A Resource Allocation Model for Hospital Administration

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A Resource Allocation Model for Hospital Administration

Praveen Kumar K M

Department of Mathematics, Brindavan College of Engg.

School of Mathematics, Reva University Rukmini Knowledge Park, Kattigenahalli, Yelahanka,

Bangalore, Karnataka, Bangalore.

Harish Babu G A Department of Mathematics, Reva ITM, School of Mathematics, Reva University

Rukmini Knowledge Park, Kattigenahalli, Yelahanka Bangalore, Karnataka, Bangalore.

Abstract This paper is devoted to the application of goal programming to medical care planning. More specially, the paper presents a goal programming resource allocation model for hospital administration. It is possible to formulate a complex multi-year resource allocation model that serves the purpose of long-range planning for the hospital. The scope of this study is limited, however, to the planning horizon of one year. It is felt that this limited scope will allow a clearer representation of the model- development. Once it is completed for one year, the basic model can be expanded for a longer planning horizon by forecasting parameter changes.

KeywordsGoal Programming, Health Care, Resources

INTRODUCTION

In recent years, hospital administration has become a very complex management process. The great demand for hospital care is understandable in view of increased concern for health care on the part of the Indian population as a whole, increased institutional protection for health and accident, and of course increasing population. Rapidly rising salaries of medical personnel, coupled with these factors, have accelerated the increase of hospital costs. However, another important contributor to the cost increase is inefficient resource allocation and ineffective utilization of existing facilities, a result of the increased complexity of hospital operations. The administration of virtually every hospital is a unique management problem. It would be difficult to find two hospitals that offer identical services to the same type of patients through identical management processes. Hence, it is difficult to design a general model that can be applied to all hospitals. However, the basic functions of the hospital are more or less universal among all types of medical facilities. Therefore, once an aggregative resource allocation model is designed for a hospital, it can be easily modify to fit the unique characteristics of the hospital for application.

Various models have been developed and improved over the past 35 years. They can aid in improving the effectiveness of the decision-making process in an organization. Arthur

[1] gave a multiple objective nurse scheduling model. An application of linear programming in hospital resource allocation was given by Grant and Henden [2]. Stinnett and

Pattiel [4] have given a mathematical model for the efficient allocation of health-care resources. In the present study, a GP Model has been used for resource allocation in a hospital. GP is a variation of linear programming. Charnes and Cooper [6] conceptualized the name goal programming. It was applied to an analytical process that solved multiple, conflicting, non commensurate problems. A goal that is not completely achieved has an under-achievement (negative deviation) or over achievement (positive deviation) of the goal. If the objective is to exceed stated goals, the objective function will only contain a negative deviational variable, d-. If the objective is to be under the stated goal, the objective function will contain a positive deviational variable, d+.

DATA OF THE PROBLEM

In this study, Lokpriya hospital in Meerut city is selected for the model design. With no resident physicians, patients are generally admitted by their personal physicians or through the emergency ward. The hospitals emergency room is staffed by local doctors on a rotation basis according to an agreement with the hospital. The hospital has 125 beds and employs 86 employees, excluding local physicians. Tables

  1. & (2) outline the model variables and other pertinent information needed for this study. The salaries given are an average of the salaries earned by each person in the individual personnel category. The figures for each category are arbitrarily determined upon the request of the hospital administrator. The personnel classifications were made in relation to the assignment of personnel expenses within the various accounting designations utilized by the hospital. Although a number of split assignments are possible and often practiced, an attempt is made here to minimize these for the model design.

    Variable

    Position

    Desired % of total Employees

    Average Salary

    Desired % Pay Increase

    x1

    Nursing service administration

    5.81

    89,700

    6

    x2

    Medical & surgical nurse

    27.90

    88,800

    7

    x3

    Pediatric nurse

    3.49

    89,400

    8

    Variable

    Position

    Desired % of total Employees

    Average Salary

    Desired % Pay Increase

    x1

    Nursing service administration

    5.81

    89,700

    6

    x2

    Medical & surgical nurse

    27.90

    88,800

    7

    x3

    Pediatric nurse

    3.49

    89,400

    8

    Table (1) Hospital Personnel, Desired Personnel Proportions Average Salaries and Desired Pay Increases

    x4

    Obstetric nurse

    4.65

    89,100

    6

    x5

    Operating & recovery room nurse

    4.65

    91,800

    8

    x6

    Service & supply room nurse

    2.33

    82,400

    7

    x7

    Emergency room nurse

    3.49

    90,000

    9

    x8

    Intensive care nurse

    2.33

    93,600

    10

    x9

    Laboratory technician

    4.65

    81,600

    5

    x10

    Pathologist

    3.49

    92,400

    10

    x11

    Cardiologist

    1.16

    91,200

    10

    x12

    Radiologist

    3.49

    1,96,800

    10

    x13

    Dietician

    13.95

    85,000

    8

    x14

    Plant operation & maintenance

    2.33

    82,200

    5

    x15

    House keeping

    8.14

    69,600

    5

    x16

    Laundry & linen

    1.16

    70,800

    5

    X17

    Administrative service

    6.98

    90,600

    6

    x4

    Obstetric nurs

    4.65

    89,100

    6

    x5

    Operating & recovery room nurse

    4.65

    91,800

    8

    x6

    Service & supply room nurse

    2.33

    82,400

    7

    x7

    Emergency room nurse

    3.49

    90,000

    9

    x8

    Intensive care nurse

    2.33

    93,600

    10

    x9

    Laboratory technician

    4.65

    81,600

    5

    x10

    Pathologist

    3.49

    92,400

    10

    x11

    Cardiologist

    1.16

    91,200

    10

    x12

    Radiologist

    3.49

    1,96,800

    10

    x13

    Dietician

    13.95

    85,000

    8

    x14

    Plant operation & maintenance

    2.33

    82,200

    5

    x15

    House keeping

    8.14

    69,600

    5

    x16

    Laundry & linen

    1.16

    70,800

    5

    X17

    Administrative service

    6.98

    90,600

    6

    Formulation of Goal Constraints

    With the data defined in Tables (1) & (2), the G.P. model constraints for resource allocation are formulated as follows:

    1. Personnel Requirement

      The hospital presently employs 86 persons and the administrator feels that the existing personnel must be retained in order to provide satisfactory services to the patient.

      17

      – _ +

      – _ +

      x1 + d1 d1 = 86

      i = 1

      Table (2) Expenses and Reserves Expenses

      Reserves for coming year

    2. New Equipment

      Variable

      Category

      Total for past year

      Total for coming year (5%

      increase)

      Y1

      Nursing division

      12,80,000

      13,44,000

      y2

      Physicians fee (emergency ward)

      16,40,000

      17,22,000

      y3

      General services (X-ray, medical supplies, etc.)

      18,30,000

      19,21,500

      y4

      Administration

      4,60,000

      4,83,000

      y5

      Miscellaneous

      14,50,000

      15,22,500

      Variable

      Category

      Total for past year

      Total for coming year (5%

      increase)

      Y1

      Nursing division

      12,80,000

      13,44,000

      y2

      Physicians fee (emergency ward)

      16,40,000

      17,22,000

      y3

      General services (X-ray, medical supplies, etc.)

      18,30,000

      19,21,500

      y4

      Administration

      4,60,000

      4,83,000

      y5

      Miscellaneous

      14,50,000

      15,22,500

      A new x-ray equipment is required if the x-ray service is to be continued for the coming year. The new equipment is estimated to cost Rs. 2,40,000. Also it is desired to reserve Rs. 5,80,000 in the contingency fund for emergencies.

      – +

      – +

      – +

      – +

      Z1 + D2 – D2 = 2,40,000 z2 + d3 – d3 = 5,80,000

    3. Employee Pay Increase

      The administrator feels that the minimum pay increase should be 5% and the maximum should be 10% for any given personnel category. The figure before each group of variables (also see table 1) is the personnel pay increase.

      0.05 ( 81,600 x9 + 82,200 x14 + 69,600 x15 + 70,.800 x16 )

      + 0.06 (89,700 x1 + 89,100 x4 + 90,600 x17) + 0.07

      Variable

      Category

      Amount

      z1

      Radiology equipment

      2,40,000

      z2

      Contingency reserve

      5,80,000

      Variable

      Category

      Amount

      z1

      Radiology equipment

      2,40,000

      z2

      Contingency reserve

      5,80,000

      (88,800 x2 + 82,400 x6) + 0.08 (89,400 x3 + 91,800 x5

      – +

      – +

      + 85,000 x13) + 0.09 (90,000 x7 ) + 0.10 (93,600 x8 + 92,400 x10 + 91,200 x11 + 1,96,800 x12) + d4 – d4 = z3

    4. Funds For Expenses

  1. Nursing Division Fund : y1 + d – – d + = 13,44,000

    5 5

    GOALS AND THEIR PRIORITIES

  2. Fund for Physicians Fee: y2 + d – – d + = 17,22,000

    6 6

    The administrator must determine the goals of the hospital

    and their priorities in order to accomplish the optimum

  3. General Services fund : y3 + d – – d + = 19,21,500

    7 7

    allocation of resources. This process usually involves a

    group decision by the hospital administrator and the board of

  4. Administrative Expenses: y4 + d – – d + = 4,83,000

    8 8

    directors. The administrator lists the following goals in

    order of importance.

  5. Miscellaneous Expenses: y5 + d – – d + = 15,22,500

9 9

  • Secure the necessary manpower to provide adequate

    services to the patient. The administrator feels that the existing personnel will be sufficient to provide adequate services for the coming year.

  • Replace and / or acquire new equipment that is required to provide the services of the hospital (this figure should be in addition to funds provided by depreciation).

  • Provide adequate pay increases to all personnel in keeping with the economy and the community labor market (see table (1) for the administrators desired pay increases).

  • Provide funds for expenses.

  • Achieve the desired distribution of each personnel category (see table (1).

  • Minimize costs and breakeven in the operation.

PERSONNEL DISTRIBUTION

According to the trend of demand for hospital services, the administrator has established the desired number of employees in each personnel classification as a proportion of the total employees as shown in Table 1. If we denote ai as the desired no. of employees in the ith category as a proportion of the total no. of employees, 17 separate equations can be expressed by a general equation as:

xi ai + di +9 _ di+9= 0 ( i = 1, 2, – — – – – 17 )

For example, for the desired number of nurses in the nursing service administration, the constraint will be

– +

– +

x1 5 + d10 – d10 =0

COST MINIMIZATION

The total cost for the hospital operation is calculated in this constraint. Hence, this constraint identifies the resource

x6 = 2 x16 = 1 z1 = 2,40,000

x7 = 3 x17 = 6 z2 = 5,80,000

x8 = 2 z3 = 5,65,402 x9 = 4

d + = 1,55,77,700 x =3

27 10

requirements to achieve the set of goals presented by the

administrator. If a certain maximum resource is previously determined, it could be used so as to identify the degree of goal achievements with the given resources. In order to simplify the constraint, let bi represent the average salary figure fo the ith personnel category as shown in Table 1. (i.e.Rs.89,700.00 for the nursing service administration, etc.). Then the cost minimization constraint will be

17 5 2

The solution of the above model indicates that all the goals can be achieved at the total cost of Rs.1,55,77,700. Since cost minimization is treated as the goal with the lowest priority factor, it is impossible to minimize the cost to zero.

REFERENCES

– d

– d

27

27

bix i + yi + zi + d –

i = 1 i = 1 i = 1

+

27 = O

  1. Arthur JL and Ravindran A [1981]: A multiple objective nurse scheduling model. IIE Transactions, 13, 55-60.

  2. Grant EW and Hendon FN [1987]: An application of linear programming in hospital resource allocation. Journal of Health Care

Objective function The objective function for the model is

3.9

Market, 7, 69-72.

Franz LS et2a6l. [1989]: A mathematical model for scheduling and staffing multiclinic health regions. European Journal of Operations

Research,41(2), 277 289.

– – – – – –

– – – – – –

Min. Z = p1d1 + p2 (d2 + d3 ) + p3d4 + p4di + p5 di +

  1. Stinnett AA and Paltiel AD [1966]: Mathematical programming for

    27

    27

    p6d –

    i= 5 i = 10

    RESULTS AND DISCUSSION

    the efficient allocation of health care resources. Journal of Health Economics, 15(5), 641-653.

  2. Brien Pallas L et al. [2001]: Forecasting models for human resources in health care. Journal of Advance Nursing, 33 (1), 120- 129.

  3. Charnes A and Cooper WW [1961]: Management models and the

    The LGP problem used in the study contains 79 variables (decision and deviational), 27 constraints and 6 goals. The solution of the problem is obtained by using QSB+ software package (based on modified simplex method). The solution of the problem is as follows

    Goal Attainment Achieved/ Not achieved Manpower for service (p1) : Achieved Equipment acquisition (p2) : Achieved Employee pay increase (p3) : Achieved Expenses (p4) : Achieved Distribution of personnel (p5) : Achieved Minimize cost (p6) : Not possible

    Variables

    x1 = 5 x11 = 1 y1 = 13,44,000

    x2 = 24 x12 = 3 y2 = 17,22,000

    x3 = 3 x13 = 12 y3 = 19,21,500

    x4 = 4 x14 = 2 y4 = 4,83,000

    x5 = 4 x15 = 7 y5 = 15,22,500

    industrial applications of linear programming.Vols1&2, John Wiley and Sons, New York.

  4. Asadoorian J, Locker D [2006]: The impact of quality assurance programming: a comparison of two Canadian dental hygienist programs. Journal of Dental Education, 70(9), 965-971.

  5. Beaulieu H et al. [2000]: A mathematical programming approach for scheduling physicians in the emergency room. Health Care Management Science, 3(3), 193-200

  6. Brody GH et al. [2004]: The Strong African American Families Program: translating research into prevention programming. Child Development, 75(3), 900-917.

  7. de Guise E et al.[2005]:Overview of traumatic brain injury patients at a tertiary trauma centre. Canadian Journal of Neurology Science, 32(2), 186-193.

  8. Ferguson EL et al. [2006]: Design of optimal food-based complementary feeding recommendations and identification of key "problem nutrients" using goal programming. Journal of Nutrition, 36(9), 2399-2404.

  9. Gaweda AE et al. [2005]: Individualization of pharmacological anemia management using reinforcement learning. Neural Networks, 18(5-6), 826-834.

  10. Alois Geyer et al.: Scenario tree generation and multi-asset financial Optimization problem, Operations Research Letters, 2013, 41, 494- 498.

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