2.1. Multi-objective medical service organization selection model with stochastic demand and limited emergency management resources

In emergency management, the manager receives medical resource demand information from the relevant department and allocates corresponding medical services for dealing with this risk. The challenge here is to allocate medial resource demand and select a suitable medical service organization approach. Notably, emergency medical service organization selection is a multiple criteria decision-making problem, and a multi-objective decision model must be built to allocate medical service demand for sudden risk and to select an organization approach among other potential approaches. Meanwhile, in developing similar models, researchers have rarely simultaneously considered stochastic demand, fuzzy objective and weight factors. Our model recognizes that these phenomena must be considered to address emergency plan selection problems. The following section discusses our model in detail and presents a flowchart describing the proposed model (Fig 1). We first make the following assumptions:

1. Medical service demand is stochastic.
2. Delayed medical service costs are considerable.
3. The objective and weight are fuzzy.
4. Medical resources are limited.
5. Life is the most precious resource of all.
6. Medical organization can support multiple services.

1. Moreover, we use the following notation throughout this paper.
2. D_j: Demand for the jth medical service
3. q: The number of objectives

1. k: The number of negative objectives
2. m: The number of constraints
3. n: The number of emergency medical organizations
4. m_t: Medical services of the tth medical organization, t = 1,2,…,n
5. p_tj: The price of the tth medical organization’s jth service, j = 1,2,…,m_t
6. V_tj: The maximum capacity for the tth medical organization to execute the jth service
7. R_tj: The probability of satisfaction with the j-th product offered by the t-th supplier
8. x_tj: The purchase quantity of a tth medical organization jth service user, x_tj∈x, x = {x₁₁, x₁₂,…, }
9. y_tj: Decision variable determining if the tth medical organization delivering the jth service is selected or not where y_tj = 1 when the tth medical organization is selected to deliver the jth service and where y_tj = 0 otherwise
10. c_tj: The unit delay cost for the jth service supplied by the tth medical organization
11. E(x_tj): The penalty function for the jth service supplied by the tth medical organization
12. f: Objective function f = {f₁,f₂,…,f_q}
13. Then, a new general multi-objective model can be written as follows:

(1) (2) with the following constraints: (3) where f(1), …, f(k) are the negative objectives or criteria-like cost, response time, etc. f(k+1), …, f(q) are positive objectives or criteria such as the degree of social satisfaction, service quality and so on; X_tj, t = 1,…, n j = 1,…,m_t are nonnegative decision variables and b_i, i = 1,…, m are independent continuous random variables with given distributions while a_itj represents the coefficient of the t-th decision variable in the i-th constraint, and β_i is the i-th pre-assigned probability level 0<β_i≤1, i = 1, …, m.

Meanwhile, to solve an emergency medical organization selection problem, we assume that the objective includes the cost f₁, degree of social satisfaction f₂, response time f₃ and service quality level f₄ together with the major constraint that medical services can mostly satisfy stochastic demand. Each emergency medical organization has its own unit cost, social satisfaction history, response time record and service quality data.

However, we assume that life is the most precious resource of all and that costs must been considered. Moreover, medical services are limited. At the same time, in real situations, many factors that shape medical service demand cannot been fully addressed. According to such conditions, we must define a penalty function. Notably, delayed medical services result in losses. Diverse conditions can result in different costs. With a scenario analysis we construct two penalty functions as follows:

1. When a delayed medical service can be provided by another organization, the penalty function is equal to zero, i.e., (4)
2. When a delayed medical service can immediately be replenished, the penalty function is linear as follows: (5)

The objective function for costs can be written as follows: (6) which should minimize the cost of the product.

The objective function for the degree of social satisfaction is defined as: (7) which should maximize the number of reliable units.

The aggregate performance measure for the response time objective function is defined as: (8) which should minimize the response time.

The objective function for service quality is defined as: (9) which should maximize service quality.

In turn, the final form of the multi-objective model for medical service organization selection is as follows: (10) (11) (12) (13)

S.t.

(14)(15)(16)

Generally, managers do not have exact and complete information related to decision-making criteria and constraints that are fuzzy and stochastic in nature. A new fuzzy multi-objective medical service organization selection model is thus developed to address this problem.

In the new model, ~ denotes the fuzzy environment. ≳ used in the objectives and constraints denotes a fuzziness of ≥ i.e., approximately greater than or equal to, and ≲ linguistically denotes “essentially smaller than or equal to.”

2.2. A new fuzzy multi-objective medical organization selection model considering multiple services and stochastic demand for emergency management

Thus, by applying c_it and σ_i where c_it>0 and 0<σ_i<β_i as predetermined values set by the decision-maker, satisfaction constraints of the decision-maker can be stated as follows:

The decision-maker is fully satisfied if (17)

The decision-maker is almost satisfied if (18)

The decision-maker is not satisfied if (19) then, the equivalent deterministic constraints for (17)–(19), respectively, are (20) (21) (22) where is the inverse of the cumulative distribution function of random variable b_i, i = 1,…,m.

Using the Bellman–Zadeh approach [18, 19], the fuzzy set objective functions f_p and constraints g_i are defined as (23) where are the membership functions of objectives and constraints and where are the degree to which x belongs to objectives and constraints. The fuzzy set objectives and constraints are thus uniquely determined by their membership function and the range of the membership function is a subset of nonnegative real numbers whose value is finite and usually finds a place in interval [0,1].

From (23), it is possible to obtain the solution proving the maximum degree (24) (25)

To obtain (24) and (25), it is necessary to build membership functions ,

p = 1,…, q, i = 1,…, m from the corresponding f_p(x), g_i(x), x∈L, p = 1,…, q, i = 1, …, m. This function is satisfied through the use of membership functions (26) for maximized objective functions or through the use of membership functions as shown in Fig 2. (27) for minimized objective functions as shown in Fig 2.

The construction of (26) or (27) involves solving the following problems: (28) (29) min_x∈Lf_p(x), max_x∈Lf_p(x) are obtained by solving the multi-objective problem as a single objective each time using only one objective where x∈L means that solutions must satisfy constraints. Since for every objective function f_p(x) its value changes from min_x∈Lf_p(x) to max_x∈Lf_p(x), it may be considered a fuzzy number with the membership function as shown in (26) and (27). (30) where k_i(x) and h_i(x) are defined as (31) (32) (33) (34) (35) (30) is a membership function of fuzzy values of linguistic variables that reflect constraints of a qualitative nature.

2.3. Decision-making processes

First, the max–min operator used by Zimmermann [20, 21] for fuzzy multi-objective problems is discussed. Then, the convex (weighted additive) operator that enables DMs to assign different weights to various criteria is described.

In fuzzy programming modeling, when using Zimmermann’s approach [22], a fuzzy solution is given by the intersection of all fuzzy sets representing either fuzzy objective or fuzzy constraints. The fuzzy solution for all fuzzy objectives and fuzzy constraints may be written as (36)

The optimal solution (x*) is given by (37) where μ_D(x), and represent membership functions of the solution, objective functions and constraints.

Under real conditions, different objectives and constraints are of unequal importance to DM and other patterns, and thus weight should be considered. The fuzzy weighted additive model can address this problem as described below.

The weighted additive model is widely used in vector-objective optimization problems; the basic premise is to use a single utility function to express the overall preference of DM to draw out the relative importance of criteria [23]. In this case, multiplying each membership function of fuzzy goals by its corresponding weight and then adding the results together generates a linear weighted utility function.

The fuzzy model proposed by Bellman and Zadeh and Sakawa and the weighted additive model developed by Tiwari are written as [24–26] (38) (39) where and are the weighting coefficients denoting the relative importance of fuzzy goals and fuzzy constraints.

(40)(41)(42)(43)

If w_j denotes the fuzzy weight of the j-th objective or constraint, let , j = 1, …, p+m be a trapezoidal fuzzy number, or let j = 1, …, p+m be a triangular fuzzy number. Then, by utilizing the α−cut approach for , as a trapezoidal fuzzy number, the proposed model can be written in the form of a weighted max–min deterministic-crisp nonlinear programming model as follows: (44) s.t.: (45) (46) (47) (48) (49) (50) (51) (52)

It should be noted that w_j, j = 1,2,…,m+p become decision variables in addition to η_p, κ_i and x_t, t = 1, …, n. Constraint (48) ensures that the relative weights should add up to one. Additionally, fuzzy weights reflect the uncertain relative importance of objectives where the sum of all fuzzy weights should be one, i.e., , and where the sum of the lowest values of all fuzzy weights should be less than one, i.e., . On the other hand, if for any j-th objective or constraint is considered a triangular fuzzy number, then w_j1 and w_j2 should be replaced with w_j0 for the j-th objective or constraint.

Assumptions

• Three medical services are purchased from three medical organizations. Factors such as price, satisfaction, response time, service quality and capacity can be obtained from historical data.
• Demands for three items are separately normally distributed fuzzy random variables with mean values of 100, 120 and 140, respectively, and variances of 25, 49 and 36, respectively; the three organizations’ services are limited.
• Objectives include the cost (f₁), degree of social satisfaction (f₂), response time (f₃) and service quality (f₄). All experimental data are included in Table 1 and Table 2. In Table 1, information on objectives and constraints is provided. As the only stochastic constraint, medical services must almost satisfy demand. Meanwhile, data on decision makers’ relative weights of fuzzy goals and constraints are shown in Table 2.

From Eq (44)–(52) and relative experimental data we find that our numerical example has interesting implications for medical service organization schemes as illustrated in Table 3, which shows that the value of the penalty function does not always influence medical service organization selection. Regarding scenario descriptions, under one scenario, it is easy to find alternatives; under the other scenario, although a medical service can be purchased, there is a loss. Meanwhile, the loss is linearly related to the purchase volume. These scenarios serve as simplified descriptions of real situations. Thus, an interesting result occurs when a selected organization incurs a higher penalty than an unselected one. In Table 3 x₂₂ and x₂₃, we purchase more medical services at higher losses rather than incurring no loss. This phenomenon is usually inconsistent with what people anticipate. However, when costs are not the sole factor, we know that this is common. In our model, decisions are influenced by costs, degrees of social satisfaction and so on. This finding means that we must simultaneously consider multiple criteria requirements. A factor cannot always influence the final result. According to this rule, a penalty function cannot can occasionally impact medical service organization selection decisions due to other factors.

Meanwhile, we use a sensitivity analysis to investigate the changes in optimal decision values regarding medical service Item 1 when only one parameter in the dataset changes while others remain unchanged. Relative data are shown in Tables 1 and 2. The computational results are illustrated in Tables 4–7.

From Tables 4–7 sensitivity analysis demonstrates that, when only one parameter is changed, the optimal decision is sometimes not influenced when applying current experimental data to the multi-objective problem. Moreover, this result simultaneously validates the result shown in Table 3 that the sole parameter occasionally fails to affect decisions.

The model proposed here is reasonable to apply under defined conditions and can solve the decision-making problem of selecting medical organization approaches with uncertain information. As the variables used in the model are different from real conditions, the model’s validity is limited to a certain extent. Therefore, it is necessary to adjust the variables used according to real conditions. In addition, this work mainly presents a theoretical analysis that needs to be verified with real data.