Inventory Management

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Inventory Management

Claudio Arbib

Università dell’Aquila __________

Part I: periodic orders

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1. Introduction

• Generally speaking,

producing a good involves the use of resources

available in limited amounts

• A resource is normally regarded as a stock

– in input, to feed production during some time span

– in output, to be delivered to customers

• Managing stocks normally asks for the solution of

decision problems

Main cost sources

• transport

• inventory

• workforce

Decisions

• how much is to be purchased/produced

• when one has to purchase/produce

Data & constraints

• process parameters

• demand

• logistic infrastructure

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Problem outlook

• Generally speaking,

producing a good involves the use of resources

available in limited amounts

• A resource is normally regarded as a stock

– in input, to feed production during some time span

– in output, to be delivered to customers

• Managing stocks normally asks for the solution of

decision problems

Example A plant makes use of two resource types.

Let b₁₁ = 10 tons b₂₁ = 9 tons be the amount of resource 1 and 2 available on day 1

Process parameters:

The process consumes

a₁ = 2 units of resource 1 a₂ = 3 units of resource 2 per unit of finished product.

The daily production capacity of the plant is q_max = 2 tons

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Problem outlook

• Let x_t be the amount produced on day t

• Then x₁ < b₁₁/a₁ x₁ < b₂₁/a₂ 0 < x₁ < q_max

Example A plant makes use of two resource types.

Let b₁₁ = 10 tons b₂₁ = 9 tons be the amount of resource 1 and 2 available on day 1

Process parameters:

The process consumes

a₁ = 2 units of resource 1 a₂ = 3 units of resource 2 per unit of finished product.

The daily production capacity of the plant is q_max = 2 tons 5

3 2

• The solution x₁^* = 2 fully exploits the production capacity of day 1

• At the beginning of day 2 stock levels therefore are

b₁₂ = b₁₁ – a₁x₁^* b₂₂ = b₂₁ – a₂x₁^*

6 3

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Problem outlook

• Let x_t be the amount produced on day t

• Then x₁ < b₁₁/a₁ x₁ < b₂₁/a₂ 0 < x₁ < q_max

• Iterating the procedure one gets

x₂ < b₁₂/a₁ x₂ < b₂₂/a₂ 0 < x₂ < q_max

• The solution x₁^* = 2 fully exploits the production capacity of day 1

• At the beginning of day 2 stock levels therefore are

b₁₂ = b₁₁ – a₁x₁^* b₂₂ = b₂₁ – a₂x₁^*

6 3

3 1 2

• The max production level is x₂^* = 1, which does not

exploits the production capacity of day 2

• To maximize production one then has to replenish with at least r = a₂q_max – b₂₂ within t = 1

3 5

3 2

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2. Delivery costs

• Summarizing, we were able to focus on the following decisions:

– purchase amount r > 3 – purchase day t < 1

• Delivery costs are normally formed by two terms:

• A fixed cost s₀, to be paid even if one moves a single gram of resource

• A variable cost which

increases with the ordered amount r of resource, for example

• To further narrowing on this issue we must examine the cost sources

• Let’s begin with delivery costs

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• Delivery costs are normally formed by two terms:

• A fixed cost s₀, to be paid even if one moves a single gram of resource

• A variable cost which

increases with the ordered amount r of resource, for example

cost

quantity

0 10 20 30 40 50 60

s₀

s0 + s¹r

ks⁰+ s¹r for (k – 1)d < r< kd

s₀ + s₁r for 0 < r < d s₀ + s₁d + s₂(r – d) for r > d, s₂ < s₁

s₀for any r < D

Delivery costs

ks₀ for (k – 1)d < r < kd

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Delivery costs

• Also specific aspects of the logistic infrastructure

– supplier multiplicity – link capacity

– …

have an impact on costs

• Heterogeneous and/or still non-critical resources can be loaded onto a single vehicle up to its capacity, so

avoiding fixed cost duplication

Supplier 2 R₁, R₃ Customer

Supplier 1 R₁, R₂

Supplier 3 R₂

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Delivery costs

• Also specific aspects of the logistic infrastructure

– supplier multiplicity – link capacity

– …

have an impact on costs

• Heterogeneous and/or still non-critical resources can be loaded onto a single vehicle up to its capacity, so

avoiding fixed cost duplication

Customer Supplier 1

R₁, R₂

Supplier 2 R₁, R₃ Supplier 3

R₂

freight consolidation

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Delivery costs

• It can be then convenient to synchronize replenishment so as to share transport resources and obtain economy of scale

0 25 50 75 100

mon tue wed thu fri sat

resource 1 resource 2 Inventory levels

0 25 50 75 100

purchase 1 purchase 2

Replenishment (today for tomorrow) four

deliveries

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Delivery costs

0 25 50 75 100

purchase 1 purchase 2

Replenishment (today for tomorrow) four

deliveries

0 25 50 75 100

resource 1 resource 2 Inventory levels

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0 25 50 75 100

resource 1 resource 2

Delivery costs

0 25 50 75 100

resource 1 resource 2

0 25 50 75 100

purchase 1 purchase 2 Replenishment (today for tomorrow)

Inventory levels

just three deliveries

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3. Inventory costs

• However, holding a material good entails a cost, that can be basically viewed at in terms of lost return

• Think to a bank account: every year, interests give you an extra, whose value, say U, increases with

– the permanence of money on account – the official interest rate s_t

on every day t of the period T

U =

S

^s_t^v_t^b_t

tÎT

1

|T|

0 25 50 75 100

resource 1 value (€) Stock level and economic value

g₀(t)b_t

Inventory cost for one resource unit on day t

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Inventory costs

• The less stock is maintained in the system;

the less is the material latency in the system

(also called lead time);

0 25 50 75 100

lun mar mer gio ven sab

Scorta 1 Valore Livello e valore economico della scorta

(€)

0 25 50 75 100

lun mar mer gio ven sab

Scorta 1 Valore Livello e valore economico della scorta

(€)

Production or logistic system

resources products

the less are inventory costs.

• Inventory costs therefore depend on:

– the economic value of each resource

– the average amount hold or hidden in the system within the planning period

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Drawbacks

• Usually, the inventory cost behavior is opposite to that delivery costs

0 20 40 60 80 100

inv entory purchase

high inventory level two deliveries

per week

0 20 40 60 80 100

inv entory purchase

low inventory level three deliveries per week

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0 20 40 60 80 100 120

1 2 3 4 5 6

inventory

replenishment total

Drawbacks

• In a given planning period T

– the inventory cost decreases – the delivery cost increases

with the frequency f = 1/Dt of resource replenishment in the period

minimum cost

Replenishment frequency

cost

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4. Periodic orders

• In order to model the problem of minimizing the total cost (inventory + delivery) let us begin with assuming that:

– a single resource is consumed at a fixed absorption ratio a₀ up to some limit b_s (safety stock)

magazine

Stock level (b₀= 10)

safety stock b_s

consuming process

(a₀ = 2)

− the stock level b(t) in the magazine linearly decreases with time

− the average inventory in the interval [0, DT]

corresponds to the blue area = ò₀^DT b(t)dt stock level b(t)

DT ^{time t}

b(t) = b0 –a

0t

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stock level DT

time

safety stock b_s

Periodic orders

• In order to model the problem of minimizing the total cost (inventory + delivery) let us begin with assuming that:

– a single resource is consumed at a fixed absorption ratio a₀ up to some limit b_s (safety stock)

Halving the replenishment period (doubling frequency)

=

Halving variable inventory + Doubling deliveries

Dt 2

− purchase occurs periodically with fixed period Dt at a cost of s₀ euros per delivery, independently on the amount delivered

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s(Dt) = s₀^× g(Dt) = g₀^×

ò

₀^T^b(t)dt

T Dt

(E

^conomic

O

^rder

Q

^uantity

)

• One can then express the

inventory and delivery costs in a given time horizon T as:

g 0(t), s(r)

t, r

s₀ g₀

• Assume that

– the holding cost g₀(t) of a

resource unit in a unit time (e.g.

one day) does not change with time: g₀(t) = g₀

– the cost s(r) of a single delivery does not depend on the amount r delivered: s(r) = s₀

Dt b 0–b s= a 0Dt

b_s b(t) = b

0 – a

0t

Area = a₀Dt²/2 + b_sDt

g₀ ^T (a₀Dt² + 2b_sDt) g2Dt₀T(a₀Dt/2 + b_s)

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(E

^conomic

O

^rder

Q

^uantity

)

cost

time

• There exists a single optimal solution Dt ^*, independent on T and b_s, that can be computed through the first derivative:

• The total management cost equals

s(Dt) = s₀^× g(Dt) =

ò

T Dt

• One can then express the inventory and delivery costs in a given time horizon T as:

Dt^* = Ö ²^s₀^/a₀^g₀

g₀T(a₀Dt/2 + b_s)

C(Dt) = T( _Dt^s⁰ + ^a⁰^g⁰^Dt + g₀b_s)

2 a₀g₀ s₀

0 = –

2 Dt²

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Variable absorption ratio

• In some cases (e.g. perishable goods, obsolescence etc.) the constant absorption rate assumption can be too simplistic.

• Example. In a food industry the amount of usable product decreases with the time t that has passed from its purchase.

Hence the amount necessary to obtain a unit of finished product increases with t, or in other words, the stock absorption rate

is an increasing function of time: a = a(t)

Call r = b₀ the stock of fresh semi-finite material acquired at t₀ = 0. After t hours, the stock level will be

b(t) = b₀– ^ta(t)dt

0

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Inventory cost

• In general, the perishing rate of a product has an exponential behavior. Suppose a(t) = a₀e^lt, with a(0) = a₀ (initial absorption rate) and l > 0. One then has

Call r = b₀ the stock of fresh semi-finite material acquired at t₀ = 0. After t hours, the stock level will be

b(t) = b₀– a₀ (e^lt – 1) l

stock level

time

Dt

safety stock b_s acquired lot b₀

b(t) = b₀– a(t)dt

0 t

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Inventory cost

Let Dt be the replenishment period. To avoid useless holding, one has to choose the replenishment b₀ so that the residual stock at time Dt equals the safety stock b_s

b(t) = b₀– a₀ (e^lt – 1) l

stock level

time

Dt

b_s = b₀– a₀ (e^lDt – 1) Þ

l b₀ = b_s + (ea₀ ^lDt – 1) l

b(t) = b_s + (ea₀ ^lDt – e^lt) l

• In general, the perishing rate of a product has an exponential behavior. Suppose a(t) = a₀e^lt, with a(0) = a₀ (initial absorption rate) and l > 0. One then has

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Inventory cost

The inventory cost depend on the period Dt. In order to find it, integrate b(t) within the interval [0, Dt]

stock level

time

= b_sDt + ^a⁰_l^Dt e^lDt – _l^a⁰₂ (e^lDt – 1) b(t) = b_s + (ea₀ ^lDt – e^lt)

l

0 Dt

a₀

ò

^(b_s ^{+ e}l ^lDt^{)dt –} ^a_l⁰

ò

^e^lt ^dt

• In general, the perishing rate of a product has an exponential behavior. Suppose a(t) = a₀e^lt, with a(0) = a₀ (initial absorption rate) and l > 0. One then has

Dt

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Inventory cost

• Calculations give the total area as a function of Dt

= b_sDt + ^a⁰_l^Dt e^lDt – _l^a⁰₂ (e^lDt – 1)

T Dt

stock level

time

safety stock b_s

Dt

acquired lot b₀

This area is repeated T/Dt times in the planning period T Area = T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1)

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Inventory cost

• Calculations give the total area as a function of Dt

stock level

time

safety stock b_s

Dt

acquired lot b₀

The inventory cost is proportional to the area and to g₀ Area = T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1)

g(Dt) = g₀T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1)

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Total cost

The total cost is formed by

• the inventory cost

• the delivery costs (proportional to s₀

and to the number of deliveries in the plannig period)

• the cost due to product losses by perhishing (proportional to the product value v₀ and to

the amount perished during Dt)

amount (= b_s) actually in the magazine after Dt hours

stock level

Dt

amount (= b₀ – a₀Dt) absorbed by the process at a rate a₀ after Dt hours amount (= b₀ – a₀Dt – b_s) perished in Dt hours (with b₀ – b_s = a₀(e^lDt – 1)/l)

g(Dt) = g₀T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1)

s(Dt) = s₀T/Dt

p(Dt) = v₀(b₀ – a₀Dt – b_s)

See slide

(e^lDt – lDt – 1)

v₀a₀ l

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Total cost

The total cost is formed by

• the inventory cost

• the delivery costs

• the cost due to product losses by perishing C(Dt) =

= g₀T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1) + ^s⁰ +

g₀Dt ^v⁰_l^a⁰ (e^lDt – lDt – 1)

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Optimal replenishment period

• Like in the case of constant absorption rate, the optimal replenish ment period Dt^* is computed through the first derivative of C(Dt)

C(Dt) =

= g₀T b_s + e^a_l⁰ ^lDt – _l^a₂_Dt⁰ (e^lDt – 1) + ^s⁰ +

g₀Dt ^v⁰_l^a⁰ (e^lDt – lDt – 1) 0 = a₀ e^lDt – _l₂¹_Dt₂ – e^lDt + e^lDt – ^s⁰ +

a₀g₀Dt² 1

lDt

1

l²Dt² ^v⁰ (e^lDt – 1)

g₀T

1 – + + e^lDt = –^{1 + l}

2s₀/a₀g₀ l²Dt² 1

lDt

1 l²Dt²

v₀ Tg₀ v₀

g₀T

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Optimal replenishment period

Optimal replenishment period

0,00 200,00 400,00 600,00 800,00 1000,00 1200,00

1 2 3 4 5 6 7 8 9 10

l = 0,2 a₀ = 5 g₀ = 2 s₀ = 10.000 v₀ = 5.000 T = 1.000

1 – _lDt¹ + + e¹ ^lDt = ^{1 + l}_l²₂^s_Dt⁰^/a₂ ⁰^g⁰ –

l²Dt²

v₀ Tg₀ v₀

g₀T

Dt* ~ 3,467

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5. Limits of EOQ

• EOQ does not allow us to

1) express the delivery cost as a function of the amount of replenishment and/or of the distance from supplier

s₀

s₀for any r < D

assumed by EOQ

ks₀ for (k – 1)d < r < kd

more realistic

cost

amount 0 10 20 30 40 50 60

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Limits of EOQ

2) express the impact of resource value oscillation in time on inventory costs

assumed by EOQ

resource value

possible value oscillation

resource value

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Limits EOQ

3) take into account fluctuations of the resource absorption rate due, for instance, to demand variation

stock level

time

Dt

assumed by EOQ

stock lelve

time

Dt

possible demand fluctuation

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Limits of EOQ

4) optimize the simultaneous management of more resource types

optimal replenishment of resource 1

10 deliveries

sub-optimal replenishment of resource 1

6 deliveries

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Limits of EOQ

• The first of those four limits:

can sometimes be easily dealt with

In fact, for a fixed absorption rate (b(t) = b₀ – a₀t), if the delivery cost has the form s(r) = s₀ + s₁r

cost

quantity 0 10 20 30 40

s₀

s₀+ s¹r

s(Dt) = [s₀ + s₁(b₀ – b_s)] =

= [s₀ + s₁a₀Dt] =

= s₀ + Ts₁a₀

T Dt

its sum during T is

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Limits of EOQ

• The first of those four limits:

can sometimes be easily dealt with

The cost term Ts₁a₀ does not depend on period Dt and does not affect the first derivative of the total cost.

Hence, it does not affect the value Dt^* of the optimal replenishment period.

Note that this is not true when the absorption is not constant.

cost

quantity 0 10 20 30 40

s₀

s₀+ s¹r

s(Dt) = s_Dt^T ₀ + Ts₁a₀

Inventory Management