# Supply Chain Disruption Assessment Based on the Perspective of Trade-off in Newsvendor Model

## Abstract

This paper centers on supply chain assessment, following the cost-income principle and taking the cost trade-off into account. The analysis tool is newsvendor model and its perspective—finding the critical point, which in tradition model stands for the demarcation point of profit but in this paper the is the least costs considering disruption costs and expected revenues. In this way, this paper tries to find out the optimum method to assess supply chain risks.

### Keywords

Supply chain disruption Risk assessment Newsvendor model## 1 Introduction

Nowadays, facing the complicated and variable commercial environment as well as have been doing efforts in lean management for quicker response and lower cost, supply chain are tend to vulnerable and liable to affected by various risks. Supply chain risks, their impact and management are receiving much attention among practitioners and academicians alike.

The topic of risk management will continue to be important to researchers and supply management professionals. The twin areas of risk assessment and identification, and risk mitigation (approaches and theories) will continue to be of interest as the outsourcing trend continues to be a dominant strategy in firms.

Wakolbinger and Cruz T. Wakolbinger [1] summarized that the risks supply chain faced can be classified into two types: supply–demand coordination risks and disruption risks. Moreover, the disruption risks are the most vital and most notably type because the fact that disruption can bring about huge losses and prevention cost. Zhang Song [2] emphasized disruptions and figured out that cost-income principle must be followed when build the fortification models, that is to say, the model concerned not only the cost of disruption, but also the expected cost of lost revenues.

This paper aims at supply chain disruption assess, analyzing two types of costs which possibly be related to disruption risks. The two costs respectively stand for two kinds of attitudes towards risks—risk averse and risk appetite. Based on newsvendor model and its trade-off idea, the paper establishes one model can weigh the two costs, namely the two risk attitudes, and therefore get the optimal assessment solution. In the end, by means of the conclusion, it is useful to ranking risks concerning importance.

## 2 Theory Review of Supply Chain Risk Management (SCRM) and Newsvendor Model

Christopher S. Tang [3] gave an integrated definitions developed by others that “the management of supply chain risks through coordination or collaboration among the supply chain partners so to ensure profitability and continuity”. We can infer from the description that the objectives of SCRM fall into two aspects: the one is to enhance profit, and the other one is to lower the disruptions existing supply chain. To reduce the disruptions means to must pay much costs for precautions and controls, but on the other hand, to protect profit will ask cost-control. Contemporary researches are increasingly beginning to focus on the tradeoff between the two constraints. Based on the definition of SCRM, in general, SCRM is an issue dealing with the identification, assessment, analysis and treatment by minimizing, monitoring and controlling the probability and impact of uncertainty disruptions in order to economically effective management. Furthermore, the risk identification and assessment stage is fundamental and critical to the success of managing supply chain risks [3, 4]. The study present by Jyri P.P. Vilko and Jukka M. Hallikas [5] conducted a preliminary research concepts and findings concerning the identification and analysis of supply chain risks.

This paper aims at bring forth new approach of risks assessment by applying the newsvendor model. Taking a wide view of risks assessment theory, it revolves mainly around two aspects: (1) The probability of risk events; (2) The consequences and losses if risk events happening. After Mitchell [6] and his theory of “Risk = P(loss)*Loss”(In this equality, Risk is the assessment result; P means the probability or possibility; Loss is consequences), industry and academic circles develops the assessment theory from the two aspects: Ding Weidong et al. [7] provided a fuzzy factor technique to evaluate risks; Ericsson developed a series of tools named ERMET (Ericsson Risk Management Evaluation) [8]; Meng Kedeng [9] built a evaluation model based on grey relational analysis combining the fuzzy assessment. Most of the researches take two types variables into consideration, namely probabilities and results, so do the newsvendor model built in the paper.

Traditionally, newsvendor models is mostly assumed to be risk-neutral and insensitive to profit variations with the objective of expected profit maximization or expected cost minimization. Recently, the vulnerability and risks in the supply chains remind the managers of the tradeoff expected profit for downside protection against possible losses [10]. In this respect, Werner.Jammernegg and Peter Kischka [11, 12, 13] specially promoted this issues. They formulated the newsvendor model of the same kind that tradeoff between service level (target value) and resulting losses by the target. Yan Qin et al. [14] enriched the theory by considering the attitude of decision-maker towards the risks, devoted to the analysis newsvendor model with various risk preferences, including, but not limited to, risk-averse and risk-seeking preferences. Besides, they also reviewed and directed the future research in newsvendor problem, modeled how the buyer’s risk profile moderates the newsvendor order quantity decision. Anastasios Xanthopoulos et al. [15] develop a newsvendor model for both risk neutral and risk-averse decision-makers and can be applicable for different types of disruptions related among others to the supply of raw materials, the production process, and the distribution system, as well as security breaches and natural disaster.

To sum up, the new direction of SCRM and newsvendor problem will be of tradeoff value target and losses may resulting in. Risk attitude of decision-maker is also a crucial point to be regarded.

## 3 Newsvendor Model for Supply Chain Disruption Assessment

The thought of traditional newsvendor model is that: when the demand is Stochastic, the managers expect to achieve profit maximization or loss minimization by optimum order quantity. In fact, the order quantity is a ratio based on a “critical point”, which makes the optimal probability of target function. In addition, traditional newsvendor managers bear the risk neutral attitude towards the risks.

There is the probabilities assessment in the evaluation of supply chain risks. No matter by means of qualitative expert evaluation method or quantitative methods emerging continuously in the academic circles, it is expected to make a best assessment of supply chain risks. By virtue of thought of newsvendor model, it is a feasible approach to get the risks assessment on the foundation of an optimal “critical point ratio” weighing against costs, and the target function is to make the risks prevention cost and risks response cost minimum. Besides, the weight of two costs, at the same time, is also the weight of two attitudes—risk averse and risk appetite.

### 3.1 Model Foundation

This model considers risks probability as a continuous variable. To assess the probability of a risk occurring, it introduces two types of costs to weigh against. That is called opportunity cost and disruption cost in this model. The former is the cost spends on preventing risks from happening, and the latter is the loss to response the consequences after risks occur.

Assume a certain risk’s probability is stochastic and obeys a known distribution.

Evaluate an occurrence probability of the risk is P.

If P is higher than the actual probability. That is to say, risk averse decision overestimates the risk giving rise to an overdone prevention and emergence action, which generates opportunity cost.

If P is lower than the actual probability. That is to say, risk appetite decision underestimates the risk binging about a potential disruption point in supply chains, which a liable to generate disruption cost.

Assume a certain risk is named NI. Its actual occurrence probability is r, obeying a distribution with density function ф(r), namely \( \int_0^{\mathrm{ r}} {\varphi (r)dr=1} \). In one risk assessment process, its calculating probability is p.

- 1.
Opportunity Cost L

When risks are overestimated (p ≥ r), opportunity cost happens and the loss is:

(p−r) · L, so its expectation value is \( \int_0^p {L\cdot (p-r)} \varphi (r)dr \);

- 2.
Disruption Cost C1

When risks are underestimated (p < r), disruption cost happens and its loss is:

(r−p) · C1, so its expectation value is \( \int_p^{\infty } {C_1} (r-p)\varphi (r)dr \).

### 3.2 Optimal Decision

Target function is min E[C(P)].

Here follows the differentiation method inference process of newsvendor model:

Therefore, p is solved from the arithmetic expression above, and be denoted as P*, then P* is the stationary point of E[C(P)].

And because \( \frac{{{d^2}E[C(P)]}}{{d{p^2}}}=L\phi (p)+C_1\phi (p)>0 \), it is clear that p* is the limited minimum point of E[C(P)], minimum point of the model.

### 3.3 Model Analysis

- 1.
Opportunity cost L

This part of cost in reality reflects that manager is risk averse attitude, which means the loss of potential profits.

Because of the averse of risk, manager takes vigorous prevention and emergence measures so as to be more defensive to disruptions in supply chain. Correspondingly, the overprotection needs more cost input, so the opportunity cost is come into being.

This kind of cost is inversely proportional to the result p*. The higher some certain kinds of risks’ opportunity cost is, the lower their optimal assessment probability is.

- 2.
Disruption cost C1

When manager’s preference towards risk is risk appetite, the cost input to prevent risk from occurring is much more than the risk averse manager. But in contrast, there are more risk events happen, and can bring more disruption cost.

The optimal risk assessment will be a tradeoff between the two types of costs.

- 3.
Density function

This is the general rule of the occurrence of risk. And generally normal distribution is the most universal used one. Its density function can be showed as following:where \( \mu \) is mean value, and \( \sigma \) is standard deviation.$$ \varphi \left( \mathrm{ r} \right)=\frac{1}{{\sigma \sqrt{{2\pi }}}}{e^{{-\frac{{{{{\left( {r-\mu } \right)}}^2}}}{{2{\sigma^2}}}}}},\quad -\infty < r<+\infty, $$

## 4 Conclusions

This paper builds a newsvendor model for supply chain disruption assessment, which applies the tradeoff idea of newsvendor to this model. This model considers both opportunity cost and disruption cost, between which is a cost-income principle tradeoff. When a risk’s assessment is higher than optimum, disruption cost will descend whereas opportunity cost ascend; when the risk’s assessment is lower than the optimal one, disruption cost will ascend while opportunity cost descend. Among which, the optimum is the “critical point” deducted by the disruption assessment in newsvendor model. The “critical point” can minimize the expectation loss of these both costs. At the same time, the two cost stand for two opposite attitudes and preferences towards risks. Opportunity cost is on behalf of risk averse; disruption cost stand for risk appetite. That can extend the traditional risk neutral newsvendor.

Simultaneously, this model can be also used to rank a series of risk events Ni (i = 1, 2, 3…) may happen in every link of a supply chain according to their importance. Pi* can represent the probability of risk Ni, whose expectation loss is the least one. Taking another look at it, the lower Pi* is, the more likely the risk event causing cost, the more attention should be paid, the more vital the risk event. In contrast, when Pi* is lower, the risk event is not that important than the former.

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