Multiple strategies for supply chain to recover from major disruptions

4.1 Crisp programming models

To better solve these uncertain programming models, we equivalently convert them into their crisp programming models according to theorems 1, 2 and 3. The crisp one of EVM is as follows: min ∫ 0 1 ( ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S π s Φ 1 i - 1 ( α ) e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S π s Φ 2 i - 1 ( α ) Φ 1 i 1 - 1 ( α ) c i y is - ∑ i = 1 I ∑ s = 1 S π s Φ 3 i - 1 ( 1 - α ) x is ) d α(16) s . t . Eqs . ( 2 ) to ( 3 ) , ∑ j = 1 J v ijts ≤ C i ( 1 - z is ) ( 1 - Φ 1 it - 1 ( β ) ) C i y is + e is , i ∈ I , t ∈ T , s ∈ S ,(17) s jts ≤ τ Φ 4 j - 1 ( 1 - γ ) , j ∈ J , t ∈ T , s ∈ S ,(18) Φ 4 j - 1 ( η ) ≤ ∑ i = 1 I v ijts + s jts + u jts , j ∈ J , t ∈ T , s ∈ S ,(19) u jts ≤ λ Φ 4 j - 1 ( 1 - δ ) , j ∈ J , t ∈ T , s ∈ S ,(20) Eqs . ( 8 ) to ( 13 ) . The crisp programming model of CCP is as follows: min C ¯(21) s . t . ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S π s Φ 1 i - 1 ( α ) e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S π s Φ 2 i - 1 ( α ) Φ 1 i 1 - 1 ( α ) c i y is - ∑ i = 1 I ∑ s = 1 S π s Φ 3 i - 1 ( 1 - α ) x is ≤ C ¯ ,(22) Eqs . ( 2 ) to ( 3 ) , Eqs . ( 8 ) to ( 13 ) , Eqs . ( 17 ) to ( 20 ) .

In this section, more concrete formulations are presented by introducing two common uncertainty distributions (linear and normal).

We assume that ec i ˜ , rc i ˜ , f i ˜ , d j ˜ , and a it ˜ are independent linear uncertain variables L ( a 1 i , b 1 i ) , L ( a 2 i , b 2 i ) , L ( a 3 i , b 3 i ) , L ( a 4 j , b 4 j ) , and L ( a 1 it , b 1 it ) , respectively. It follows from definitions 3 and 6 that the crisp equivalent model of EVM under the assumption of linear uncertainty distribution is as follows: min ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S 1 2 π s ( a 1 i + b 1 i ) e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S 1 6 π s ( a 2 i ( 2 a 1 i 1 + b 1 i 1 ) + b 2 i ( a 1 i 1 + 2 b 1 i 1 ) ) c i ( z is - x is ) - 1 2 ∑ i = 1 I ∑ s = 1 S π s ( a 3 i + b 3 i ) x is(23) s . t . Eqs . ( 2 ) to ( 3 ) , ∑ j = 1 J v ijts ≤ ( 1 - ( ( 1 - β ) a 1 it + β b 1 it ) z is ) C i * ( 1 - x is ) + e is ( 1 - z is ) , i ∈ I , t ∈ T , s ∈ S ,(24) s jts ≤ τ ( γ a 4 j + ( 1 - γ ) b 4 j ) , j ∈ J , t ∈ T , s ∈ S ,(25) ( ( 1 - η ) a 4 j + η b 4 j ) ≤ ∑ i = 1 I v ijts + s jts + u jts , j ∈ J , t ∈ T , s ∈ S ,(26) u jts ≤ λ ( δ a 4 j + ( 1 - δ ) b 4 j ) , j ∈ J , t ∈ T , s ∈ S ,(27) Eqs . ( 8 ) to ( 13 ) .

Similarly, the crisp equivalent model of CCP under the assumption of linear uncertainty distribution is as follows: min C ¯(28) s . t . ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S π s ( ( 1 - α ) a 1 i + α b 1 i ) e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S π s ( ( 1 - α ) a 2 i + α b 2 i ) * ( ( 1 - α ) a 1 i 1 + α b 1 i 1 ) c i ( z is - x is ) - ∑ i = 1 I ∑ s = 1 S π s ( α a 3 i + ( 1 - α ) b 3 i ) x is ≤ C ¯ ,(29) Eqs . ( 2 ) to ( 3 ) , Eqs . ( 8 ) to ( 13 ) , Eqs . ( 25 ) to ( 28 ) .

Assume that ec i ˜ , rc i ˜ , f i ˜ , d j ˜ , and a it ˜ are independent normal uncertain variables N ( e 1 i , σ 1 i ) , N ( e 2 i , σ 2 i ) , N ( e 3 i , σ 3 i ) , N ( e 4 j , σ 4 j ) , and N ( e 1 it , σ 1 it ) , respectively. It follows from definitions 4 and 6 that the crisp equivalent model of EVM under the assumption of normal uncertainty distribution is as follows: min ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S π s e 1 i e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S π s ( e 2 i e 1 i 1 + 9.87 σ 2 i σ 1 i 1 π 2 ) * c i ( z is - x is ) - ∑ i = 1 I ∑ s = 1 S π s e 3 i x is(30) s . t . Eqs . ( 2 ) to ( 3 ) , ∑ j = 1 J v ijts ≤ ( 1 - ( e 1 it + 3 σ 1 it π ln β 1 - β ) z is ) C i ( 1 - x is ) + e is ( 1 - z is ) , i ∈ I , t ∈ T , s ∈ S ,(31) s jts ≤ τ ( e 4 j - 3 σ 4 j π ln γ 1 - γ ) , j ∈ J , t ∈ T , s ∈ S ,(32) ( e 4 j + 3 σ 4 j π ln η 1 - η ) ≤ ∑ i = 1 I v ijts + s jts + u jts , j ∈ J , t ∈ T , s ∈ S ,(33) u jts ≤ λ ( e 4 j - 3 σ 4 j π ln δ 1 - δ ) , j ∈ J , t ∈ T , s ∈ S ,(34) Eqs . ( 8 ) to ( 13 ) .

In the same way, the crisp equivalent model of CCP under the assumption of normal uncertainty distribution is as follows: min C ¯(35) s . t . ∑ i = 1 I ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s tc ij v ijts + ∑ j = 1 J ∑ t = 1 T ∑ s = 1 S π s ( sc j s jts + uc j u jts ) + ∑ i = 1 I ∑ s = 1 S π s ( e 1 i + 3 σ 1 i π ln α 1 - α ) e is + ∑ i = 1 I ∑ n = 1 N ∑ s = 1 S π s ( e 2 i + 3 σ 2 i π ln α 1 - α ) * ( e 1 i 1 + 3 σ 1 i 1 π ln α 1 - α ) c i ( z is - x is ) - ∑ i = 1 I ∑ s = 1 S π s ( e 3 i - 3 σ 3 i π ln α 1 - α ) x is ≤ C ¯ ,(36) Eqs . ( 2 ) to ( 3 ) , Eqs . ( 8 ) to ( 13 ) , Eqs . ( 30 ) to ( 34 ) .

5.1 Experimental design

We consider a two-tier supply chain that includes 10 facilities and 30 customers. Some necessary data are given as follows. Here, we need to emphasize that all uncertain variables are regarded as normal with normal uncertainty distributions. The demand of each customer d _j is drawn uniformly from [1000,10000]. The capacity of each facility C _j is drawn uniformly from [1000,10000]. For the cost parameters, the unit transportation cost from facility i to customer j tc _ij is Euclidean distance between two sites. The unit outsourcing cost sc _j and unit penalty cost uc _j are drawn uniformly from [0,10]. The excepted value e_1i of unit expansion cost ec _j and e_2i of unit recovery cost rc _j are drawn uniformly from [20,30, 20,30]. Beyond that, we set the time horizon as t = 5, and the expected value e_1it of unavailable capacity a _it is shown in Table 2. The fixed income of closing disrupted facility i is correlated to its available capacity, so we denote expected value as e_3i = κ _i  (1 - a_i1) C _i and κ _i is drawn uniformly from (0,1]. The standard deviation of these uncertain variables are set as σ_1i = 2,σ_2i = 2,σ_3i = 2,σ_4j = 2 and σ_1it = 0.01. For disruption scenario, the possibility of scenario π _s is drawn uniformly from [0,1] and then normalized such that the total possibility of all scenarios is equal 1.

Section 3 presents a risk-neutral model (EVM) and a risk-averse model (CCP) in accordance with two different decision criteria. In the latter model, confidence level α is introduced to quantify the risk performance of decision-makers and a larger α means decision-makers are more risk-averse. To test how the optimal objective value is affected by risk attitude, we consider 56 disruption scenarios and implement a set of numerical examples under two common uncertainty distributions. The normal distributions of uncertain variables are described in section 5.1. Here, we assume linear uncertainty distributions as follows: ec i ˜ ∼ L ( ec i - dc , ec i + dc ) , rc i ˜ ∼ L ( rc i - dc , rc i + dc ) , f i ˜ ∼ L ( f i - df , f i + df ) , d j ˜ ∼ L ( d j - dd , d j + dd ) , a it ˜ ∼ L ( a it - da , a it + da ) . The value of ec _i , rc _i , f _i , d _j , a _it are the same as the expected value of corresponding uncertain variables in section 5.1. Besides, we set dc = 1, df = 10, dd = 10, da = 0.01. Other parameters in the presented model are set as β, γ, η, δ = 0.9, and λ = τ = θ = 0.1.

Figures 1 and 2 report the optimal objective value of two models under linear and normal uncertainty distributions, respectively. The expected value of the objective value is also described in these figures to indicate the relationship of two models, although it is not influenced by the variation of α. We can observe that the objective value of CCP increases while confidence level α becomes larger. This result means that a decision-maker who is more risk-averse prefers to mitigate risk at a higher expense. Besides, we also notice that the objective values of the two models are the same as α = 0.5, representing a neutral attitude to risk.

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Fig. 1

The performance of two models under linear distribution.

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Fig. 2

The performance of two models under normal distribution

As mentioned in section 3, every facility may be disrupted when facing potential threats. In the following numerical examples, 10 facilities yield 1024 disruption scenarios that may result in a large-scale model size and increase the computational burden. Identifying scenarios is possible but difficult and time-consuming. Meanwhile, simultaneous failure of more than two facilities, in reality, has minimal likelihood (Peng et al., 2011). Thus, considering excessive disruption scenarios is unnecessary and advisable. References that adopt the scenario-based approach usually identify a subset of disruption scenarios to conduct analysis. Therefore, we choose 11, 56 and 176 scenarios to design a series of experiments, and the results are shown in Table 2. Here 11 disruption scenarios include 1 normal scenario and 10 scenarios in which only one facility is disrupted. 56 scenarios contain above 11 scenarios and 45 scenarios in which 2 facilities are disrupted simultaneously. These 56 scenarios plus 120 scenarios in which 3 facilities are disrupted simultaneously constitute 176 disruption scenarios. Besides, all confidence levels β, γ, η, δ and α are set to 0.9. From Table 3, we know that the optimal objective values of two models, whether expected cost or optimistic cost reduce with the decrease of scenario number and the difference is noticeable. This result means that identifying the disruption scenario is a critical factor to influence decision making. Thus, in reality, decision-makers need to comprehensively consider various factors to carefully identify scenarios of concern, such as the protection capability of the facility (capability for fire resistance or seismic resistance), the geographical location of the facility (whether the facility is more likely to be disrupted), and the importance of these facilities, etc.

In this study, we adopt outsourcing and capacity expansion strategies for supply chains to recover from disruptions and related parameters, respectively, which are τ and θ. Another critical parameter is λ, which confines the amount of unmet demand. To determine how the variations of these key parameters affect the optimal objective values, we design a series of experiments, and the results are shown in Tables 4 and 5. Similarly, we set all confidence levels β, γ, η, δ and α to 0.9 and consider 56 disruption scenarios.

It can be seen from each row that the objective values decrease slightly when θ increases under fixed λ and τ. They remain unchanged with the growth of θ while the upper limit of outsourcing strategy or unmet demand is larger. This condition implies that the capacity expansion strategy does not always work on reducing expected and optimistic costs. We also note that the objective values significantly decrease as τ increases. Furthermore, the difference is smaller when λ or θ takes a bigger value. Nevertheless, the outsourcing strategy is feasible to reduce the objective values. Besides, the increase of λ also results in a rapid drop of objective values first, followed by a slighter decrease. This condition indicates that decision-makers may reduce total cost at the expense of lower demand satisfaction. Moreover, from the preceding observations, we also know that capacity expansion strategy is useful to deal with the risk posed by facility disruptions, while outsourcing and unmet demand are not allowed, or their upper limit is relatively small. When unmet demand is allowed, both two strategies can respond effectively to disruption risks, but outsourcing strategy is superior to capacity expansion strategy.

In the presented models, confidence levels in constraints are the other parameters identified subjectively by decision-makers. In this section, we design a variety of numerical examples to test how the optimal results change with the variation of these parameters, aiming to provide some meaningful and practical insights for decision-making in reality. In these experiments, the default values of confidence levels β, γ, η, and δ are all set to 0.9, and only one parameter value is changed in each experiment. Furthermore, we consider 56 disruption scenarios and two situations including full demand satisfaction λ = 0.0, τ = 0.15, θ = 0.15 and unmet demand λ = 0.1, τ = 0.05, θ = 0.05.

The sensitivity analysis results are reported in Figures 3–6, in which the first two figures are results of EVM and the latter two are results of CCP. The objective value increases or stays the same, while one of the parameters grows. Specifically as δ increases, the objective value increases slightly when λ = 0.1 and remains unchanged when λ = 0.0. The reason is that the constraint (7) does not work when unmet demand is not allowed. Similarly, the growth of γ also results in a slight increase in objective value. Moreover, the objective value increases mostly when β increases, followed by η in these figures. From these observations, we know that some confidence levels are also important factors that influence costs (expected and optimistic costs), such as β and η. Therefore, for outsourcing constraint (5) and unmet demand constraint (7), decision-makers can set higher confidence levels to ensure that the solutions always satisfy the constraints. Meanwhile, it is also an opportunity for decision-makers to significantly reduce costs at the expense of lower confidence levels β or η (constraint 4 or 6) according to their conservativeness level (or risk preference) in reality.

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Fig. 3

Expected cost as λ = 0, τ = 0.15, θ = 0.15

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Fig. 4

Expected cost as λ = 0.1, τ = 0.05, θ = 0.05

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Fig. 5

Optimistic cost as λ = 0, τ = 0.15, θ = 0.15

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Fig. 6

Optimistic cost as λ = 0.1, τ = 0.05, θ = 0.05

In the preceding subsections, we implement a series of computational experiments to verify the feasibility of the presented models. Furthermore, the effects of crucial parameters, such as disruption scenarios, the upper limit of recovery strategies, and confidence levels on the optimal solutions are analyzed. From the obtained results, we can draw managerial insights that can provide decision support for practical applications.

In subsection 5.3, we discuss how the disruption scenarios can influence the objective values. The results show that the total cost increases obviously when we consider more disruption scenarios. For example, the expected value increases by 50% (from 311093 to 468157) and 108% (from 311093 to 648219) under 56, 176 scenarios compared with 11 scenarios, respectively. This result means that major disruptions (more facilities to be disrupted simultaneously) have a devastating effect on supply chains, from which huge costs are needed in the recovery process. To alleviate these expenditures, managers should pay close attention to supply chain disruption management and plan for disruption in advance. Proactive strategies such as fortifying facilities with more investment and pre-positioning emergency inventory in the planning stage may be conducive for enterprises to reduce the likelihood of disruption events or mitigate the subsequent results.

From the results in subsection 5.4, we can see that the capacity expansion strategy is not always so effective in reducing recovery costs. Its advantage is reflected obviously while we claim for higher satisfaction rate of customer demand and lower outsourcing rate. Furthermore, outsourcing strategy is proved to be a useful measure for the supply chain to recover from major disruptions and is more effective under a higher satisfaction rate of customer demand. Thus, in practical applications, managers can outsource part of products to improve the service level of the supply chain after disruption, if enterprises cannot hire more human resources or provide more production equipment. However, some precautions should be taken against the risk caused by product outsourcing, such as information disclosure, etc.

Moreover, computational results show that the optimistic value of total cost is always greater than the expected value when α > 0.5. The difference between them increases as this confidence level increases because decision-makers are more averse to taking risks. These two metrics report the expected cost and maximum possible cost during the recovery process, which can provide decision support for enterprises. Managers should adopt practical approaches to mitigate and control disruption risks, if these values, mostly optimistic value, exceed the budget or expectation of enterprise.