Equations for Inventory Management

Equations for Inventory Management Chapter 1 Stocks and inventories Empirical observation for the amount of stock held in a number of locations: N 2 AS(N 2 ) = AS(N 1 ) N 1 where: N 2 = number of planned future facilities N 1 = number of existing facilities AS(N i ) = aggregate stock with N i facilities Chapter 3 Economic order quantity The variables used here, and throughout the book, are: Q = order quantity D = demand UC = unit cost RC = reorder cost HC = holding cost T = cycle length VC = variable cost per unit time TC = total cost per unit time ROL = reorder level LT = lead time Qo = optimal order quantity To = optimal cycle length VCo = optimal variable cost per unit time TCo = optimal total cost per unit time ž Economic order quantity: 2 RC D Qo = HC ž Optimal stock cycle length: 2 RC To = Qo/D = D HC

230 Equations for Inventory Management ž Variable cost per unit time: VC = RC D Q ž Optimal value of variable cost per unit time: VCo = HC Qo = 2 RC D Qo + HC Q 2 = 2 RC HC D ž Total cost per unit time: TC = UC D + VC ž Optimal cost per unit time: TCo = UC D + VCo ž Change of variable cost moving away from the EOQ: VC VCo = 1 [ Qo 2 Q + Q ] Qo ž Reorder level: Reorder level = lead time demand stock on order ROL = LT D n Qo Chapter 4 Models for known demand Model for finite replenishment rate, P ž Optimal order quantity: 2 RC D P Qo = HC P D ž Optimal time cycle time: 2 RC P To = HC D P D ž Optimal variable cost: ž Optimal total cost: VCo = P D 2 RC HC D P TCo = UC D + VCo

Model for planned shortages and backorders ž Optimal order quantity: Equations for Inventory Management 231 SC = shortage cost per unit per unit time Qo = 2 RC D (HC + SC) HC SC ž Optimal amount to be backordered: 2 RC HC D So = SC (HC + SC) ž Time during which demand is met: T 1 = (Qo So)/D ž Time during which demand is backordered: T 2 = So/D ž Cycle time; T = T 1 + T 2 Model for shortages with lost orders R = revenue Z = proportion of demand met ž Cost of each unit of lost sales including loss of profits: LC = DC + SP UC ž Optimal revenue: Ro = Z [D LC 2 RC HC D] Model for constraints on space AC = additional cost related to the storage area (or volume) used by each unit of the item. S i = amount of space occupied by one unit of item i. ž The total holding cost per unit per unit time: HC + AC S i

232 Equations for Inventory Management ž Optimal order quantities: 2 RC i D i Q i = HC i + AC S i Model for constraint on investment ž Optimal order quantities: UL = upper limit on the total average investment Model for discrete variable demand 2 UL HC Q i = Qo i N UC VCo i ž Test for the point where it is more expensive to order for N + 1 periods than to order for N periods: N (N + 1) D N+1 > 2 RC HC ž Confirming that it is more expensive to order for N + 2 periods than to order for N periods: N (N + 2) [D N+1 + D N+2 ] > 4 RC HC ž Variable cost per period: i=1 VC N = RC N + HC Chapter 5 Models for uncertain demand Model for the newsboy problem ž Test for the optimal order size: Prob(D Qo) > 2 SP = selling price SV = scrap value UC SV SP SV ž Expected profit with buying Q units: Q EP(Q) = SP D Prob(D) + Q D=0 N i=1 D i > Prob(D Qo + 1) D=Q+1 Prob(D) Q UC

Model for discrete demand with shortages ž Test for the optimal stock level: Equations for Inventory Management 233 A = Actual stock level SC Prob(D Ao) Prob(D Ao 1) HC + SC Approach to intermittent demand ž Service level = 1 Prob(shortage) = 1 [Prob(there is a demand) Prob(demand > A)] Joint calculation of order quantity and reorder level with shortages ž Calculation for order quantity: Q = 2 D [ ] HC RC + SC (D ROL) Prob(D) ž Calculation for reorder level: HC Q SC D = Model for order quantity with shortages ž Order quantity: [ Q = 2 D HC RC + SC Model for uncertain lead time demand ž Safety stock: ž Reorder level: D=ROL D=ROL D=ROL Prob(D) (D ROL) Prob(D) SS = Z standard deviation of lead time = Z σ LT ROL = lead time demand + safety stock = LT D + Z σ LT Model for service level with uncertain lead time ž Service level = Prob (LT D < ROL) = Prob(LT < ROL/D) ]

234 Equations for Inventory Management Model for periodic review method ž Target stock level: TSL = D (T + LT) + Z σ (T + LT) Chapter 6 Sources of information Accounting information ž Cost of products sold = opening stock + net purchases closing stock ž Value of stock = number of units in stock unit value ž Average cost = Total cost of units Number of units bought ž Closing stock = opening stock + purchases sales ž Gross profit = sales revenue cost of units sold Chapter 7 Forecasting demand Value of demand in a time series Linear relationship Actual demand = underlying pattern + random noise dependent variable = a + b independent variable y = a + bx x = value of the independent variable y = value of the dependent variable a = intercept, where the line crosses the y axis b = gradient of the line. ž Equations for linear regression: b = n (x y) x y n ( ) 2 x 2 x a = y x n b n ž coefficient of determination = (coefficient of correlation) 2

Equations for Inventory Management 235 Multiple regression y = a + b 1 variable 1 + b 2 variable 2 + b 3 variable 3 + b 4 variable 4... Exponential smoothing ž New Forecast = α latest demand + (1 α) previous forecast ž α is the smoothing constant (usually between 0.1 and 0.2) ž Tracking signal = sum of forecast errors mean absolute deviation seasonal value ž Seasonal index = deseasonalized value ž Demand = (underlying value + trend) seasonal index + noise Chapter 8 Planning and stocks Stock and planning Stock at end Stock at Production Demand Backorders Backorders of this = end of last + during met during from earlier + met in later period period this period this period periods periods Chapter 9 Material requirements planning ž Basic calculation Gross requirements = number of units made amount of material for each unit Net requirements = gross requirements current stock stock on order ž Batching rule to find N Where: N (N + 1) D N+1 > 2 RC HC N = the period number in a cycle D N+1 = demand in period N + 1 of a cycle

236 Equations for Inventory Management Chapter 10 Just-in-time ž Number of kanbans to maintain smooth operations demand in the cycle Number of kanbans = size of each container D (TP + TD) K = C Where: C = number of units held in each container TP = time container spends in production part of a cycle (waiting, being filled and moving to the store of work in progress) TD = time container spends in demand part of a cycle (waiting, being emptied and moving to the store of work in progress). Totalcyclelength = TP + TD ž Number of kanbans with safety factor SF = safety factor (generally less than 0.1) K < ž Maximum stock of work in progress D (TP + TD) (1 + SF) C Maximum stock level = K C = D (TP + TD) (1 + SF)