Inventory management with advance demand information and flexible shipment consolidation

We start our discussion with the first case in which \(L_d \le T\) and \(L_s \le T\), also illustrated in Fig. 2, where \(t_n\) \((t_n=nT,n\in {\mathbb {N}})\) represents the nth shipment day.

The TCL \(M(t_{n})\) is composed of the following parts. First, the remaining units \(K(t_{n-1})\) of the previous shipment day are included in the TCL. These units were backordered due to a lack of stock at the warehouse or ordered units when the due date was not reached, and there was not sufficient capacity available to ship them earlier. Since \(L_s\le T\), all backordered units at time \(t_{n-1}\) can be shipped at \(t_n\), and all ordered units before \(t_{n-1}\) must at the latest be shipped by \(t_n\). Second, all orders and demands during the interval \((t_{n-1},t_n]\) can be shipped at \(t_n\) if there is sufficient stock and shipment capacity available. Otherwise, backordered or ordered units will be left behind, indicating that we must subtract \(K(t_n)\). These considerations lead to the following expression for the TCL at \(t_n\)This expression reveals that we must characterize the stochastic process describing the remaining units at each shipment point \(K(t_n)\). As mentioned above, there are two reasons why units cannot be shipped. First, if there is a lack of stock at the warehouse, then even demanded units cannot be shipped, and \(IL(t_n)^-\) units are backordered. Second, all units that are ordered but not demanded must wait until the subsequent shipment dispatches if there is not enough reserved capacity at \(t_n\) available. The total number of units, which exceeds the capacity of the primary transportation option, is given as \((Cap-D(t_{n-1},t_n)-K(t_{n-1}))^-\), but since all demanded units must be shipped by the alternative transportation option, at maximum, all orders \(D(t_n-L_d,t_n)\) remain at the warehouse and must wait until the next shipment. Doing so yieldsIt is easy to see that the following limit holds,which shows that, in the case of high capacity, the shipment policy strives to ship all units earlier because only in the case of a lack of stock at the warehouse backorders must wait until the next dispatch time. In contrast, no early deliveries will occur in the case of small available capacity, as seen in (7).For the remaining analysis, we split the time interval \((t_{n-1},t_n]\) into several subintervals, and we express the number of backorders at time \(t_n\) based on the inventory position at time \(t_n-L_s\) because \(t_n-L_s\) is the last time when a replenishment order can be placed that arrives before or at \(t_n\). Reformulating (5) leads toAlthough the distributions of the demand and the inventory position are known and independent, the conditional distribution \(P(K(t_n) = j \mid K(t_{n-1}) = i)\) has to be considered; therefore, the pmf of \(K(t_n)\) cannot be computed directly by applying (8). When computing \(K(t_{n-1})\), the number of backorders is obtained based on the inventory position at \(t_{n-1}-L_s\), which is why we must also include information about the demand and the inventory position before time \(t_{n-1}\), and we replace \(IP(t_n-L_s)\) withWe obtainTo obtain \(K(t_n)\), we first have to compute \(K(t_{n-1})\), which can be specified asThis expression means that the number of remaining units \(K(t_n)\) depends on the number of remaining units of all previous SCL cycles. We assume that the impact of these quantities decreases the further we look back into the past; therefore, we only include the information of the previous two SCL cycles to determine \(K(t_n)\). However, we have to deal with \(K(t_{n-2})\). We suggest replacing it with a constant value and providing more information about this approximation at the end of this section.

Hence, the TCL given in (4) can be modified, and we replace in (12) the expressions for \(K(t_n)\) and \(K(t_{n-1})\) with (10) and (11) to compute the probability distribution for the TCL:As the demand follows a Poisson process, the demand during a specific time interval is Poisson distributed. Further, \(IP(t_{n-1}-L_s)\) is uniformly distributed between \(R+1\) and \(R+Q\) (Axsäter 2015). Since all random variables are independently distributed, convolutions can be used to calculate the pmf of M. For the determination of \(K(t_{n-2})\), we rely on an approximation presented at the end of this subsection, and we perform convolutions to obtain the probability mass function of the TCL \(M(t_n)\). Due to our simplifying assumptions, the obtained distribution is an approximation, the performance of which is tested in a numerical study in section 6.

We obtain three additional cases when \(L_d \le T\) and \(L_s>T\) based on the length of \(L_s\), as shown in Table 2. In the second case, the supply lead time is in the range of \(T < L_s\le T+L_d\), which is why the time \(t_{n-1}-L_d\) is before \(t_n-L_s\), as illustrated in Fig. 3. The sequence of the latter time points changes if \(T+L_d< L_s\le 2T\), which legitimates the third case. The fourth case occurs if \(L_s>2T\) because \(t_n-L_s\) is before \(t_{n-2}\). The main difference from the first case is that the time \(t_n-L_s\) of the last warehouse replenishment order, which will arrive at the latest by \(t_n\), is before the previous shipment day at \(t_{n-1}\), which is why we cannot assure that all backorders at an arbitrary shipment date can be satisfied by the following shipment day. This change in sequences leads to different time intervals during \(t_{n-1}-L_s\) and \(t_n\) and therefore to different formulas of \(K(t_n)\) and \(K(t_{n-1})\), which are shown in Appendix.

Now, the focus is on all seven cases in which \(L_d > T\). To express the differences between \(L_d \le T\) and \(L_d > T\), we investigate the fifth case with \(T< L_d \le 2T, T<L_s\le 2T\), as illustrated in Fig. 4 in more detail. The other situations can be handled similarly. The shipment policy only allows for shipping an ordered unit one shipment date earlier to the production facilities. As long as \(L_d \le T\), this assumption is fulfilled automatically. However, it is not true for \(L_d >T\). Orders during \((t_{n+1}-L_d,t_n)\) cannot be shipped at \(t_n\) because their official shipment date is at \(t_{n+2}\). The earliest shipment date for these orders is \(t_{n+1}\). Therefore, the demand during \(t_{n+1}-L_d\) and \(t_n\) is directly added to the number of remaining units at \(t_n\), which can be obtained byThis property can also be applied for determining \(K(t_{n-1})\), where the orders \(D(t_n-L_d,t_{n-1})\) will be considered on the shipment day at \(t_{n}\). Doing so yieldsThe TCL is the sum of demands during the considered SCL cycle plus \(K(t_{n-1})\) minus \(K(t_n)\), as given in (4).When we still consider the range \(T<L_d\le 2T\) for the demand lead time, the sequence of time points changes depending on the length of \(L_s\), which is why the sixth and seventh cases arise. The time \(t_n-L_s\) is between \(t_{n-1}-L_d\) and \(t_{n-2}\) if \(2T<L_s\le T+L_d\) and accordingly between \(t_{n-1}-L_s\) and \(t_{n-1}-L_d\) if \(T+L_d<L_s\). Similarly, all ranges for the remaining cases can be obtained. All formulas for \(K(t_n)\) and \(K(t_{n-1})\) are shown in Appendix.

While we have derived expressions for \(K(t_n)\) and \(K(t_{n-1})\), it remains to determine \(K(t_{n-2})\) to be able to compute the pmf of the TCL \(M(t_n)\). As mentioned before, we neglect the remaining units of prior shipment days before \(t_{n-2}\) and use a fixed value \({\overline{K}}\) for \(K(t_{n-2})\), which is updated in an iterative procedure. The idea is to replace the number of remaining units with its expectation. Therefore, we start with a given value of \({\overline{K}}\), determine the probability distribution of \(K(t_n)\) and use it to compute the expectation of \(K(t_n)\). We also consider a factor that reflects the capacity utilization of the vehicle when we update the value for \({\overline{K}}\). In general, the obtained value for \({\overline{K}}\) is not an integer, rendering the computation of the convolutions impossible. As a solution, we determine the pmf for the TCL and the remaining units at \(t_n\) two times. First, we use the rounded-down value \(\lfloor {\overline{K}} \rfloor\) and the second time the rounded-up value \(\lceil \; {\overline{K}} \; \rceil\). Both results are merged as follows:As a starting value for the iterative procedure, we have chosen \((Cap- \lambda T-\frac{1}{2}\lambda L_d)^-\), which is obtained by replacing the random demand in the expression with the total number of units, which exceeds the capacity of the primary transportation option, with its expectation. Furthermore, the remaining units \(K(t_{n-1})\) are replaced with 50% of the expected number of ordered units with a due date after the shipment time. In summary, we provide a sketch of the algorithm to compute the expected shipment costs.

Observing a specific unit on its way through the system, we recognize four essential points in time. First, a unit is available for satisfying a demand at the warehouse exactly \(L_s\) time units after the replenishment order was placed, thus at time \(t_{a}= t_r+L_s\). The following important event is the time when the facility order for this unit arrives at the warehouse, which is given as \(t_{o} = t_r+\Omega (S)\), where \(\Omega (S)\) represents the time until the \(S\)th facility order occurs. The third interesting time is when the status of the unit changes from an order to a demand, which occurs at \(t_{d} = t_r+\Omega (S)+L_d\). Finally, there is the time \(t_{s}\) when the unit is shipped to the production facility, depending on the available reserved transportation capacity.

Which types of costs occur and to what extent depend on the sequence of the events. For the derivation of the formulas, we have to distinguish all possible situations, which are denoted with \(i\in \{A,B,C,D,E,F,G\}\). Situation A is illustrated in Fig. 5, and we explain all of the cases in the following in more detail.

The first three situations (A,B,C) are related to a situation in which the unit is available when the order occurs, which means that \(t_{a} < t_{o}\). Otherwise, the unit is ordered while there is no available stock for the considered unit at the warehouse (D,E,F,G). We further differentiate between the situations in which prequalification and qualification occur at different points in time (D,E,F) and at the same time (G), whereby the latter case only occurs if the unit is available after it is demanded. If the unit is available before demand occurs, then the unit is prequalified at time \(\max (t_a,t_o)\) and qualified at time \(t_d\). In the following, \(t_n \in {\mathbb {N}}\) denotes an arbitrary shipment day. In summary, we provide all of the cost expressions for situations \(i \in \{A,B,C,D,E,F,G\}\) in Table 3.

Obviously, for a given S, T and Cap, the inventory cost depends on the random variables \(\Omega (S)\) and V and the available reserved transportation capacity at \(t_n\). However, we only know the pmf of the TCL at \(t_n\) after all demands and orders of the considered SCL cycle occurred, which again is why we rely on an approximation. We denote the inventory cost for case \(S>0\) in the following analysis with \(C(\Omega (S),V,M(t_n))\). Denoting the joint density function of \(\Omega (S)\), V and \(M(t_n)\) with f(x, y, z), we can obtain the expected inventory cost bywhich means that we can split the derivation into several parts, where each part corresponds to one of the aforementioned situations. For the following analysis, we neglect the dependency between \(M(t_n)\) and the other random variables such that we obtainTherefore, we can reformulate (19) aswhere \(C_i(x,y)\) and \(f_i(x,y)\) with \(i\in \{A,B,C,D,E,F,G\}\) represent the inventory cost depending on \(\Omega (S)\) and V and the joint distribution function of \(\Omega (S)\) and V, respectively.

Since the cost expressions can easily be derived from Table 3, it remains to determine the functions \(f_i(x,y), i \in \{A,B,C,D,E,F,G\}\). It can be shown that the functions \(f_i(x,y)\) are positive on different domains (Table 4). Further, in the range where they are unequal to zero, the functions have the form \(g^S(x)u(y)\) for all situations i, \(i\in \{A,B,C,D,E,F,G\}\). In the following, we show the derivation for situation A, whereas the remaining derivations are given in Appendix.

Since \(T = V + L_d + t_r+ \Omega (S) - t_{n-1}\) (Fig. 5), we obtain for \(x > L_s\) and \(y \le (T-L_d)^+\)Since V is uniformly distributed between [0, T] (Tijms 2003), the upper bound of y cannot be smaller than 0. Thus, \(f_A(x,y)\) is given as the partial derivative with respect to x and ySince demand follows a Poisson process, \(\Omega (S)\) is Erlang distributed with the parameters S and \(\lambda\) and is independent of the shipment delay V (Tijms 2003). Therefore, the expectation of the inventory cost for each situation can be computed, and the results forare given in Tables 5 and 6. Since the range for y depends on T and \(L_d\), we must consider the cases \(L_d \le T\) and \(L_d>T\) separately. The derivation is again reported in Appendix.

If \(S = 0\), then the considered unit is ordered by the production facility at the same time that the warehouse orders it at the outside supplier. Thus, no safety stock is kept at the warehouse. Focusing on \(S<0\), the warehouse orders the considered unit at the outside supplier after the next \(|S|\)th facility orders occur. Due to \(L_d \le L_s\), both cases imply that the considered unit is always demanded before it is available for shipment. Therefore, warehouse stock-keeping costs arise for the time interval V, whereas waiting costs occur for the time interval \(V+L_s-L_d+\Omega (|S|)\). As \(t_a\ge t_d\), the flexible delivery option cannot be used. Therefore, inventory cost \({\tilde{C}}(\Omega (S),V)\) for case \(S\le 0\) is independent of \(M(t_n)\), and the expectation of this cost can be obtained byIn summary, the expected inventory cost of the system is given by

First, we focus on a given capacity and on the influence of an increasing demand lead time on the expected total cost. We compute the average marginal relative decrease in the expected total cost when the demand lead time \(L_d\) is stepwise increased by two time units, while the other parameters are fixed. The results are presented in Table 8 and indicate that large cost reductions can be achieved if customers are willing to place orders in advance. In general, it can be seen that the longer the demand lead time, the greater the total cost reductions. The maximal marginal relative cost reduction when \(L_d\) is increased from 0 to 2 is 20.29%, whereas we can achieve a maximum decrease of the expected cost when \(L_d\) is increased from 0 to 8 of 35.57%. Further, it can be observed that an increase in the demand lead time of two time units can have a different effect depending on the starting point. For example, coming from the situation where \(L_d=0\) to a situation where \(L_d=2\), on average, the expected total cost per time unit can be decreased from 62.01 to 55.54 (10.43%), whereas the expected total cost per time unit can be reduced from 55.54 to 52.91 (4.74%) when increasing the demand lead time from 2 to 4. Thus, the marginal value of the ADI decreases with an increasing amount of information.

There are two possible sources for the cost reduction. First, as already observed in (Hariharan and Zipkin 1995), a longer demand lead time reduces the effective lead time \((L_s-L_d)\) and therefore the safety stock. Second, due to the flexible shipment policy, a longer demand lead time results in more orders available for earlier shipments. Thus, better utilization of the transportation capacity can be attained, and fewer emergency deliveries are necessary. Since early deliveries are only allowed when enough free capacity is available, it is clear that the reserved capacity must have an impact on the benefit of ADI in our setting. It can be observed that the marginal relative cost reduction decreases drastically with increasing \(L_d\) for the situation \(Cap=5\) (Table 8). The mentioned effect is not so large for \(Cap=20\). Although the marginal relative cost reduction also decreases as \(L_d\) increases, it decreases much less, and even when the demand lead time is increased from 2 to 4, 6.61% of the expected total cost can still be saved. In situations with a small reserved transportation capacity, the capacity is already fully utilized on many shipping days when \(L_d=2\), so further information will not result in additional early deliveries in many cases. With a higher reserved capacity, there is a higher probability of unused capacity for orders, even if information is already available. Since the reserved capacity seems to be an important variable, we will later also determine the optimal reserved capacity.

To investigate the proportion of the average total cost decrease caused by reducing the safety stock at the warehouse and by early shipments, respectively, we computed the expected total cost for the same 540 examples when flexible deliveries are not allowed at all but ADI is still available.

Additionally, we are interested in the question of whether cost can be reduced even more if more early shipments are allowed. An evident and simple policy to investigate is to dispatch all orders regardless of the remaining transportation capacity. Thus, the primary transportation option may be already exhausted, but additional orders will also be shipped to the facilities by using the alternative option. Our analysis can be easily adapted to obtain formulas for the computation of the expected total cost because remaining units will only occur when the warehouse is running out of stock.

In Fig. 6, we depict the average total cost for all three policies as a function of \(L_d\). It can be seen that a shipment policy without flexible deliveries performs worst. For such a policy, we observe an almost linear cost decrease with increasing demand lead time of approximately 0.7% due to a reduction in the effective supply lead time, resulting in less stock at the warehouse.

Significant cost savings can be obtained with the introduction of flexible deliveries. However, shipping all orders one shipment day ahead and neglecting the available capacity can further be improved by our shipment policy where the reserved capacity is taken into account when deciding about the TCL. Then, an increase in the demand lead time from 0 to 2 results in a decrease of the average total cost by more than 10%, indicating that a reduction of approximately 9.5 percentage points is caused by adapting the time-based SCL policy and allowing for flexible deliveries. We can conclude that the more significant part of the cost reduction is induced by the flexible delivery option and not by reducing the safety stock.

With increasing \(L_d\), this effect is reduced until we reach a point at which additional ADI will only decrease the stock because the flexible delivery option is used to its extent. This is shown in Fig. 7, where we illustrate the marginal relative cost reduction of all three policies.

We can also observe that with increasing demand lead, the cost difference of the two flexible delivery concepts is increasing. The marginal relative total cost reduction for the simple policy is even lower than for the shipment policy without flexible deliveries, because for large demand lead times the alternative transportation option has to be used too often, which prevents further cost reductions.

Moreover, to enable early deliveries, even more safety stock is kept at the warehouse compared to a situation without flexible deliveries. This situation is illustrated in Table 9, where the optimal policy parameters \(R^*\) and \(T^*\) are presented for different values of the reserved transportation capacity, demand lead time and order rate. The remaining parameters are fixed at \(h=1, w=2, e=2, c_1 = 20, c_2 = 2\frac{\alpha }{Cap},\) and \(L_s=10\). While the demand lead time \(L_d\) has a significant impact on the numerical value of the reorder level, the influence on the optimal length of the SCL cycle is much less. This condition holds for both situations, with and without flexible deliveries. The optimal SCL cycle length is influenced by the shipment costs, as well as by the relationship between the average demand and the capacity. An increasing order rate raises the reorder level and reduces the SCL cycle length to utilize the reserved transportation capacity and to avoid expensive additional shipments.

We are further interested in the influence of flexible deliveries on the optimal capacity of the primary transportation option for a given length of the SCL cycle. In Table 10, it can be seen that flexible deliveries lead to equal or larger reorder levels for a fixed and given SCL cycle length. This means that more safety stock is required to enable flexible deliveries. Additionally, more reserved transportation capacity \(Cap^*\) is needed in situations with large values of T to exploit the benefit of flexible deliveries fully. However, the influence of the flexible deliveries on the optimal capacity to be reserved is negligible.