We consider a multiproduct risk-averse newsvendor under the law-invariant coherent measures of risk. We first establish several fundamental properties of the model regarding the convexity of the problem, the symmetry of the solution, and the impact of risk aversion. Specifically, we show that for identical products with independent demands, increased risk aversion leads to decreased orders. For a large but finite number of heterogeneous products with independent demands, we derive closed-form approximations for the optimal order quantities. The approximations are as simple to compute as the classical risk-neutral solutions. We also show that the risk-neutral solution is asymptotically optimal as the number of products tends to be infinity, and thus risk aversion has no impact in the limit. For a risk-averse newsvendor with dependent demands, we show that positively (negatively) dependent demands lead to lower (higher) optimal order quantities than independent demands. Using a numerical study, we examine the convergence rates of the approximations and develop additional insights into the interplay between dependent demands and risk aversion.
For general law invariant coherent measures of risk, we derive an equivalent representation of a risk-averse newsvendor problem as a mean-risk model. We prove that the higher the weight of the risk functional, the smaller the order quantity. Our theoretical results are confirmed by sample-based optimization.
[Purpose] The purpose of this study was to examine the effects of cervical traction
treatment, cranial rhythmic impulse treatment, a manual therapy, and McKenzie exercise, a
dynamic strengthening exercise, on patients who have the neck muscle stiffness of the
infrequent episodic tension-type (IETTH) headache and frequent episodic tension-type
headache(FETTH), as well as to provide the basic materials for clinical interventions.
[Subjects] Twenty-seven subjects (males: 15, females: 12) who were diagnosed with IETTH
and FETTH after treatment by a neurologist were divided into three groups: (a cervical
traction group (CTG, n=9), a cranial rhythmic contractiongroup (CRIG, n=9), and a McKenzie
exercise group (MEG, n=9). An intervention was conducted for each group and the
differences in their degrees of neck pain and changes in muscle tone were observed.
[Results] In the within-group comparison of each group, headache significantly decreased
in CTG. According to the results of the analysis of the muscle tone of the upper
trapezius, there was a statistically significant difference in MEG on the right side and
in CRIG on the left side. According to the results of the analysis of the muscle tone of
the sternocleidomastoid muscle, there was a statistically significant difference in MEG on
the right side and in CRIG on the left side. [Conclusion] In the comparison of the
splenius capitis muscle between the groups, there was a statistically significant
difference on the right side. Hence, compared to the other methods, cervical traction is
concluded to be more effective at reducing headaches in IETTH and FETTH patients.
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