data.

**2.** Estimation of task times by purpose

**3. Results**

**nurse is responsible**

nurse is responsible.

given patient.

220 Advances in Discrete Time Systems

A multiple regression analysis model is generally formulated as

$$\mathbf{x} = \beta\_0 + \beta\_1 \mathbf{x}\_1 + \dots + \beta\_p \mathbf{x}\_p + \varepsilon \tag{1}$$

where *y* is a dependent variable, there are *p* independent variables {*xk* } that have fixed ef‐ fects and *ε* is margin of error. As opposed to this, Multilevel Analysis (Jones, 1991; Gold‐ stein, 2003) is a method of estimating parameters (coefficients) in which, by assuming random effects resulting from a given phenomenon *j*, it is possible to take into account inter‐ nal correlations between data with the same value for *j* in relation to the correlation {*β<sup>k</sup>* } be‐ tween an intercept *β*0 in (1) and independent variables.

Because the time study survey periods were continuous, the same patients were included in different shifts in the data set relating to 425 patients subject to analysis. This means that there were elements that affected the care time, which is a variable dependent on patient identity (i.e., there were internal correlations in the data), although it was difficult to treat these ele‐ ments as regular fixed effects, as in the case of the patient's bodily strength or personality.

In order to estimate parameters having variable effects that explained these internal correla‐ tions, we constructed, for the purposes of this study, a model for the estimation of task time devoted to patients for whom the nurse was responsible using Multilevel Analysis, intro‐ ducing randomness for each 'patient' with respect to the intercept.

#### *3.1.2. Optimal model*

Rather than a model in which the actual care times are regressed in their original dimen‐ sions, we judged that the optimal model was one in which logistic conversion was per‐ formed with regard to the dependent variable.

Our reasons were as follows.

Because it is a mathematical model for explaining task times, when the estimated expected value *y* is negative, there is a marked lack of conformance of the times. For this reason, it is necessary to perform logarithmic conversion such that the time value that is the dependent variable does not appear as a negative value.

In the later creation of the virtual ward environment, we used this model for estimation of task time devoted to 'patients for whom the nurse is responsible,' which used normal random num‐ bers (see 3.2.4. below). Because the distribution of the raw data values (here, the distribution of the raw care times) was reproduced using normal random numbers, it was necessary at the point where random number values were generated to convert the raw data so that it showed a data distribution close to a normal distribution (see 3.3.1. for details) (Fig. 4).

*uj* ~ *N* (0, 0.674) *β*0*ij* =*β*<sup>0</sup> + *uj* + *eij*

nurse responsible. Here, *yij*

is margin of error (chance variability).

**Figure 5.** Data distributions after conversions.

With regard to the method of entering independent variables, we found as a result of repeat‐ ed investigation that a model using 3x3 items in which 'shift' and 'nursing intensity' were confounded, as in 'night shift, nursing intensity A,' was optimal (equation 0.4→4). This nu‐ merical formula is a model for the estimation of nursing time devoted to one patient by the

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

and E are respectively 'night shift,' 'day shift' and 'evening shift,' and superscript *A*, *B*, and *C* show nursing intensity. In this model, the value taken by a given independent variable is dichotomous, either (1) or (0), as, for example, in 'night shift nursing intensity is A (1)' or 'is not A (0),' so that when they are looked at as a whole, they form a categorical variable group in which if 'is (1)' appears with respect to an independent variable in a particular place, any other independent variables are necessarily 'is not (0).' For this reason, the independent var‐ iable 'evening shift, nursing intensity C' becomes the intercept itself, and in the case of 'day shift is nursing intensity A (1),' the parameter 'day shift, nursing intensity A' added to the

intercept becomes the care time prescribed for 'day shift, nursing intensity A'. *β*0*ij*

tercept (evening shift, nursing intensity C), *u* is variability depending on the patient, and *eij*

are *i* th care times on nursing occasions for the *j* th patient, N, D,

is the in‐

http://dx.doi.org/10.5772/51014

223

*eij* ~ *N* (0, 0.794)

**Figure 4.** Conversions for the purpose of recreating actuality using normal random numbers.

We found parameters for a logistic conversion that would give real upper and lower limits by minimizing AIC (Akaike's Information Criterion) after performing logistic conversion. The data set used for this analysis included some extremely small time values, so we fixed the low‐ er limit at zero. AIC for the case where the upper limit was *a* was calculated as follows:

$$\text{AIC} \le \text{nlog}\frac{\wedge\_2}{\sigma} - 2\text{nlog}a + 2\sum\_{i=1}^{n} \log \left| y\_i(a - y\_i) \right| + 2(p + 1) \tag{2}$$

Here, *σ* ^2 is a maximum likelihood estimator, *yi* are individual time values, *n* is a sample number, and *p* is an estimated parameter number.

This means that the smaller the AIC value the closer to a normal distribution; we used MLwiN ver. 1.1 for model analysis.

The constant *a* that ultimately produced the smallest AIC was 300, and we used the con‐ version method

$$\left[\log \mathbb{L}\right] y / \left(\text{300} - y\right) \text{\AA} \tag{3}$$

The result was that a data conversion close to normal distribution became possible, as seen in Fig. 5.

$$\begin{aligned} \mathbf{M} \mathbf{g} \mathbf{L} \boldsymbol{y}\_{ij} / (\mathbf{G} \mathbf{O} \mathbf{O} - \mathbf{y}\_{ij}) \mathbf{I} &= \boldsymbol{\beta}\_{0ij} + \boldsymbol{\beta}\_{NA} \mathbf{N}\_{ij}^{A} + \boldsymbol{\beta}\_{NB} \mathbf{N}\_{ij}^{B} + \\ \boldsymbol{\beta}\_{NC} \mathbf{N}\_{ij}^{C} &+ \boldsymbol{\beta}\_{DA} \mathbf{D}\_{ij}^{A} + \boldsymbol{\beta}\_{DB} \mathbf{D}\_{ij}^{B} + \boldsymbol{\beta}\_{DC} \mathbf{D}\_{ij}^{C} + \boldsymbol{\beta}\_{EA} \mathbf{E}\_{ij}^{A} + \boldsymbol{\beta}\_{EB} \mathbf{E}\_{ij}^{B} \end{aligned} \tag{4}$$


*uj* ~ *N* (0, 0.674) *β*0*ij* =*β*<sup>0</sup> + *uj* + *eij eij* ~ *N* (0, 0.794)

necessary to perform logarithmic conversion such that the time value that is the dependent

In the later creation of the virtual ward environment, we used this model for estimation of task time devoted to 'patients for whom the nurse is responsible,' which used normal random num‐ bers (see 3.2.4. below). Because the distribution of the raw data values (here, the distribution of the raw care times) was reproduced using normal random numbers, it was necessary at the point where random number values were generated to convert the raw data so that it showed a

We found parameters for a logistic conversion that would give real upper and lower limits by minimizing AIC (Akaike's Information Criterion) after performing logistic conversion. The data set used for this analysis included some extremely small time values, so we fixed the low‐

er limit at zero. AIC for the case where the upper limit was *a* was calculated as follows:

*i*=1 *n* log{*yi*

This means that the smaller the AIC value the closer to a normal distribution; we used

The constant *a* that ultimately produced the smallest AIC was 300, and we used the con‐

The result was that a data conversion close to normal distribution became possible, as

*<sup>B</sup>* <sup>+</sup> *<sup>β</sup>DCDij*

*<sup>A</sup>* <sup>+</sup> *<sup>β</sup>NBNij*

*<sup>C</sup>* <sup>+</sup> *<sup>β</sup>EAEij*

*B* +

(*a* − *yi*

)} + 2(*p* + 1) (2)

*<sup>B</sup>* (4)

are individual time values, *n* is a sample

log *y* / (300− *y*) (3)

*<sup>A</sup>* <sup>+</sup> *<sup>β</sup>EBEij*

data distribution close to a normal distribution (see 3.3.1. for details) (Fig. 4).

**Figure 4.** Conversions for the purpose of recreating actuality using normal random numbers.

^2 <sup>−</sup>2*n*log*<sup>a</sup>* + 2*<sup>Σ</sup>*

variable does not appear as a negative value.

222 Advances in Discrete Time Systems

AIC=*n*log*σ*

number, and *p* is an estimated parameter number.

MLwiN ver. 1.1 for model analysis.

*βNCNij*

is a maximum likelihood estimator, *yi*

log *yij* / (300− *yij*) =*β*0*ij* + *βNANij*

*<sup>A</sup>* <sup>+</sup> *<sup>β</sup>DBDij*

*<sup>C</sup>* <sup>+</sup> *<sup>β</sup>DADij*

Here, *σ* ^2

version method

seen in Fig. 5.

With regard to the method of entering independent variables, we found as a result of repeat‐ ed investigation that a model using 3x3 items in which 'shift' and 'nursing intensity' were confounded, as in 'night shift, nursing intensity A,' was optimal (equation 0.4→4). This nu‐ merical formula is a model for the estimation of nursing time devoted to one patient by the nurse responsible. Here, *yij* are *i* th care times on nursing occasions for the *j* th patient, N, D, and E are respectively 'night shift,' 'day shift' and 'evening shift,' and superscript *A*, *B*, and *C* show nursing intensity. In this model, the value taken by a given independent variable is dichotomous, either (1) or (0), as, for example, in 'night shift nursing intensity is A (1)' or 'is not A (0),' so that when they are looked at as a whole, they form a categorical variable group in which if 'is (1)' appears with respect to an independent variable in a particular place, any other independent variables are necessarily 'is not (0).' For this reason, the independent var‐ iable 'evening shift, nursing intensity C' becomes the intercept itself, and in the case of 'day shift is nursing intensity A (1),' the parameter 'day shift, nursing intensity A' added to the intercept becomes the care time prescribed for 'day shift, nursing intensity A'. *β*0*ij* is the in‐ tercept (evening shift, nursing intensity C), *u* is variability depending on the patient, and *eij* is margin of error (chance variability).

**Figure 5.** Data distributions after conversions.

According to the results of estimation using this model, given a patient with 'day shift, nurs‐ ing intensity A,' adding the parameter value 2.695 of 'day shift, nursing intensity A' to the intercept -3.834, then adding variability due to the individual patient and margin of error, gives the care time for this patient. Returning to the time dimension by using the reverse logistic conversion shows it to be about 72.8 minutes. In the same way, in the case of 'day shift, nursing intensity B' the time is about 39.4 minutes, and in the case of 'day shift, nurs‐ ing intensity C' about 20 minutes.

were present for the number of days shown by the patterns of change. The patterns were

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

225

For example, in Fig. 7, pattern No. 5 was chosen and 'CCCBBBA' was inserted horizontally, one letter per cell, starting in bed number 1 on day number 1. At the beginning of the simu‐ lation there were no patients at all, and patterns of nursing intensity were allocated for the number of allocated patients on the ward. The pattern of nursing intensity differed depend‐ ing on the number of days for which a patient was on the ward, so as days elapsed, patients

inserted in column direction.

**Figure 6.** Determination of number of patients on ward.

**Figure 7.** Layout of patterns of change in nursing intensity.

#### **3.2. Estimation of number of patients for whom one nurse is responsible, and care times**

Next, we determined the number of patients for whom one nurse was responsible and esti‐ mated the total time spent on those patients. At this point we embarked on the construction of an algorithm using a Monte Carlo Simulation.

A Monte Carlo Simulation is a method of obtaining approximate solutions to problems when simulating the processes of chance phenomena by carrying out numerical value cal‐ culations using random numbers. In this study, we used normal random numbers and created algorithms for them using MATLABR2012a. This simulation was conducted with respect to the day shift.

#### *3.2.1. Bed matrix*

We went through the process of recreating the actual bed occupancy status, which changes daily as a result of the admission and discharge of patients.

Since the ward studied was a 50-bed ward, we created a matrix for use in the simulation (hereafter 'bed matrix') consisting of vertical columns of 50 cells representing the beds, and on the horizontal time axis (representing days elapsed) we used rows containing enough cells to cover a long time period (for reasons explained later, we used rows of 1,000 cells in this study). Each cell in the bed matrix represents one bed-day.

#### *3.2.2. Determination of number of patients on ward*

By randomly determining the daily number of patients on the ward from the average num‐ ber of patients on the ward already calculated and its standard deviation, we recreated the changes in the actual number of patients on the ward. In the case of Fig. 6, for example, the number of patients on the ward over a seven-day period is randomly divided up and shad‐ ed cells show patients on the ward.

#### *3.2.3. Determination of patients on ward*

Having determined the number of patients on the ward each day, we simulated the state of affairs relating to patients on the ward whose condition underwent change by inserting the patterns of change in nursing intensity by number of patients on the ward. For this purpose we apportioned patterns randomly chosen from among the total of 281 patterns of change in nursing intensity found under 2.3.1. At this time, the patients on the ward were present for the number of days shown by the patterns of change. The patterns were inserted in column direction.

**Figure 6.** Determination of number of patients on ward.

According to the results of estimation using this model, given a patient with 'day shift, nurs‐ ing intensity A,' adding the parameter value 2.695 of 'day shift, nursing intensity A' to the intercept -3.834, then adding variability due to the individual patient and margin of error, gives the care time for this patient. Returning to the time dimension by using the reverse logistic conversion shows it to be about 72.8 minutes. In the same way, in the case of 'day shift, nursing intensity B' the time is about 39.4 minutes, and in the case of 'day shift, nurs‐

**3.2. Estimation of number of patients for whom one nurse is responsible, and care times**

Next, we determined the number of patients for whom one nurse was responsible and esti‐ mated the total time spent on those patients. At this point we embarked on the construction

A Monte Carlo Simulation is a method of obtaining approximate solutions to problems when simulating the processes of chance phenomena by carrying out numerical value cal‐ culations using random numbers. In this study, we used normal random numbers and created algorithms for them using MATLABR2012a. This simulation was conducted with

We went through the process of recreating the actual bed occupancy status, which changes

Since the ward studied was a 50-bed ward, we created a matrix for use in the simulation (hereafter 'bed matrix') consisting of vertical columns of 50 cells representing the beds, and on the horizontal time axis (representing days elapsed) we used rows containing enough cells to cover a long time period (for reasons explained later, we used rows of 1,000 cells in

By randomly determining the daily number of patients on the ward from the average num‐ ber of patients on the ward already calculated and its standard deviation, we recreated the changes in the actual number of patients on the ward. In the case of Fig. 6, for example, the number of patients on the ward over a seven-day period is randomly divided up and shad‐

Having determined the number of patients on the ward each day, we simulated the state of affairs relating to patients on the ward whose condition underwent change by inserting the patterns of change in nursing intensity by number of patients on the ward. For this purpose we apportioned patterns randomly chosen from among the total of 281 patterns of change in nursing intensity found under 2.3.1. At this time, the patients on the ward

ing intensity C' about 20 minutes.

224 Advances in Discrete Time Systems

respect to the day shift.

*3.2.1. Bed matrix*

of an algorithm using a Monte Carlo Simulation.

daily as a result of the admission and discharge of patients.

this study). Each cell in the bed matrix represents one bed-day.

*3.2.2. Determination of number of patients on ward*

ed cells show patients on the ward.

*3.2.3. Determination of patients on ward*

**Figure 7.** Layout of patterns of change in nursing intensity.

For example, in Fig. 7, pattern No. 5 was chosen and 'CCCBBBA' was inserted horizontally, one letter per cell, starting in bed number 1 on day number 1. At the beginning of the simu‐ lation there were no patients at all, and patterns of nursing intensity were allocated for the number of allocated patients on the ward. The pattern of nursing intensity differed depend‐ ing on the number of days for which a patient was on the ward, so as days elapsed, patients began to be discharged. In cases where the total number of patients given by change in pat‐ terns of nursing intensity for a given day fell below the specified minimum number of pa‐ tients for the ward, we apportioned new patients randomly from the patterns of change in nursing intensity. At this point, we had reached the stage where the set number of days per patient and patients on the ward were shown by nursing intensity in the bed matrix.

*3.2.5. Assignment of patients to nurses and total task times devoted by nurses to patients for whom*

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

227

On the basis of the statistical values already found for numbers of patients for whom nurses are responsible, we simulated the allocation of responsibility for these ward patients to indi‐ vidual nurses. The following two points suggested themselves as factors in determining the

patients for whom nurses are responsible in the actual ward environment:

**1.** Number of patients for whom the nurse is responsible

**2.** Severity of the conditions of those patients

**Figure 8.** Method of calculating care time for each patient.

**Figure 9.** Task times devoted to patients for whom nurse is responsible.

When the number of patients for whom a given nurse is responsible is large, or when they include patients whose condition is very severe, no further patients can be assigned to that nurse, and the quantity of tasks is distributed so that, for example, new patients are appor‐ tioned among other nurses with a relatively small number of patients in their charge or nurses whose patients have relatively mild conditions. Consequently, in the virtual environ‐ ment, first, at the point where patients were randomly admitted to the ward, we totaled the

*they are responsible*

#### *3.2.4. Calculation of care time for each patient*

On the basis of this nursing intensity, we calculated the care time believed necessary using the previously constructed model for estimating task time devoted to patients for whom the nurse is responsible. From equation (4) we saw that care time is not simply a function of nursing in‐ tensity but is the sum of (i) a quantity depending on nursing intensity, (ii) variability depend‐ ing on the individual patient, and (iii) other chance variability. The average of each of the latter two items was 0, and they were the parts that varied according to a normal distribution with certain variances. For the calculation of care time, first we took as the basis an estimated value for care time corresponding to nursing intensity, then generated a normal random number with a certain estimated variance and an average of 0 as the common value for all the days the patient spent in hospital and added that number. We then added a normal random number with another estimated variance and an average of 0 as chance variability.

In Fig. 8, the right-hand side is an example of the bed matrix when there are patients on the ward. The figure shows the method of calculating care time for a given patient *j*, represented by the lightly shaded cells. This patient is on the ward for three days. On the first day, nurs‐ ing intensity for this patient is B. On the second day it is C and on the third day also C, at which point the patient is discharged. Care time on the first day has the coefficient for day shift, nursing intensity B, of '‒3.834+1.945.' Next, we take the value 0.21, randomly generat‐ ed from the normal distribution average 0, variance 0.674, as the individual variability for patient *j*. The care necessary for this day has a total of ‒1.23, including the randomly gener‐ ated value 0.45, generated from the normal distribution with average 0 and variance 0.794. We now perform reverse conversion of the logistic conversion previously performed and ex‐ press the result in the time dimension: 68.1 minutes. Next, the care time for the second day has a total of ‒1.81, including the coefficient for day shift nursing intensity C, '‒3.834+1.207,' the variable effect depending on patient *j* 0.21, and the randomly determined value 0.6. The care time ultimately obtained is calculated as 42.1 minutes. The variable effect depending on patient *j* has the same value from when patient *j* is admitted up to the time the patient is discharged. For another patient, *k*, a value for *k* is allotted which is also the same from ad‐ mission to discharge. Further, the required care time on a given day changes at random dai‐ ly with regard to every patient, and is allotted randomly each day. Consequently, on the third day, in spite of the fact that nursing intensity for the same patient *j* is the same, care time ultimately differs from that on the second day. By means of the above operations, a bed matrix of the kind shown in Fig. 9 is created, showing care time devoted to each patient for whom a given nurse is responsible.

#### *3.2.5. Assignment of patients to nurses and total task times devoted by nurses to patients for whom they are responsible*

On the basis of the statistical values already found for numbers of patients for whom nurses are responsible, we simulated the allocation of responsibility for these ward patients to indi‐ vidual nurses. The following two points suggested themselves as factors in determining the patients for whom nurses are responsible in the actual ward environment:


began to be discharged. In cases where the total number of patients given by change in pat‐ terns of nursing intensity for a given day fell below the specified minimum number of pa‐ tients for the ward, we apportioned new patients randomly from the patterns of change in nursing intensity. At this point, we had reached the stage where the set number of days per

On the basis of this nursing intensity, we calculated the care time believed necessary using the previously constructed model for estimating task time devoted to patients for whom the nurse is responsible. From equation (4) we saw that care time is not simply a function of nursing in‐ tensity but is the sum of (i) a quantity depending on nursing intensity, (ii) variability depend‐ ing on the individual patient, and (iii) other chance variability. The average of each of the latter two items was 0, and they were the parts that varied according to a normal distribution with certain variances. For the calculation of care time, first we took as the basis an estimated value for care time corresponding to nursing intensity, then generated a normal random number with a certain estimated variance and an average of 0 as the common value for all the days the patient spent in hospital and added that number. We then added a normal random number

In Fig. 8, the right-hand side is an example of the bed matrix when there are patients on the ward. The figure shows the method of calculating care time for a given patient *j*, represented by the lightly shaded cells. This patient is on the ward for three days. On the first day, nurs‐ ing intensity for this patient is B. On the second day it is C and on the third day also C, at which point the patient is discharged. Care time on the first day has the coefficient for day shift, nursing intensity B, of '‒3.834+1.945.' Next, we take the value 0.21, randomly generat‐ ed from the normal distribution average 0, variance 0.674, as the individual variability for patient *j*. The care necessary for this day has a total of ‒1.23, including the randomly gener‐ ated value 0.45, generated from the normal distribution with average 0 and variance 0.794. We now perform reverse conversion of the logistic conversion previously performed and ex‐ press the result in the time dimension: 68.1 minutes. Next, the care time for the second day has a total of ‒1.81, including the coefficient for day shift nursing intensity C, '‒3.834+1.207,' the variable effect depending on patient *j* 0.21, and the randomly determined value 0.6. The care time ultimately obtained is calculated as 42.1 minutes. The variable effect depending on patient *j* has the same value from when patient *j* is admitted up to the time the patient is discharged. For another patient, *k*, a value for *k* is allotted which is also the same from ad‐ mission to discharge. Further, the required care time on a given day changes at random dai‐ ly with regard to every patient, and is allotted randomly each day. Consequently, on the third day, in spite of the fact that nursing intensity for the same patient *j* is the same, care time ultimately differs from that on the second day. By means of the above operations, a bed matrix of the kind shown in Fig. 9 is created, showing care time devoted to each patient for

patient and patients on the ward were shown by nursing intensity in the bed matrix.

with another estimated variance and an average of 0 as chance variability.

*3.2.4. Calculation of care time for each patient*

226 Advances in Discrete Time Systems

whom a given nurse is responsible.

**Figure 9.** Task times devoted to patients for whom nurse is responsible.

When the number of patients for whom a given nurse is responsible is large, or when they include patients whose condition is very severe, no further patients can be assigned to that nurse, and the quantity of tasks is distributed so that, for example, new patients are appor‐ tioned among other nurses with a relatively small number of patients in their charge or nurses whose patients have relatively mild conditions. Consequently, in the virtual environ‐ ment, first, at the point where patients were randomly admitted to the ward, we totaled the number of patients for whom each nurse was responsible and specified that if the number was 0, priority would be given to the assignment of patients to that nurse. We also control‐ led assignment of patients so that, as far as possible, each nurse was responsible for no few‐ er than 4 and no more than 7 patients. In addition, we carried out weighting such that extra patient responsibility was first given to nurses who were devoting a relatively small amount of task time to the patients for whom they had already been assigned responsibility.

and variance of the original data. In order to recreate the distribution of the original data using normal random numbers, it is necessary to think of a conversion in which a distribu‐ tion based on the conversion of the original is as close as possible to a normal distribution and, after having generated random numbers that have the post-conversion average and variance, it is then necessary to perform reverse conversion to obtain a value. At the same time, it is necessary to ensure that the randomly created values do not have a negative val‐ ue. For these reasons, we decided in this study to perform the following logistic conversion

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

with respect to task times by purpose *y* in order to solve problems 1), 2)→ 1 ,2 above.

^2 <sup>−</sup>2*n*log(*<sup>a</sup>* <sup>−</sup>*b*) <sup>+</sup> <sup>2</sup>∑

sample number and *p* is an estimated parameter number.

the following expression.

As in formula (2), *σ*

**Figure 10.** Conversion data plots.

AIC=*n*log*σ*

We used AIC (Akaike's Information Criteria) to evaluate whether distribution of the postconversion values was close to a normal distribution. We looked for an upper limit value *a* and a lower limit value *b* that would minimize AIC values. AIC were calculated by means of

> *i*=1 *n*

log{(*yi* −*b*)(*a* − *yi*

^2 is a maximum likelihood estimator, *yi* are individual time values, *n* is a

log (*y* −*b*) / (*a* − *y*) (5)

)} + 2(*p* + 2) (6)

http://dx.doi.org/10.5772/51014

229

In this way, we determined which patients a given nurse was responsible for. The shaded cells in the example shown in Fig. 9 show the task times devoted by one nurse to the pa‐ tients in her charge. The total, 165 minutes, is the 'task time devoted to patients' by that nurse. At this stage, we have created a virtual simulation of the approximate amount of time a nurse devotes in reality to all the patients for whom she is responsible.

### **3.3. Estimation of task time by purpose**

Next, on the basis of 'care time devoted to all patients for whom the nurse is responsible' by a single nurse found in 3.2.5, we randomly generated 'task time devoted to other patients,' 'time devoted to other tasks' and 'rest time' for the day shift and proceeded to simulate ac‐ tual task times by purpose. As mentioned in 3.1.2, it is necessary to bear the following points in mind when carrying out these simulations.

**1.** It is a prerequisite that the random numbers generated should follow a normal dis‐ tribution

In this study we have employed a Monte Carlo simulation using normal random numbers, and it is a prerequisite that the distribution for the generation of random numbers should be a normal distribution.

**2.** Negative values are to be avoided among randomly generated values

Since the units in the results obtained are times, task times that have negative values are not realistic and will prevent the simulation results from conforming to reality.

**3.** Covariance among the four task times by purpose is to be maintained

The generation of random values in which covariance among the variates is not taken into consideration makes it impossible to recreate the characteristic feature that they vary in rela‐ tion to one another.

Bearing these points in mind, we carried out the following operations.

#### *3.3.1. Logistic conversion*

As Fig. 3 shows, a certain amount of skew with regard to all four of the items 'tasks per‐ formed for patients for whom the nurse is responsible,' 'tasks performed for other patients,' 'other tasks,' and 'rest time' and a degree of unevenness (probably attributable to sampling limitations) were observed. It is not possible to recreate the distribution exhibited by these original data even if one generates normal random numbers using only the average values and variance of the original data. In order to recreate the distribution of the original data using normal random numbers, it is necessary to think of a conversion in which a distribu‐ tion based on the conversion of the original is as close as possible to a normal distribution and, after having generated random numbers that have the post-conversion average and variance, it is then necessary to perform reverse conversion to obtain a value. At the same time, it is necessary to ensure that the randomly created values do not have a negative val‐ ue. For these reasons, we decided in this study to perform the following logistic conversion with respect to task times by purpose *y* in order to solve problems 1), 2)→ 1 ,2 above.

$$\log \mathbf{f}(y-b) / (a-y)\mathbf{j} \tag{5}$$

We used AIC (Akaike's Information Criteria) to evaluate whether distribution of the postconversion values was close to a normal distribution. We looked for an upper limit value *a* and a lower limit value *b* that would minimize AIC values. AIC were calculated by means of the following expression.

$$\text{AIC} \equiv n \log \hat{\sigma}^2 - 2n \log(a - b) + 2 \sum\_{i=1}^{n} \log \left\{ (y\_i - b)(a - y\_i) \right\} + 2(p + 2) \tag{6}$$

As in formula (2), *σ* ^2 is a maximum likelihood estimator, *yi* are individual time values, *n* is a sample number and *p* is an estimated parameter number.

**Figure 10.** Conversion data plots.

number of patients for whom each nurse was responsible and specified that if the number was 0, priority would be given to the assignment of patients to that nurse. We also control‐ led assignment of patients so that, as far as possible, each nurse was responsible for no few‐ er than 4 and no more than 7 patients. In addition, we carried out weighting such that extra patient responsibility was first given to nurses who were devoting a relatively small amount

In this way, we determined which patients a given nurse was responsible for. The shaded cells in the example shown in Fig. 9 show the task times devoted by one nurse to the pa‐ tients in her charge. The total, 165 minutes, is the 'task time devoted to patients' by that nurse. At this stage, we have created a virtual simulation of the approximate amount of time

Next, on the basis of 'care time devoted to all patients for whom the nurse is responsible' by a single nurse found in 3.2.5, we randomly generated 'task time devoted to other patients,' 'time devoted to other tasks' and 'rest time' for the day shift and proceeded to simulate ac‐ tual task times by purpose. As mentioned in 3.1.2, it is necessary to bear the following points

**1.** It is a prerequisite that the random numbers generated should follow a normal dis‐

In this study we have employed a Monte Carlo simulation using normal random numbers, and it is a prerequisite that the distribution for the generation of random numbers should be

Since the units in the results obtained are times, task times that have negative values are not

The generation of random values in which covariance among the variates is not taken into consideration makes it impossible to recreate the characteristic feature that they vary in rela‐

As Fig. 3 shows, a certain amount of skew with regard to all four of the items 'tasks per‐ formed for patients for whom the nurse is responsible,' 'tasks performed for other patients,' 'other tasks,' and 'rest time' and a degree of unevenness (probably attributable to sampling limitations) were observed. It is not possible to recreate the distribution exhibited by these original data even if one generates normal random numbers using only the average values

of task time to the patients for whom they had already been assigned responsibility.

a nurse devotes in reality to all the patients for whom she is responsible.

**2.** Negative values are to be avoided among randomly generated values

realistic and will prevent the simulation results from conforming to reality. **3.** Covariance among the four task times by purpose is to be maintained

Bearing these points in mind, we carried out the following operations.

**3.3. Estimation of task time by purpose**

in mind when carrying out these simulations.

tribution

228 Advances in Discrete Time Systems

a normal distribution.

tion to one another.

*3.3.1. Logistic conversion*

The upper limit *a* and lower limit *b* for each task time by purpose that minimized AIC are as shown in Table 4.

It will be seen from scrutiny of the post-conversion data plots (Fig. 10) that in each of the variables, data distribution is closer to a normal distribution than before conversion (Fig. 3).


**Table 4.** Upper and lower limits of conversion functions.

#### *3.3.2. Simulation of task times by purpose using covariance*

Using the above conversion data, we generated random numbers that maintained cova‐ riance among the 4 variables (the third point to be borne in mind). Here, *X* is 'tasks per‐ formed for patients for whom the nurse is responsible,' *Y* is 'tasks performed for other patients,' *Z* is 'other tasks,' and *W* is 'rest.' In this study, we constructed the simulation algo‐ rithms on the basis of the hypothesis that 'tasks performed for patients for whom the nurse is responsible' would exert an influence on the allocation of task times to other tasks, and therefore designed the model so that *X* would exert an influence on *Y*, *Z,* and *W*.

If the equation

$$\begin{aligned} Y &= \alpha + \beta X + \varepsilon\\ \varepsilon &\sim N\{0, \sigma^2\} \end{aligned} \tag{7}$$

If coefficient values with regard to *X* are calculated in the same way for *Z* and *W*, with a

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

In this way, it is possible to maintain covariance between (*X*, *Y*), (*X*, *Z*), and (*X*, *W*). That is to say, it is possible to recreate changes in *Y*, *Z*, and *W* that occur when *X* changes, but because *ε*1,*ε*2,*ε*3 are independent normal random numbers, covariance between (*X*, *Y*), (*X*, *Z*), (*X*, *W*) is 0, and it is not possible to recreate changes based on the residual in 'other tasks' and 'rest' that occur when 'tasks performed for other patients' changes. So in order to maintain cova‐ riance between*ε*1,*ε*2,*ε*3 we decided to find the covariance structure of (*ε*1,*ε*2,*ε*3) by means of

(10)

231

http://dx.doi.org/10.5772/51014

(11)

(12)

*Y* =*α*<sup>1</sup> + *β*1*X* + *ε*<sup>1</sup> *Z* =*α*<sup>2</sup> + *β*2*X* + *ε*<sup>2</sup> *W* =*α*<sup>3</sup> + *β*3*X* + *ε*<sup>3</sup>

*ε*<sup>1</sup> =*Y* −(*α*<sup>1</sup> + *β*1*X* ) *ε*<sup>2</sup> =*Z* −(*α*<sup>2</sup> + *β*2*X* ) *ε*<sup>3</sup> =*W* −(*α*<sup>3</sup> + *β*3*X* )

Substituting the various coefficients calculated as a result of the above into formula (10)

Using residuals with these regression coefficients and covariances, we generated random numbers. Fig. 11 shows on the same plots the values obtained after logistic conversion of the

*Y* =1.04−1.77*X* + *ε*<sup>1</sup> *Z* = −1.04−0.64*X* + *ε*<sup>2</sup> *W* =1.26−0.22*X* + *ε*<sup>3</sup>

and generate 3-dimensional normal random numbers with this covariance.

Covariance structure of (*ε*1, *ε*2, *ε*3) is shown in Table 5.

**Table 5.** Covariance structure of residuals.

given covariance, the result is

the equations

gives the result

is used when generating *Y* with a given covariance, expected value E(*Y*), variance Var(*Y*) and covariance Cov(*X*, *Y*) become

$$\begin{aligned} \text{E(Y)} &= \alpha + \beta E(X) \\ \text{Var}(Y) &= \beta^2 \text{Var}(X) + \sigma^2 \\ \text{Cov}(X, Y) &= \beta \text{Var}(X) \end{aligned} \tag{8}$$

and the unknown quantities *α*,*β*,*σ* <sup>2</sup> are found by means of

$$\begin{aligned} \beta &= \text{Cov}(X, Y) / \text{Var}(X) \\ \alpha &= E(Y) - \beta E(X) \\ \sigma^2 &= \text{Var}(Y) - \beta^2 \text{Var}(X) \end{aligned} \tag{9}$$

If coefficient values with regard to *X* are calculated in the same way for *Z* and *W*, with a given covariance, the result is

$$\begin{aligned} Y &= \alpha\_1 + \beta\_1 X + \varepsilon\_1 \\ Z &= \alpha\_2 + \beta\_2 X + \varepsilon\_2 \\ W &= \alpha\_3 + \beta\_3 X + \varepsilon\_3 \end{aligned} \tag{10}$$

In this way, it is possible to maintain covariance between (*X*, *Y*), (*X*, *Z*), and (*X*, *W*). That is to say, it is possible to recreate changes in *Y*, *Z*, and *W* that occur when *X* changes, but because *ε*1,*ε*2,*ε*3 are independent normal random numbers, covariance between (*X*, *Y*), (*X*, *Z*), (*X*, *W*) is 0, and it is not possible to recreate changes based on the residual in 'other tasks' and 'rest' that occur when 'tasks performed for other patients' changes. So in order to maintain cova‐ riance between*ε*1,*ε*2,*ε*3 we decided to find the covariance structure of (*ε*1,*ε*2,*ε*3) by means of the equations

$$\begin{aligned} \varepsilon\_1 &= Y - (\alpha\_1 + \beta\_1 X) \\ \varepsilon\_2 &= Z - (\alpha\_2 + \beta\_2 X) \\ \varepsilon\_3 &= W - (\alpha\_3 + \beta\_3 X) \end{aligned} \tag{11}$$

and generate 3-dimensional normal random numbers with this covariance.

Substituting the various coefficients calculated as a result of the above into formula (10) gives the result

$$\begin{aligned} Y &= 1.04 - 1.77X + \varepsilon\_1 \\ Z &= -1.04 - 0.64X + \varepsilon\_2 \\ W &= 1.26 - 0.22X + \varepsilon\_3 \end{aligned} \tag{12}$$

Covariance structure of (*ε*1, *ε*2, *ε*3) is shown in Table 5.


**Table 5.** Covariance structure of residuals.

The upper limit *a* and lower limit *b* for each task time by purpose that minimized AIC are as

It will be seen from scrutiny of the post-conversion data plots (Fig. 10) that in each of the variables, data distribution is closer to a normal distribution than before conversion (Fig. 3).

Using the above conversion data, we generated random numbers that maintained cova‐ riance among the 4 variables (the third point to be borne in mind). Here, *X* is 'tasks per‐ formed for patients for whom the nurse is responsible,' *Y* is 'tasks performed for other patients,' *Z* is 'other tasks,' and *W* is 'rest.' In this study, we constructed the simulation algo‐ rithms on the basis of the hypothesis that 'tasks performed for patients for whom the nurse is responsible' would exert an influence on the allocation of task times to other tasks, and

therefore designed the model so that *X* would exert an influence on *Y*, *Z,* and *W*.

E(*Y* )=*α* + *βΕ*(*X* ) Var(*Y* )=*β* <sup>2</sup>

Cov(*X* , *Y* )=*β*Var(*X* )

*β* =Cov(*X* , *Y* ) / Var(*X* )

*α* =*Ε*(*Y* )−*βΕ*(*X* ) *σ* <sup>2</sup> =Var(*Y* )−*β* <sup>2</sup>

*Y* =*α* + *βX* + *ε ε* ∼ *N* (0, *σ* <sup>2</sup>

is used when generating *Y* with a given covariance, expected value E(*Y*), variance Var(*Y*)

Var(*X* ) + *σ* <sup>2</sup>

are found by means of

Var(*X* )

) (7)

(8)

(9)

shown in Table 4.

230 Advances in Discrete Time Systems

If the equation

and covariance Cov(*X*, *Y*) become

and the unknown quantities *α*,*β*,*σ* <sup>2</sup>

**Table 4.** Upper and lower limits of conversion functions.

*3.3.2. Simulation of task times by purpose using covariance*

Using residuals with these regression coefficients and covariances, we generated random numbers. Fig. 11 shows on the same plots the values obtained after logistic conversion of the original data (red dots) and the random numbers generated (blue dots). It can be seen that they have almost identical distributions. This shows that it is possible to generate random numbers that maintain the correlation structure between the variates (covariance).

Fig. 12 shows together the original data (red dots) and the values obtained after performing reverse conversion on the logistic conversions of the random numbers generated (blue dots). It can be seen that the distribution of the original data relating to time dimensions has been almost exactly recreated. The black dots represent what we judged to be anomalies in the original data. These data relate to anomalous tasks, such as receiving training or attending meetings, that were performed in the afternoons, with only the mornings being spent on

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

233

Following the above procedure, we randomly generated individual task times while main‐ taining associations between 'tasks performed for patients for whom the nurse is responsi‐ ble,' 'tasks performed for other patients,' 'other tasks,' and 'rest.' At this point, we had completed construction of a virtual ward environment for the purpose of simulating actual 'time devoted to tasks performed for patients for whom the nurse is responsible,' 'time de‐ voted to tasks performed for other patients,' 'time devoted to other tasks,' and 'rest time' for

Using the virtual ward environment we had constructed, we simulated long-term ward task times. On the first day of simulation the situation was that all patients were admitted to the ward at once, so none would be discharged for some time. As days passed in the simulation, gradually some patients began to be discharged. We disregarded simulation results ob‐ tained up to the point where it seemed that a stable situation had eventually been reached, with a balance between admissions and discharges. From that point, we specified 1,000 days of simulation. For the purposes of the simulation, we also specified that from the point of

The result of the simulation was as follows: the total number of patients was 44,846; totals by nursing intensity were: A=11,193 (25.0%), B=26,469 (59.0%), and C=7,184 (16.0%). The largest cohort of patients comprised those subject to nursing intensity B, the next largest those subject to nursing intensity A, and the smallest those subject to nursing intensity C. Unsurprisingly, this trend reflected almost exactly the trend in the 281 nursing intensity pat‐ terns we had established, where the frequency of nursing intensity A was 3,287 (24.2%), that

Fig. 13 is a graph showing changes in number of patients by nursing intensity. The vertical axis shows number of patients and the horizontal axis days elapsed. The upper panel is a graph showing nursing intensity. The daily number of patients at nursing intensity C, the lowest level of severity, is smallest, and the number at B, the intermediate level, is highest. The lower panel shows cumulative totals by nursing intensity. It will be seen, first, that al‐ most all patients on the ward are accounted for by nursing intensity A and B, and, second, that once the number of patients has risen, it remains for some time at the higher level.

ward duties. We therefore decided to exclude them from the analysis.

one nurse, and for conducting test experiments.

**3.4. Test experiments in the virtual ward environment**

view of the work system, every day was a weekday.

*3.4.1. Changes in patient numbers by nursing intensity*

of B 8,010 (59.1%), and that of C 2,265 (16.7%).

**Figure 11.** Random number values and logistic conversion values.

**Figure 12.** Time dimensions and random number values.

Fig. 12 shows together the original data (red dots) and the values obtained after performing reverse conversion on the logistic conversions of the random numbers generated (blue dots). It can be seen that the distribution of the original data relating to time dimensions has been almost exactly recreated. The black dots represent what we judged to be anomalies in the original data. These data relate to anomalous tasks, such as receiving training or attending meetings, that were performed in the afternoons, with only the mornings being spent on ward duties. We therefore decided to exclude them from the analysis.

Following the above procedure, we randomly generated individual task times while main‐ taining associations between 'tasks performed for patients for whom the nurse is responsi‐ ble,' 'tasks performed for other patients,' 'other tasks,' and 'rest.' At this point, we had completed construction of a virtual ward environment for the purpose of simulating actual 'time devoted to tasks performed for patients for whom the nurse is responsible,' 'time de‐ voted to tasks performed for other patients,' 'time devoted to other tasks,' and 'rest time' for one nurse, and for conducting test experiments.

#### **3.4. Test experiments in the virtual ward environment**

original data (red dots) and the random numbers generated (blue dots). It can be seen that they have almost identical distributions. This shows that it is possible to generate random

numbers that maintain the correlation structure between the variates (covariance).

**Figure 11.** Random number values and logistic conversion values.

232 Advances in Discrete Time Systems

**Figure 12.** Time dimensions and random number values.

Using the virtual ward environment we had constructed, we simulated long-term ward task times. On the first day of simulation the situation was that all patients were admitted to the ward at once, so none would be discharged for some time. As days passed in the simulation, gradually some patients began to be discharged. We disregarded simulation results ob‐ tained up to the point where it seemed that a stable situation had eventually been reached, with a balance between admissions and discharges. From that point, we specified 1,000 days of simulation. For the purposes of the simulation, we also specified that from the point of view of the work system, every day was a weekday.

#### *3.4.1. Changes in patient numbers by nursing intensity*

The result of the simulation was as follows: the total number of patients was 44,846; totals by nursing intensity were: A=11,193 (25.0%), B=26,469 (59.0%), and C=7,184 (16.0%). The largest cohort of patients comprised those subject to nursing intensity B, the next largest those subject to nursing intensity A, and the smallest those subject to nursing intensity C. Unsurprisingly, this trend reflected almost exactly the trend in the 281 nursing intensity pat‐ terns we had established, where the frequency of nursing intensity A was 3,287 (24.2%), that of B 8,010 (59.1%), and that of C 2,265 (16.7%).

Fig. 13 is a graph showing changes in number of patients by nursing intensity. The vertical axis shows number of patients and the horizontal axis days elapsed. The upper panel is a graph showing nursing intensity. The daily number of patients at nursing intensity C, the lowest level of severity, is smallest, and the number at B, the intermediate level, is highest. The lower panel shows cumulative totals by nursing intensity. It will be seen, first, that al‐ most all patients on the ward are accounted for by nursing intensity A and B, and, second, that once the number of patients has risen, it remains for some time at the higher level.

#### *3.4.2. Task times devoted to patients for whom the nurse is responsible from the point of view of nursing intensity*

The quantity of care time per patient necessary when patients for whom the nurse was re‐ sponsible were at nursing intensity A was on average 93.5 minutes. For patients on nursing intensity B the average time was 57.1 minutes, and for those on nursing intensity C the aver‐ age time was 31.6 minutes.

*3.4.4. Changes in task times by purpose*

**Figure 14.** Changes in task times by purpose.

able to take the 60 minutes of rest prescribed by law.

(Horizontal axis: minutes. Vertical axis: number of persons).

We simulated changes in task times by purpose for an individual nurse when the number of nurses on a single day shift was increased gradually from 8 to 15. Assuming that the num‐ ber of nurses actually on duty on a single day shift is 8, a 1,000 day simulation is equivalent to 8,000 nurse-shifts; assuming that the number is 9, a 1,000 day simulation is equivalent to 9,000 nurse-shifts, and so on. Figures 15-18 show the distribution of the number of patients for whom nurses are responsible when the total task times for the entire ward are shared by 8 to 15 nurses, and the frequency distribution of task times per nurse in each of those cases

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

235

Let us look first at the situation where the number of nurses specified is lowest. Almost ev‐ ery nurse is responsible for between 5 and 7 patients. The largest number of nurses has over‐ all task times of between 560 minutes (9.3 hours) and 570 minutes (9.5 hours). The average time for the whole group is 530 minutes. It can be seen that there are some nurses whose

for whom the nurse is responsible,' the largest number of nurses have times corresponding to the median of 312 minutes, or about 5 hours. Time spent on 'tasks performed for other patients' is about 30 minutes, and the number of nurses who spend about 60 minutes on 'other tasks' stands out. With regard to 'rest time,' it will be seen that almost no nurses were

overall task time exceeds 10 hours (over 600 minutes). Next, with regard to 'patients

#### *3.4.3. Changes in task times by purpose for the whole ward*

The upper panel in Fig. 14 shows changes in task times by purpose for the ward as a whole. The horizontal axis shows number of days elapsed and the vertical axis shows times. 'Task times devoted to patients for whom the nurse is responsible' show large fluctuations, while 'rest times' display nowhere near as large a range of variation. Further, it can be seen that when 'task times devoted to patients for whom the nurse is responsible' decrease, 'task times devoted to other patients' and 'times devoted to other tasks' increase; and when 'task times devoted to patients for whom the nurse is responsible' increase, 'task times devoted to

**Figure 13.** Changes in patient numbers by nursing intensity.

other patients' and 'times devoted to other tasks' decrease.

The lower panel in Fig. 14 shows cumulative totals of task times by purpose. In spite of the fact that 'task times devoted to patients for whom the nurse is responsible' and 'task times devoted to other patients' fluctuate widely, total task times as a whole are kept to an almost uniform level.

#### *3.4.4. Changes in task times by purpose*

*3.4.2. Task times devoted to patients for whom the nurse is responsible from the point of view of*

The quantity of care time per patient necessary when patients for whom the nurse was re‐ sponsible were at nursing intensity A was on average 93.5 minutes. For patients on nursing intensity B the average time was 57.1 minutes, and for those on nursing intensity C the aver‐

The upper panel in Fig. 14 shows changes in task times by purpose for the ward as a whole. The horizontal axis shows number of days elapsed and the vertical axis shows times. 'Task times devoted to patients for whom the nurse is responsible' show large fluctuations, while 'rest times' display nowhere near as large a range of variation. Further, it can be seen that when 'task times devoted to patients for whom the nurse is responsible' decrease, 'task times devoted to other patients' and 'times devoted to other tasks' increase; and when 'task times devoted to patients for whom the nurse is responsible' increase, 'task times devoted to

*nursing intensity*

234 Advances in Discrete Time Systems

age time was 31.6 minutes.

*3.4.3. Changes in task times by purpose for the whole ward*

**Figure 13.** Changes in patient numbers by nursing intensity.

an almost uniform level.

other patients' and 'times devoted to other tasks' decrease.

The lower panel in Fig. 14 shows cumulative totals of task times by purpose. In spite of the fact that 'task times devoted to patients for whom the nurse is responsible' and 'task times devoted to other patients' fluctuate widely, total task times as a whole are kept to We simulated changes in task times by purpose for an individual nurse when the number of nurses on a single day shift was increased gradually from 8 to 15. Assuming that the num‐ ber of nurses actually on duty on a single day shift is 8, a 1,000 day simulation is equivalent to 8,000 nurse-shifts; assuming that the number is 9, a 1,000 day simulation is equivalent to 9,000 nurse-shifts, and so on. Figures 15-18 show the distribution of the number of patients for whom nurses are responsible when the total task times for the entire ward are shared by 8 to 15 nurses, and the frequency distribution of task times per nurse in each of those cases (Horizontal axis: minutes. Vertical axis: number of persons).

Let us look first at the situation where the number of nurses specified is lowest. Almost ev‐ ery nurse is responsible for between 5 and 7 patients. The largest number of nurses has over‐ all task times of between 560 minutes (9.3 hours) and 570 minutes (9.5 hours). The average time for the whole group is 530 minutes. It can be seen that there are some nurses whose overall task time exceeds 10 hours (over 600 minutes). Next, with regard to 'patients

**Figure 14.** Changes in task times by purpose.

for whom the nurse is responsible,' the largest number of nurses have times corresponding to the median of 312 minutes, or about 5 hours. Time spent on 'tasks performed for other patients' is about 30 minutes, and the number of nurses who spend about 60 minutes on 'other tasks' stands out. With regard to 'rest time,' it will be seen that almost no nurses were able to take the 60 minutes of rest prescribed by law.

As the number of nurses on duty progressively increases, the number of patients for whom each nurse is responsible gradually decreases, until a situation is reached in which some nurses are responsible for 0 patients, and where the ward's nurse requirement can be said to be satisfied.

**Figure 17.** Tasks times when number of nurses is 13.

**Figure 18.** Tasks times when number of nurses is 15.

**4.1. Changes in numbers of patients by nursing intensity**

We carried out an evaluation of whether the simulation algorithms we constructed might be

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

237

We were able to judge whether the cohort of patients on the ward reflected the real-world sit‐ uation, in which patients still requiring assistance are in a majority. The reasons are as follows.

**4. Observations**

incompatible with reality.

**Figure 15.** Tasks times when number of nurses is 8.

In addition, while time spent on 'tasks performed for patients for whom the nurse is respon‐ sible' decreases along with this variation in the number of patients for whom a nurse is re‐ sponsible, more nurses are able to increase the time they spend on 'task times devoted to other patients' and 'time spent on other tasks,' while the number able to take a rest period approaching 60 minutes will be seen to have increased.

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data http://dx.doi.org/10.5772/51014 237

**Figure 17.** Tasks times when number of nurses is 13.

As the number of nurses on duty progressively increases, the number of patients for whom each nurse is responsible gradually decreases, until a situation is reached in which some nurses are responsible for 0 patients, and where the ward's nurse requirement can

In addition, while time spent on 'tasks performed for patients for whom the nurse is respon‐ sible' decreases along with this variation in the number of patients for whom a nurse is re‐ sponsible, more nurses are able to increase the time they spend on 'task times devoted to other patients' and 'time spent on other tasks,' while the number able to take a rest period

be said to be satisfied.

236 Advances in Discrete Time Systems

**Figure 15.** Tasks times when number of nurses is 8.

**Figure 16.** Tasks times when number of nurses is 11.

approaching 60 minutes will be seen to have increased.

**Figure 18.** Tasks times when number of nurses is 15.

### **4. Observations**

We carried out an evaluation of whether the simulation algorithms we constructed might be incompatible with reality.

#### **4.1. Changes in numbers of patients by nursing intensity**

We were able to judge whether the cohort of patients on the ward reflected the real-world sit‐ uation, in which patients still requiring assistance are in a majority. The reasons are as follows.

The ward studied was a surgical ward, so for almost all patients the period immediately fol‐ lowing surgery or related tests was when their condition was at its severest. After a few days they would emerge from the acute stage (when nursing intensity was A) and enter a period during which they received intravenous drugs or other active treatments as their wounds fol‐ lowed the healing process (the period of nursing intensity B). This follow-up period was the longest. As soon as the outlook was such that the patient could be sent home or return to work (the period of nursing intensity C), discharge followed within 1 to 2 days. In our simulation re‐ sults also, almost all the patients on the ward were at nursing intensity A or B.

**4.5. Changes in task time by purpose**

When task time available for completion of 'tasks performed for patients for whom the nurse is responsible' is insufficient, the nurse is unable to carry out 'tasks for other patients' and 'other tasks.' But when the number of nurses increases and adequate care time can be devoted to patients for whom a nurse is responsible, time can be found to spend on aspects of nursing care such as tasks performed for 'other patients' and 'other tasks' that have had to be neglected before the increase in nursing staff. Our simulation appeared to reflect this reality. This is also a reflection of the fact that, as noted under 4.3, there are tasks that are omitted because working time is limited. Specific examples are given below. There are occa‐ sions when a nurse is so busy providing care to patients for whom she is responsible that she is unable to respond to a call from another patient, even one she is responsible for. On occasion, under these conditions, if a nurse passing along a corridor discovers a patient whose intravenous drip is leaking, if that patient is not one for whom she is responsible the series of tasks involved in dealing with a leaking intravenous drip assume a low priority for her and she must call the nurse who is responsible for the patient in question. If the nurses continue to be fully occupied with patient care, they are unable to tidy up the ward or put things in order. As a result, the ward declines into a state where in an emergency staff must look for a wheelchair that is not in its proper place, or they trip over or bump into things that are in places that should be empty, or they find that when they need to fix a drip in place quickly the tape they need has run out, or that there are not enough specimen contain‐ ers when specimens are needed for urgent tests, or when pressure of work slackens a little and they set out to update their records they find that the necessary forms have run out. However, when the patients on the ward are in a relatively settled state and care time re‐ quirements are met, if a nurse discovers a patient with a leaking intravenous drip in a corri‐ dor she will undertake the series of measures necessary to replace it, even if the patient is not one for whom she is responsible, and will fully carry out administrative and manage‐

Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data

http://dx.doi.org/10.5772/51014

239

ment tasks such as tidying the ward and putting things in order.

formed to reality.

**5. Conclusions**

tors governing tasks.

We observed in our simulation results also that when the number of nurses on a shift was increased, there was a decrease in the number of nurses who spent a very large amount of time on 'tasks performed for patients for whom the nurse is responsible' (unbalanced work‐ loads were resolved) and at the same time there was an increase in the number of nurses who spent a large amount of time on 'tasks performed for other patients' and 'other tasks.'

We concluded from the above that the simulation algorithms constructed in this study con‐

We created a formula for the nursing times provided on the basis of time study data ob‐ tained through a short-term survey and patient condition information, and quantified fac‐

In addition, it seemed that not only nurses but also all medical professionals agreed that they had a sense that on occasion, after the number of patients in a severe condition in‐ creased, that state of affairs would continue for some time, and then the patients would all recover at once.

#### **4.2. Task times devoted to patients for whom the nurse is responsible, by nursing intensity**

It is, of course, entirely natural that care time required increases with severity of nursing inten‐ sity. The results we obtained conformed to this observation and thus seemed to reflect reality.

#### **4.3. Changes in task time by purpose for the whole ward**

There are some tasks that need to be performed when spare time becomes available, but, be‐ cause there is a fixed limit on task time, tasks are in fact omitted. Our simulation appeared to reflect this reality. The reasons are as follows.

An increase in the amount of care time devoted to the patients for whom a nurse is responsi‐ ble means that the quantity of care, in the form of treatment and observation of the patient, is greater. But on closer examination it appears that the number of drugs prescribed increas‐ es, many tests have to be carried out, extra treatments and prescriptions are added, tasks such as changing dressings increase in number, or the procedures involved become more complicated. This affects the usage quantities of documents, medicines, and other materials managed by the ward. The result is that management task time also expands. In theory, therefore, it seems that if 'task time devoted to patients for whom the nurse is responsible' increases, 'time devoted to other tasks' that have no direct connection with patients should also increase, and as a consequence overall task time (shift time) ought to increase. The re‐ sults of our simulation show, however, that when 'task time devoted to patients for whom the nurse is responsible' increases 'task time devoted to other tasks' is reduced and overall task time does not increase very much. No extension of time devoted to 'other tasks' or of overall task time was observed.

#### **4.4. Work time per nurse**

The 9-hour work shift prescribed by law comprises 8 hours of working time and 1 hour of rest time. There was no marked deviation from this time in our simulation results.

#### **4.5. Changes in task time by purpose**

The ward studied was a surgical ward, so for almost all patients the period immediately fol‐ lowing surgery or related tests was when their condition was at its severest. After a few days they would emerge from the acute stage (when nursing intensity was A) and enter a period during which they received intravenous drugs or other active treatments as their wounds fol‐ lowed the healing process (the period of nursing intensity B). This follow-up period was the longest. As soon as the outlook was such that the patient could be sent home or return to work (the period of nursing intensity C), discharge followed within 1 to 2 days. In our simulation re‐

In addition, it seemed that not only nurses but also all medical professionals agreed that they had a sense that on occasion, after the number of patients in a severe condition in‐ creased, that state of affairs would continue for some time, and then the patients would all

It is, of course, entirely natural that care time required increases with severity of nursing inten‐ sity. The results we obtained conformed to this observation and thus seemed to reflect reality.

There are some tasks that need to be performed when spare time becomes available, but, be‐ cause there is a fixed limit on task time, tasks are in fact omitted. Our simulation appeared

An increase in the amount of care time devoted to the patients for whom a nurse is responsi‐ ble means that the quantity of care, in the form of treatment and observation of the patient, is greater. But on closer examination it appears that the number of drugs prescribed increas‐ es, many tests have to be carried out, extra treatments and prescriptions are added, tasks such as changing dressings increase in number, or the procedures involved become more complicated. This affects the usage quantities of documents, medicines, and other materials managed by the ward. The result is that management task time also expands. In theory, therefore, it seems that if 'task time devoted to patients for whom the nurse is responsible' increases, 'time devoted to other tasks' that have no direct connection with patients should also increase, and as a consequence overall task time (shift time) ought to increase. The re‐ sults of our simulation show, however, that when 'task time devoted to patients for whom the nurse is responsible' increases 'task time devoted to other tasks' is reduced and overall task time does not increase very much. No extension of time devoted to 'other tasks' or of

The 9-hour work shift prescribed by law comprises 8 hours of working time and 1 hour of

rest time. There was no marked deviation from this time in our simulation results.

**4.2. Task times devoted to patients for whom the nurse is responsible, by nursing**

**4.3. Changes in task time by purpose for the whole ward**

to reflect this reality. The reasons are as follows.

overall task time was observed.

**4.4. Work time per nurse**

sults also, almost all the patients on the ward were at nursing intensity A or B.

recover at once.

238 Advances in Discrete Time Systems

**intensity**

When task time available for completion of 'tasks performed for patients for whom the nurse is responsible' is insufficient, the nurse is unable to carry out 'tasks for other patients' and 'other tasks.' But when the number of nurses increases and adequate care time can be devoted to patients for whom a nurse is responsible, time can be found to spend on aspects of nursing care such as tasks performed for 'other patients' and 'other tasks' that have had to be neglected before the increase in nursing staff. Our simulation appeared to reflect this reality. This is also a reflection of the fact that, as noted under 4.3, there are tasks that are omitted because working time is limited. Specific examples are given below. There are occa‐ sions when a nurse is so busy providing care to patients for whom she is responsible that she is unable to respond to a call from another patient, even one she is responsible for. On occasion, under these conditions, if a nurse passing along a corridor discovers a patient whose intravenous drip is leaking, if that patient is not one for whom she is responsible the series of tasks involved in dealing with a leaking intravenous drip assume a low priority for her and she must call the nurse who is responsible for the patient in question. If the nurses continue to be fully occupied with patient care, they are unable to tidy up the ward or put things in order. As a result, the ward declines into a state where in an emergency staff must look for a wheelchair that is not in its proper place, or they trip over or bump into things that are in places that should be empty, or they find that when they need to fix a drip in place quickly the tape they need has run out, or that there are not enough specimen contain‐ ers when specimens are needed for urgent tests, or when pressure of work slackens a little and they set out to update their records they find that the necessary forms have run out. However, when the patients on the ward are in a relatively settled state and care time re‐ quirements are met, if a nurse discovers a patient with a leaking intravenous drip in a corri‐ dor she will undertake the series of measures necessary to replace it, even if the patient is not one for whom she is responsible, and will fully carry out administrative and manage‐ ment tasks such as tidying the ward and putting things in order.

We observed in our simulation results also that when the number of nurses on a shift was increased, there was a decrease in the number of nurses who spent a very large amount of time on 'tasks performed for patients for whom the nurse is responsible' (unbalanced work‐ loads were resolved) and at the same time there was an increase in the number of nurses who spent a large amount of time on 'tasks performed for other patients' and 'other tasks.'

We concluded from the above that the simulation algorithms constructed in this study con‐ formed to reality.
