Basic EOQ with stochastic parameters

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Russian: Экономичный размер заказа при случайном спросе

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.
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