Equations

Equations for Inventory Management

Chapter 1 Stocks and inventories

Empirical observation for the amount of stock held in a number of locations:

AS(N2) = AS(N1) ×√

N2

N1

where:

N2 = number of planned future facilitiesN1 = 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 = order quantity Qo = optimal order quantityD = demand

UC = unit costRC = reorder costHC = holding cost

T = cycle length To = optimal cycle lengthVC = variable cost per unit time VCo = optimal variable cost per unit timeTC = total cost per unit time TCo = optimal total cost per unit time

ROL = reorder levelLT = lead time

ž Economic order quantity:

Qo =√

2 × RC × DHC

ž Optimal stock cycle length:

To = Qo/D =√

2 × RCD × HC

230 Equations for Inventory Management

ž Variable cost per unit time:

VC = RC × DQ

+ HC × Q2

ž Optimal value of variable cost per unit time:

VCo = HC × Qo = 2 × RC × DQo

= √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:

VCVCo

= 12

×[

QoQ

+ QQo

]

ž Reorder level:

Reorder level = lead time demand − stock on order

ROL = LT × D − n × Qo

Chapter 4 Models for known demandModel for finite replenishment rate, P

ž Optimal order quantity:

Qo =√

2 × RC × DHC

×√

PP − D

ž Optimal time cycle time:

To =√

2 × RCHC × D

×√

PP − D

ž Optimal variable cost:

VCo = √2 × RC × HC × D ×

√P − D

P

ž Optimal total cost:TCo = UC × D + VCo

Equations for Inventory Management 231

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:

So =√

2 × RC × HC × DSC × (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


ž 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∑

i=1

VCoi

Model for discrete variable demand

ž Test for the point where it is more expensive to order for N + 1 periods than toorder for N periods:

N × (N + 1) × DN+1 >2 × RC

HC

ž Confirming that it is more expensive to order for N + 2 periods than to orderfor N periods:

N × (N + 2) × [DN+1 + DN+2] >4 × RC

HC

ž Variable cost per period:

VCN = RCN

+HC ×

N∑i=1

Di

2

Chapter 5 Models for uncertain demandModel for the newsboy problem

SP = selling price

SV = scrap value

ž Test for the optimal order size:

Prob(D ≥ Qo) >UC − SVSP − SV

> Prob(D ≥ Qo + 1)

ž Expected profit with buying Q units:

EP(Q) = SP × Q∑

D=0

D × Prob(D) + Q×∞∑

D=Q+1

Prob(D)

− Q × UC


Model for discrete demand with shortages

A = Actual stock level

ž Test for the optimal stock level:

Prob(D ≤ Ao) ≥ SCHC + SC

≥ Prob(D ≤ Ao − 1)

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

(D − ROL) × Prob(D)

]

ž Calculation for reorder level:

HC × QSC × D

=∞∑

D=ROL

Prob(D)

Model for order quantity with shortages

ž Order quantity:

Q =√√√√2 × D

HC×

[RC + SC ×

∞∑D=ROL

(D − ROL) × Prob(D)

]

Model for uncertain lead time demand

ž Safety stock:

SS = Z × standard deviation of lead time = Z × σ × √LT

ž Reorder level:

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)


Model for periodic review method

ž Target stock level:

TSL = D × (T + LT) + Z × σ × √(T + LT)

Chapter 6 Sources of informationAccounting 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 unitsNumber of units bought

ž Closing stock = opening stock + purchases − sales

ž Gross profit = sales revenue − cost of units sold

Chapter 7 Forecasting demandValue 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

n ×∑

x2 −(∑

x)2

a =∑

y

n− b ×

∑x

n

ž coefficient of determination = (coefficient of correlation)2


Multiple regression

y = a + b1 × variable 1 + b2 × variable 2 + b3 × variable 3 + b4 × 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 errorsmean absolute deviation

ž Seasonal index = seasonal valuedeseasonalized 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 periodthis 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 >2 × RC

HC

Where:

N = the period number in a cycleDN+1 = demand in period N + 1 of a cycle


Chapter 10 Just-in-time

ž Number of kanbans to maintain smooth operations

Number of kanbans = demand in the cyclesize of each container

K = D × (TP + TD)

C

Where:

C = number of units held in each containerTP = 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 <D × (TP + TD) × (1 + SF)

C

ž Maximum stock of work in progress

Maximum stock level = K × C = D × (TP + TD) × (1 + SF)

Equations

Technology

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