# Basic EOQ with stochastic parameters

## Notations

• $\, Q$ – order quantity;
• $\, EOQ$ – economic order quantity ;
• $\, n$ – number of orders per year;
• $\,D$, $\, D_{i}$ – annual demand;
• $\, C$ – cost of unit of resource;
• $\, h$ – inventory costs per year (%);
• $\, H$ – marginal inventory costs per year;
• $\, p$ – rate of production;
• $\, d$ – demand rate;
• $\, L$ – lead-time;
• $\, I$ – inventory;
• $\, s$ – standard deviation of demand;
• $\, T$ – delivery time;
• $\, SL$ – standard deviation;
• $\, \alpha$ – deficit risk;
• $\, P_{sl}$ – service level;
• $\, ROP$ – reorder point;
• $\, SS$ – safety stock;
• $\, \bar{C}$ – surplus costs;
• $\, \underline{C}$ – shortage costs;
• $\, P$ – profit.

Economic order quantity: $EOQ=\sqrt{\frac{2DS}{H} }$.

Standard normal distribution:

$p(z)=\frac{1}{\sqrt{2\pi } } \exp \left(-\frac{z^{2} }{2} \right)$ ,

where $z=\frac{x-\bar{x}}{s}$ - deviation from mean; s – standard deviation, x – demand, $\bar{x}$ – average demand.

safety stock: $\, SS = zSL$;

reorder point: $\, ROP = dL+SS$;

standard deviation during delivery time:

$SL=s\sqrt{L}$ – with constant leadtime L.

or $SL=\sqrt{s^{2} L+d^{2} s_{l}^{2}}$ – when varies with mean L and standard deviation sl;

the number of un satisfied customers:

$\, E(P_{sl} )=(1-P_{sl} )Q$ or $E(z)=S_{L} \left(\frac{1}{\sqrt{2\pi } } \exp (-\frac{z^{2} }{2} )-z\alpha \right)$;

order quantity:

$Q=d(L+T)+zs\sqrt{L+T} -I$ ;

risk:

$\alpha =\frac{C_{u} }{C_{u} +C_{H} }$.

optimal order quantity in one-period model [1]:

$\, P_{Q} =Pd-s(zC_{u} +(C_{u} +C_{H} )L(z))$,

where $\,L(z)=\frac{1}{\sqrt{2\pi } } \exp \left(-\frac{z^{2} }{2} \right)-z\alpha$.

## References

1. Методы оптимизации управления и принятия решений: примеры, задачи, кейсы: учебное пособие. -- М.: Дело, 2007.