Inventory management

Inventory management

Giovanni Righini

Università degli Studi di Milano

Terminology and classification

Inventory systems

In the supply chain there are several points wherestocksare kept for different purposes.

These are all goods and products that do not undergo transformation, assembly or similar operations. Stocks are placed

in the production system (e.g. between a machine and another), in the distribution system (e.g. travelling stocks, stocks in the shops),

between a facility and another within the supply chain (e.g. warehouses, wholesalers).

Costs

Costsdue to stocks are mainly of three types:

costs for purchasing;

costs for holding the inventory; costs for obsolescence.

Stocked goods can be classified on the basis of: their value for unit of weight or volume; the frequency with which they are requested; the predictability/uncertainty of their demand.

Terminology and classification

Classification

Inventory systems can be classified with four main criteria: number of inventory locations,

number of product types,

deterministic or non-deterministic,

1/1/D/C systems: data

Single-point, single-product, deterministic inventory systems with continuous replenishment are characterized by:

a demand d ;

a replenishment rate r (r >d); a replenishment period Tr;

a period T ;

a maximum level M;

a maximum stock-out level s.

During the replenishmant periods the level of the inventory increases at a rate r₋d ; during the other periods if decreases at a rate d .

The period T and the order quantity q that is replenished in each replenishment period are linked with d and r through the relation

q=dT =rTr.

Continuous replenishment

1/1/D/C systems: graphical representation

I(t) Tr [-d] [r-d] M -s 0 T1 T2 T3 T4 T t

1/1/D/C systems: costs

We indicate the overallcost function(per unit of time) as follows:

µ(q,s) = 1

T(k +cq+hIT+us+v ST)

dove

k is the fixed cost for each replenishment operation; c is the price of the product;

h is the obsolescence cost per unit of product and per unit of

time;

u is the unitary stock-out cost;

v is the unitary stock-out cost per unit of product and per unit of

time;

I is the average inventory level; S is the average stock-out level.

Continuous replenishment

1/1/D/C systems: analysis

The following relations hold:

Tr =T1+T2 T =T1+T2+T3+T4 s+M= (r₋d)Tr I = 1 T M(T2+T3) 2 S= 1 T s(T1+T4) 2 s= (r−d)T1 M = (r₋d)T2 M =dT3 s=dT4.

From them we obtain

µ(q,s) =kd q +cd+ h[q(1−dr)−s]2 2q(1−dr) +usd q + vs2 2q(1−dr) .

1/1/D/C systems: solution

By computing the partial derivatives ofµ(q,s)wth respect to the two variables q and s and imposing they are null, we obtain:

q∗₌ s h+v v ( 2kd h(1−d/r)− (ud)2 h(h+v)) and s∗₌(hq∗−ud)(1−d/r) h+v

In the case with no stock-out allowed (s=0), they reduce to:

q∗₌

s

2kd

Discrete replenishment

1/1/D/D systems: graphical representation

We can study them as special cases of 1/1/D/C systems when

r _{→ ∞}. I(t) [-d] M -s 0 T t

1/1/D/D systems: solution

In the case with no stock-out allowed (s=0) we have

q∗₌

r

2kd

h

The corresponding value of the minimum cost is

µ(q∗_{) =}√_2kdh+cd.

Discounts on the price of the product

Types of discount

The optimal amount to purchase at every replenishment operation may also depend on other factors, like the possibility of obtaining discounts on larger quantities.

We consider two different types of discounts: discount on the whole quantity;

Discount on the whole quantity

The purchase cost is a(q) =ciq for qi−1≤q≤qi with i=1, . . . ,n.

The uitary cost (price) ci decreases when i increases:

qi >qi−1 ∀i=1, . . . ,n

ci <ci−1 ∀i =1, . . . ,n

With this type of discount it may happen that for two quantities q′_and q′′_{with q}′_<q′′_{we have a(q}′_)>a(q′′_).

Discounts on the price of the product

Discount on the whole quantity

For each price range i =1, . . . ,n:

we determine the value of the EOQqˆi as usual;

we set q∗ i =    qi−1 ifqˆi <qi−1 ˆ qi if qi−1≤qˆi ≤qi qi ifqˆi >qi

we compute the corresponding costµ(q∗ i ).

Finally we choose the price range for which the cost turns out to be minimum.

Incremental discounts

The purchase cost is given by a(q) =ai−1+ci(q−qi−1)for

qi−1≤q≤qi with i =1, . . . ,n.

The unit cost (price) ci decreases when i increases:

qi >qi−1 ∀i=1, . . . ,n,

ci <ci−1 ∀i =1, . . . ,n.

With this type of discount the cost function a(q)is monotonically increasing with q.

Discounts on the price of the product

Incremental discounts

for each price range we have

µi(q) = [k +ai−1+ci(q−qi−1)]

d

q +

p

2[ai−1+ci(q−qi−1)]. For each price range i =1, . . . ,n:

we computeqˆi =

q

2d[k+ai−1−ciqi−1]

pci ;

we discardqˆi if it falls outside the range[qi−1,qi].

Finally we select the price range for which the cost turns out to be minimum.