The solution of Nurse Scheduling Problem with Different Scheduling Models in Hospital using Linear and Nonlinear Programming

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 6, Issue 11, November 2016)

The solution of Nurse Scheduling Problem with Different

Scheduling Models in Hospital using Linear and Nonlinear

Programming

Manmohan Patidar

1

, Sanjay Choudhary

2

1_{Department of Mathematics, Govt. N.M.V. Hoshangabad} 2

Head, Department of Mathematics, Govt. N.M.V. Hoshangabad (M.P.), India

Abstract-- The purpose of the current investigation is the determination of an optimal number of nurses in a multispecialty Hospital of Bhopal. There are different models present in our study and each models minimization, scheduling, and allocation of nurses to perform by using the mathematical optimization method.

Keyword: Mathematical optimization, Linear

programming, Nonlinear Programming, Quadratic programming, Nurse Scheduling, Nurse staffing.

I. INTRODUCTION AND REVIEW

Hospital's fundamental principal goal is to assure continuous ward watching duty with the proper nursing skills in the staffs. The preparation and scheduling are performed to avoid the extra non-essential charge for a hospital — attendees preferences are night shift and rest days to perform a technique of mathematical linear programming model as a useful tool to determine obstacles of optimization in healthcare. The limited number of nurses are required 24 hours a day in a hospital. This paper represents the nurses scheduling problems determined by linear programming. This paper represents Nurses scheduling at a multi-specialty Hospital in Bhopal by applying linear programming. The procedure for nurse allocation should be well-organized to ensure a continuous and satisfactory level of patient supervision services while supporting the internal policies and authoritative requirements. This situation becomes hard when some of the factors such as the admission of the patient, nurse skills or permission to practice, the variety of condition as well as unexpected accidents. The individual holiday or work shift choices requirements of the nurses add a new figure to the problems in scheduling. Operational Research is a technique produced to provide useful tools for decision-making modes — a set of models and simulation methods included in it. And mathematical optimization, i.e., linear,

There are several grades of nurses like a junior nurse, an expert nurse and some of the nurses' strength to be trained to handle particular medical situations or experienced in a particular field such as intensive supervision. Because of the various training and specializations, the particular kind of nurse is to be staffed forwards needing those skills. Many others researcher as B.Satish Kumar, S. Naresh Kumar, S.Kumaraghuru[1], Lorraine Trilling, Alain Guinet, Dominique Le Magny[2], Mohsen Bayati, Erfan Kharazmi, Mehdi Javananbakht, Aboozar Sadegi, Masud Arefnejad, Sajad Vahedi, Firooz Esmaeilzadeh[3], worked in this field. We[4] also work in Nurse scheduling initially in the present paper applying the above concept the results obtained in details in linear as well as nonlinear programming.

II. LINEAR PROGRAMMING

Linear programming methods examined as a mathematically based decision-making tool. Such a method needs two basic kinds of constraints, objective and functions, that is explained to create a closed-form solution. Under a standard Operations research problem, Z is the objective function expressed to define the maximum value while minimizing value with the provided constraints.

Model (A): Linear Programming model

MIN Z =

∑

7

1 =

i

D

_,Subject to Di ≥ xi, where Di is the

number of Nurses Schedule for Day one, Day two, Day three, etc. Also, Xi is the Demand of Nurses needed on a particular Day one, Day two, Day three, etc. And i = 1, 2, 3… 7.

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International Journal of Emerging Technology and Advanced Engineering

D 1 D 2 D 3 D4 D 5 D6 D7

DAY MON TUE WED THU FRI SAT SUN

NUMBERS OF NURSES

230 200 240 125 235 300 100

Each nurse works six days and then needs one day leave, revolving this pattern regularly. Minimize the number of nurse's staff in the hospital

Let D1 be the representation of nurses starts working from

Monday own wards (Mon –Saturday)

Tuesday own wards (Tue– Sunday)

Let D3 be the representation of nurses’ starts working from

Wednesday own wards (Wed – Monday)

Thursday own wards (Thu– Tuesday)

Friday

Own wards (Fri –Wednesday)

Let D6 be the representation of nurses’ starts working from

Saturday own wards (Sat – Thursday)

Let D7 be the representation of nurses starts working

from Sunday own wards (Sun– Friday)

Nurses required in the hospital on Monday: 230, Tuesday: 200, Wednesday: 240, Thursday: 125, Friday: 235, Saturday: 300 and Sunday 100 sequentially.

Di represents the number of Nurses who starts Six-day working on Day i.

The objective function is D1+D2+D3+D4+D5+D6+D7.

Suppose the constraint for Monday staffing level of 230. They start their working shift on Monday (D1), they

continuously work from Monday to Saturday for six days. These start their turn on Tuesday (D2), they have work for

continuously six days, that is, Tuesday to Sunday. Similarly, those who start their work on Wednesday (D3),

they have work Wednesday to Monday and so on. Formulate the following table.

Discussions provide a total tabulation of

Min Z =

∑

7

1 =

i

D

_,

Subject to Constraints

D1+D2+D3+D4+D5+D6 ≥ 230,

D2+D3+D4+D5+D6+D7≥ 200,

D1+ D3+D4+D5+D6+D7≥ 240,

D1+D2 +D4+D5+D6+D7≥ 125,

D1+D2+D3+D5+D6+D7 ≥ 235,

D1+D2+D3+D4+D6+D7 ≥ 300,

D1+D2+D3+D4+D5+D7 ≥ 100,

D1 ≥0 , D2 ≥0 ,D3 ≥0, D4 ≥0, D5 ≥0, D6 ≥0, D7 ≥0.

The above model of linear programming has seven variables. Therefore we cannot determine this manually, so the model is solved by using LINGO software .which gives 300 as a Global optimal solution.

D1 D2 D3 D4 D5 D6 D7

Start shift Monday * * * * * * -

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International Journal of Emerging Technology and Advanced Engineering

Model (B): Linear Programming Model:

∑∑

4

1 =

4

1 =

=

Z

M IN

i j

Sij

,

Sij = Number of Nurse on Category i and Working on Shift type j.

i = 1 to 4 and j = 1 to 4.

Problem Definition and solutions

In our study in a Hospital, there are four categories of Nurses are required in a hospital first Emergency room

nurse, second Operating room nurse, third medical specialty nurse, an orthopedic nurse. Nurse's working time in an hour divided into four different Shifts.Minimize the number of nurses required to hire for the hospital.

1. First shift 8:00 am to 2:00 pm 2. Second shift 2:00 pm to 8:00 pm 3. Third shift 8:00 pm to 2:00 am 4. Fourth shift 2:00 am to 8:00 am.

Each nurse has to Work in two consecutive shifts per day. Details of nurses demand in a hospital as follows:

Categories shift S1 Emergency room

nurse

S2 Operating room

nurse

S3 Medical specialty

nurse

S4 Orthopedic nurse

First shift 14 12 12 5

Second shift 20 18 10 8

Third shift 25 10 11 6

Fourth shift 10 8 14 4

Solving the shifting problem table possibility Shift of nurses:

S1 I II III IV

S2 I II III IV

S3 I II III IV

S4 I II III IV First shift * * - - * * - - * * - - * * - - Second shift * - - * * - - * * - - * * - - * Third shift - * * - - * * - - * * - - * * - Fourth shift - - * * - - * * - - * * - - * *

Objective function:

∑∑

4 4

=

Z

M IN

Sij

Constraints:

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International Journal of Emerging Technology and Advanced Engineering

Second Operating room nurse: S21+S22≥12, S21+S24≥18, S22+S23≥10, S23+S24≥8,

Third Medical specialty nurse S31+S32≥12, S31+S34≥10, S32+S33≥11, S33+S34≥14,

Fourth Orthopedic nurse S41+S42≥5,

S41+S44≥8, S42+S43≥6, S43+S44≥ 4,

S11≥0, S12≥O, S13≥0, S14≥0, S21≥0, S22≥0 S23≥0 S24≥0,

S31≥0, S32≥0, S33≥0, S34≥0, S41≥0, S42≥0, S43≥0, S44≥0.

Solved the above model by using LINGO 18.0 software and the results obtained as given below:

Objective Value: 113.00 Total Solver: 11

Solution Table:

Categories Shift S1 S2 S3 S4

Shift type one 20 18 12 8

Shift type two 15 2 0 2

Shift type three 10 8 14 4

Shift type four 0 0 0 0

Model (C): Nonlinear model:

* Ci = Z

M IN

∑∑

4 1 =

i j

Sij,

Where Ci= Charges of Nurse per shifts.

Sij = Number of Nurse on Category i and Working on Shift type j.

i = 1 to 4 and j = 1 to 4.

Problem Definition and solutions

In our study in a Hospital, there are four categories of Nurses are required in a hospital first Emergency room nurse, second Operating room nurse, third medical specialty nurse, an orthopedic nurse.

Nurse's working time in an hour divided into four different Shifts. Minimize the number of nurses with minimum charges required to hire for the hospital.

1. First shift 8:00 am to 2:00 pm 2. Second shift 2:00 pm to 8:00 pm 3. Third shift 8:00 pm to 2:00 am 4. Fourth shift 2:00 am to 8:00 am.

Each nurse has to Work in two consecutive shifts per day. Details of nurses demand in a hospital as follows:

Categories shift S1 Emergency room

nurse

S2 Operating room

nurse

S3 Medical specialty

nurse

S4 Orthopedic nurse

First shift 14 12 12 5

Second shift 20 18 10 8

Third shift 25 10 11 6

Fourth shift 10 8 14 4

Categories Nurse Charges Minimum Charges Maximum

Emergency room nurse 250 350

Operating room nurse 300 400

Medical specialty nurse 400 450

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International Journal of Emerging Technology and Advanced Engineering

Solving the shifting problem table possibility Shift of nurses:

S1 I II III IV

S2 I II III IV

S3 I II III IV

S4 I II III IV First shift * * - - * * - - * * - - * * - - Second shift * - - * * - - * * - - * * - - * Third shift - * * - - * * - - * * - - * * - Fourth shift - - * * - - * * - - * * - - * *

Objective function:

*

Ci

=

Z

M IN

∑∑

4

1 =

4

1 =

i j

Sij

,

Where Ci= Charges of Nurse per shifts.

Sij = Number of Nurse on Category i and Working on Shift type j, i = 1 to 4 and j = 1 to 4.

Constraints:

First Emergency room nurse: S11+S12≥14,

S11+S14≥20, S12+S13≥25, S13+S14≥10, Second Operating room nurse:

S21+S22≥12, S21+S24≥18,

S22+S23≥10, S23+S24≥8, Third Medical specialty nurse

S31+S32≥12, S31+S34≥10, S32+S33≥11, S33+S34≥14, Fourth Orthopedic nurse

S41+S42≥5, S41+S44≥8, S42+S43≥6, S43+S44≥ 4,

S11≥0, S12≥O, S13≥0, S14≥0, S21≥0, S22≥0 S23≥0 S24≥0,

S31≥0, S32≥0, S33≥0, S34≥0, S41≥0, S42≥0, S43≥0, S44≥0.

Solved the above model by using LINGO 18.0 software and the results obtained as given below:

C1 C2 C3 C4

Min. charges 250 300 400 250 Categories Shift S1 S2 S3 S4 Shift type one 9 17 12 7 Shift type two 25 0 0 1 Shift type three 0 10 13 5 Shift type four 11 1 1 1

11250 8400 10400 3500 Total 33550

Local optimal solution found.

Objective value: 33550.00 Total solver iterations: 8 Elapsed runtime seconds: 0.08

Model Class: QP

III. CONCLUSION

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International Journal of Emerging Technology and Advanced Engineering

The requirement for quality software resolutions is essential for various reasons. In selective, it is necessary to ultimately use the work with time, to slightly manage the load of the work and trying to fulfill individual decisions. A high-quality program can head to a higher satisfied and more powerful workforce.

REFERENCES

[1] B.Satish Kumar, S. Naresh Kumar, S.Kumaraghuru. ―Linear Programming Applied to Nurses Shifting Problems." International

Journal of Science and Research (IJSR), Vol.3, No.3 (2014) PP.171-73.

[2] Lorraine Trilling, Alain Guinet, Dominique Le Magny. ―Nurse scheduling using integer linear programming and constraint programming."Elsevier, 3(2006) PP.651-656.

[3] Mohsen Bayati,Erfan Kharazmi,Mehdi Javananbakht,Aboozar Sadegi,Masud Arefnejad,Sajad Vahedi,Firooz Esmaeilzadeh― Optimization the number of nurses in the emergency department using linear programming technique‖.Journal of Health Management & Informatics(JHMI),Vol.1,No.2(2014) PP.41-45.