Formula Sheet
Inventory Management
• Cu = underage cost = marginal gain of extra unit sold
• Co = overage cost = marginal cost of unsold unit
• Expected underage cost of (n+1)st unit = Cu x Pr (Demand > n)
• Expected overage cost of (n+1)st unit = Co x Pr (Demand ≤ n)
𝐶
• Critical Fractile = Pr (Demand ≤ n) = 𝐶 +𝑢𝐶
•
•
𝑢
𝑜
Under normally distributed demand, optimal ordering quantity (n*) = 𝜇 + 𝑧 ∗ 𝜎
Service Level (S) = Pr (Demand ≤ n)
Economic Order Quantity (EOQ) Model
• Q = Quantity in each order
• D = Demand rate
• H = Holding cost
• S = Ordering cost
• Q/D = Time between two orders
• D/Q = Order frequency per unit time
2𝐷𝑆
𝐻
•
𝑄∗ = √
•
Holding cost per unit time =
•
Order or setup cost per unit time =
•
Total cost per unit time = TC(Q) =
•
ROP* = DL + z𝜎√𝐿
𝐻𝑄
2
𝑆𝐷
𝑄
𝐷𝑆
𝑄𝐻
+ 2
𝑄
Economic Production Quantity (EPQ) Model
•
2𝐷𝑆
𝑄∗ = √
𝐷
𝐻(1− )
𝑃
•
Total cost = TC(Q) =
𝐷𝑆
𝑄
+
𝑄𝐻
(1
2
𝐷
− 𝑃)
Forecasting
• 𝐹𝑡 = 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑜𝑚𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
• 𝐹𝑡−1 = 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑎𝑠𝑡 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
• 𝐴𝑡−1 = 𝑎𝑐𝑡𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑎𝑠𝑡 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
• 𝛼 = 𝑠𝑚𝑜𝑜𝑡ℎ𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑎𝑛𝑡
𝐴
+𝐴
+ 𝐴 +⋯+ 𝐴𝑡−𝑛
• Simple Moving Average, 𝐹𝑡 = 𝑡−1 𝑡−2 𝑛𝑡−3
•
•
•
•
•
•
Weighted Moving Average, 𝐹𝑡 = 𝑤1 𝐴𝑡−1 + 𝑤2 𝐴𝑡−2 + ⋯ + 𝑤𝑛 𝐴𝑡−𝑛 ; ∑𝑛𝑖=1 𝑤𝑖 = 1
Exponential Smoothing, 𝐹𝑡 = 𝐹𝑡−1 + 𝛼(𝐴𝑡−1 − 𝐹𝑡−1 )
Linear Regression: Y = a + bX; Y: dependent variable, X: independent variable
̂𝑖 = 𝑎 + 𝑏𝑋𝑖 ; Minimize squared error 𝑚𝑖𝑛 ∑𝑛𝑖=1(𝑌𝑖 − 𝑌
̂𝑖 )2 =
Linear Regression Analysis: Forecast 𝑌
𝑛
2
∑𝑖=1(𝑌𝑖 − (𝑎 + 𝑏𝑋𝑖 ))
Least Squared Method: Optimal coefficients a* and b* that minimize the sum of the squared errors:
∑𝑛 𝑋𝑖 𝑌𝑖 −𝑛𝑋̅𝑌̅
∗̅ ̅
̅
̅
𝑏 ∗ = 𝑖=1
𝑛
2
̅̅̅̅
2 , a* = 𝑌 − 𝑏 𝑋 , 𝑋 : average of 𝑋𝑖 , 𝑌 : average of 𝑌𝑖
∑𝑖=1 𝑋𝑖 −𝑛𝑋̅
Measuring Forecasting Accuracy:
o Error = Actual – Forecast
o Mean absolute deviation (MAD) = ∑|𝐴𝑐𝑡𝑢𝑎𝑙 − 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡| / n
o Tracking signal (TS) = ∑(𝐴𝑐𝑡𝑢𝑎𝑙 − 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡) / MAD
Appendix: Standard Normal Distribution Table :
F(z) = Pr( N(0,1) ≤ 𝑧 )