Inventory management

Transcription

1 Inventory management Giovanni Righini Università degli Studi di Milano Logistics

2 Terminology and classification Inventory systems In the supply chain there are several points where stocks are 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).

3 Terminology and classification Costs Costs due 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.

4 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, replenishment modality (continuous or discrete).

5 Continuous replenishment 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 T r ; 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 = rt r. The value of q (or T ) is a decision variable.

6 Continuous replenishment 1/1/D/C systems: graphical representation M I(t) [r-d] [-d] T 1 T 2 T 3 T 4 0 t T r -s T The extreme values M and s can also be decision variables.

7 Continuous replenishment 1/1/D/C systems: costs We indicate the overall cost function (per unit of time) as follows: dove µ(q, s) = 1 (k + cq + hit + us+vst) T 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.

8 Continuous replenishment 1/1/D/C systems: analysis The following relations hold: T r = T 1 + T 2 T = T 1 + T 2 + T 3 + T 4 s + M = (r d)t r I = 1 T M(T 2 +T 3 ) 2 s(t 1 +T 4 ) 2 S = 1 T s = (r d)t 1 M = (r d)t 2 M = dt 3 s = dt 4. From them we obtain µ(q, s) = kd q + cd + h[q(1 d r ) s]2 2q(1 d r ) + usd q + vs 2 2q(1 d r ).

9 Continuous replenishment 1/1/D/C systems: solution By computing the partial derivatives of µ(q, s) wth respect to the two variables q and s and imposing they are null, we obtain: q h+v 2kd = ( v 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 2kd = h(1 d/r).

10 Discrete replenishment 1/1/D/D systems: graphical representation We can study them as special cases of 1/1/D/C systems when r. M I(t) [-d] 0 t -s T

11 Discrete replenishment 1/1/D/D systems: solution Then we have µ(q, s) = kd q q = h(q s)2 + cd + 2q + usd q + vs2 2q from which h+v ( 2kd v h (ud)2 h(h + v) ) s = (hq ud). h+v In the case with no stock-out allowed (s = 0) we have 2kd q = h The corresponding value of the minimum cost is µ(q ) = 2kdh+cd. The optimal value q is also called Economic Order Quantity (EOQ).

12 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; incremental discount.

13 Discounts on the price of the product Discount on the whole quantity costo 0 q 0 q 1 q 2 q 3 q The purchase cost is a(q) = c i q for q i 1 q q i with i = 1,...,n. The uitary cost (price) c i decreases when i increases: q i > q i 1 i = 1,...,n c i < c i 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 ).

14 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 EOQ ˆq i as usual; we set qi = ˆq i q i 1 q i if ˆq i < q i 1 if q i 1 ˆq i q i if ˆq i > q i we compute the corresponding cost µ(q i ). Finally we choose the price range for which the cost turns out to be minimum.

15 Discounts on the price of the product Incremental discounts costo 0 q 0 q 1 q 2 q 3 q The purchase cost is given by a(q) = a i 1 + c i (q q i 1 ) for q i 1 q q i with i = 1,...,n. The unit cost (price) c i decreases when i increases: q i > q i 1 i = 1,...,n, c i < c i 1 i = 1,...,n. With this type of discount the cost function a(q) is monotonically increasing with q.

16 Discounts on the price of the product Incremental discounts Since µ(q) = kd pc(q)q q + c(q)d + 2, that is µ(q) = (k + a(q)) d q + p 2 a(q), for each price range we have µ i (q) = [k + a i 1 + c i (q q i 1 )] d q + p 2 [a i 1 + c i (q q i 1 )]. For each price range i = 1,...,n: 2d[k+ai 1 c we compute ˆq i = i q i 1 ] pc i ; we discard ˆq i if it falls outside the range [q i 1, q i ]. Finally we select the price range for which the cost turns out to be minimum.