# Dual Market Facility Network Design under Bounded Rationality
## 3 Mathematical Model Formulation
### Newsvendor Model under Bounded Rationality  
The assumption of perfect rationality for decision makers is a typical tactic in a majority of normative analysis. In order to obtain bounded rationality, we followed the approach of [13], where  
Figure 3: Flow-chart of the algorithm. multi-nomial logit choice model is applied and assumed that decision makers selects alterative \(j\), with probability. \[\phi_{j}=\frac{e^{\frac{u_{j}}{p}}}{\sum\limits_{j}e^{\frac{u_{j}}{p}}} \tag{6}\]  
Correspondingly, the following density function gives the logit choice probabilities over a continuous domain:  
\[\phi(y)=\frac{e^{\frac{u(y)}{p}}}{\int_{y}e^{\frac{u(y)}{p}}} \tag{7}\]  
Thus, the selection of agents is a random variable. The superior options are selected frequently using the above logit structure. The best alternative depends on the mode of the selection distribution. The magnitude of cognitive and computational restrictions suffered by the decision makers is elucidated by \(\beta\) parameter. This can be observed when selection distribution approximates the uniform distribution over the options in \(\beta\rightarrow\infty\) limit. In some utmost instances, decision makers randomize the options with equal probabilities when they failed to make any informed choices. In the other scenario, when \(\beta\rightarrow\infty\) the choice distribution in (1) completely focused on utility maximizing options. The selection of perfect rational decision maker corresponds with this choice. Therefore, we can consider the magnitude of \(\beta\) as the extent of bounded rationality. The logit choice framework to the newsvendor problem is employed here. The profit function is as stated in Equation (1) is  
\[\pi(x)=p\,E(\min(D,x))-cx\]  
This is uniquely maximized at,  
\[x^{*}=F^{-1}(1-c/p) \tag{8}\]  
where, \(F\) is the cumulative distribution function of demand D. All other symbols are as used before. This solution is selected when the decision maker is perfectly rational, although the newsvendor's ordering amount is dependent on noise and it called the behavioral solution in bounded rationality. The bounded rational newsvendor can order any amount of product within the range of minimum possible and maximum possible demand realizations in the domain \(S\). Then, the Equation (9) gives the probability density function associated with behavioral solution:  
\[\phi(y)=\frac{e^{\frac{\pi(x)}{p}}}{\int_{s}e^{\frac{\pi(x)}{p}}dv}=\frac{e^{ \frac{pE(\min(D,x))-cx}{p}}}{\int_{s}e^{\frac{pE(\min(D,y))-cx}{p}}dv} \tag{9}\]  
Now, if the demand \(D\) is uniformly distributed in in the interval of _[a, b]_, then the behavioral solution would follow truncated normal distribution over _[a, b]_ with mean \(\mu\) and standard deviation \(\sigma\)[13]. \[\mu=b-\frac{c}{p}(b-a) \tag{10}\]  
\[\sigma^{2}=\beta\frac{b-a}{p} \tag{11}\]  
The expected value of this truncated normal distribution would be,  
\[E(x)=\mu-\sigma\frac{\phi((b-\mu)/\sigma)-\phi((a-\mu)/\sigma)}{\phi((b-\mu)/ \sigma)-\phi((a-\mu)/\sigma)} \tag{12}\]  
where, \(\phi(. )\) denotes cumulative normal distribution and \(\phi(. )\) denotes normal density function. As an example in the dual market scenario, each market decision maker would be subjected to bounded rationality.