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An allocation model based on improved Fuzzy-DEA is proposed for Tangshan medical health resource. The membership function is particularly introduced to characterize the non-strict partial order relation of the fuzzy numbers, and the internal form of the unit efficiency values is converted into point values. The variance analysis index system is then used to find factors leading to non-DEA effectiveness and the allocation plan is recommended accordingly: the relevant departments should coordinate the inter-regional resources and optimize the medical staffing level.

The centralized pre-hospital command mode is still adopted by 120 emergency centers in Tangshan at present, following the principle of “patient willingness, immediate emergency, and service ability”, and uniformly dispatches and directs medical emergency services. The demand for pre-hospital emergency care has gradually expanded with the improvement of the economic level. The lack of first-aid sites, unreasonable site selection, and the fact that the emergency radius cannot effectively cover the demand points, and the problem of first-aid in the far city has been particularly highlighted. In addition, the lack of professional input and inefficient first aid further hindered the development of pre-hospital emergency medical services [

DEA (Data Envelopment Analysis) measures the relative efficiency of decision units through linear programming and has been widely used in the analysis of the relative efficiency of health services. Nunamaker first applied DEA to evaluate the efficiency of health resource allocation in 17 hospitals; (Nunamaker T R. Measuring routine nursing service efficiency [J]. Health Service Research, 1983, 18(1): 83-208). Rona proposed that the factor restricting the efficiency of pre-hospital emergency medical services is that the internal structure of supply is misalignment of supply structure rather than insufficient supply on the surface (analysis of the status quo and existing problems of Nanning emergency medical service supply from the perspective of public products); Wan Haohua proposed the evaluation of the nature of standby points around the emergency center in Nanchang. Strengthen the weak areas and improve the service capacity of on-call service points [

For the solution of fuzzy DEA, Guo et al. proposed a multiplier model based on fuzzy number comparison rules, converting fuzzy constraints into definite constraints; Leon et al. established the envelope model of the same solution idea; Wu Haiping 14, Zeng Xiangyun et al. In the L_R interval DEA model established by Wu Wenjiang et al., most models based on comparison rules require that the number of intervals follow a strict partial order relationship, and equations can be equivalent to clear numbers. However, the actual interval fuzzy number is often obtained based on empirical knowledge, and the upper and lower limits of the index cannot be obtained. Based on this, this paper converts the precise output elements into fuzzy numbers, and introduces the membership function to represent its non-strict partial order relationship. By introducing the telescopic coefficient, the membership function was selected to quantitatively characterize the degree of constraint relaxation, and the upper and lower limits of unit efficiency and the best point value were obtained. The model was used to evaluate the efficiency of pre-hospital emergency resource allocation in 16 hospitals in Tangshan City.

This article collects information on the number of hospitalized ambulances, medical personnel, drivers, and the total amount of medical treatment for 16 pre-hospital service units in Tangshan City according to the hospital’s medical record. The data spans from March 1, 2014 to March 2017. April 7. Considering the generality and simplicity of the indicators, and avoiding the mixing of economic and quantitative indicators [

The basic situation of the visiting units is shown in

Considering that the input-output index has ambiguity due to its own nature, external subjective cognition, and incomplete information, it limits the accuracy of the sample. The indicators are represented by fuzzy numbers so that the evaluation model is transformed into a fuzzy DEA model.

The output indicators are obscured in this paper, α-level cur set of fuzzy number A ˜ is noted as A α = { x ∈ R | μ A ˜ ( x ) ≥ α } ( A α = [ A α L , A α R ] ). Among them, Confidence level α ∈ [ 0 , 1 ] , A α L and A α R denote the upper and lower

index | 120 Call Center | The eighth hospital | Second Hospital | Women and children hospital | worker hospital | North China University of Technology Affiliated Hospital | Kai Luan hospital | Kai ping hospital |
---|---|---|---|---|---|---|---|---|

Input indicators | ||||||||

Number of cars | 1 | 1 | 2 | 1 | 6 | 2 | 4 | 2 |

Number of doctors | 2 | 9 | 8 | 25 | 10 | 6 | 14 | 36 |

Number of nurses | 1 | 6 | 5 | 5 | 11 | 5 | 14 | 7 |

Number of drivers | 2 | 4 | 4 | 5 | 12 | 8 | 10 | 5 |

Output indicators | ||||||||

General treatment | 9 | 8109 | 3821 | 2045 | 7942 | 10207 | 4405 | 7227 |

Li Kang hospital | Ren Min hospital | Tang Gang hospital | Railway hospital | Xie He hospital | Zhong Yi hospital | ninth hospital | Er wuwu hospital | |

Number of cars | 1 | 4 | 1 | 1 | 2 | 1 | 2 | 2 |

Number of doctors | 1 | 35 | 4 | 4 | 3 | 10 | 21 | 18 |

Number of nurses | 1 | 10 | 4 | 6 | 4 | 7 | 20 | 22 |

Number of drivers | 1 | 7 | 4 | 4 | 3 | 5 | 16 | 13 |

Output indicators | ||||||||

General treatment | 1099 | 11,073 | 7764 | 7963 | 9616 | 1017 | 3009 | 2901 |

boundaries of the internal. n 1 and n 3 represent the boundary of observation data floating up and down 10%, N α L = n 1 + ( n 2 − n 1 ) α , N α R = n 3 − ( n 3 − n 2 ) α . X ˜ j = ( x ˜ 1 j , ⋯ , x ˜ m j ) T and Y ˜ j = ( y ˜ 1 j , ⋯ , y ˜ l j ) T indicate the input and output indexes of the decision unit D M U j ( j = 1 , 2 , ⋯ , n ) .

DEA [

_ A ˜ ( x ) = { L ( ( a − x ) / T 1 ) x ∈ [ a _ , a ] 1 x = a R ( ( x − a ) / T 2 ) x ∈ [ a , a ¯ ] 0 o t h e r

Among them, L ( x ) and R ( x ) are strictly monotonically decreasing function of the left and right branches respectively, the fuzzy number can be written as A ˜ = ( a _ , a , a ¯ ) , a Is the median or kernel of the fuzzy number A ˜ , a _ and a ¯ being its upper and lower limits. T 1 = a − a _ and T 2 = a ¯ − a are called left and right shape. When the scale is 0, the fuzzy number degenerates to an exact number, that is, the exact number is a special fuzzy number whose left and right expansion values are 0. The geometric image is shown in

The traditional fuzzy DEA solution method is to transform it into a deterministic planning equation [

DEA draws the weights of inputs and outputs based on the sample data itself. The CCR model, originally founded by Charnes, Cooper, and Rhodes, assumes

that each production unit is always in the stage of constant returns to scale, but this assumption is not true in actual production. The impact of scale was ruled out based on the variable-scale BCC model. Based on the assumptions of minimumness, invalidity, and convexity, when the unit indicator is an L-R interval number, there are production possible sets:

T = { ( X ˜ , Y ˜ ) | X ˜ ≥ ∑ j = 1 n λ j X ˜ j , Y ˜ ≤ ∑ j = 1 n λ j Y ˜ j , Δ ∑ j = 1 n λ j = Δ , ∀ λ j ≥ 0 , j = 1 , 2 , ⋯ , n , Δ = 0 o r 1 }

Under the premise of output assumptions, this paper will reduce the input of medical resources and optimize the allocation of primary medical resources as the main way to improve the efficiency of medical services. The input-oriented BBC model is selected:

min θ s .t . { ∑ j = 1 n λ j x ˜ j ≤ θ x ˜ 0 ∑ j = 1 n λ j y ˜ j ≥ y ˜ r k λ j ≥ 0 , j = 1 , ⋯ , n s − ≥ 0 ; s + ≥ 0 ;

According to the partial order relation theorem of L-R fuzzy number [

D i ( x ) = { 1 ∑ j = 1 n λ j x ˜ j ≤ θ x ˜ 0 1 − ( ∑ j = 1 n λ j x ˜ j − θ x ˜ 0 ) / d i θ x ˜ 0 ≤ ∑ j = 1 n λ j x ˜ j ≤ θ x ˜ 0 + d i 0 ∑ j = 1 n λ j x ˜ j ≥ θ x ˜ 0 + d i

Among them, d i ( d i ∈ ℝ , i = 1 , 2 , ⋯ , n ) is the relaxation coefficient of the partial order relation of fuzzy numbers. By introducing this constant index, the strict partial order relationship between fuzzy numbers is relaxed in the form of fuzzy constraints. Let D = D 1 ∩ D 2 ∩ ⋯ ∩ D n , so that all fuzzy numbers are relaxed.

If θ 0 is the optimal solution when the fuzzy indicator degenerates into a clear number, the degree of relaxation is D i ( x ) = 1 . If the degree of membership of the fuzzy constraint D ( x ) is properly reduced, the constraint condition will relax accordingly, and the target value can be further optimized, which means that the value θ of the objective function can be reduced. When the constraint relaxes to the widest, θ 1 is the optimal value of the fuzzy index when it can accept the constraint edge value, and the fuzzy constraint condition D ( x ) also approaches 0 accordingly, so the target point value extends to the elastic variation range min { θ 0 , θ 1 } ≤ θ ≤ max { θ 0 , θ 1 } , and the target set belongs to The function can be expressed as:

G ( x ) = { 0 θ ≤ max { θ 0 , θ 1 } ( θ − min { θ 0 , θ 1 } ) / d 0 min { θ 0 , θ 1 } ≤ θ ≤ max { θ 0 , θ 1 } 1 θ ≥ min { θ 0 , θ 1 }

Among them, d 0 = max { θ 0 , θ 1 } − min { θ 0 , θ 1 } , let fuzzy decision sets be D F = D ∩ G , The best decision can be expressed as:

max x ∈ X ( D ( x ) Λ G ( x ) ) = m a x { λ | D ( x ) ≥ λ , G ( x ) ≥ λ , λ ∈ [ 0 , 1 ] }

The interval values of the efficiency of the decision unit can then be converted into point values by solving the following equation:

max ψ { 1 − ( ∑ j = 1 n λ j x ˜ j − θ x ˜ 0 ) / d i ≥ ψ j = 1 , 2 , ⋯ , n ( θ − min { θ 0 , θ 1 } ) / d 0 ≥ ψ ψ ≥ 0 x j ≥ 0 θ ≥ 0

θ i = ψ d 0 + min { θ 0 , θ 1 } .

Definition The model’s evaluation of efficiency is:

When θ i = 1.0 , the decision unit D M U i is relatively effective;

When θ i < 1.0 , D M U i is relatively ineffective.

Applying the model proposed in the previous section and selected indicators, the efficiency of pre-hospital emergency resource allocation in 16 hospitals in Tangshan area was evaluated. The conventional method of taking cut-offs and the CCR model based on the non-strict partial ordering of fuzzy numbers were applied to their efficiency values. Solve. For the cut-off solution method, different confidence level values α are selected, and finally the fuzzy effective ranges

[ ( θ * ) α L , ( θ * ) α R ] and optimal values θ of the Tangshan City Hospital are respectively obtained (θ is the efficiency values obtained by the method in this paper).

Thus, the effective range and optimal effectiveness of the 16 hospitals are shown in

The ranking of medical resource allocation efficiency in 16 hospitals was: [120 Call Center , Maternal and Child Hospital, Workers Hospital Affiliated Hospital of North China University of Science and Technology, Kaiping Hospital, Likang Hospital, Ren min Hospital, Railway Hospital, Ninth Hospital, No. 255 Hospital (Parallel)], Eighth Hospital , Second Hospital, [Kai Luan Hospital, Tangshan Hospital (Parallel)], Chinese Medicine Hospital.

As can be seen from

The utilization efficiency of hospital resources in the center of the urban area is generally higher than that of other hospitals. The reason is that high-level technicians and advanced equipment are focused on the urban areas. The infrastructure of the medical institutions in townships and townships is primitive, and the average technicians are far below. Urban area. The medical arrangements in each region are divided and there is a hierarchical relationship.

Special hospitals such as the second hospital are inefficient, because their professional service functions limit the type of disease they see. In urban areas, the prevalence of chronic non-communicable diseases among the residents of the urban areas has led to a drastic reduction in the population served by the urban residents. The fuzzy DEA visually shows the resources and efficiency of each medical institution, and the relevant personnel can control the entry point for improving efficiency.

Comparing the two methods, the traditional method of interception defines the upper and lower limits of the efficiency of the decision-making unit. The relative efficiency of the method to solve this method is higher than the traditional method but the deviation is not large, and effectively compensates for the requirement of the fuzzy number, possessing a strict partial order relationship.

α | 0.2 | 0.4 | 0.6 | 0.8 | 1 | θ |
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DMU_{1} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{2} | [9.19e-3, 1.08e-2] | [4.71e-3, 5.33e-3] | [3.21e-3, 3.49e-3] | [2.46e-3, 2.57e-3] | [2.00e-3, 2.01e-3] | 0.00334 |

DMU_{3} | [5.31e-3, 6.26e-3] | [2.72e-3, 3.08e-3] | [1.86e-3, 2.01e-3] | [1.42e-3, 1.48e-3] | [1.16e-3, 1.16e-3] | 0.00193 |

DMU_{4} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{5} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{6} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{7} | [7.82e-4, 1.82e-3] | [4.09e-4, 8.94e-4] | [2.84e-4, 5.85e-4] | [2.2e-4, 4.30e-4 | [1.84e4, 3.38e-4] | 0.00043 |

DMU_{8} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{9} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{10} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{11} | [1.22e-2, 1.44e-2] | 6.28e-3, 7.09e-3 | 4.28e-3, 4.64e-3 | 3.28e-3, 3.42e-3 | 2.68e-3, 2.68e-3 | 0.00043 |

DMU_{12} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{13} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{14} | [7.82e-4, 1.51e-3] | [4.09e-4, 7.41e-4 | [2.84e-4, 4.85e-4] | [2.22e-4, 3.56e-4] | [1.84e-4, 2.80e-4] | 0.00038 |

DMU_{15} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

DMU_{16} | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | 1 |

According to J.Cohen, an indicator system can be used to assess the effect of variance analysis [

η 2 = S S b e t w e e n g r o u p s S S o v e r a l l

That means the proportion of the sum of squares between the groups after the experimental treatment in the sum of squares of the overall population. The larger the value, the greater the effect of the experiment reflected. η 2 = 0.01 as the smaller effect, η 2 = 0.06 denotes the medium effect, and η 2 = 0.14 presents that the effects are significant. The meaning is clear and easy to understand, so it was adopted by SPSS as an index of the size of the relevant experimental effect in the general linear model module (Chinese version shows “partial Eta square”; English version is translated as “Partial Eta Squared”), as can be seen from

non-DEA effective unit | Partial Eta Squared |
---|---|

Number of cars | 0.170 |

Number of doctors | 1.000 |

Number of nurses | 1.000 |

Number of drivers | 0.388 |

General treatment | 1.000 |

that the Eighth Hospital, the Second Hospital, the Kailuan Hospital, the Tangshan Hospital, and the Chinese Medicine Hospital are basically in the same urban traffic environment. The increase in the number of vehicles does not have a significant impact on the patient’s transportation. However, the changes in the allocation of medical staff will greatly affect the speed of the treatment of patients and the efficiency of the distribution of medical resources. The total diagnosis and treatment reflect the total amount of patients treated, so the relevant departments should coordinate the allocation of resources between regions, and optimize service levels. This is the best entry point for improving the efficiency of the hospital.

Sang, Q.Z., Yao, F.Z., Hao, S. and Zhang, Z.N. (2018) On Analysis of Tangshan Pre-Hospital Service Efficiency Based on Improved Fuzzy-DEA. Open Access Library Journal, 5: e4720. https://doi.org/10.4236/oalib.1104720