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Chapter 10
Inventory management problems
Inventories are stockpiles of items (raw materials, components and finished goods) wait- ing to be processed, transported or used. The motivations for stockpiling such items are for improving the service level, reducing the overall logistics costs, coping with ran- domness in customer demands, and so on. However, keeping inventories can be very expensive. It is therefore crucial to solve inventory management problems: that is, for each stocking point in the supply chain, to decide when to reorder (for a single or mul- tiple products) and how much to order, so that the expected (annual) cost is minimized while achieving a certain service level.
Among the relevant involved costs, there are:
- Procurement costs associated with the acquisition of goods; they can be fixed costs or variable costs (i.e. dependent on the amount acquired): e.g. (fixed) reorder costs, (variable) purchasing or manufacturing costs, transportation costs.
- Inventory holding costs incurred when materials are stocked for a period of time: e.g. opportunity (or capital) costs, i.e. the return on investment the firm would have realized if money had been invested in a profitable economic activity instead of inventory; warehousing costs, which may include space and equipment costs, maintenance costs, taxes, or fees to pay for storing the goods.
- Shortage costs are paid when customer orders are not met; for example:
- lost sales costs: they are related to lost profit and to a negative effect on future sales;
- back order costs: they are due to the fact that, in case of a delayed sale, often a penalty has to be paid.
- Obsolescence costs.
Inventory management models can be classified according to various criteria:
- Deterministic versus stochastic models: in a deterministic model, demands, costs and lead times are assumed to be known in advance, while in a stochastic model
108 10.1. Single stocking point: single commodity, constant demand
some data are uncertain.
- Fast-moving items versus slow-moving items: fast-moving items are manufactured and sold regularly, so we have to decide when to replenish stocking points and how much to order each time; slow-moving items (e.g. spare parts of complex machineries) have a very low demand (e.g. few units every 20 years), and the main issue is to decide how many items to produce and store at the beginning of the planning horizon.
- Number of stocking points: usually optimal inventory policies can be derived an- alytically only for the case of a single stocking point.
- Number of commodities: inventory management models are different depending on the fact that they deal with single or with multiple commodities.
- Instantaneous resupply versus noninstantaneous resupply: there are differences depending on the fact that a stocking point can be replenished almost instanta- neously or not (e.g. at a constant rate via a manufacturing process).
Hereafter we shall present some basic inventory management models, with different characteristics according to the criteria that have been listed before.
10.1 Single stocking point: single commodity under con-
stant demand rate
The model is based on the following assumptions:
- we are dealing with a single stocking point for a single commodity, under a con- stant demand rate, say d;
- the planning horizon is infinite.
Since d is a constant, a natural policy is to replenish the stocking point on a periodic basis. To this end, let:
- T be the time lapse between two consecutive orders (period ),
- q be the order size, i.e. the amount of product to order each time.
The figure below provides a graphical representation of the relationship among demand, period and order size.
d d d d d d d d d d d d d d d
q q q q
T T T
time unit
Clearly it is: q = d · T.
110 10.1. Single stocking point: single commodity, constant demand
rate equal to d, since items are picked up at a rate d. Also, s + m = (r d) T (^) r , i.e.:
m = (r d)T (^) r s = r T |{z} (^) r q
d T |{z} (^) r q r
s = q
d r
s
Therefore, m depends on q and s, i.e. q and s are the variables to be determined, whereas T and m can then be computed by means of the formulas:
T =
q d
and m = q
d r
s
Recall that the optimal values of variables q and s are the ones that minimize the total average cost per unit of time, which is given by the following function:
μ(q, s) =
T
(K + c q + h I T¯ + u s + v S T¯ | {z } average cost per period
In this function, the holding cost h· I¯ ·T and the shortage costs (u·s and v · S¯ ·T ) depend on the average inventory level I¯ and on the average shortage level S¯, given respectively by:
T
Z T
0
I (^) + (t) | {z } “positive part” of I(t)
dt =
T
m(T 2 + T 3 ) 2
• S¯ =
T
Z T
0
I (^) (t) | {z } “negative part” if I(t)
dt =
T
s(T 1 + T 4 ) 2
See Figure 10.1 for a graphical illustration of the subintervals T 1 , T 2 , T 3 and T 4. In order to express T 1 , T 2 , T 3 and T 4 as functions of variables q and s, observe that:
8
<
:
s = (r d)T (^1) m = (r d)T (^2) m = d T (^3) s = d T (^4)
and so:
T 1 = s/(r d) T 2 = m/(r d) T 3 = m/d T 4 = s/d.
Therefore:
I^ ¯ =^1
T
m(T 2 + T 3 ) 2
T
m^2 ( (^) r ^1 d + (^1) d ) 2
d 2 q
m 2 d + r (^) d d (r d)
m^2 2 q
r r d
m 2 2 q
r d r
m^2 2 q(1 dr ) | {z } as a function of q and m
[q(1 dr ) s] 2 2 q(1 dr )
| {z } as a function of q and s
Chapter 10. Inventory management problems 111
In a similar way we can derive:
S^ ¯ = s^
2 2 q(1 dr )
By replacing (10.1) and (10.2) in μ(q, s) we get:
μ(q, s) = K
d q
h[q(1 dr ) s] 2 2 q(1 dr )
u s d q
v s 2 2 q(1 dr )
This is a nonlinear, yet convex, function. Therefore, a (global) minimum point (q ⇤^ , s ⇤^ ) of μ(q, s) exists and can be obtained by solving:
8
<
:
@q
μ(q, s) = 0
@ @s
μ(q, s) = 0
stationary point.
We get (no proof provided): 8
<
:
q ⇤^ =
r h + v v
s 2 Kd h(1 dr )
(u d) 2 h(h + v)
s ⇤^ =
(h q ⇤^ u d)(1 dr ) h + v
i.e., in each period, the optimal amount to order to minimize the total average unit cost is q ⇤^ , and the optimal time interval to order is T ⇤^ = q ⇤^ /d.
Furthermore, in each period the optimal shortage level (to satisfy the demand by mini- mizing μ(q, s)) is s ⇤^ , while the optimal maximum inventory level is m ⇤^ = q ⇤^ (1 dr ) s ⇤^.
Notice that, in the simple inventory scenario under consideration, we have been able to determine the optimal inventory policy in an analytical way, i.e. by means of the formulas that have been derived for q ⇤^ , T ⇤^ , s ⇤^ and m⇤^.
An example
Golden Food distributes tinned foodstuff in Great Britain. Consider a warehouse in Birmingham where:
- the demand rate d for tomato purée is 400 pallets a month (so, it is constant each month);
- the value of a pallet is c = £ 2 , 500 ;
- the annual interest rate is p = 14.5 %, including warehousing costs;
- issuing an order costs K = £ 30 ;
Chapter 10. Inventory management problems 113
t
I(t) q ⇤^ = m⇤ slope (^) = (^) d
T ⇤^ T ⇤
Figure 10.2: Inventory level as a function of time in the EOQ model
The inventory level, as a function of time, in EOQ becomes the one depicted in Fig- ure 10.2.
An example
Al-Bufeira motors manufactures spare parts for aircraft engines in Saudi Arabia. A certain component Y02PN has the following supply requirements:
- the demand is d = 220 units per year;
- the unit production cost is c = $1200;
- the set-up cost (for producing the component) is K = $800;
- the annual interest rate is p = 18 % (including warehousing costs);
- no shortages are allowed (i.e. s = 0).
Therefore the holding cost is:
h = 0. 18 · $1200 = $216 per year per unit.
Since replenishment is achieved via internal production, we can assume r! + 1 , and so the EOQ model can be applied (observe that d is a constant per year). According to this model we can therefore derive:
8
<
:
q ⇤^ =
r 2 Kd h
r 2 · 800 · 220 216 = 40. 37 units
T ⇤^ =
= 0. 18 years = 66. 8 days
Therefore, to minimize its cost Al-Bufeira Motors has to manufacture ⇠ 40 units of the component Y02PN every ⇠ 67 days.
Notice that, in practice, q ⇤^ and T ⇤^ must be suitably rounded in order to apply the optimal policy.
114 10.2. Single stocking point: multicommodity, capacity constraints
10.2 Single stocking point: multicommodity with capacity
constraints
Let us consider now inventory problems where different types of commodity must be stocked and a limit is placed on the total investment in inventories, or on the common space. Assume that, for each product j, the assumptions made for the EOQ model can be applied (see Section 10.1.1). In other words assume that, for each product j, the demand rate d (^) j is constant, the replenishment rate r (^) j goes to infinity, and that shortage is not allowed.
Let:
- n be the number of commodities in stock;
- q (^) j be the amount of commodity j to reorder at each replenishment, j = 1,... n.
Under the EOQ assumption, the average unit cost for commodity j is:
μ (^) j (q (^) j ) = K (^) j d (^) j q (^) j
h (^) j q (^) j , j = 1,... n,
where K (^) j is the fixed reorder cost, c (^) j is the unit value of item j and h (^) j = p (^) j c (^) j is the holding cost for item j.
Therefore, the average global unit cost (for all commodities in stock) is:
μ(q 1 , q 2 ,... qn ) =
X^ n
j=
μ (^) j (q (^) j ).
Assume now that there is a side, linear, constraint (capacity constraint), usually repre- senting a budget constraint or a warehouse constraint, of form:
X^ n
j=
a (^) j q (^) j b
where a (^) j , j = 1,... n, and b are constants. Then, the inventory management problem that we want to solve can be expressed as:
min
X^ n
j=
K (^) j d (^) j q (^) j
h (^) j q (^) j
X^ n
j=
a (^) j q (^) j b
q 1 , q 2 ,... qn 0
This is a nonlinear (hard) model, yet it is possible to suggest a simple heuristic for the special case of identical interest rates, i.e. p (^) j = p, j = 1,... n.
116 10.2. Single stocking point: multicommodity, capacity constraints
model yearly demand value yearly interest price Preppie knapsack d 1 = 150, 000 units c 1 = $30 h 1 = 20 % c (^1) Yuppie suitcase d 2 = 100, 000 units c 2 = $45 h 2 = 20 % c (^2)
Table 10.
An example
New Frontier distributes Knapsacks and suitcases, whose most successful models are the Preppie knapsack and the Yuppie suitcase. Table 10.1 summarizes the relevant information about demand, value and interest rate of the two products. Notice that the interest rate is at 20 % for both products. In addition, the reorder cost is the same, too: K 1 = K 2 = $250.
The company imposes the following additional constraint: the average capital invested in inventory can not exceed $75, 000 , i.e. 302 q 1 + 452 q 2 75 , 000. In fact, since
I^ ¯ = [q(1^ ^
d r )^ ^ s]^
2 2 q(1 dr )
is the average inventory level under the inventory policy presented in Section 10.1, in the special case of EOQ the average inventory level is I¯ = q/ 2.
So, q 1 / 2 is the average inventory level for Preppie knapsacks, q 2 / 2 is the average inven- tory level for Yuppie suitcases, and therefore 30 q 2 1 + 45 q 2 2 is the average capital invested in inventories. It is assumed, for precaution, that the overall average inventory level is the sum of the average inventory levels for the two products.
Therefore, we can apply the previously presented heuristic:
Step 1:
q 10 =
r 2 · 250 · 150 , 000
- 2 · 30
= 3, 535. 53 units
q 20 =
r 2 · 250 · 100 , 000
- 2 · 45
= 2, 357. 02 units
It is 30 q (^01) 2
q 20 2
Step 2: ⇤^ = 0. 2
q 1 (0.2) =
s 2 · 250 · 150 , 000 (0.2 + 0.2) · 30
= 2, 500 units
q 2 (0.2) =
s 2 · 250 · 100 , 000 (0.2 + 0.2) · 45
⇡ 1 , 666 units
Chapter 10. Inventory management problems 117
t
I(t)
I S
`
q
q
t (^) `
t (^) `
Figure 10.3: Example of utilization of the reorder point policy
Therefore, according to the heuristic we have to maintain in stock 2500 Preppie knap- sacks and about 1666 Yuppie suitcases.
10.3 The reorder point policy
The reorder point policy is a stochastic model„ since it assumes that the demand is uncertain. Precisely, it is assumed that the demand to the stocking point is a random variable. The inventory level I(t) is kept under observation in an almost continuous way; when it reaches a reorder point , then a constant quantity q is reordered. Such a quantity q will be delivered after a reorder time, or lead time, t (^) , which is assumed to be constant. Figure 10.3 shows an example of the policy at work.
The reorder size q may be computed as follows:
q =
r 2 K d¯ h
which is the EOQ formula, where d¯ is the expected demand rate in the time unit. In order to develop the model, we will make the assumption that the expected demand rate d¯ and the standard deviation (^) d are constant in each unit of time.
In order to determine the reorder point , we want that the inventory level I(t) be nonnegative, during t (^) , with a given probability ↵. This is equivalent to impose that, during the interval t (^) , the total demand does not exceed with probability ↵.
Let d be the random variable denoting the total demand in the lead time; then
d = (^) |{z}d (^1) first time unit
- d 2 +... + d (^) t (^) ` |{z} last time unit
By assuming an identical demand distribution in each unit of time, and assuming d 1 , d 2 ,... d (^) t (^) independent, if t (^) is sufficiently large then from the Central limit theorem
Chapter 10. Inventory management problems 119
First, compute the holding cost (the time unit is a month):
h = p c = 0. 2 · AC4 = AC 0. 8 /year per item = AC 0. 067 /month per item.
Then, compute the reorder size:
q =
r 2 K d¯ h
r 2 · 30 · 45
- 067 ⇡ 201 items.
Since z (^) ↵ = 2, to guarantee the required service level we have to set
` = 45 · 1 + 5 · 1 · 2 = 45 + 10 = 55 units, and so I (^) S = 55 45 = 10 units.
That is, we must reorder when the inventory level reaches ` = 55 units; the correspond- ing safety stock is I (^) S = 10 units (to take into account variations around the average demand in the reorder period).
10.4 The slow-moving item model
When the demand is very low (e.g. spare parts of complex machineries), the main issue is to determine how many items to purchase at the beginning of the planning horizon. Consider a scenario in which:
- c is the purchase cost of an item at the beginning of the planning horizon;
- u is the purchase cost of an item during the planning horizon (u > c due to storage penalty);
- holding costs are disregarded;
- n is the number of items to be purchased at the beginning of the planning horizon (decision variable);
- m is the number of items demanded in the planning horizon (unknown).
Given m and n, the total purchase cost is:
C(n, m) =
c n if n m c n + μ(m n) if n < m.
However, m is unknown. Let P (m) be the probability that m items will be demanded during the planning horizon. Then, the expected cost if n items will be purchased can be expressed as:
C(n) =
X^ +^1
m=
C(n, m) P (m) = c n + u
+X 1
m=n+
(m n) P (m).
120 10.4. The slow-moving item model
The problem to be solved is:
min C(n) n 0 , integer
To solve the problem in an analytical way, let F (m) = Prob(d m), where d denotes the demand in the planning horizon, which is a discrete random variable. That is, F (m) is the probability that m units (or less) will be demanded during the planning horizon. Then, the following two relationships hold:
C(n 1) = c (n 1) + u
X^ +^1
m=n
(m n + 1) P (m) =
= c n c + u
X^ +^1
m=n
(m n) P (m) + u
+X 1
m=n
P (m) =
= C(n) c + u Prob(d n) = = C(n) c + u Prob(d > n 1) = = C(n) c + u [1 F (n 1)]
C(n + 1) = c (n + 1) + u
X^ +^1
m=n+
(m n 1) P (m) =
= c n + c + u
+X 1
m=n+
(m n 1) P (m) =
= c n + c + u
+X 1
m=n+
(m n) P (m) u
+X 1
m=n+
P (m) =
= C(n) + c u Prob(d n + 1) = = C(n) + c u Prob(d > n) = = C(n) + c u [1 F (n)]
Now, the minimum expected cost is achieved if n ⇤^ items are purchased at the beginning of the planning horizon, where:
( C(n ⇤^ ) C(n ⇤^ 1) C(n ⇤^ ) C(n ⇤^ + 1)
By exploiting (10.5) and (10.6), we derive:
C(n ⇤^ ) c(n ⇤^ ) c + u [1 F (n ⇤^ 1)] () F (n ⇤^ 1) u^ ^ c u
C(n ⇤^ ) c(n ⇤^ ) + c u [1 F (n ⇤^ )] () F (n ⇤^ ) u^ ^ c u
122 10.4. The slow-moving item model
- 0 0.0009 0. m Prob(d = m) F (m) - 1 0.0064 0. - 2 0.0223 0. - 3 0.0521 0. - 4 0.0912 0. - 5 0.1277 0. - 6 0.1490 0. - 7 0.1490 0. - 8 0.1304 0. - 9 0.1014 0.
- 10 0.0710 0.
- References G. Ghiani, G. Laporte, and R. Musmanno (2004): Chapter Table 10.2: The cumulative probability distribution of d
