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# Equations

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### Equations

1. 1. Equations for Inventory ManagementChapter 1 Stocks and inventoriesEmpirical observation for the amount of stock held in a number of locations: N2 AS(N2 ) = AS(N1 ) × N1where: N2 = number of planned future facilities N1 = number of existing facilitiesAS(Ni ) = aggregate stock with Ni facilitiesChapter 3 Economic order quantityThe variables used here, and throughout the book, are: Q= order quantity Qo = optimal order quantity D= demand UC = unit cost RC = reorder cost HC = holding cost T= cycle length To = optimal cycle length VC = variable cost per unit time VCo = optimal variable cost per unit time TC = total cost per unit time TCo = optimal total cost per unit timeROL = reorder level LT = lead timež Economic order quantity: 2 × RC × D Qo = HCž Optimal stock cycle length: 2 × RC To = Qo/D = D × HC
2. 2. 230 Equations for Inventory Managementž Variable cost per unit time: RC × D HC × Q VC = + Q 2ž Optimal value of variable cost per unit time: 2 × RC × D √ VCo = HC × Qo = = 2 × RC × HC × D Qož Total cost per unit time: TC = UC × D + VCž Optimal cost per unit time: TCo = UC × D + VCož Change of variable cost moving away from the EOQ: VC 1 Qo Q = × + VCo 2 Q Qož Reorder level: Reorder level = lead time demand − stock on order ROL = LT × D − n × QoChapter 4 Models for known demandModel for ﬁnite replenishment rate, Pž Optimal order quantity: 2 × RC × D P Qo = × HC P−Dž Optimal time cycle time: 2 × RC P To = × HC × D P−Dž Optimal variable cost: √ P−D VCo = 2 × RC × HC × D × Pž Optimal total cost: TCo = UC × D + VCo
3. 3. Equations for Inventory Management 231Model for planned shortages and backorders SC = shortage cost per unit per unit timež Optimal order quantity: 2 × RC × D × (HC + SC) Qo = HC × SCž Optimal amount to be backordered: 2 × RC × HC × D So = SC × (HC + SC)ž Time during which demand is met: T1 = (Qo − So)/Dž Time during which demand is backordered: T2 = So/Dž Cycle time; T = T1 + T2Model for shortages with lost orders R = revenue Z = proportion of demand metž Cost of each unit of lost sales including loss of proﬁts: LC = DC + SP − UCž 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. Si = amount of space occupied by one unit of item i.ž The total holding cost per unit per unit time: HC + AC × Si
4. 4. 232 Equations for Inventory Managementž Optimal order quantities: 2 × RCi × Di Qi = HCi + AC × SiModel for constraint on investment UL = upper limit on the total average investmentž Optimal order quantities: 2 × UL × HC Qi = Qoi × N UC × VCoi i=1Model for discrete variable demandž Test for the point where it is more expensive to order for N + 1 periods than to order for N periods: 2 × RC N × (N + 1) × DN+1 > HCž Conﬁrming that it is more expensive to order for N + 2 periods than to order for N periods: 4 × RC N × (N + 2) × [DN+1 + DN+2 ] > HCž Variable cost per period: N HC × Di RC i=1 VCN = + N 2Chapter 5 Models for uncertain demandModel for the newsboy problem SP = selling price SV = scrap valuež Test for the optimal order size: UC − SV Prob(D ≥ Qo) > > Prob(D ≥ Qo + 1) SP − SVž Expected proﬁt with buying Q units:   Q ∞ EP(Q) = SP ×  D × Prob(D) + Q× Prob(D) − Q × UC D=0 D=Q+1
5. 5. Equations for Inventory Management 233Model for discrete demand with shortages A = Actual stock levelž Test for the optimal stock level: SC Prob(D ≤ Ao) ≥ ≥ Prob(D ≤ Ao − 1) HC + SCApproach to intermittent demandž Service level = 1 − Prob(shortage) = 1 − [Prob(there is a demand) × Prob(demand > A)]Joint calculation of order quantity and reorder level with shortagesž Calculation for order quantity: ∞ 2×D Q= × RC + SC × (D − ROL) × Prob(D) HC D=ROLž Calculation for reorder level: ∞ HC × Q = Prob(D) SC × D D=ROLModel for order quantity with shortagesž Order quantity: ∞ 2×D Q= × RC + SC × (D − ROL) × Prob(D) HC D=ROLModel for uncertain lead time demandž Safety stock: √ SS = Z × standard deviation of lead time = Z × σ × LTž Reorder level: √ ROL = lead time demand + safety stock = LT × D + Z × σ × LTModel for service level with uncertain lead timež Service level = Prob (LT × D < ROL) = Prob(LT < ROL/D)
6. 6. 234 Equations for Inventory ManagementModel for periodic review methodž Target stock level: √ TSL = D × (T + LT) + Z × σ × (T + LT)Chapter 6 Sources of informationAccounting informationž Cost of products sold = opening stock + net purchases − closing stockž Value of stock = number of units in stock × unit value Total cost of unitsž Average cost = Number of units boughtž Closing stock = opening stock + purchases − salesž Gross proﬁt = sales revenue − cost of units soldChapter 7 Forecasting demandValue of demand in a time series Actual demand = underlying pattern + random noiseLinear relationship 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: n× (x × y) − x× y b= 2 n× x2 − x y x a= −b× n nž coefﬁcient of determination = (coefﬁcient of correlation)2
7. 7. Equations for Inventory Management 235Multiple regressiony = a + b1 × variable 1 + b2 × variable 2 + b3 × variable 3 + b4 × variable 4 . . . .Exponential smoothingž New Forecast = α × latest demand + (1 − α) × previous forecastž α is the smoothing constant (usually between 0.1 and 0.2) sum of forecast errorsž Tracking signal = mean absolute deviation seasonal valuež Seasonal index = deseasonalized valuež Demand = (underlying value + trend) × seasonal index + noiseChapter 8 Planning and stocksStock 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 periodsChapter 9 Material requirements planningž Basic calculation Gross requirements = number of units made × amount of material for each unit Net requirements = gross requirements – current stock – stock on orderž Batching rule to ﬁnd N 2 × RC N × (N + 1) × DN+1 > HCWhere: N = the period number in a cycleDN+1 = demand in period N + 1 of a cycle
8. 8. 236 Equations for Inventory ManagementChapter 10 Just-in-timež Number of kanbans to maintain smooth operations demand in the cycle Number of kanbans = size of each container D × (TP + TD) K= CWhere: 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 + TDž Number of kanbans with safety factor SF = safety factor (generally less than 0.1) D × (TP + TD) × (1 + SF) K< Cž Maximum stock of work in progress Maximum stock level = K × C = D × (TP + TD) × (1 + SF)