Equations for Inventory Management
Chapter 1 Stocks and inventories
Empirical observation for the amount of stock held in a number of locations:
N2
AS(N2 ) = AS(N1 ) ×
N1
where:
N2 = number of planned future facilities
N1 = number of existing facilities
AS(Ni ) = aggregate stock with Ni facilities
Chapter 3 Economic order quantity
The variables used here, and throughout the book, are:
Q=
D=
UC =
RC =
HC =
T=
VC =
TC =
ROL =
LT =
ž
order quantity
demand
unit cost
reorder cost
holding cost
cycle length
variable cost per unit time
total cost per unit time
reorder level
lead time
Qo = optimal order quantity
To = optimal cycle length
VCo = optimal variable cost per unit time
TCo = optimal total cost per unit time
Economic order quantity:
Qo =
ž
2 × RC × D
HC
Optimal stock cycle length:
To = Qo/D =
2 × RC
D × HC
230
ž
Equations for Inventory Management
Variable cost per unit time:
VC =
ž
RC × D HC × Q
+
Q
2
Optimal value of variable cost per unit time:
2 × RC × D √
= 2 × RC × HC × D
Qo
VCo = HC × Qo =
ž
Total cost per unit time:
ž
Optimal cost per unit time:
TC = UC × D + VC
TCo = UC × D + VCo
ž
Change of variable cost moving away from the EOQ:
VC
1
Qo
Q
= ×
+
VCo
2
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:
Qo =
ž
2 × RC × D
×
HC
To =
2 × RC
×
HC × D
P
P−D
Optimal variable cost:
VCo =
ž
P
P−D
Optimal time cycle time:
ž
Optimal total cost:
√
2 × RC × HC × D ×
TCo = UC × D + VCo
P−D
P
Equations for Inventory Management
Model for planned shortages and backorders
SC = shortage cost per unit per unit time
ž
Optimal order quantity:
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:
T1 = (Qo − So)/D
ž
Time during which demand is backordered:
T2 = So/D
ž
Cycle time;
T = T1 + T2
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.
Si = amount of space occupied by one unit of item i.
ž
The total holding cost per unit per unit time:
HC + AC × Si
231
232
ž
Equations for Inventory Management
Optimal order quantities:
Qi =
2 × RCi × Di
HCi + AC × Si
Model for constraint on investment
UL = upper limit on the total average investment
ž
Optimal order quantities:
Qi = Qoi ×
2 × UL × HC
UC ×
N
VCoi
i=1
Model for discrete variable demand
ž
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 >
HC
ž
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 ] >
HC
ž
Variable cost per period:
RC
+
VCN =
N
HC ×
N
Di
i=1
2
Chapter 5 Models for uncertain demand
Model for the newsboy problem
SP = selling price
SV = scrap value
ž
Test for the optimal order size:
Prob(D ≥ Qo) >
ž
UC − SV
> Prob(D ≥ Qo + 1)
SP − SV
Expected profit with buying Q units:


Q
∞
EP(Q) = SP × 
D × Prob(D) + Q×
Prob(D) − Q × UC
D=0
D=Q+1
Equations for Inventory Management
Model for discrete demand with shortages
A = Actual stock level
ž
Test for the optimal stock level:
Prob(D ≤ Ao) ≥
SC
≥ 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:
∞
2 × D
× RC + SC ×
Q=
(D − ROL) × Prob(D)
HC
D=ROL
ž
Calculation for reorder level:
∞
HC × Q
=
Prob(D)
SC × D
D=ROL
Model for order quantity with shortages
ž
Order quantity:
∞
2 × D
× RC + SC ×
Q=
(D − ROL) × Prob(D)
HC
D=ROL
Model for uncertain lead time demand
ž
Safety stock:
SS = Z × standard deviation of lead time = Z × σ ×
ž
√
LT
Reorder level:
ROL = lead time demand + safety stock = LT × D + Z × σ ×
Model for service level with uncertain lead time
ž
Service level = Prob (LT × D < ROL) = Prob(LT < ROL/D)
√
LT
233
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
ž
ž
Cost of products sold = opening stock + net purchases − closing stock
ž
ž
Closing stock = opening stock + purchases − sales
Value of stock = number of units in stock × unit value
Total cost of units
Average cost =
Number of units bought
Gross profit = sales revenue − cost of units sold
Chapter 7 Forecasting demand
Value of demand in a time series
Actual demand = underlying pattern + random noise
Linear 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:
b=
n×
(x × y) −
x×
y
2
n×
x2 −
x
a=
ž
n
y
−b×
x
n
coefficient of determination = (coefficient of correlation)2
Equations for Inventory Management
235
Multiple regression
y = a + b1 × variable 1 + b2 × variable 2 + b3 × variable 3 + b4 × variable 4 . . . .
Exponential smoothing
ž
ž
New Forecast = α × latest demand + (1 − α) × previous forecast
ž
Tracking signal =
sum of forecast errors
mean absolute deviation
ž
ž
Seasonal index =
seasonal value
deseasonalized value
α is the smoothing constant (usually between 0.1 and 0.2)
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
N × (N + 1) × DN+1 >
Where:
N = the period number in a cycle
DN+1 = demand in period N + 1 of a cycle
2 × RC
HC
236
Equations for Inventory Management
Chapter 10 Just-in-time
ž
Number of kanbans to maintain smooth operations
demand in the cycle
size of each container
D × (TP + TD)
K=
C
Number of kanbans =
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
D × (TP + TD) × (1 + SF)
C
Maximum stock of work in progress
Maximum stock level = K × C = D × (TP + TD) × (1 + SF)