Optimization of spares varieties in the uncertain systems

Abstract

Stocking strategy for spare parts has a pivotal influence on the productivity and efficiency of industrial plants. Considering the situation of lacking historical statistics causing by uncertainty, this paper proposes a method to optimize spares varieties based on uncertainty theory and uncertain data envelopment analysis (DEA) model. Firstly, a recursive hierarchy structure is established to construct an evaluation system to meet the requirement of avoiding attributes’ redundancy. Then, an uncertain spares optimization model (USOM), which is based on uncertain DEA model, is developed to optimize spares varieties. Furthermore, the uncertainty theory is utilized to convert the USOM into an equivalent deterministic model for simplification. Finally, a numerical example is given to illustrate the performance of this model. The results show that the stocking strategy obtained from the proposed decision model can satisfy the purpose of saving resources and prompting continuous operation.

Keywords

Spares varieties uncertain systems optimization data envelopment analysis

1 Introduction

Since the objective existence of failure, parts in a certain system always need to be repaired or replaced in the life cycle. Obviously, spares are important for the availability of the systems. For many industrial plants, spare parts inventories which serve for the maintenance planning are important management strategies (Joanne [1], Siddiqi and Weck [2]). At present, most of storage solutions are reserving spares in the warehouse. Nevertheless, there are some problems in inventory strategy. On one hand, if too few varieties of spare parts are reserved in the warehouse, the equipment may not be repaired once the damage occurs because of the lack of spare parts, which may cause severe consequences. For example, if a weapon system could not recover from fault timely, it might lead to combat failed and casualties. On the other hand, however, considering it may lead to plenty of unnecessary storage and management, it is not feasible to reserve all varieties of spare parts in the warehouse. To make a balance between high costs and low availability, a good planning, which can determine the part to reserve by giving priorities to spares, needs to be made.

Various approaches have been developed to make spares schemes during the last decades. For distinguishing high priority and low priority, the primary key is the selection of characteristics which have great influences on items. Many scholars have proposed different indexing systems to determine spares varieties. For example, Nahmias [3] used a one-dimensional factor based on the total dollar usage. Duchessi et al. [4] proposed a two-dimensional scheme to realize spares classification. Zhou and Fan [5] and Ng [6] considered lead time, average unit cost and the dollar usage as influencing factors. Petrovic [7] proposed a model with six factors, including essentiality, price, availability of required system, weight and volume of the item, market availability of the spares, and efficiency of repair. Other classification criteria were also proposed by many scholars (Krupp [8], Rad et al. [9], Zhang et al. [10], Fuller et al. [11]). Unfortunately, some parts will lead to non-strategic stocks if the decisive characteristics are not included in the indexing system. Such as, a part that have a high price and a short mean time between failure (MTTR) should not be identified as a spare part when we only consider the price, but it requires attention when we consider the both factors. In addition, the redundancy among indices may cause a large deviation in actual results. So in this paper, we will take into consideration of both independence and comprehensiveness to construct an indexing system.

Specifying optimal inventory policies for spares requires the use of forecasting techniques. Traditional spare parts optimization methods have greatly improved the performance of support system. However, those models have suffered limitations in actual situations. The traditional ABC classification method [12], mainly based on the percentage of total annual dollar usage to classify those spare parts into three groups, just considers a single factor. So comprehensively consideration of multiple factors should be raised. Ng [6] and Teunter et al. [13] suggested to use a multi-criteria ABC classification method that includes multiple parameters. But the ABC method based on historical data and experiences may not meet requirements of complex components. The analytic hierarchy process [14], which includes multiple criteria, is dependent on a single scalar measure of importance by subjectively rating the criteria. Genetic algorithms [15] and artificial neural networks [16] are heuristic and therefore cannot provide the optimal solution. Data Envelopment analysis (DEA) is a nonparametric approach to evaluate the relative efficiency of decision making units (DMUs) proposed by Charnes et al. in 1978 [17] and Banker et al. in 1984 [18]. Data envelopment analysis has been proved an effective method to evaluate the relative efficiency of the decision making units (DMUs) with multiple inputs and outputs. Its advantages cannot be underestimated in terms of simplifying algorithms, avoiding subjective factors and reducing errors. Meanwhile, it has no limitations to the input and output. So DEA has become an attractive tool in many fields, such as economy [19], group decision making [20, 21], automatic clustering algorithm [22], case retrieval method [23] and so on. Cooper et al. proposed Imprecise Data Envelopment Analysis (IDEA) method, to deal with imprecise information on input and output levels, which results in an efficiency interval rather than a single value [24]. Wen & Li extended the traditional DEA models to a fuzzy framework based on credibility measure [25]. Recently, Zhou et al. proposed multi-objective Fuzzy DEA model for evaluating DMUs with type-2 fuzzy input and output variables [26]. Hatami-Marbini et al. investigated the role of pre-determined positive “ as decision variables bound to differentiate between strongly and weakly efficient DMUs in Fuzzy DEA approaches [27].

Over the past few decades, great efforts have been made to determine the indexing system of spares varieties’ optimization. However, the emergence of various uncertain factors may lead to the uncertainty of evaluation indices. The probability theory is the main mathematical system for modeling indeterminacy. Regattieri et al. [28] had proved that a better approach for predicting intermittent demand was weighted moving average. For more accurate prediction, Godoy et al. [29] presented a graphic technique which considered a reliability threshold and a stochastic lead time to enhance spare parts ordering decision-making. These papers addressed the problems based on enough historical data, where the mean value was chosen to make decision. However, with the increase of equipment complexity and environmental uncertainty, there is few statistical data to be used under certain conditions, so it became particularly difficult to make any schemes for optimizing spares varieties. In this case, we must invite some experts to evaluate the belief degree that the event will happen. Liu [30] proposed the uncertainty theory which can deal with human’s belief degree mathematically in 2007. Then the belief degree can be regarded as uncertainty distribution of the uncertain variable, and deal with it via Liu’s [30] theory. In order to deal with the situation of lacking data, all the indices are regarded as uncertain variables in this paper, thus producing an uncertain spares optimization model (USOM) based on DEA model.

This paper is organized as follows. Section 2 briefly introduces some basic concepts and properties about uncertain variables. Section 3 establishes an evaluation system and gives a classification of uncertain parameters, constant parameters and qualitative parameters. In Section 4, the USOM is developed for a relatively simple and representative structure—a two-site operating organization. Then a simplified equivalent deterministic model of the USOM are presented. Especially, a simplified model is given when all the uncertain variables obey zigzag distributions. In Section 5, a numerical example is given to illustrate the proposed method. Finally, the general conclusions are included in Section 6.

2 Preliminaries

Uncertainty theory has become a branch of axiomatic mathematics since it was founded by Liu in 2007 [30]. It is used for modeling belief degrees that each event will happen when no samples are available. Some basic axioms, definitions and theorems, which are vital remainders of this paper, will be presented in this section. Let be a nonempty set, and a σ-algebra over. Each element $Λ \in L$ is called an event. Each event Λ will obtain a number $M {Λ} \in [0, 1]$ to indicate the belief degree. The uncertain measure $M$ must satisfy the following axioms that was suggested by Liu [31, 32]:

Axiom 1 (Normality Axiom) $M {Γ} = 1$ for the universal set Γ.

Axiom 2 (Duality Axiom) $M {Λ} + M {Λ^{C}} = 1$ for any event Λ.

Axiom 3 (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, . . ., we have $M {⋃_{i = 1}^{\infty} Λ_{i}} ⩽ \sum_{i = 1}^{\infty} M {Λ_{i}} .$ (1)

Axiom 4 (Product Axiom) let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty spaces for k = 1, 2, . . .. The product uncertain measure $M$ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = {min}_{k = 1}^{\infty} M_{k} {Λ_{k}} .$ (2) where Λ_k are arbitrarily chosen events from $L_{k}$ for k = 1, 2, . . ., respectively.

Definition 2.1. (Liu [30]) Let Γ be a nonempty set, and $L$ a σ-algebra over Γ, and $M$ an uncertain measure. Then the triplet $(Γ, L, M)$ is called an uncertainty space.

Definition 2.2. (Liu [30]) An uncertain variable ξ is a measurable function from an uncertainty space $(Γ, L, M)$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ (3) is an event.

Definition 2.3. (Liu [30]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ ⩽ x}$ (4) for any real number x.

Example 2.1. An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution $Φ (x) = {\begin{matrix} 0, & if x ⩽ a \\ \frac{x - a}{2 (b - a)}, & if a < x ⩽ b \\ \frac{(x + c - 2 b)}{2 (c - b)}, & if b < x ⩽ c \\ 1 & if x > c \end{matrix}$ (5) denoted by Z (a, b, c) where a, b, c are real numbers with a < b<c. The figure of Z (a, b, c) is shown as follows:

Fig. 1

Zigzag Uncertainty Distribution.

Definition 2.4. (Liu [32]) The uncertain variables ξ₁, ξ₂, . . . ξ_n are said to be independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = {min}_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ (6) for any Borel sets B₁, B₂, . . . , B_n.

Definition 2.5. (Liu [30]) An uncertainty distribution Φ (x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1 .$ (7)

Definition 2.6. (Liu [30]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) (α ∈ (0, 1)) is called the inverse uncertainty distribution of ξ.

Example 2.2. The inverse uncertainty distribution of zigzag uncertain variable Z (a, b, c) is $Φ^{- 1} (α) = {\begin{matrix} (1 - 2 α) a + 2 α b, & if α < 0.5 \\ (2 - 2 α) b + (2 α - 1) c, & if α ⩾ 0.5 . \end{matrix}$ (8)

Theorem 2.1. (Liu [30]) Let ξ₁, ξ₂, . . . ξ_n be independent uncertain variables with regular uncertainty distribution Φ₁, Φ₂, . . . Φ_n, respectively. If f is a strictly increasing function, then the uncertain variable ξ = f (ξ₁, ξ₂, . . . , ξ_n) has an inverse uncertainty distribution $Φ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{n}^{- 1} (α)) .$ (9)

Theorem 2.2. (Liu [30]) Let ξ₁, ξ₂, . . . , ξ_n be independent uncertain variables with regular uncertainty distribution Φ₁, Φ₂, . . . Φ_n, respectively. If f is a strictly decreasing function, then the uncertain variable ξ = f (ξ₁, ξ₂, . . . , ξ_n) has an inverse uncertainty distribution

$\begin{matrix} Φ^{- 1} (α) = & f (Φ_{1}^{- 1} (1 - α), Φ_{2}^{- 1} (1 - α), \dots, \\ Φ_{n}^{- 1} (1 - α)) . \end{matrix}$ (10)

Theorem 2.3. (Liu [30]) Let ξ₁, ξ₂, . . . , ξ_n be independent uncertain variables with regular uncertainty distribution Φ₁, Φ₂, . . . Φ_n, respectively. If the function f (ξ₁, ξ₂, . . . , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, . . . , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, . . . , ξ_n, then the uncertain variable ξ = f (ξ₁, ξ₂, . . . , ξ_n) has an inverse uncertainty distribution

$\begin{matrix} Φ^{- 1} (α) = & f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$ (11)

3 The establishment of evaluation system

Let us consider a weapon system made up of n components, each one performing a given function. As we know, all the components of the equipment possess a certain failure rate. Those spares that play important roles in normal function should be configured for emergency use. But from the overall statistical analysis, we would find that not so many spares on the equipment need to be reserved during a long period. Generally speaking, most of the products do not malfunction, and the occurrence of failure parts only account for a small part of the components. This phenomenon not only reflects the contingency of the failure, but also implies that the failure has great relationship with the using environment, operational time, equipment availability and so on. Whether sufficient spares are available to restore the fight capacity is our most pressing problem. So the target of the analysis is to establish an optimal spare parts allocation in terms of which spares have to be kept in storage.

Due to the uncertainty of the failure, it becomes particularly difficult to confirm the required spares varieties at present. We extract a number of parameters from every kind of influencing factors, and they are regarded as evaluation parameters to confirm the spares varieties. The factors are identified in four respects: operation, design, maintenance & supportability and expense. For the purpose of avoiding the redundancy among all the parameters, we extract some direct parameters. The details are shown in Table 1.

Table 1
The influencing factors and parameters that determine spares varieties

Influencing factors Parameters Direct parameters

Operation Using environment Using environment

Operational readiness The time before executing next task(TAT)

Time between failures Time between failures

The number of equipment Working hours

The number of equipment

Design Reliability Single installation number

Maintenance Failure consequence

Fault severity Replacing ability

Single installation number

Replacing ability

Maintenance &support Repair level Maintenance time

Maintenance time Lead time

Equipment availability Support delay time

Support delay time The number of suppliers

Standard part or not Standard part or not

Support probability

Lead time

The number of suppliers

Expense Acquisition cost Cost

holding cost

Shortage cost

Influencing factors	Parameters	Direct parameters
Operation	Using environment	Using environment
	Operational readiness	The time before executing next task(TAT)
	Time between failures	Time between failures
	The number of equipment	Working hours
		The number of equipment
Design	Reliability	Single installation number
	Maintenance	Failure consequence
	Fault severity	Replacing ability
	Single installation number
	Replacing ability
Maintenance &support	Repair level	Maintenance time
	Maintenance time	Lead time
	Equipment availability	Support delay time
	Support delay time	The number of suppliers
	Standard part or not	Standard part or not
	Support probability
	Lead time
	The number of suppliers
Expense	Acquisition cost	Cost
	holding cost
	Shortage cost

Some of the above parameters which derive from the four influencing factors are qualitative indicators, such as using environment, failure consequence, replacing ability, standard part or not. These qualitative indicators cannot be received directly. So there are some concrete scalars to express various performance levels. The regulations for these qualitative indicators are shown in Table 2.

Table 2

Evaluation criterion of qualitative parameters

parameter\Score	1	3	5	7	9
Using environment	Particularly harsh environment	Sever environment	General environment	Mild environment	Gentle environment
Fault consequence	Major consequence	Between major C. and serious C.	Serious consequence	Between serious C. and general C.	General consequence
Replacing ability	Cannot be replaced	Can be replaced
Standard part or not	Non-standard part	Standard part

The others are quantitative parameters. Usually, if there are enough historical data, we could use the probability distributions to reflect the characteristic of dynamic parameters. When there are no enough samples, they must be evaluated by the experts and regarded as uncertain variables. In this complex evaluation system, the quantitative parameters are divided into two categories: uncertain parameters and constant parameters, their details are in Table 3. Then we will employ the uncertain variable to model the system.

Table 3

Uncertain parameters and constant parameters

Sign	uncertain parameters	Sign	constant parameters
T	TAT	N	Single installation number
F	Time between failures	Z	The number of equipment
W	Working hours	L	Lead time
M	Maintenance time	S	The number of suppliers
D	Support delay time	C	Cost

4 Modeling USOM based on uncertain DEA

DEA method works by dividing indices in indexing system into inputs and outputs. Because inputs and outputs are difficult to measure in an accurate way, many researchers proposed to model DEA with different theories, such as probability theory [33], chance-constrained programming [34 –36], fuzzy theory [37], uncertainty theory [38]. In this paper, we originally consider an optimization model that contains uncertain variables for the optimization of spares varieties. We develop the basic model for a relatively simple and representative structure—a two-site operating organization, which is formed by bases and sites. In addition, some limitations in actual conditions are ignored in studies. So some assumptions are put forward before modeling:

Spare parts are divided into standard parts and non-standard parts.

All the uncertain parameters subject to uncertain distributions.

All the items are repairable. In large industrial enterprises, most of the items are expensive and durable, the repairable items are more important than consumable or non-repairable items, so we make this simplification.

All the maintenance tasks can be done at bases or depots.

In addition, assuming that there are n DMUs, the relative symbols and notations are introduced as follows:

DMU_k: the kth DMU, k = 1,2, . . . , n;

DMU₀: target DMU;

${\tilde{T}}_{k}$ : the uncertain TAT of DMU_k;

${\tilde{F}}_{k}$ : the uncertain time between failures of DMU_k;

${\tilde{W}}_{k}$ : the uncertain working hours of DMU_k;

${\tilde{M}}_{k}$ : the uncertain maintenance time of DMU_k;

${\tilde{D}}_{k}$ : the uncertain support delay time of DMU_k;

S_k: the number of suppliers of DMU_k;

E_k: the using environment of DMU_k;

G_k: the failure consequence of DMU_k;

A_k: the replacing ability of DMU_k;

P_k: standard part or not of DMU_k;

N_k: the single installation number of DMU_k;

Z_k: the number of equipment of DMU_k;

L_k: the lead time of DMU_k;

C_k: the cost of DMU_k;

Φ_{T
_k}: the uncertainty distribution of ${\tilde{T}}_{k}$ ;

Ψ_{F
_k}: the uncertainty distribution of ${\tilde{F}}_{k}$ ;

Φ_{W
_k}: the uncertainty distribution of ${\tilde{W}}_{k}$ ;

Ψ_{M
_k}: the uncertainty distribution of ${\tilde{M}}_{k}$ ;

Ψ_{D
_k}: the uncertainty distribution of ${\tilde{D}}_{k}$ ;

$M$ : the uncertain measure;

α: the belief degree between 0 and 1;

λ_k: the weight of DMU_k;

$s_{i}^{-}$ : the slack of each ith input;

$s_{j}^{+}$ : the slack of each jth output.

DEA method can be regarded as a production process with multiple inputs and outputs. As is known, all the manufactures hope to produce maximum outputs with the least inputs. Therefore, this principle is also reflected in DEA model. The DMU will be more effective than other DMUs if it has a smaller input as well as a larger output. Various spare parts are regards as DMUs in our case, which should be evaluated. An index should be classified as an input if more attention need to be paid to make it smaller. Conversely an index should be selected as an output, if the larger the index is, the more important the demand is. For example, the number of suppliers is regarded as input, since more attentions have to be paid to parts with fewer suppliers. Lead time is considered as an output, it will take a long time to be received when a part has a longer lead time, which reduces the availability of the systems. Similarly, other indices can be classified in accordance with the principle. As a result, the inputs vector and outputs vector are: $\begin{matrix} X_{k} = {{\tilde{T}}_{k}, {\tilde{F}}_{k}, S_{k}, E_{k}, G_{k}, A_{k}, P_{k}}, k = 1, 2, . . ., n . \\ Y_{k} = {{\tilde{W}}_{k}, N_{k}, Z_{k}, {\tilde{M}}_{k}, L_{k}, {\tilde{D}}_{k}, C_{k}}, k = 1, 2, . . ., n . \end{matrix}$

Then the USOM is: ${\begin{matrix} max θ = \sum_{i = 1}^{7} s_{i}^{-} + \sum_{j = 1}^{7} s_{j}^{+} \\ subject to : \\ M {\sum_{k = 1}^{n} X_{ki} λ_{k} ⩽ X_{0 i} - s_{i}^{-}} ⩾ α, i = 1, 2 \\ \sum_{k = 1}^{n} X_{ki} λ_{k} ⩽ X_{0 i} - s_{i}^{-}, i = 3, 4, 5, 6, 7 \\ M {\sum_{k = 1}^{n} Y_{kj} λ_{k} ⩾ Y_{0 j} + s_{j}^{+}} ⩾ α, j = 1, 4, 6 \\ \sum_{k = 1}^{n} Y_{kj} λ_{k} ⩾ Y_{0 j} + s_{j}^{+}, j = 2, 3, 5, 7 \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} ⩾ 0, k = 1, 2, . . ., n \\ s_{i}^{-} ⩾ 0, i = 1, 2, . . ., 7 \\ s_{j}^{+} ⩾ 0, j = 1, 2, . . ., 7 \end{matrix}$ (12)

where $X_{k 1} = {\tilde{T}}_{k}$ , $X_{k 2} = {\tilde{F}}_{k}$ , X_k3 = S_k, X_k4 = E_k, X_k5 = G_k, X_k6 = A_k, X_k7 = P_k, $Y_{k 1} = {\tilde{W}}_{k}$ , Y_k2 = N_k, Y_k3 = Z_k, $Y_{k 4} = {\tilde{M}}_{k}$ , Y_k5 = L_k, $Y_{k 6} = {\tilde{D}}_{k}$ , Y_k7 = C_k.

Ranking criterion: The closer to 0 the θ is, the more efficient the DMU₀ is ranked.

The θ is the sum of all the input slacks and output slacks for the target DMU. θ should be close to 0, either the inputs are small, or the outputs are larger, or both. Thus the closer to 0 the objective is, the greater potential the DMU is to be reserved. Next we need to evaluate all the DMUs by solving (12), and then we can rank all the parts according to the ranking criterion. Finally, a reasonable spares scheme can be made by considering the actual capital situation.

The above uncertain spares optimization model, which is difficult to solve by traditional methods, is a nonlinear programming model. When inputs and outputs are uncertain variables, the uncertain model can be simplified to an equivalent deterministic model by following Liu’s uncertainty theory. All the inputs and outputs that have been illustrated in Section 3 are independent from each other. Now assuming the uncertainty distributions of ${\tilde{T}}_{k}, {\tilde{F}}_{k}, {\tilde{W}}_{k}, {\tilde{M}}_{k}, {\tilde{D}}_{k} (k = 1, 2, . . ., n)$ are Φ_{T
_k}, Φ_{F
_k}, Ψ_{W
_k}, Ψ_{M
_k}, Ψ_{D
_k}, and the inverse functions of these uncertain variables are $Φ_{T_{k}}^{- 1}, Φ_{F_{k}}^{- 1}, Ψ_{W_{k}}^{- 1}, Ψ_{M_{k}}^{- 1}, Ψ_{D_{k}}^{- 1}$ .

Theorem 4.1. Assume ${\tilde{T}}_{1}, {\tilde{T}}_{2}, . . ., {\tilde{T}}_{n}$ are independent uncertain variables defined on uncertain space $(Γ, L, M)$ , in which ${\tilde{T}}_{0}$ is any variable belongs to ${\tilde{T}}_{1}, {\tilde{T}}_{2}, . . ., {\tilde{T}}_{n}$ . The uncertainty distributions of ${\tilde{T}}_{1}, {\tilde{T}}_{2}, . . ., {\tilde{T}}_{n}$ are Φ₁, Φ₂, . . . Φ_n. Then $M {\sum_{k = 1}^{n} {\tilde{T}}_{k} λ_{k} ⩽ {\tilde{T}}_{0} - s_{1}^{-}} ⩾ α$ (13) holds if and only if

$\sum_{k = 1, k \neq 0}^{n} λ_{k} Φ_{k}^{- 1} (α) + λ_{0} Φ_{0}^{- 1} (1 - α) ⩽ Φ_{0}^{- 1} (1 - α) - s_{1}^{-} .$ (14)

Proof. We consider the equation $M {\sum_{k = 1}^{n} {\tilde{T}}_{k} λ_{k} ⩽ {\tilde{T}}_{0} - s_{1}^{-}} ⩾ α$ (15)

Equation (15) can be rewrite as $M {\sum_{k = 1, k \neq 0}^{n} {\tilde{T}}_{k} λ_{k} - (1 - λ_{0}) {\tilde{T}}_{0} ⩽ - s_{1}^{-}} ⩾ α .$ (16)

Because $- (1 - λ_{0}) {\tilde{T}}_{0}$ is an uncertain variable which is decreasing with respect to ${\tilde{T}}_{0}$ , its inverse uncertainty distribution is $ϒ_{T_{0}}^{- 1} (α) = - (1 - λ_{0}) Φ_{0}^{- 1} (1 - α), 0 < α < 1 .$ (17)

For each 1 ⩽ k ⩽ n and k ≠ 0, ${\tilde{T}}_{k} λ_{k}$ is an uncertain variable which is increasing with respect to ${\tilde{T}}_{k}$ , its inverse uncertainty distribution is $ϒ_{T_{k}}^{- 1} (α) = λ_{k} Φ_{k}^{- 1} (α), 0 < α < 1 .$ (18)

It follows from the operational law that the inverse uncertainty distribution of the sum $\sum_{k = 1, k \neq 0}^{n} {\tilde{T}}_{k} λ_{k} - (1 - λ_{0}) {\tilde{T}}_{0}$ is

$\begin{matrix} ϒ^{- 1} (α) & = \sum_{k = 1}^{n} ϒ_{T_{k}}^{- 1} (α) \\ = \sum_{k = 1, k \neq 0}^{n} λ_{k} Φ_{k}^{- 1} (α) - (1 - λ_{0}) Φ_{0}^{- 1} (1 - α), \\ 0 < α < 1 . \end{matrix}$ (19)

Similarly, we may derive the result immediately for the uncertain variables, thus getting the other equivalent constraints. Then the USOM (12) can be converted to the crisp model as follows: ${\begin{matrix} max θ = \sum_{i = 1}^{7} s_{i}^{-} + \sum_{j = 1}^{7} s_{j}^{+} \\ subject to : \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} Φ_{X_{ki}}^{- 1} (α) + λ_{0} Φ_{X_{0 i}}^{- 1} (1 - α) \\ ⩽ Φ_{X_{0 i}}^{- 1} (1 - α) - s_{i}^{-}, i = 1, 2 \\ \sum_{k = 1}^{n} X_{ki} λ_{k} ⩽ X_{0 i} - s_{i}^{-}, i = 3, 4, 5, 6, 7 \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} Ψ_{Y_{kj}}^{- 1} (1 - α) + λ_{0} Ψ_{Y_{0 j}}^{- 1} (α) \\ ⩾ Ψ_{Y_{0 j}}^{- 1} (α) + s_{j}^{+}, j = 1, 4, 6 \\ \sum_{k = 1}^{n} Y_{kj} λ_{k} ⩾ Y_{0 j} + s_{j}^{+}, j = 2, 3, 5, 7 \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} ⩾ 0, k = 1, 2, \dots, n \\ s_{i}^{-} ⩾ 0, i = 1, 2, \dots, 7 \\ s_{j}^{+} ⩾ 0, j = 1, 2, \dots, 7 \end{matrix}$ (20) which is a linear programming, where $X_{k 1} = {\tilde{T}}_{k}$ , $X_{k 2} = {\tilde{F}}_{k}$ , X_k3 = S_k, X_k4 = E_k, X_k5 = G_k, X_k6 = A_k, X_k7 = P_k, $Y_{k 1} = {\tilde{W}}_{k}$ , Y_k2 = N_k, Y_k3 = Z_k, $Y_{k 4} = {\tilde{M}}_{k}$ , Y_k5 = L_k, $Y_{k 6} = {\tilde{D}}_{k}$ , Y_k7 = C_k. Thus it can be easily solved by many traditional methods [39, 40]. Particularly, when all the independent uncertain variables follow zigzag distributions $({\tilde{T}}_{k} \sim Z (a_{T_{k}}, b_{T_{k}}, c_{T_{k}}))$ respectively, the Equation (20) can be simplified as:

${\begin{matrix} max θ = \sum_{i = 1}^{7} s_{i}^{-} + \sum_{j = 1}^{7} s_{j}^{+} \\ subject to : \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} [(2 - 2 α) b_{X_{ki}} + (2 α - 1) c_{X_{ki}}] \\ + λ_{0} [(1 - 2 α) a_{X_{0 i}} + 2 α b_{X_{0 i}}] \\ ⩽ [(1 - 2 α) a_{X_{0 i}} + 2 α b_{X_{0 i}}] - s_{i}^{-}, i = 1, 2 \\ \sum_{k = 1}^{n} X_{ki} λ_{k} ⩽ X_{0 i} - s_{i}^{-}, i = 3, 4, 5, 6, 7 \\ \sum_{k = 1, k \neq 0}^{n} λ_{k} [(1 - 2 α) a_{Y_{kj}} + 2 α b_{Y_{kj}}] \\ + λ_{0} [(2 - 2 α) b_{Y_{0 j}} + (2 α - 1) c_{Y_{0 j}}] \\ ⩾ [(2 - 2 α) b_{Y_{0 j}} + (2 α - 1) c_{Y_{0 j}}] + s_{j}^{+}, j = 1, 4, 6 \\ \sum_{k = 1}^{n} Y_{kj} λ_{k} ⩾ Y_{0 j} + s_{j}^{+}, j = 2, 3, 5, 7 \\ \sum_{k = 1}^{n} λ_{k} = 1 \\ λ_{k} ⩾ 0, k = 1, 2, \dots, n \\ s_{i}^{-} ⩾ 0, i = 1, 2, \dots, 7 \\ s_{j}^{+} ⩾ 0, j = 1, 2, \dots, 7 . \end{matrix}$ (21)

5 A numerical example

In order to illustrate the practicality of the presented model, this paper considers a supply support system serving for eight airplanes. Selecting ten LRUs as the analyzing objects of the part lists, we assume that these ten LRUs are in the working state all the time during the execution of the mission. According to the characteristic of each index, parameters can be classified into two categories: constant parameters (qualitative parameters and quantitative parameters) and uncertain parameters. The qualitative values can be obtained by experts brainstorming, and quantitative values are obtained by historical data, as is shown in Table 4. The uncertainty distribution function of uncertain variables can be obtained from experts, and we may invite some domain experts to evaluate the belief degree that each event will happen. The relative distributions which using mean values are shown in Table 5.

Table 4
Qualitative values

Spares number The number of suppliers Using environment The failure consequence The replacing ability Standard part or not The single installation number the number of equipment Lead time (day) Cost (million)

LRU 1 6 7 7 3 3 12 8 16 0.7

LRU 2 6 1 5 3 1 4 8 8 0.9

LRU 3 8 9 7 3 1 2 8 8 0.5

LRU 4 2 7 7 1 3 1 8 8 3.7

LRU 5 2 5 9 1 3 1 8 8 9.5

LRU 6 5 3 1 1 1 3 8 7 0.7

LRU 7 7 7 5 1 3 19 8 7 0.4

LRU 8 15 7 5 1 3 25 8 7 1.2

LRU 9 4 7 7 3 3 8 8 10 0.2

LRU 10 6 5 3 3 3 4 8 10 0.3

Spares number	The number of suppliers	Using environment	The failure consequence	The replacing ability	Standard part or not	The single installation number	the number of equipment	Lead time (day)	Cost (million)
LRU 1	6	7	7	3	3	12	8	16	0.7
LRU 2	6	1	5	3	1	4	8	8	0.9
LRU 3	8	9	7	3	1	2	8	8	0.5
LRU 4	2	7	7	1	3	1	8	8	3.7
LRU 5	2	5	9	1	3	1	8	8	9.5
LRU 6	5	3	1	1	1	3	8	7	0.7
LRU 7	7	7	5	1	3	19	8	7	0.4
LRU 8	15	7	5	1	3	25	8	7	1.2
LRU 9	4	7	7	3	3	8	8	10	0.2
LRU 10	6	5	3	3	3	4	8	10	0.3

Table 5

Distributions of uncertain variables

Spares number	TAT(h)	Time between failures(h)	Working hours(h)	Maintenance time(h)	The support delay time(h)
LRU 1	Z(10.22, 11, 11.78)	Z(48904, 49103, 50122)	Z(58970, 60724, 61067)	Z(13.1, 14.5, 16.2)	Z(6.53, 7.3, 7.83)
LRU 2	Z(7.24, 7.9, 8.46)	Z(131087, 134081, 134795)	Z(150900, 159089, 160897)	Z(20.9, 22.0, 23.2)	Z(9.39, 9.9,11.2)
LRU 3	Z(0.495, 0.5, 0.525)	Z(78029, 78495, 79147)	Z(229867, 239800, 240959)	Z(15.8, 17.6, 19.7)	Z(3.59, 3.7, 3.87)
LRU 4	Z(19.72, 20, 20.68)	Z(48790, 50466, 51289)	Z(676550, 678006, 679459)	Z(3.06, 3.5, 3.93)	Z(10.3, 11.4, 12.6)
LRU 5	Z(4.35, 4.5, 4.72)	Z(153034, 154077, 159133)	Z(149660, 149800, 149987)	Z(10.2, 11.0, 11.8)	Z(7.21, 7.8, 8.56)
LRU 6	Z(2.63, 2.7, 2.75)	Z(69378, 70290, 71356)	Z(130065, 132079, 134017)	Z(2.36, 2.5, 2.66)	Z(10.9, 12.1, 13.8)
LRU 7	Z(5.85, 5.9, 5.92)	Z(125728, 130839, 135675)	Z(4752680, 4765000, 4788507)	Z(9.70, 10.5, 11.65)	Z(13.5, 15.6, 16.6)
LRU 8	Z(17.8, 18.6, 19.3)	Z(92135, 93675, 94296)	Z(158630, 158900, 159083)	Z(15.5, 16.7, 17.9)	Z(12.7, 13.6, 14.8)
LRU 9	Z(1.86, 1.9, 1.98)	Z(164590, 165009, 167342)	Z(356220, 367284, 378264)	Z(16.3, 17.9, 19.6)	Z(7.6, 8.7, 9.5)
LRU 10	Z(1.73, 1.8, 1.87)	Z(132840, 134200, 134507)	Z(500183, 529099, 545l57)	Z(4.28, 4.5, 4.87)	Z(5.24, 5.3, 5.66)

Table 6 shows the results of evaluating all the DMUs by uncertain spares optimization model in section 4 when we set the confidence level α = 0.8. According to ranking criterion, when α = 0.8, the DMUs can be ranked as follows: DMU2, DMU4, DMU5, DMU7, DMU8, DMU9, DMU10, DMU1, DMU6, DMU3. Among these, DMU2, DMU4, DMU5, DMU7, DMU8, DMU9 and DMU10 are efficient. This phenomenon indicates that there are about 70% percent of the DMUs are effective, it will not be an efficient scheme when reserving so many kinds of spares varieties.

Table 6

Results of evaluating the DMUs with α= 0.8

	DMU1	DMU2	DMU3	DMU4	DMU5	DMU6	DMU7	DMU8	DMU9	DMU10
θ	6.6983*10⁴	0	4.4212*10⁵	0	0	6.8856*10⁴	0	0	0	0

We have considered all the possible influencing factors, but results showed that seven LRUs are of equal importance. Therefore, another ranking order based on the results showed above is proposed. Usually a higher value of component leads to a higher cost, it is unnecessary to store an expensive component in inventory. So we give the second ranking criterion based on unit costs. Then the new sequence is as follows: DMU9, DMU10, DMU7, DMU2, MU8, DMU4, DMU5, DMU1, DMU6, DMU3. Obviously, DMU9 has the highest priority to be stored. Then the company can generate inventory strategy according to ranking order with the limitations of specific funding considered.

6 Conclusion

The contributions of this paper are summarized as follows: (a) We extracted a reasonable evaluation system which considers both redundancy and comprehensiveness. (b) The uncertain spares optimization model has been established based on uncertainty DEA under uncertainty environment. (c) The USOM has be converted to a linear programming when the inputs and outputs have some particularly characters. Especially, the simplified model was presented when uncertain variables obey zigzag distributions. (d) A ranking criterion according to cost in the numerical example has been given when multiple effective DMUs exist simultaneously. The timely supply of spare parts is the key to ensure the equipment system to recover the readiness state efficiently. Therefore, the management of spare parts have both advantages of saving resources and prompting continuous operation. However, there is room for improvement. In the future, the sensitivity and stability of the model need to be further studied. Moreover, this paper cannot handle with conditions where both aleatory uncertainty and epistemic uncertainty exist. In future research, uncertain random variables are needed to introduced into USOM to deal with this situation.

Compliance with ethical standards

Funding

This study was funded by National Natural Science Foundation of China (Nos.71671009, 61871013, 61573041 and 61573043).

Conflict of interest

Author Meilin Wen declares that she has no conflict of interest. Author Yubing Chen declares that he has no conflict of interest. Author Yi Yang declares that she has no conflict of interest. Author Rui Kang declares that he has no conflict of interest. Author Miaomiao Guo declares that she has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

Joanne

, Maintenance spare parts logistics: Special characteristics and strategic choices, International Journal of Production Economics 71 (2001), 125–133, 2001.

Siddiqi

and De Weck

OL.

, Spare. Parts Requirements for Space Missions with Reconfigurability and Commonality, Journal of Spacecraft and Rockets 44 (2007), 147–155.

Nahmias

, Production & Operations Analysis, McGraw-Hill 2008.

Duchessi

, Tayi

G.K.

and Levy

J.B.

, A conceptual approach for managing of spare parts, International Journal of Physical Distribution & Materials Management 18 (1988), 8–15.

Zhou

and Fan

, A note on multi-criteria ABC inventory classification using weighted linear optimization, European Journal of Operational Research 182 (2007), 1488–1491.

W.L.

, A simple classifier for multiple criteria ABC analysis, European Journal of Operational Research 177 (2007), 344–353.

Petrovic

, et al., SPARTA II: Further development in an expert system for advising on stocks of spare parts, International Journal of Production Economics 24 (1992), 291–300.

Krupp

J.A.G.

, Are ABC codes an obsolete Technology?, APICS–The Performance Advantage 4 (1994), 34–35.

Rad

, Shanmugarajan

, Wahab

M.I.M.

, Classification of critical spares for aircraft maintenance, 8th International Conference on Service Systems and Service Management-Proceedings of ICSSSM’11, 2011.

10.

Zhang

Z.Y.

, Kang

, Qv

L.L.

, Wang

X.W.

, Method of Determining Spares Varieties Based on AHP and DEA, 8th International Conference on Reliability, Maintainability & Safety, 2009.

11.

Fuller

J.B.

, O’Connor

and Rawlinson

, Tailored: The next advantage, Harvard Business Review 71 (1993), 87–98.

12.

Hadi-Vencheh

, An improvement to multiple criteria ABC inventory classification, European Journal of Operational Research 201 (2010), 962–965.

13.

Teunter

, Babai

and Syntetos

, ABC classification: Service levels and inventory costs, Production and Operations Management 19 (2010), 343–352.

14.

Gajpal

P.P.

, Ganesh

L.S.

and Rajendran

, Criticality analysis of spare parts using the analytic hierarchy process, International Journal of Production Economics 35 (1994), 293–297.

15.

Guvenir

H.A.

and Erel

, Multi-criteria inventory classification using a genetic algorithm, European Journal of Operational Research 105 (1998), 29–37.

16.

Partovi

F.Y.

and Anandarajan

, Classifying inventory using an artificial neural network approach, Computers and Industrial Engineering 41 (2002), 389–404.

17.

Charnes

, CooPer

W.W.

and Rhodes

, Measuring the Efficiency of Decision Making Units, European Journal of Operational Research 6 (1978), 429–444.

18.

Banker

R.D.

, Charnes

and Cooper

W.W.

, Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science 30 (1984), 1078–1092.

19.

Charnes

, Cooper

W.W.

and Li

, Using DEA to evaluate relative efficiencies in the economic performance of Chinese-key cities, Socio-Economic Planning Sciences 23 (1989), 325–344.

20.

Liu

, Song

, Xu

, et al., Group decision making based on DEA cross-efficiency with intuitionistic fuzzy preference relations, Fuzzy Optimization and Decision Making 1, 2018.

21.

Liu

, Xu

, Chen

, et al., Group decision making with interval fuzzy preference relations based on DEA and stochastic simulation, Neural Computing and Applications 2017.

22.

Balavand

, Kashan

A.H.

and Saghaei

, Automatic clustering based on Crow Search Algorithm-Kmeans (CSA-Kmeans) and Data Envelopment Analysis (DEA), International Journal of Computational Intelligence Systems 11 (2018), 1322–1337.

23.

Zheng

, Wang

Y.M.

and Zhang

, A case retrieval method combined with similarity measurement and DEA model for alternative generation, International Journal of Computational Intelligence Systems 11 (2018), 1123–1141.

24.

Cooper

W.W.

, Kyung

S.P.

and Gang

, Idea and ar-idea: Models for dealing with imprecise data in dea, Management Science 45 (1999), 597–607.

25.

Wen

M.L.

and Li

H.S.

, Fuzzy data envelopment analysis (DEA): Model and ranking method, method, Journal of Computational and Applied Mathematics 223 (2009), 872–878.

26.

Zhou

, Pedrycz

, Kuang

, et al., Type-2 fuzzy multi-objective dea model: An application to sustainable supplier evaluation, Applied Soft Computing, S1568494616301946, 2016.

27.

Hatami-Marbini

, Agrell

P.J.

, Fukuyama

, et al., The role of multiplier bounds in fuzzy data envelopment analysis, Annals of Operations Research 250 (2017), 249–276.

28.

Regattieri

, Gamberi

, Gamberini

and Manzini

, Managing lumpy demand for aircraft spare parts, J Air Transp Manag 11 (2005), 426–431.

29.

Godoy

D.R.

, Rodrigo

and Knights

, Critical spare parts ordering decisions using conditional reliability and stochastic lead time, Reliability Engineering & System Safety 119 (2013), 199–206.

30.

Liu

, Uncertainty theory, 2nd Edition, Springer-Verlag, Berlin, 2007.

31.

Liu

, Uncertainty theory, 4th Edition, Springer-Verlag, Berlin, 2015.

32.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

33.

Sengupta

J.K.

, Efficiency measurement in stochastic input-output systems, International Journal of Systems Science 13 (1982), 273–287.

34.

Olesen

and Petersen

N.C.

, Chance constrained efficiency evaluation, Management Science 141 (1995), 442–457.

35.

Cooper

W.W.

, Huang

Z.M.

and Li

S.X.

, Satisfying DEA models under chance constraints, Annals of Operations Research 66 (1996), 279–295.

36.

Cooper

W.W.

, Huang

Z.M.

, Lelas

, et al., Chance constrained programming formulations for stochastic characterizations of efficiency and dominance in DEA, Journal of Productivity Analysis 9 (1998), 53–79.

37.

Kao

and Liu

S.T.

, Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems 119 (2000), 149–160.

38.

Wen

M.L.

, Guo

L.H.

, Kang

, Yang

, Data envelopment analysis with uncertain inputs and outputs, Journal of Applied Mathematics, 2014.

39.

Zhou

X.G.

, Cao

B.Y.

and Wu

, Global optimization method for linear multiplicative programming, Acta Mathematicae Applicatae Sinica, English Series 3 (2015), 325–334.

40.

Zuo

X.D.

, Xue

S.J.

and Luo

, Lei and A Way. to Find All the Optimal Solutions in Linear Programming, Journal of Systems Engineering and Electronics 11 (2000), 11–16.