Reliability-oriented extended importance measures in uncertain inventory systems for mechanical products

Abstract

This article mainly analyzes the applications of importance measures in inventory systems’ reliability. Previous importance measures are mostly developed based on Birnbaum importance; however, they are not applicable to the identification of weakness and optimization in inventory systems by considering the customer requirements for mechanical products. Based on the three-echelon inventory system structure, the corresponding reliability is analyzed under the Poisson distribution. Then three importance measures, namely, composite importance measure, mean absolute deviation, and differential importance measure, are extended to the three-echelon inventory systems. By calculating and comparing the importance values of various components, the key parameters in inventory systems can be identified to achieve the minimal cost and optimal inventory system. Finally, a numerical example is used to demonstrate the proposed method.

Keywords

Reliability importance measure inventory system cost component

Introduction

The inventory optimization has always been the focus for enterprise managers. It has achieved the minimal cost and efficiency optimization. However, research mostly focuses on two-echelon inventory systems and neglects the application of importance measures in the inventory system, which represent an important tool in reliability engineering to identify the weakness and optimize the system performance. Nowadays, the study about the application of the system reliability and importance measures in an inventory system has made certain achievements.

Almaktoom¹ introduced a method of quantifying the reliability of an inventory management system. Also, a novel, reliability-based robust design optimization model has been developed to optimally allocate and schedule time while considering uncertainty associated with inventory movement. Jin et al.² proposed an integrated product–service model to ensure the system availability by concurrently allocating reliability, redundancy, and spare parts for a variable fleet. To reduce the additional rework cost, Lukinskiy and Lukinskiy³ proposed evaluation of stock management strategies’ reliability at dependent demand. In this article, there is a suggested methodical approach that allows calculating the safety stock quantity at the dependent demand in view of supply chain’s reliability requirements. Manna et al.⁴ focused on an imperfect production inventory model with production system reliability under two-layer supply chain management. Meyer et al.⁵ analyzed the reliability of a single-stage production-storage system in meeting a constant known demand. Considering reliability as a decision variable, Sarkar⁶ constructed an integrated profit function which is maximized by control theory. Selçuk and Ağral⁷ jointly model the reliability and inventory problems to minimize the sum of holding and emergency shipment costs subject to a limited reliability improvement budget and a target service level. Shah and Vaghela⁸ incorporated reliability as a decision variable in the model to maximize the profit function with respect to the reliability parameter.

Based on the differences in reliability evaluation,^9–12 the importance measures have been used to evaluate the effect of components on the system reliability. For example, Borgonovo^13,14 introduced a new sensitivity measure by relating the differential importance measure (DIM) and applied it to the economic order quantity (EOQ) inventory model. Ramirez-Marquez and Coit¹⁵ proposed the composite importance measure (CIM) and mean absolute deviation (MAD) for multi-state systems with multi-state components. Nguyen et al.¹⁶ presented a joint predictive maintenance and inventory strategy for systems with complex structure and multiple non-identical components. Considering the transition rates of component states, Si et al.¹⁷ presented the definition, evaluation method, and characteristics of integrated importance measures of component states. For the semi-Markov processes and renewal function, Dui et al.^18–21 discussed the influence of importance measure on system performance and its applications in aviation and other fields.

The applications of the system reliability and importance measures in the inventory models have gained some attention. However, they are used little to optimize the multiple-level inventory systems, especially the three-level inventory systems. Hence, combining the customer requirements under the Poisson distribution and three-level inventory system structure, this article extends the CIM, MAD, and DIM to applications in the three-level inventory systems.

The rest of the article is organized as follows. Section “Reliability analysis in inventory systems” analyses reliability in inventory systems. The inventory models and formulas of reliability in three-echelon inventory are briefly described in the same section. Section “Importance measures for inventory systems” derives three importance measures: CIM, MAD, and DIM, and the computational methods about various parameters of the cost function. An application is presented to illustrate the importance values and changes with parameters in section “Numerical example.” Section “Conclusion and future work” gives the conclusions of this article.

Reliability analysis in inventory systems

Inventory systems

Assume that there is a three-echelon inventory system for mechanical products, which contains m suppliers, n distribution centers, and p retailers, as shown in Figure 1.

Figure 1.

A three-level supply chain network.

In Figure 1, supplier i represents one of m suppliers. Similarly, distribution center j represents one of n suppliers and retailer q represents one of p retailers. Then we can obtain the corresponding reliability block diagram, as shown in Figure 2.

Figure 2.

Reliability block diagram.

In Figure 2, S_i represents one of m suppliers, D_j represents one of n suppliers, and R_k represents one of p retailers.

The inventory control policy of the supplier is (Q, s), in which if the stock is less than s, we will order Q quantity. The control policy of distribution center i is (S_i, s_i), in which if the stock of distribution center i is less than s_i, we will order S_i – s_i quantity. We have the following assumptions:

The customers’ demand for mechanical products meets an independent Poisson distribution. The demand intensity of the Poisson distribution of the distribution center i is $λ_{i}$ .

i_k is a safety stock factor of the distribution center.

Holding cost per item per unit time is C_h.

Ordering cost per order is C_o.

Order lead time is L.

Shortage cost due to no spare parts on stock is C_s.

Planning horizon is T.

Supplier cost

In the inventory control policy of the suppliers, using the linear regression model, we can obtain $y = Q, x_{i} = ((T - L) / T_{i}) (S_{i} - s_{i})$ . Then the multiple linear regression equation is

{\begin{matrix} Q_{1} = b_{0} + b_{1} x_{1} + \dots + b_{m} x_{m} \\ Q_{2} = b_{0} + b_{1} x_{1} + \dots + b_{m} x_{m} \\ ⋮ \\ Q_{m} = b_{0} + b_{1} x_{1} + \dots + b_{m} x_{m} \\ Q_{m + 1} = b_{0} + b_{1} x_{1} + \dots + b_{m} x_{m} \end{matrix}

According to the existing data, we can get the value of $b_{0}, b_{1}, \dots, b_{m}$ . Then $y = b_{0} + b_{1} x_{1} + b_{2} x_{2} + \dots + b_{m} x_{m}$ , where b₀ is a constant and $b_{1}, \dots, b_{m}$ are the regression coefficients. Let $y = s, x_{i} = (L / T_{i}) (S_{i} - s_{i})$ ; then we can get $s = c_{0} + c_{1} x_{1} + \dots + c_{n} x_{n}$ , where c₀ is a constant and $c_{1}, \dots, c_{n}$ are the regression coefficients.

Distribution center cost

Ordering cost

The ordering cost of distribution centers is the product of ordering cost per order and order number given as follows: $C_{io} (λ_{i} / (S_{i} - s_{i}))$ .

Holding cost

The holding cost per unit time of all distribution centers is $C_{ih} (((S_{i} + s_{i}) / 2) - λ_{i} L_{i})$ .

Shortage cost

The shortage cost of distribution centers is $C_{is} (λ_{i} / (S_{i} - s_{i})) \sum_{D_{i} L_{i} = s_{i}}^{+ \infty} (D_{i} L_{i} - s_{i}) f (D_{i} L_{i})$ .

Thus, the cost of distribution center is

\begin{matrix} C_{i} = C_{io} \frac{λ_{i}}{S_{i} - s_{i}} + C_{ih} (\frac{S_{i} + s_{i}}{2} - λ_{i} L_{i}) \\ + C_{is} \frac{λ_{i}}{S_{i} - s_{i}} \sum_{D_{i} L_{i} = s_{i}}^{+ \infty} (D_{i} L_{i} - s_{i}) f (D_{i} L_{i}) \end{matrix}

(1)

Taking the derivative of S_i and Q_i successively, we can get

Q_{i} = S_{i} - s_{i} = \sqrt{\frac{2 λ_{i} (C_{io} + C_{is} \sum_{D_{i} L_{i} = s_{i}}^{+ \infty} (D_{i} L_{i} - s_{i}) f (D_{i} L_{i}))}{C_{ih}}}

(2)

\sum_{D_{i} L_{i} = s_{i}}^{+ \infty} f (D_{i} L_{i}) = \frac{C_{ih}}{C_{is} \frac{λ_{i}}{S_{i} - s_{i}}} = \frac{C_{ih} (S_{i} - s_{i})}{C_{is} λ_{i}}

(3)

The calculation steps are as follows:

Calculate Q_i, as an estimated value, $Q_{i} = \sqrt{2 λ_{i} c_{i 0} / c_{ih}}$ ;

Substituting Q_i into formula (3), we can get s_i;

Substituting s_i into formula (2), we can calculate Q_i;

Perform the second and third steps repeatedly to determine the value of Q_i, until the two values of Q_i are approximately equal.

Reliability in three-level inventory systems

In inventory management, a supply chain system is required to supply the customers’ demand. The system reliability can be understood as the probability that the inventory of the supply chain system can meet a given customers’ demand. The demand may be fixed or vary depending on the conditions in different inventory cycles.

From Figure 2, we know that the system is composed of three subsystems: m suppliers, n distribution centers, and p retailers.

If the demand is constant, then the reliability of the inventory system is given by

R_{d} = \Pr (Φ (X) ⩾ d)

(4)

Assume that x_i is one of the suppliers (i = 1, 2, …, m) and the demand for mechanical products follows the Poisson distribution. Thus, the reliability of x_i is $R_{i} = \sum_{Q_{i}}^{\infty} λ^{k} e^{- λ} / k!$ and the reliability of a supplier is

R_{s_{i}} = 1 - Π_{i = 1}^{m} (1 - \sum_{Q_{i}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})

(5)

Assuming that x_j is one of the distribution centers (j = 1, 2, …, n), the reliability of a distribution center is

R_{d_{j}} = 1 - Π_{j = 1}^{n} (1 - \sum_{Q_{j}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})

(6)

Given that x_q is one of p retailers (q = 1, 2, …, p), the reliability of a retailer is

R_{r_{q}} = 1 - Π_{q = 1}^{p} (1 - \sum_{Q_{q}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})

(7)

Thus, the reliability of a three-echelon supply chain system is

R = R_{s} R_{d} R_{r} = [1 - Π_{i = 1}^{m} (1 - \sum_{Q_{i}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})] [1 - Π_{j = 1}^{n} (1 - \sum_{Q_{j}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})] [1 - Π_{q = 1}^{p} (1 - \sum_{Q_{q}}^{\infty} \frac{λ^{k} e^{- λ}}{k!})]

(8)

Importance measures for inventory systems

CIM for inventory systems

For a constant demand, the CIM generalization for Birnbaum importance measure can be expressed as

\begin{matrix} {BM}_{i}^{CIM} = \frac{\sum_{j = 1}^{w_{i}} | \Pr (Φ (X) ⩾ d | X_{i} = b_{ij}) - \Pr (Φ (X) ⩾ d) |}{w_{i} - 1} \\ = \frac{\sum_{j = 1}^{w_{i}} | \Pr (Φ (X) ⩾ Q - b_{ij}) - \Pr (Φ (X) ⩾ Q) |}{w_{i} - 1} \\ = \frac{\sum_{j = 1}^{w_{i}} | \sum_{Q - b_{ij}}^{\infty} \frac{λ^{k} e^{- λ}}{k!} - \sum_{Q}^{\infty} \frac{λ^{k} e^{- λ}}{k!} |}{w_{i} - 1} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q - b_{ij}}^{Q} \frac{λ^{k} e^{- λ}}{k!}}{w_{i} - 1} \end{matrix}

(9)

Based on the inventory systems, the CIMs of the supplier, distribution center, and retailer are

CI M_{S} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!}}{w_{i} - 1}

(10)

CI M_{d} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!}}{w_{i} - 1}

(11)

CI M_{r} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{q} - b_{ij}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!}}{w_{i} - 1}

(12)

For a varying demand $d_{m}$

\begin{matrix} {BM}_{i}^{CIM} = \frac{\sum_{j = 1}^{w_{i}} \sum_{w} | \Pr (Φ (X) ⩾ d_{w} | X_{i} = b_{ij}) - \Pr (Φ (X) ⩾ d_{w}) | {T'}_{w}}{w_{i} - 1} \\ = \frac{\sum_{j = 1}^{w_{i}} \sum_{w} | \Pr (Φ (X) ⩾ Q_{w} - b_{ij}) - \Pr (Φ (X) ⩾ Q_{w}) | {T'}_{w}}{w_{i} - 1} \\ = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{w} - b_{ij}}^{Q_{w}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}}{w_{i} - 1} \end{matrix}

(13)

where w_i is the number of the states of component i, b_ij is the demand or capacity corresponding to state j of component i, and $T'_{w}$ is given by $T'_{w} = T_{w} / \sum_{i = 1}^{k} T_{i}$ .

Based on the inventory systems, the CIMs of the supplier, distribution center, and retailer are

CI M_{s} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}}{w_{i} - 1}

(14)

CI M_{d} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}}{w_{i} - 1}

(15)

CI M_{r} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{q} - b_{ij}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}}{w_{i} - 1}

(16)

MAD for inventory systems

For a constant demand, the MAD is

\begin{matrix} I^{MAD} (i) = \sum_{j = 0}^{w_{i}} \Pr (X_{i} = b_{ij}) | \Pr \\ (Φ (X) ⩾ d | X_{i} = b_{ij}) - \Pr (Φ (X) ⩾ d) | \\ = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q - b_{ij}}^{Q} \frac{λ^{k} e^{- λ}}{k!} \end{matrix}

(17)

Based on the inventory systems, the MADs of the supplier, distribution center, and retailer are

I_{i}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!}

(18)

I_{j}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!}

(19)

I_{q}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{q} - b_{ij}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!}

(20)

2. For varying demand $d_{m}$

\begin{matrix} I^{MAD} (i) = \sum_{j = 0}^{w_{i}} \Pr (X_{i} = b_{ij}) \sum_{w} | \Pr (Φ (X) ⩾ d_{w} | X_{i} = b_{ij}) \\ - \Pr (Φ (X) ⩾ d_{w}) | {T'}_{w} \\ = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q - b_{ij}}^{Q} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w} \end{matrix}

(21)

Based on the inventory systems, the MADs of the supplier, distribution center, and retailer are

I_{i}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}

(22)

I_{j}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}

(23)

I_{q}^{MAD} = \sum_{j = 0}^{w_{i}} \frac{λ^{b_{ij}} e^{- λ}}{b_{ij}!} \sum_{Q_{q} - b_{ij}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!} {T'}_{w}

(24)

where $Q_{i}, Q_{j}, and Q_{q}$ are the supplies of the supplier, distribution center, and retailer during T_w, respectively; w_i is the number of states of component i; b_ij is the demand or capacity corresponding to state j of component i; and $T'_{w}$ is given by $T'_{w} = T_{w} / \sum_{i = 1}^{k} T_{i}$ .

DIM for inventory systems

The DIM generalization can be expressed as

\begin{matrix} {DIM}_{i}^{CIM} = \frac{{BM}_{i}^{CIM} d x_{i}}{\sum_{j} {BM}_{j}^{CIM} d x_{j}} \\ = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q - b_{ij}}^{Q} \frac{λ^{k} e^{- λ}}{k!} d x_{i}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q - b_{ij}}^{Q} \frac{λ^{k} e^{- λ}}{k!} d x_{i}} \end{matrix}

(25)

Then the DIM of every part is

DI M_{S} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!} d x_{i}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!} d x_{j}}

(26)

DI M_{d} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!} d x_{i}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!} d x_{j}}

(27)

DI M_{r} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{q} - b_{qj}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!} d x_{q}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q_{q} - b_{qj}}^{Q_{q}} \frac{λ^{k} e^{- λ}}{k!} d x_{j}}

(28)

When a small change is the same for different distribution centers x_i and x_j, $δ x_{i} = δ x_{j}$ , we have

{DIM}_{i}^{CIM} = \frac{{BM}_{i}^{CIM}}{\sum_{j} {BM}_{j}^{CIM}} = \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!}}

When the distribution centers x_i and x_j change by the same fraction of their nominal values, $δ x_{i} / x_{i} = δ x_{j} / x_{j}$ , we get

{DIM}_{i}^{CIM} = \frac{{BM}_{i}^{CIM} x_{i}}{\sum_{j} {BM}_{j}^{CIM} x_{j}} \frac{\sum_{j = 1}^{w_{i}} \sum_{Q_{i} - b_{ij}}^{Q_{i}} \frac{λ^{k} e^{- λ}}{k!} x_{i}}{\sum_{j} \sum_{j = 1}^{w_{i}} \sum_{Q_{j} - b_{ij}}^{Q_{j}} \frac{λ^{k} e^{- λ}}{k!} x_{j}}

When the metrics of distribution centers x_i and x_j are different, we can use $δ x_{i} / x_{i} = δ x_{j} / x_{j}$ to normalize their effects.

Numerical example

There is a three-level inventory, with three suppliers, two distribution centers, and three retailers, as shown in Figure 3. The parametric data of the supplier, distribution center, and retailer are shown in Table 1.

Figure 3.

A three-level inventory model.

Table 1.

Parametric data.

	Supplier					Distribution center					Retailer
	Q_i	λ		b_ij		Q_j	λ		b_ij		Q_q	λ		b_ij
1	6	5	6	1	2	12	9	10	2	2	12	10	11	3	1
2	9	6	7	1	3	14	11	12	2	3	15	12	12	3	2
3	10	8	9	2	3						18	15	14	3	3
Sum	25					26					45

Then we can calculate the values of reliability and importance, as shown in Tables 2 –4.

Table 2.

Results of reliability.

Constant demand								Varying demand
S			D		R			S			D		R
S ₁	S ₂	S ₃	D ₁	D ₂	R ₁	R ₂	R ₃	S ₁	S ₂	S ₃	D ₁	D ₂	R ₁	R ₂	R ₃
0.384	0.1527	0.2825	0.197	0.2187	0.3033	0.2279	0.2512	0.4862	0.2237	0.3606	0.2608	0.2242	0.374	0.2862	0.2198

Table 3.

The importance values for a constant demand.

CIM			MAD			DIM
S	D	R	S	D	R	S	D	R
0.8189	0.5768	0.6672	0.0527	0.0029	0.0036	0.1683	0.8862	0.3229
0.6425	0.669	0.6514	0.0445	0.0017	0.0008	0.5749	0.1138	0.376
0.8656		0.7076	0.0183		0.0001	0.256		0.301

CIM: composite importance measure; MAD: mean absolute deviation; DIM: differential importance measure.

Table 4.

The importance values for a varying demand.

CIM			MAD			DIM
S	D	R	S	D	R	S	D	R
0.7933	0.7061	0.6682	0.0361	0.0029	0.0002	0.052	0.5439	0.2692
0.6246	0.7088	0.6871	0.0353	0.0017	0.0012	0.4592	0.4561	0.3148
0.8751		0.6742	0.0129		0.0002	0.4889		0.4161

CIM: composite importance measure; MAD: mean absolute deviation; DIM: differential importance measure.

When the demand is a constant, let $T_{1} = 2, T_{2} = 3, T = 5$ , and we can get the importance values for a constant demand.

For a varying demand, let $T_{1} = 2, T_{2} = 3, T = 5$ , and we can get the importance values for a varying demand.

From Tables 2 –4, we can apparently see whether λ is a constant or not, and the results of reliability and importance are different. For example, in Table 2, for a constant demand, the order of retailers’ reliability is $R_{r 1} > R_{r 3} > R_{r 2}$ , while for a varying demand the order is $R_{r 1} > R_{r 2} > R_{r 3}$ . For a constant demand in Table 3, the order of retailers’ DIM is $DI M_{r 2} > DI M_{r 1} > DI M_{r 3}$ ; however, for a varying demand in Table 4, the order is $DI M_{r 3} > DI M_{r 2} > DI M_{r 1}$ .

In order to better analyze the dynamic effects of various parameter changes on CIM, MAD, and DIM, we set each parameter within an interval and analyze the changes when the parameters change in their intervals, while others remain constant. Then we get the graphs in which importance changes with a certain varying parameter, as shown in Figure 4.

Figure 4.

Importance degree changes with parameters: (a) importance degree changes with Q; (b) MAD changes with Q; (c) importance degree changes with b₁₁; and (d) importance degree changes with λ.

From Figure 4(a), we can see that CIM and DIM show similarities in the change law, and they all increase first and then decrease as Q increases. Because the value of MAD is too small to see its fluctuation, we specially depict MAD and Q, as shown in Figure 4(b). Apparently, its change law is similar to CIM and DIM.

Meanwhile, from Figure 4(a) and (b), we also know that the order of importance in general is $CIM > DIM > MAD$ . At the beginning, the values of CIM, MAD, and DIM are lower and overlap each other; then they increase at different speeds, the value of Q is 23 approximately, and the importance reaches the maximum.

In Figure 4(c), we can see that the value of importance is constantly increasing as b₁₁ increases. Before b₁₁ is around 9, they remain almost equal. Then the CIM grows at the fastest speed, and when b₁₁ is more than 28, the curve change is rather slow and tends to be stable. DIM maintains a steady change rate when b₁₁ is greater than or equal to 26. On the contrary, MAD tends to decrease after it achieves the maximum.

In Figure 4(d), when λ is increasing, the values of CIM, MAD, and DIM all climb up and then decline. The CIM and DIM reach the maximum when λ is equal to 16. MAD reaches its maximum when λ is 11, then it constantly descends and finally becomes stable.

Conclusion and future work

Based on the existing theoretical research results of importance measures and the inventory control models, the CIM, MAD, and DIM are introduced into a three-echelon inventory model. The main conclusions are as follows:

By comparing and analyzing the calculation results of importance values with the changes of various parameters, the importance grades of these parameters and the effect of each parameter on the system are identified.

An optimal inventory control and management are carried out to optimize the whole inventory system by identifying and evaluating the critical parameters in a three-level inventory control model.

This work can be extended to the analysis of integrated importance measure into inventory systems. The integrated importance measure is related to the transition rates of components. When considering the component failure rate or repair rate, we can use the measure to identify the most important component in the lifecycle for the inventory systems.

Footnotes

Appendix 1 Acknowledgements

H.D. and L.C. conceived and designed the experiments and proposed the idea of this paper; C.L. performed the experiments and analyzed the data. All authors have contributed to the editing and proofreading of this paper.

Handling Editor: Zhaojun Li

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the financial support for this research from the National Natural Science Foundation of China (Nos. 61807031, 61401403, and 71501173) and Scientific and Technological Research Project of Henan Province, China (No. 132102210560).

ORCID iD

Hongyan Dui

References

Almaktoom

AT.

Stochastic reliability measurement and design optimization of an inventory management system. Complexity 2017; 8: 1–9.

Jin

Taboada

Espiritu

et al . Allocation of reliability–redundancy and spares inventory under Poisson fleet expansion. IISE T 2017; 49: 737–751.

Lukinskiy

Evaluation of stock management strategies reliability at dependent demand. Transp Telecommun J 2017; 18: 53–56.

Manna

Dey

Mondal

SK.

Two layers supply chain in an imperfect production inventory model with two storage facilities under reliability consideration. J Indus Prod Eng 2018; 35: 57–73.

Meyer

Rothkopf

Smith

SA.

Reliability and inventory in a production-storage system. Manage Sci 1979; 25: 799–807.

Sarkar

An inventory model with reliability in an imperfect production process. Appl Math Comput 2012; 218: 4881–4891.

Selçuk

Ağralı

Joint spare parts inventory and reliability decisions under a service constraint. J Oper Res Soc 2013; 64: 446–458.

Shah

Vaghela

CR.

Imperfect production inventory model for time and effort dependent demand under inflation and maximum reliability. Int J Syst Sci 2018; 5: 60–68.

Cai

Liu

Fan

et al . Multi-source information fusion based fault diagnosis of ground-source heat pump using Bayesian network. Appl Energy 2014; 114: 1–9.

10.

Liu

Lin

et al . Bayesian reliability and performance assessment for multi-state systems. IEEE T Reliab 2015; 64: 394–409.

11.

Feng

MSW

Chen

MSY

et al . Cooperative game approach based on agent learning for fleet maintenance oriented to mission reliability. Comput Indus Eng 2017; 112: 221–230.

12.

Zhu

Guo

et al . Optimal design of redundant structures by incorporating various costs. IEEE T Reliab 2018; 67: 1084–1095.

13.

Borgonovo

Apostolakis

GE.

A new importance measure for risk-informed decision making. Reliab Eng Syst Safe 2001; 72: 193–212.

14.

Borgonovo

Differential importance and comparative statics: an application to inventory management. Int J Prod Econ 2008; 111: 170–179.

15.

Ramirez-Marquez

Coit

DW.

Composite importance measures for multi-state systems with multi-state components. IEEE T Reliab 2005; 54: 517–529.

16.

Nguyen

Grall

Joint predictive maintenance and inventory strategy for multi-component systems using Birnbaum’s structural importance. Reliab Eng Syst Safe 2017; 168: 249–261.

17.

Dui

Zhao

et al . Integrated importance measure of component states based on loss of system performance. IEEE T Reliab 2012; 61: 192–202.

18.

Dui

Zuo

et al . Semi-Markov process-based integrated importance measure for multi-state systems. IEEE T Reliab 2015; 64: 754–765.

19.

Dui

Chen

Generalized integrated importance measure for system performance evaluation: application to a propeller plane system. Eksploatacja I Niezawodnosc—Maint Reliab 2017; 19: 279–286.

20.

Dui

et al . An importance measure for multistate systems with external factors. Reliab Eng Syst Safe 2017; 167: 49–57.

21.

Dui

Yam

RCM

. Importance measures for optimal structure in linear consecutive-k-out-of-n systems. Reliab Eng Syst Safe 2018; 169: 339–350.