In this section, we recall some results we presented in [43] where FOHPNs have been used to model, simulate and control inventory management systems (IMS). In particular, independent demand systems are considered, that is, inventory systems where the demand of each product is considered as independent from the demand of all the other products. The following are the two main policies in this regard: fixed-order quantity system (FOQS) and periodic review systems (PRS) [78]. FOQS place an order of fixed size whenever stock falls to a certain level. A continuous monitoring of stock levels is necessary for such systems. They are mainly used when the demand is low and irregular and the value of the items is expensive. On the contrary, PRS place orders of varying size at regular intervals to raise the stock level to a specified value. They are mainly used when the demand is high and regular and the value of the items is low. The operating cost of PRS is lower than the one of FOQS.
We also present the most important models of IMS characterized by a fixed-order quantity—
namely, FOQS with finite lead time, finite replenishment rate and back order—and the basic periodic review model.
3.5.1 FOQSWITHFINITELEADTIME
Let us consider an FOQS with a finite lead time and a fixed reorder level. If we assume a continuous and constant demand, the stock level of an item can be represented by the pattern shown in Figure 3.7a, where
• L(τ)is the stock level at the generic time instantτ
• Q is the fixed order quantity
D
LT T T
ROL L(τ)
τ L(0)
Q
p5
UC Q+RC LT
Q
Q D
t1 Lmax–ROL
Lmax Lmax
Lmax–ROL–Q p1
p1 p3
p2
Q
t3 t4
t5 p4
SC D t2
(a) (b)
FIGURE 3.7 FOQS with finite lead time: (a) regular pattern and (b) net model.
• LT is the lead time, that is, the delay between placing an order and receiving the goods in stock
• T is the cycle time, that is, the time between two consecutive replenishment
• D is the demand, that is, the number of units to be supplied from stock in a given time period (it coincides with the constant slope, taken as positive, of the curves in Figure 3.7a).
In FOQS, we have a new order each time the stock level falls to the reorder level ROL.
The FOHPN model for this kind of systems is depicted within the grey rectangle in Figure 3.7b.
The marking m1of continuous place p1represents the stock level, while the marking m1of comple- mentary place p1 represents the available space in the storage area. By construction, at each time instantτ, it is true that m1+m1=Lmax, where Lmaxrepresents the maximum capacity of the storage area which is in general much greater than the order quantity Q.
When the stock level is different from zero, that is, when m1 >0, transition t2 may fire at its MFS D, thus reducing the marking of p1 with a constant slope D. As soon as the marking of p1
falls to the reorder level ROL (i.e., the marking of p1goes over Lmax−ROL), discrete transition t1is enabled and fires after LT time units. When t1fires, the ordered quantity Q is received in the stock;
thus, this firing produces an increasing of Q units in p1and a decreasing of the same quantity in the complementary place p1.
3.5.1.1 Management Costs Evaluation
Let us now discuss how an appropriate FOHPN module can be used to compute the total costs related to the IMS module previously described.
Let us first introduce the following notation:
• UC is the unit cost, that is, the price charged by the suppliers for one unit of the item.
• RC is the reorder cost, that is, the cost of placing a routine order for the item and might include allowances for drawing up an order, computer time, correspondence, telephone costs, receiving, use of equipment, expediting delivery and quantity checks.
• HC is the holding cost, that is, the cost of holding one unit of an item in stock for one period of time.
• Lavis the average stock level.
• This the time held, that is, the interval of simulation.
• SC is the shortage cost: if the demand for an item cannot be met because stocks have been exhausted, there is usually some associated shortage cost.
• Nscis the number of lost sales.
The total cost C during an interval of time Thmay be calculated as follows:
C=UCãQãN+RCãN+HCãLavãTh+SCãNsc. (3.10) Thus, it is the sum of four different components: the unit cost component UCãQãN, the reorder cost component RCãN, the holding cost component HCãLavãTh and the shortage cost component SCãNsc.
Note that the fixed-order quantity policy assumes that in nominal conditions (when the demand is constant), the system is designed so that D=Q/T=ROL/LT holds, and no shortage occurs. On the contrary, if we have a stochastic demand, shortage may occur, thus increasing the total cost.
All the previously mentioned costs can be computed via an appropriate FOHPN shown in Figure 3.7b. The net within the grey rectangle models the FOQS and has already been discussed.
The content of continuous place p5 is equal to the sum of the unit cost component and the reorder cost component. In fact, the firing of timed transition t1 corresponds to a new order and the weight of the arc from t1to p5is equal to UCãQ+RC.
The holding cost component can be computed from the knowledge of the marking in p1, being Lav =1/ThTh
0 m1(τ)dτ. We assume that this measure is directly available when simulating the net with a HPN tool.
The rest of the FOHPN has been added so as to compute the shortage cost component, which is null unless unforeseen demand occurs. In normal operating condition, place p2is marked as shown in Figure 3.7b. A shortage occurs when place p1becomes empty and the marking of its complementary place p1 reaches the value Lmax, thus enabling the firing of immediate transition t3 that moves the token from p2to p3. While p3is marked, transition t5fires at its MFS D, thus increasing the marking of continuous place p4, which represents at each instantτthe shortage cost. As soon as a new order arrives, m1=Q and immediate transition t4fires, re-establishing the normal operating condition.
3.5.2 FOQSWITHFINITEREPLENISHMENTRATES
In the previous section, we considered a framework typical for the wholesalers: a large delivery of an item instantaneously raises the stock level and then the demand reduces it. In this section instead, we consider the stock of finished goods at the end of a production line. If the rate of production is greater than the demand, goods will accumulate at a finite rate while the line is operating. This gives a situation where the instantaneous replenishment is replaced by a finite replenishment rate.
A similar pattern is met with stocks of work in progress between two machines: the first machine builds up stock at a finite rate while the second machine creates demand to reduce it.
If the rate of production is greater than the demand, the stock level rises at a rate which is the difference between production and demand. If we call the rate of production P, stocks will build up at a rate P−D, as shown in Figure 3.8a. Stock will continue to accumulate as long as production continues. After some time, PT, a decision is made to stop production. Then, stock is used to meet demand and declines at a rate D. After some further time, DT, all stock has been used and production must start again. Thus, a decision must be taken at some point to stop production of this item and transfer facilities to making other items [78]. The resulting variation in stock level is shown in Figure 3.8a, where A=(P−D)ãPT and L, Q, T have the same physical meaning as in the previous section.
The FOHPN modelling this system is shown in Figure 3.8b within the grey rectangle. The marking of continuous place p1denotes the stock level, while p1is its complementary place and at each time instant m1+m1=A. When the discrete place p5is marked, as in Figure 3.8b, continuous transitions t1 and t2 may fire at their MFS, P and D, respectively. Assuming P> D, the fluid content of p1
(resp., p1) increases (resp., decreases) with a constant slope equal to P−D. As soon as m1 =A, transition t6fires thus moving the token from p5to p6. This disables t1, and the stock level decreases with a slope equal to the demand D, until p1gets empty, and the content of its complementary place p1is equal to A, thus producing the firing of transition t5. Then the cycle repeats unaltered.
L(τ) Q A
P-D D SUC
A A P
p5 p1
t5 t6
t1 t2
ε ε
t4
t7
p1
p2
t3 p4 p3
p6
A A
A
A D
SC D
p7 UC P
PT DT τ
(a) T (b)
FIGURE 3.8 FOQS with finite replenishment rate: (a) regular pattern and (b) net model.
3.5.2.1 Management Costs Evaluation
The total cost C during an interval of time Thmay be calculated as
C=UCãPãTP+SUCãNsu+HCãLavãTh+SCãNsc, (3.11) where
TPis the length of the time interval during which production occurs SUC is the set-up cost required each time production is started
Nsuis the number of set-ups, that is, the number of production cycles and the other variables have already been defined in the previous section
Thus, the total cost is given by the sum of four cost components: the unit cost component UCãQãTP, the set–up cost component SUCãNsu, the holding cost component HCãLavãThand the shortage cost component SCãNsc.
As in the previous case, the total cost can be computed via an appropriate FOHPN module shown in Figure 3.8b. The content of continuous place p7is equal to the sum of the unit cost component (UCãQãTP) and the set-up cost component (SUCãNsu). In fact, the continuous transition t1fires during all time intervals of production, while discrete transition t6fires whenever a new production cycle occurs.
The computation of both holding and shortage cost components is similar to that presented in Section 3.5.1.1. There are just two minor changes: the weights of the arcs in the self loops p1, t4and p1, t3 are equal toε 0 and A, respectively. In fact, shortages occur as soon as m1 = A, that is, m1 =0, and, due to the finite replenishment rate, stop as soon as m1 >0. Finally, as in the previous case, the marking of continuous place p4represents the shortage cost at each instantτ.
3.5.3 FOQSWITHBACK–ORDERS
All models considered in the previous sections assumed a constant demand. This is due to the fact that shortages are very expensive and must be avoided. However, there are situations where planned shortages are beneficial, for example, when the cost of keeping an item in stock is higher than the gross profit made from selling it. We have shortages when the customer demand for an item cannot be met immediately. At this point, the customer can either wait for an item to come into stock, in which case it is met by a back order, or she/he can withdraw her/his order and go to another supplier, in which case there are lost sales [78]. We will examine the first case.
Under the assumption of continuous and constant demand, the stock level of an item varies with a typical pattern shown in Figure 3.9a. Note that here back orders are shown as negative stock levels.
L(τ)
τr,1 τs,1 τs,2
τr,2
τ Q–S
Q–S Q–S Q–S
Q–S
S
L(0) D Q
p2
t1
S p4
p1
p1
p5 t4
t2
t5
p6
p7 p9 p8
∞
∞BO ΔT ε Δt–ε
t6
t3p3 D
D UC Q+RC
(a) (b)
FIGURE 3.9 FOQS with back orders: (a) regular pattern and (b) net model.
Let L(0)be the initial stock level. At timeτ¯=L(0)/D, the stock gets empty and the following demand cannot be met, thus producing shortages. When shortages reach a certain value S, a new order Q is immediately supplied: S units are used to satisfy the unmet demand, while Q−S units are stored in the stock. And the process repeats periodically.
The FOHPN model for this kind of systems is shown in Figure 3.9b within the grey rectangle.
As in the previous cases, the fluid content of continuous place p1 represents the stock level, while place p1 is its complementary place and at each time instant m1+m1 =Q−S. As soon as p1gets empty, that is, m1=Q−S, the immediate transition t4fires thus marking discrete place p5. Once p5
is marked, continuous transition t3 can fire with a firing speed equal to the demand D. As soon as p3, whose fluid content represents the amount of unmet demand, is equal to S, transition t1fires and both p3 and p5 get empty, while an amount of fluid equal to Q−S is supplied to continuous place p1. At this point, the demand can be satisfied and the process repeats cyclically.
3.5.3.1 Management Costs Evaluation The total cost C for FOQS with back orders is
C=UCãQãN+RCãN+HCãLavãTh−BO N
i=0 τr,i
τs,i
L(τ)dτ (3.12)
where the first three terms are the same as in Equation 3.10, and the last term represents the back- order cost component. Here BO is a cost for unit of product and for unit of time (the time the customer has been waiting). Such a cost only increases during all the time intervals in which the stock level L(τ)is negative, that is, during the time intervals(τs,i,τr,i), for all i∈N, whereτs,iand τr,irepresent the beginning and the end, respectively, of the shortage in the i-th period of simulation (see Figure 3.9a).
Even in this case, the total cost can be easily computed by an appropriate FOHPN module, as shown in Figure 3.9b. The content of continuous place p2 represents the sum of both the unit cost component and the reorder level cost component, while the holding cost component can be computed by evaluating the average marking of place p1, as already discussed in Section 3.5.1.
Finally, in order to compute the back-order cost, we have introduced the FOHPN module in the right, outside the grey rectangle. It provides a tool for integrating in p7 the fluid content of contin- uous place p3, which exactly coincides with the back-order level. Thus, the marking of p7 will be
L(τ)
τ L(0)
Lmin
Lmax
UC p4 RC
p3
p1 p2 t4
t3
t5 p5
p6
p7
t6
t7 p1
t1
Δt–ε t2 D
D SC Lmax
Lmax
Lmax
Lmax D
T
ε
∞
(a) (b)
FIGURE 3.10 PRS: (a) regular pattern and (b) net model.
m7=BOãTãn
k=1m3(kãT), where n=Th/T andT is the length of an appropriate finite step of integration.
3.5.4 PERIODICREVIEWSYSTEMS
PRS place orders of varying size at regular intervals to raise the stock level to a specified value. The operating cost of these systems is lower and they are in general applied when there is a high, regular demand of low-value items. The regular pattern of stock level is shown in Figure 3.10a.
The physical meaning of variables is the same as in the previous cases apart from Lmin which denotes the minimum stock level during each time period T. Note that while in the case of FOQS, what triggers a new order is the fact that the level of the stock reaches a given value (ROL); in this case, the new order is triggered by the fact that a time period T has elapsed.
The corresponding FOHPN model is shown in Figure 3.10b. The content of place p1, as in the previous cases, represents the stock level, while transition t2models the constant demand D. Place p1is the complementary place of p1and at each time instantτthe sum of their markings is equal to Lmax. When discrete place p3 is marked, continuous transition t1is enabled and it fires at an infinite speed transferring all the content of place p1 in place p1. We assume that the firing speed of t1 is infinite to simplify the model and to stress out that the time needed for the transferring of goods may be neglected with respect to the time period T.
Finally, the delay of transition t3 isε ∼= 0, that is, it fires after a very short time period, thus enabling t4, whose time delay is equal to T−ε. And the process proceeds unaltered.
3.5.4.1 Management Costs Evaluation
Also in the case of PRS, the total cost C during an interval of time Th may be calculated by an FOHPN module similar to that used in Figure 3.7. However, in the module used for PRS and shown in Figure 3.10b, the unitary cost component depends on the continuous firing of transition t1whose firing represents the replenishment of the stock.
3.5.5 STOCHASTICDEMAND
In the previous section, we assumed that the demand is continuous and constant. This assumption simplifies the model but it is not suitable for real applications. To provide more realistic models, we can use FOHPNs to simulate exponentially distributed stochastic demand. In such a case, we need to substitute all continuous transitions that represent the demand by exponentially distributed timed transitions with an appropriate value of the average firing rate.
p1 p1 p1 p1 L(0)
ROL L(τ)
τ
α λ α
t1 t1
D
(c) (b)
(a)
FIGURE 3.11 (a) Model of a constant demand. (b) Model of a stochastic demand. (c) Stock level of the IMS in Figure 3.7 with stochastic demand.
Thus, to each subnet of the kind in Figure 3.11a, we replace the subnet in Figure 3.11b. In particular, for both nets p1models the stock (its marking represents the stock level), while p1models the available space of the stock (it is the complementary place of p1). Continuous transition t1 in Figure 3.11a models the constant demand and is replaced by the exponentially distributed timed transition t1 in Figure 3.11b. Finally, to emphasize the stochastic effect of the demand, we assume that the arcs between p1, p1 and the stochastic transition t2 have weightα, while the firing rate of transition t2 isλ = D/α so that the expected value of the stochastic demand is equal to D.
Figure 3.11c shows a sample pattern of the stock level of an FOQS with finite lead time in the case of stochastic demand, where shortage occurs.