A (Q, R) inventory replenishment model with two delivery modes

European Journal of Operational Research xxx (2014) xxx–xxx

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A (Q, R) inventory replenishment model with two delivery modes Cinthia Pérez, Joseph Geunes ⇑ Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, United States

a r t i c l e

i n f o

Article history: Received 7 November 2012 Accepted 23 February 2014 Available online xxxx Keywords: Inventory planning and control Supply chain management Lead time management Inventory theory

a b s t r a c t Although splitting shipments across multiple delivery modes typically increases total shipping costs as a result of diseconomies of scale, it may offer certain benefits that can more than offset these costs. These benefits include a reduction in the probability of stockout and in the average inventory costs. We consider a single-stage inventory replenishment model that includes two delivery modes: a cheaper, less reliable mode, and another, more expensive but perfectly reliable mode. The high-reliability mode is only utilized in replenishment intervals in which the lead time of the less-reliable mode exceeds a certain value. This permits substituting the high-reliability mode for safety stock, to some degree. We characterize optimal replenishment decisions with these two modes, as well as the potential benefits of simultaneously using two delivery modes. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction In inventory replenishment, the three main criteria of cost, quality, and delivery reliability are of paramount importance in evaluating supplier performance, where delivery reliability is typically a function of the supplier’s lead-time performance, as noted by Minner (2003). All else being equal, lower cost shipping modes imply both longer and less reliable delivery lead-time performance. Using such suppliers therefore tends to necessitate at least an occasional use of quick-response suppliers who can fill the gaps when the low-cost supplier’s lead time is too long. Even if a single supplier is preferred for sourcing, a variety of different delivery modes are typically available to shippers, depending on the product type. That is, a wide range of package carriers and third-party logistics providers may be used to fulfill orders. Because of the uncertainty in delivery performance of different modes, it is not immediately clear which mode, or what mix of different modes, should be used when ordering from a supplier or from multiple suppliers. This research focuses on the benefits of delivery mode diversification as a strategy to reduce inventory-related costs. We consider a system in which a ‘‘buyer’’ uses two different delivery modes for inventory replenishment. These modes may correspond to entirely different suppliers, or they may imply different delivery modes employed by a single supplier. In particular, we consider an inventory replenishment model where the buyer uses a continuous review ðQ ; rÞ replenishment policy and has a constant, deterministic demand rate of k units per unit time. Our model assumes that ⇑ Corresponding author. Tel.: +1 3523921464. E-mail addresses: ccperez@ufl.edu (C. Pérez), [email protected]fl.edu (J. Geunes).

the buyer does not permit shortages, and may use two different shipping modes within each inventory replenishment cycle. One of these modes comes at a low cost, but is less reliable, reflected by a stochastic delivery lead time (with a finite upper bound). In addition, a higher cost, perfectly reliable shipping mode is available if needed. We assume that the uncertainty in the delivery time of the less reliable supply mode results to a large extent from a high utilization of this mode. Our model considers the effect that an increase in the utilization of a stressed shipping mode has on delivery performance. This is done by expressing the maximum value of the less reliable mode’s lead time as a stepwise, nondecreasing function of the order size. We can think of the less reliable supply mode as traversing an already congested route, where sending a larger quantity via this mode might require increasing the number of trucks using the route (or the frequency at which trucks utilize the route). As a result, more capacity is consumed and the expected value and variability of the travel time for the entire order quantity along the route increase. Alternatively, an increase in order quantity would also likely increase the required production time, which leads to an increase the delivery lead time for the order. For the perfectly reliable mode, we assume that it has a sufficient degree of excess capacity, and therefore, it will not be affected by the quantity ordered. Using multiple transportation modes and making dynamic decisions for deliveries within a supply chain has become more prevalent in recent years with the ability to use information technology to obtain near real-time information on the status of orders placed with suppliers. For example, Liz Claiborne uses a product from TradeBeam that enables monitoring the status of orders

http://dx.doi.org/10.1016/j.ejor.2014.02.049 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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placed with suppliers and expediting these orders when needed using an online tool (see Jain, Groenevelt, & Rudi (2010)). The use of multiple sourcing of supply has also been studied in the context of managing supply chain disruptions. Yu, Zeng, and Zhao (2009) discuss several practical situations in which firms have been able to mitigate the impacts of supply disruptions through the use of multiple sourcing strategies. They evaluate the impact of using single or dual sourcing strategies to manage the impacts of supply chain disruptions. Their model also assumes the presence of a less reliable and cheaper supplier (identified as a foreign supplier) and a more reliable but more expensive supplier (characterized as a local supplier). Babich, Aydin, Brunet, Keppo, and Saigal (2012) state that one of the main reasons for firms to diversify their supply base is to manage supply risk. Their work reviews the relationship among financing constraints, trade-credit loans and the number of suppliers. Dong and Tomlin (2012) mention the use of inventory, emergency sourcing, dual sourcing, demand management and process improvement as operational measures to manage disruption risk, and they study the interaction between business interruption insurance and operational measures to manage risk (specifically the use of inventory and emergency sourcing). Our work falls within a stream of literature on multiple sourcing inventory models. Following the characterization that Minner (2003) presented in his review of multiple-supplier inventory models, we identify two main lines of research based on the assumptions regarding supplier lead times. First, we will review inventory models with n supply modes with deterministic lead times, where the general assumption is that L1 < L2 . . . < Ln and c1 > c2 . . . > cn , where Li and ci are the lead time and unit purchasing cost for supply mode i, for i ¼ 1; . . . ; n. The main focus of most of the works that employ this assumption is to define optimal replenishment policies or to define policy parameters for a specific replenishment policy. After discussing these works, we will review models that assume multiple delivery options with stochastic lead times. These models are primarily concerned with order splitting, where an order is placed at the beginning of a replenishment cycle and is split among different suppliers. Among the studies that assume the use of supply modes with deterministic lead times, Moinzadeh and Nahmias (1988) consider a model with a continuous review policy, two supply modes and random demand; their goal is to find the order quantities and reorder points for each supply mode in order to minimize total inventory costs. Later, Moinzadeh and Schmidt (1991) presented a model based on a one-for-one ðS  1; SÞ inventory policy with two supply options, where the decision on whether to place an order with the regular or emergency supplier depends on the age of the outstanding order. Jain et al. (2010) consider a ðQ ; rÞ policy in which two transportation modes exist, each with a deterministic lead time and fixed plus variable shipping cost structure. After the manufacturing lead time, the buyer must decide on how much of the order to ship via each transportation mode under stochastic demand. In contrast with our model, these works assume deterministic lead times for both supply modes and allow inventory shortages. Several past works on multiple sourcing consider a fast emergency supplier, which can be used to avoid shortages by placing an emergency order (based on some triggering condition). For example, Tagaras and Vlachos (2001) assume an emergency order is placed when the probability of stockout is high; Chiang and Gutierrez (1996) propose an ‘indifference level’ for inventory that triggers a second order; in Johansen and Thorstenson (1998), the reorder point for the emergency supplier is a function of the time until the regular order arrives. Our model is similar in spirit to these works, except that an order is placed with the ‘emergency supplier’ only when reaching a given time point in the replenishment cycle, and only if the order from the primary (cheaper) supply mode has not arrived by this time point.

Instead of using multiple supply modes, Chiang and Chiang (1996) propose an inventory model with one supplier and multiple deliveries. They assume the use of a continuous review policy with Normally distributed demand, constant lead time and a predetermined service level. In their model an order of size Q is placed at the beginning of the replenishment cycle, which is split between n deliveries with interarrival times Li for i ¼ 1; 2; . . . ; n. We find a similar approach in Chiang (2001), who proposes the use of one supply mode and multiple deliveries for periodic review ðR; SÞ inventory systems. In both of these papers, the cost reduction, when compared with a single delivery mode, results from a reduction in cycle stock. Unlike the previous works in the literature, we propose the use of multiple supply modes with different ordering and purchasing costs and different levels of delivery reliability. Our model does not place orders with the two supply modes at the beginning of the replenishment cycle; on the contrary, we decide whether to place a second order with the more reliable and expensive supply mode at a specific time during the replenishment cycle, only if the first order has not arrived by that time. Past works on models that assume stochastic lead times and multiple delivery options are primarily concerned with order splitting. In order splitting, an order is placed at the beginning of the replenishment cycle, and is split among different suppliers. The main benefit of using order splitting is the reduction in the effective lead time (the time until the first order arrives), which leads to a reduction in the safety stock level required to meet a given service level. This effect was demonstrated by Sculli and Wu (1981) and Kelle and Silver (1990), when each order is split between two suppliers. Later, Pan, Ramasesh, Hayya, and Keith Ord (1991), based on order statistics, presented expressions to estimate the parameters for the distributions of the effective lead time and time between arrivals when two supply modes with identical leadtime distributions are used. As an example, they considered three lead-time distributions: Normal, Uniform, and Exponential, and observed a reduction in the mean and variance of the effective lead time, compared with the use of a single supply mode. Most of the relevant models that assume stochastic lead times also assume ðQ ; rÞ inventory policies, and seek the optimal order quantity, reorder point and the proportion of order splitting, as in Lau and Lau (1994) and Lau and Zhao (1993), the latter accounting for stochastic demand. These works showed that most of the savings from order splitting result from a reduction in cycle stock, which often exceeds the cost savings due to associated safety stock reduction. Ramasesh, Ord, Hayya, and Pan (1991) showed that, under the assumption of identical lead-time distributions for the different supply modes, an optimal solution is found when the order is split equally, and that the savings come from reduced holding and backordering costs. A different approach was presented by Ganeshan, Tyworth, and Guo (1999), who proposed a discounting option for the less reliable supplier, and presented exchange curves to define when it is beneficial to split the order, as well as the percentage discount necessary to make the model attractive. Our model differs from previous literature as it considers one (cheaper, less reliable) supply mode with a stochastic lead time and another with a perfectly reliable (deterministic) lead time. This more closely reflects typical cases in which a buyer uses a primary delivery mode that is not perfectly reliable, but may place expedited orders (via, e.g., overnight shipping) when stockouts become imminent. In contrast with order splitting models, our work does not place an order with two shipping modes at the beginning of every replenishment cycle. Instead, we seek the optimal reorder point and order quantity such that the long-run expected inventory ordering, holding and purchasing costs are minimized without any shortages (i.e., with a 100% service level). Our assumption of a 100% service level is intended to approximate systems with extremely high stockout costs, and we recognize that many practical

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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systems operate at near 100%, although they are not perfectly reliable (see Graves & Willems (2000) for a similar application of a 100% service level as an approximation to ensure model tractability). Note that a distinguishing feature of our model is reflected in the fact that we need not always place an order with the reliable delivery mode. That is, if the order from the unreliable supplier arrives sufficiently early in the cycle, we need not place an order with the reliable supply mode. In other words, we can ‘wait-and-see’ whether a second order is needed in each cycle, allowing us to incorporate specific information about the unreliable supplier’s performance in each cycle before determining whether a second order is needed. This feature resembles the use of expediting policies, as in the work presented by Chiang (2010), who considers a single-item continuous review expedited ordering policy with stochastic demand, order quantity Q and reorder point s. In his model, if at the end of the supplier’s manufacturing time, the inventory level falls below the expedite-up-to-level R (R < s), the firm requests that part of its order is expedited (via a fast transportation mode) to bring its inventory level up to R; the remainder of the order is delivered through the regular supply mode. In contrast to our model, where the decision time on whether to use the second supply mode may occur at any time during the replenishment cycle, Chiang’s model assumes that this expediting decision must be made after the order has been manufactured but before it has been shipped. In addition, the expedited order serves as a fraction of the original order, where we permit an additional order quantity above the amount that was ordered via the regular supply mode. In par, where s  corresponds ticular, our model sets a reorder point r ¼ ks to an amount of time that may be less than an upper limit on the lead time of the lower cost shipping mode (assuming this leadtime distribution is bounded from above). If an order is placed at  is the time at which the current on-hand inventime zero, then s tory will reach zero, i.e., the inventory ‘run-out’ time. Suppose that at the beginning of each replenishment cycle the buyer places an order using the less reliable shipping mode. If the order has not   L2 , where L2 is the lead time of the perbeen received by time s fectly reliable shipping mode, the buyer will need to place a second order with the faster, reliable shipping mode, which will be re, thereby avoiding any shortages. ceived at time s Another important attribute that differentiates our model from previous works is that we account for the impact that a variation in the utilization level of a congested supply mode has on delivery performance. We assume that the lead time of the less reliable supply mode follows a continuous distribution that is supported on a bounded interval, and that as we increase the size of the order placed with this supplier, the upper bound on its delivery time may also increase; therefore, there is a corresponding increase in the variance of the delivery time. We are interested in the degree to which using the reliable, quick-response supply mode can reduce inventory-related costs when compared with the use of a single shipping mode. We are also interested in characterizing the circumstances and system parameter values under which the use of both supply modes is beneficial. The remainder of this paper is organized as follows. Section 2 presents our model description, where expressions for the cost functions are derived. Section 3 presents a numerical analysis and shows the circumstances under which our proposed model is preferable to using only a single shipping mode, and Section 4 summarizes our work and discusses future research directions.

2.1. Single-mode model We first consider the cost of using each shipping mode independently, which will permit benchmarking the performance of the model with two delivery modes. 2.1.1. Shipping mode with uniformly distributed lead time We consider an inventory replenishment model where a buyer uses a continuous review ðQ ; rÞ policy, with constant demand rate k, where no order crossing is permitted. The buyer orders from a supplier with unlimited capacity who ships via a mode with a stochastic lead time, called shipping mode 1. Given that we treat time as continuous, we initially assume that this delivery lead time follows a Uniform distribution, where L1 is the random variable for mode 1 lead time, and sl and su denote the lower and upper limits for L1 (note that our later numerical tests will consider a more general lead-time distribution, namely a Beta distribution). We assume that the buyer desires a zero probability of stockout; therefore, Q P r and the reorder point, r, must equal ksu , when the supplier uses mode 1 exclusively. Fig. 1 shows a realization of the model, assuming without loss of generality that an order is placed at time zero. Since the lead time is Uniformly distributed, the probability of sl an order arrival before or at time t is PfL1 6 tg ¼ st where u s l

t 2 ½sl ; su . Based on this probability we can compute the expected inventory level at any time t 2 ½0; T, where t ¼ 0 is the time when an order is placed, i.e., when the inventory level is equal to r, and T ¼ Qk is the length of the replenishment cycle. To find the expected inventory level at any time t 2 ½0; T, we divide the cycle into three different intervals: ½0; sl Þ; ½sl ; su Þ; ½su ; T. The first interval starts at time zero, when the inventory level is equal to r and an order of size Q is placed. From this time until the end of the cycle, the inventory level will be depleted at a constant rate of k units per unit of time due to demand. Since the order will arrive at or after time sl , we have that PfL1 < sl g ¼ 0, and the expected inventory level during the interval ½0; sl Þ is ksu  kt. Given that the arrival time is Uniformly distributed between sl and su , at any time t 2 ½sl ; su  the expected inventory level increases (with respect to the level during the interval ½0; sl Þ) by QPfL1 6 tg; therefore, during the interval ½sl ; su Þ the expected   sl  kt. For the remaining interval, inventory level is ksu þ Q st u s l

the order of size Q has been received with probability one, and therefore, the expected inventory level is ksu þ Q  kt. As a result, the expected inventory level is:

8 ks  kt; 0 6 t < sl ; > < hu i h i Q sl Q E½IðtÞ ¼ t su sl  k þ ksu  su sl ; sl 6 t 6 su ; > : su < t 6 T: ðQ þ ksu Þ  kt;

2. Problem description This section first describes the costs associated with the use of a single shipping mode and then with the use of two shipping modes for inventory replenishment.

Fig. 1. Single-mode model, where L1  Uðsl ; su Þ.

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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As shown in Appendix A, we need Q P kðsu  sl Þ to prevent order crossing. Also, in order to prevent shortages, the size of the order has to be at least the size of the reorder point, i.e., Q P ksu . Based on the previous conditions for the size of Q, we can see that the expected inventory during t 2 ½sl ; su  will have a positive slope. Fig. 2 shows the expected inventory level as a function of time. We next determine the average inventory level in a replenishment cycle. In order to find the average total inventory per cycle, denoted by IT , we integrate the expected inventory level over the cycle, which is equal to:

IT ¼

Q 2 Q ðsu  sl Þ þ : 2 2k

To obtain the average inventory level per unit time, denoted by bI, we divide the average total inventory by T, which gives

bI ¼ Q þ 1 kðsu  sl Þ: 2 2 This model has a safety stock of 12 kðsu  sl Þ since, on average, or1 ders will be received  at time  2 ðsu þ sl Þ and the inventory position at that time is I t ¼ su 2sl ¼ r  12 kðsu þ sl Þ, which is equal to 1 kðsu  sl Þ. 2 To compute the total relevant cost per cycle we consider a fixed order cost, inventory holding cost, and variable purchasing cost. We define A1 as the cost of placing an order using shipping mode 1, h as the inventory holding cost per unit per unit time, and c1 as the unit purchasing cost when using mode 1. For this model, the average total cost per replenishment cycle is

! Q 2 Q ðsu  sl Þ þ c1 Q: TC 1 ðQ Þ ¼ A1 þ h þ 2 2k The expected cost per cycle is equal to the average total cost divided by the length of the cycle, which equals

G1 ðQÞ ¼ A1

  k Q 1 þh þ kðsu  sl Þ þ c1 k: Q 2 2

ð2:1Þ

Since the second derivative of the average cost function, ¼ 2AQ 13 k, is non-negative 8Q > 0, the function is convex for @f Q > 0 and will reach its minimum at Q such that @Q ¼ 0, which implies

@ 2 G1 @Q 2

Q 1

rffiffiffiffiffiffiffiffiffiffiffi 2kA1 : ¼ h

ð2:2Þ

The above equation implies that the order quantity corresponds to the economic order quantity (EOQ). Using Q 1 in G1 ðQ Þ we have a minimum average cost per cycle of

pffiffiffiffiffiffiffiffiffiffiffiffiffi khDs G1 ¼ 2kA1 h þ þ kc1 ; 2

where Ds ¼ su  sl . We may view the second term above, kh2Ds, as the incremental cost over that of the standard EOQ Model, due to the uncertainty in the delivery mode 1 lead time. We next consider the possibility that the degree of uncertainty in the delivery time of shipping mode 1 may be affected by the level of utilization of this mode. Therefore, we would like to account for the potential effect that a change in the utilization of mode 1 has on delivery performance. To approximate this effect, we define the upper limit on mode 1 lead time, su , as a step function of the order size Q, such that, su ¼ siu if qi 6 Q < qiþ1 for i ¼ 1; 2; . . . ; n, where qi < qiþ1 , siu < siþ1 and n is the number of u intervals for Q. Fig. 3 shows a graph of su as a function of Q. The procedure to find the optimal order size Q under the assumption that su is a stepwise function of the order size is similar to the one used by Chopra and Meindl (2007) to find the optimal order quantity under an all units quantity discount scheme. Observe from Eq. (2.2) that the optimal order size under supply mode 1 is independent of su . Consequently, as we increase the value of su , all else being equal, the value of Q that minimizes the average inventory cost will not change, but the average inventory   cost will increase by 12 kh siþ1  siu for i ¼ 1; . . . ; n  1, where n is u the number of intervals for su . Fig. 4 shows the average inventory cost function when su follows a step function of Q. Based on the previous observation, the optimal value of Q when su is a step function of the order size is either on the (unique) interval where Q 1 , obtained from (2.2), is feasible, or at one of the break points to the left of Q 1 . Appendix B presents a detailed description of the solution procedure to find Q 1 when shipping mode 1 is used exclusively. An alternative approach to account for the impact that utilization has on the delivery performance of mode 1 would be to define the lead-time upper bound of mode 1 as a linear function of Q, such ^u þ cQ , where s ^u is the base level for the lead-time that su ¼ s upper bound of mode 1, and c is the factor by which the lead time increases per unit ordered. The procedure to find the average inventory cost per unit time when su is a linear function of Q, represented by Gc1 , is similar to the one used when su is fixed. As result we obtain:

Gc1 ðQ Þ ¼ A1

Observe that when the lead-time upper bound is a linear function of the order size Q, then as we increase the quantity ordered we will have an increase in the average inventory holding cost equal to 12 hkcQ , due to the increase in safety stock. The average inventory cost per unit time is a convex function of c

Q, since

ð2:3Þ

Fig. 2. Expected inventory level as a function of time for the single-mode model, where L1  Uðsl ; su Þ.

  k Q 1 ^u þ cQ  sl Þ þ c1 k: þh þ kðs Q 2 2

@ 2 G1 @Q 2

¼ 2AQ 13k > 0 8Q > 0, the order size that minimizes the

average inventory cost per unit time is

Fig. 3. Lead-time upper bound as a stepwise function of the order quantity.

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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2.2. Dual-mode model

Fig. 4. Average inventory cost when lead-time upper bound is a function of order quantity.

Q

c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2kA1 ; ¼ hð1 þ kcÞ

and the minimum average inventory cost is given by

G1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hkDs þ kc1 : 2kA1 hð1 þ kcÞ þ 2

For the remainder, we will assume that the lead-time upper bound for delivery mode 1, su , is a step function of Q, since we believe this provides a better approximation of the effect that the order quantity has on the delivery time of a stressed supply mode. That is, a step function allows for changes at discrete quantities, as may be the case when an additional truck is required for a highly congested route or an additional production batch is required on a capacitated production line. 2.1.2. Deterministic lead time mode We next consider the cost of using a shipping mode, called mode 2, that has a deterministic lead time L2 . In this case we can use the EOQ model where Q 2 is equal to

Q 2 ¼

rffiffiffiffiffiffiffiffiffiffiffi 2kA2 ; h

A2 is the fixed cost of placing an order when using mode 2, h is the inventory holding cost per unit per unit time, and k is the constant demand rate. We recognize that a more general model would permit holding costs that depend on the variable purchase cost. However, this difference in holding cost is typically very small in practice, and the resulting model in which holding costs do not depend on purchase cost permits obtaining much more in the way of analytical results (our computational tests presented later consider the impact of holding costs that depend on variable costs, and show that this impact is quite small). Letting c2 denote the unit purchasing cost for mode 2 and Dc ¼ c2  c1 , the average cost per replenishment cycle is equal to:

G2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 2kA2 h þ kðc1 þ DcÞ:

ð2:4Þ

In order to compare the average cost of exclusively using shipping mode 1 versus mode 2, we subtract (2.4) from (2.3). Assuming that A1 6 A2 and Dc P 0 we obtain:

G1  G2 6

khDs  kDc ¼ x; 2

ð2:5Þ

where x ¼ kh2Ds  kDc. Note that when x 6 0 mode 1 should be chosen as the preferred single mode. Later we will use this expression during the computation of the average inventory cost per replenishment cycle for the dual-mode model.

We next present a model where the buyer may use two different shipping modes, implying a positive (expected) order quantity for both modes. We will then compare our proposed dual-mode model with the single-mode model and choose the one with the minimum expected cost per unit time. Our proposed model assumes that in addition to mode 1, the buyer may place a second order that will use an expedited shipping mode, called mode 2, with deterministic lead time, L2 . In this case, in order to reduce the required safety stock, the buyer may define a , where s  2 ½L2 ; su Þ, and where Q P r different reorder point r ¼ ks  is in order to avoid shortages. Note that for the dual-mode model, s strictly less than su (in other words, the expected order quantity via mode 2 is strictly positive; the case in which the expected order quantity is zero corresponds to the single-mode case with mode 1  to define the reorder point increases the probability only). Using s sl ;sg g ¼ su maxf of stockout to PfL1 P s , which is strictly greater than su sl  < su . zero, since s Given that the buyer desires a zero probability of stockout, this may require using mode 2, which will be used as a backup in cases   L2 . Note that s   L2 is the time at which the buyer where L1 > s must place a second order using mode 2 if the mode 1 order has not arrived. If this is the case, then the buyer must place an order Þ units that will be sent via mode 2. This second order for kðsu  s , which is the inventory run-out time, and must will arrive at time s contribute enough inventory to avoid stocking out in the case that L1 ¼ su , which is the worst-case delivery time for mode 1. Note that we assume that su is a step function of Q, as shown in Fig. 3. This means that the lead-time upper bound for mode 1 will increase for increasing values of the quantity ordered. In the following section we first develop the expressions for the average inventory cost for the dual-mode model, and after that we will present a procedure to find an optimal solution for our proposed model when su is a step function of Q.   L2 < sl , then the buyer must decide on Observe that if s whether to place an order via mode 2 before any information regarding the value of L1 can be obtained (i.e., before time sl ). In this case, the supplier must place an order with mode 2 (otherwise, the probability of a stockout is positive, violating the 100% service   L2 P sl , then there is a positive level). On the other hand, if s probability that no order will be placed via mode 2 in a cycle. Because there are two distinct cases based on the relative values   L2 and sl , the following subsections will consider these cases of s separately.  2 ½L2 ; sl þ L2 Þ 2.2.1. CASE 1: s   L2 , the buyer must determine whether the mode 1 At time s order has arrived to evaluate whether it is necessary to place a  2 ½L2 ; sl þ L2 Þ we have that s   L2 < sl , and, mode 2 order. When s therefore, the buyer will always place a mode 2 order. Fig. 5 shows an example of the model realization. Observe that the replenishment cycle starts when the inventory  units. During the cycle the buyer orders Q units level reaches r ¼ ks Þ units via mode 2. Therefore, the cycle will via mode 1 and kðsu  s Þ. restart at time T ¼ Qk þ ðsu  s To analyze the expected inventory level at any time t, we divide the Þ; ½s ; su Þ; ½su ; T. replenishment cycle into four intervals: ½0; sl Þ; ½sl ; s Assume without loss of generality that the cycle starts at time t ¼ 0,  and an order of size Q is placed with when the inventory level is ks mode 1. This order will be received at some time t 2 ½sl ; su ; therefore, the probability of receiving the order during the first interval is PfL1 < sl g ¼ 0. During the replenishment cycle, the inventory level will decrease at a constant rate k per unit time due to demand, and hence   kt. the expected inventory level during t 2 ½0; sl Þ is ks

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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 2 ½L2 ; sl þ L2 Þ. Fig. 5. Dual-mode model realization. Case 1: s

Þ, the expected inventory During the second interval t 2 ½sl ; s  þ QPfL1 6 tg  kt, where PfL1 6 tg ¼ stssl , as level is equal to ks u l in the single-mode model. , when A shift in the expected inventory function occurs at t ¼ s Þ shipped via mode 2 arrives, and hence the order of size kðsu  s the expected inventory level during the third interval is ksu þ QPfL1 6 tg  kt. By time t ¼ su the order from supplier 1 has arrived with probability one and the expected inventory level is Q þ ksu  kt until the end of the cycle. A summary of the expected inventory level as a function of time is presented below.

8  ks  kt; 0 6 t < sl ; > h i > > > t Q  k þ ½ks Q sl >   s s ; sl 6 t < s; < su sl u l h i h i E½IðtÞ ¼ Q s >  6 t < su ; t suQs  k þ ksu  su ls ; s > > l l > > : su 6 t 6 Qk þ ðsu  sÞ: ðQ þ ksu Þ  kt; Fig. 6 shows the expected inventory level as a function of time. Recall that because we assume Q P kðsu  sl Þ (that is, we assume no order crossing; see Appendix A), the slope of the function for t 2 ½sl ; su  is non-negative. We can integrate the expected inventory level curve over the cycle to find the average total inventory per cycle, which gives

IT ¼

Q 2 Q ðsu  sl Þ k  Þ2 : þ þ ð su  s 2 2 2k

The average inventory per cycle is the average total inventory divided by T, which equals 2 2  Þ2 bI ¼ Q þ Q kðsu  sl Þ þ k ðsu  s : Þ 2½Q þ kðsu  s

For the dual-mode model, the fixed order cost has two components: A1 , which is the fixed order cost incurred by the buyer when placing the first order via mode 1; and A, which is the additional

 2 ½L2 ; sl þ L2 Þ. Fig. 6. Expected inventory level as a function of time, Case 1: s

order cost incurred when placing an expedited order via mode 2. Note that the expedited order may be placed with the same supplier or may be planned when placing the first order; we therefore assume that A 6 A2 , since part of the associated fixed order cost is incurred when placing the first order in the cycle. The total cost per cycle includes the fixed order cost, A1 þ A, the inventory holding cost per unit per unit time, h, and the purchasing cost per unit shipped by modes 1 and 2, denoted by c1 and c2 , respectively. This implies an average total cost per cycle equal to:

Q 2 Q ðsu  sl Þ k Þ2 TC I ¼ ðA1 þ AÞ þ h þ þ ð su  s 2 2 2k

!

Þc2 Þ: þ ðQc1 þ kðsu  s The average cost per cycle is equal to the total cost per cycle divided by the cycle length T, which equals

Þ ¼ GI ðQ ; s

Þ2 Þ þ c1 Q k þ c2 k2 ðsu  s Þ kðA1 þ AÞ þ h2 ðQ 2 þ Q kðsu  sl Þ þ k2 ðsu  s : Þ Q þ kðsu  s

Þ. Without loss of generality we let A ¼ A1 þ A and b ¼ ðsu  s Observe that kb ¼ Q 2 , i.e., the quantity delivered via mode 2. Thus, the total order quantity in a cycle equals Q þ kb. Noting that Q and b are decision variables, and the average cost per unit time is given by

GI ðQ ; bÞ ¼

kA hðQ þ kbÞ k þ þ ðc1 Q 1 þ c2 kbÞ ðQ þ kbÞ 2 ðQ þ kbÞ khQ  s þs s  u l : þ ðQ þ kbÞ 2

ð2:6Þ

The first three terms of (2.6), are the average annual fixed order cost, holding cost and purchasing cost of the EOQ model if we were . to order Q þ kb units, and with a deterministic lead time equal to s The last term represents a correction factor for the average annual holding cost term to account for the fact that the order of size Q faces a stochastic lead time. To illustrate this, note that in our model, the order placed with supply mode 2 will always arrive at time s; however, the order placed with supply mode 1 may arrive at any time between sl and su . In the case that the expected lead time for , the second term the order placed with mode 1, su 2þsl , is less than s of (2.6) underestimates the average annual holding cost, and therefore, the correction factor will be positive; on the other hand, if the , the correcexpected lead time of supply mode 1 is greater than s tion factor will be negative, since we are overestimating the average annual holding cost. From Appendix C we have that GI ðQ ; bÞ is convex 8Q > 0 and b > 0 if 4kAh > x2 , where x ¼ kh2Ds  kDc (Appendix C also provides an argument as to why this condition is likely to be mild and non-restrictive in practice). Therefore we find a stationI I ary point for GI ðQ ; bÞ using the conditions @G ¼ 0 and @G ¼ 0. @b @Q

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

 2 ½sl þ L2 ; su Þ and L1 6 s   L2 ). Fig. 7. Dual-mode model realization (Case 2.1: s

As result, we have that for a given b, assuming the convexity condition is met, the optimal Q and minimum average inventory cost per cycle are:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k2 bDc 2kA  Q ðbÞ ¼ kb þ 2k2 b2  k2 bDs þ þ ; h h

ð2:7Þ

ð2:8Þ

I Using @G ¼ 0, we arrive at the following stationary point solu@b tion for b:

2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3  2   1 4 khDs khDs b ¼  kDc þ  kDc 5: 4kAh  2kh 2 2 

Substituting x ¼ kh2Ds  kDc and a ¼ 4kAh, we have

b ¼

i 1 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ax þx ; 2kh

ð2:9Þ

and the optimal value for Q can then be expressed as

Q ¼

i 1 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ax x : 2h

ð2:10Þ

Assuming the stationary point solution is feasible, because the total order quantity per cycle equals Q  þ kb , Eqs. (2.9) and (2.10) indicate that the total order quantity per cycle at optimality equals

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a  x2 Q þ kb ¼ : h 



concave functions of x. The total order quantity per cycle will attain its maximum when x ¼ 0, which results in Q  ¼ kb. This means that when the incremental holding cost due to the uncertainty in the lead time of mode 1 is equal to the additional purchasing cost per unit of mode 2, the dual-mode model will order the same quantity from both supply modes, assuming the stationary points for (2.6) are feasible, i.e., Q > 0 and b 2 ½su  sl  L2 ; su  L2 Þ. We also observe that when x < 0 we will have Q  > kb. The reason for this is that a negative value of x indicates that the average inventory cost when using shipping mode 1 exclusively is less than the average inventory cost of using shipping mode 2, and therefore, the dual mode model will increase the use of shipping mode 1. The opposite happens when x > 0, when the incremental cost of using shipping mode 2 over the standard EOQ model is smaller than the incremental cost of using shipping mode 1, and as a result kb > Q  . Using (2.9) and (2.10), the minimum average inventory cost is

GI ¼

i 1 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 a  x þ x þ kc2 : 2

ð2:12Þ

Note that when the stationary point solution b is feasible, i.e., b 2 ðsu  sl  L2 ; su  L2  we will use (2.10) and (2.12) for Q  and GI , respectively; otherwise the optimal value of b will be at an end point of the interval and we will use (2.7) and (2.8) at the appropriate value of b to determine Q  and GI , respectively. Now that we have an expression for the average inventory cost  2 ½L2 ; L2 þ sl Þ, we can derive a for the dual-mode model when s method for determining optimal order quantities for modes 1 and 2 when su is a step function of Q, such that su ¼ siu if qi 6 Q < qiþ1 for i ¼ 1; 2; . . . ; n, where qi < qiþ1 ; siu < siþ1 u , and n is the number of intervals for Q. Note that because an increase in su increases the uncertainty in the arrival time of the order placed with mode 1, we expect to observe an increase in the average inventory holding cost for increasing values of su . However, in some cases, when the stationary point solution for b is not feasible, and therefore, the optimal b is found at one of the end points of the interval (in particular, b ¼ su  L2 ), the average inventory cost may actually decrease as su increases. The reason for this is that, in this case, it is preferable to increase Þ); as a result, the size of the order placed with mode 2, (kðsu  s as we increase the value of su , an increased order quantity with supply mode 2 may actually decrease the average inventory holding cost. As in the single-mode case, our solution procedure is similar to the one used by Chopra and Meindl (2007) to find the optimal order quantity under the all units quantity discount scheme; however, because GI may increase or decrease for increasing values of su , as explained above, when we analyze an interval i, if the stationary point for Q is not feasible for that interval, we still need to evaluate the value of Q at one of the breakpoints, depending 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2k2 bDc 2kA Ds GI ðbÞ ¼ h 2k2 b2  k2 bDs þ þ b þ kh h h 2 þ kc1 :

7

ð2:11Þ

Based on 2.9, 2.10 and 2.11, we observe that the quantities shipped via modes 1 and 2 (and the total order quantity per cycle) are

 2 ½sl þ L2 ; su Þ and L1 > s   L2 ). Fig. 8. Dual-mode model realization (Case 2.2: s

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

on whether Q > qiþ1 or Q < qi . Algorithm 1, in Appendix D, provides a detailed description of the solution procedure to find the minimum average inventory cost for the dual-mode model when su is a step function of Q.  2 ½sl þ L2 ; su Þ 2.2.2. CASE 2: s   L2 the buyer must deterIn our dual-mode model, at time s mine whether or not to place a second order via mode 2 if the first order has not yet arrived. Since Case 2 considers the interval s 2 ½sl þ L2 ; su Þ, we have that s  L2 P sl , and the probability of sL2 Þ   L2 Þg ¼ su ð placing a second order is PfL1 P ðs s s . u

l

  L2 there is no need to place an expedited order When L1 6 s using mode 2, and therefore the total amount ordered in the cycle 2 sl Þ   L2 Þg ¼ ðsL is Q. This occurs with probability PfL1 6 ðs s s . Fig. 7 u

l

Based on the probability of each of these sub-cases, we can compute the expected inventory as a function of time. As in the previous section we will divide the replenishment cycle into four Þ; ½s ; su Þ; ½su ; T, where T, the cycle length, is intervals: ½0; sl Þ; ½sl ; s now a random variable, and E½T is the expected cycle length, i.e.,

E½T ¼

  Q Q   L2 g þ Þ PfL1 > s   L2 g; PfL1 6 s þ ðsu  s k k

  2 sl   L2 g ¼ sL where after substituting PfL1 6 s su sl , we have

E½T ¼

Q Þ þ ðsu  s k



The last interval of the cycle is t 2 ½su ; T. By time t ¼ su the order from supplier 1 has arrived with probability one, and we have two cases with respect to the second order:   L2 , the inventory level is Q þ ks   kt after time t ¼ su , 1. If L1 6 s ; and T ¼ Q =k þ s   L2 , the inventory level is Q þ ks  þ kðsu  s Þ  kt after 2. If L1 > s time t ¼ su , and T ¼ Q =k þ su .

l

shows an example of this case.   L2 , a second order of size kðsu  s Þ is placed at When L1 > s   L2 , which will be shipped using mode 2 and will be time t ¼ s received at time s. This occurs with probability sL2 Þ   L2 Þg ¼ su ð PfL1 > ðs s s . Fig. 8 shows an example of this case. u

 6 t < su Þ E½Iðtjs   Q ¼t k su  sl    Q sl s  s þ L2  Þ u : þ kðsu  s þ ks su  sl su  sl

 þ kðsu  s Þ E½IðtÞ ¼ Q þ ks





su  s þ L2  kt: su  sl

A summary of the expected inventory level as a function of time is presented below. Fig. 9 shows the expected inventory level as a function of time. 8   kt; ks 0 6 t < sl ; > > > Q Q sl > >  ;  k þ ½k s  ; s t½ > l 6 t < s su sl < su sl      þL2 E½IðtÞ ¼ t Q  k þ ks Q sl s s u   s s þ kðsu  s  Þ s s  6 t < su ; ; s > u u > l l > su sl >   > >  2 : Q þ ks  þ kðsu  s Þ sussþL su 6 t 6 T:  kt; u s l



su  s þ L2 : su  sl

Therefore the expected inventory for t 2 ½su ; T is:

ð2:13Þ

The replenishment cycle starts at time t ¼ 0, when the  and an order of size Q is placed, which will inventory level is ks be sent via mode 1 and received at some time t 2 ½sl ; su ; the probability of receiving the order during the first interval is PfL1 < sl g ¼ 0. During the replenishment cycle, the inventory level decreases at a constant rate of k per unit time due to demand, and hence the   kt. During the secexpected inventory level during t 2 ½0; sl Þ is ks Þ, the expected inventory level is equal to ond interval, t 2 ½sl ; s  þ QPfL1 6 tg  kt, where PfL1 6 tg ¼ stssl . If, at time t ¼ s   L2 , ks u

The assumption that the model does not allow shortages or or and Q P kðsu  sl Þ (from Appender crossing requires that Q P ks dix A). Therefore, we can see that the slope of the expected inventory function is non-negative for t 2 ½sl ; su . In order to characterize the average total inventory per replenishment cycle, we integrate the expected inventory function over the four intervals, and we obtain:

E½I ¼

Q2 Q Þ2 þ 2ðsu  s ÞL2 þ ðs   sl Þ2  þ ½ðsu  s 2k 2ðsu  sl Þ 2

þ

Þ2 þ ðsu  s ÞL2  k½ðsu  s 2ðsu  sl Þ2

ð2:14Þ

l

the buyer does not place an expedited order using mode 2, this means that the mode 1 order has been received, and the inventory ; su Þ is equal to ks  þ Q  kt, which occurs with problevel for t 2 ½s   L2 g. With probability PfL1 > s   L2 g, however, ability PfL1 6 s   L2 for the buyer places a second order via mode 2 at time t ¼ s Þ units, which will be received at time s . kðsu  s For the latter case, when the second order with mode 2 was   L2 , we can have two situations for the interplaced at time t ¼ s ; su Þ: val t 2 ½s  þ kðsu  s Þ  kt, for 1. L1 > t, where the inventory level is ks ; su Þ; t 2 ½s  þ kðsu  s Þ þ Q  kt, for 2. L1 6 t, where the inventory level is ks ; su Þ. t 2 ½s ; su Þ, the expected inventory level is: Therefore, for t 2 ½s

 6 t < su Þ ¼ PfL1 6 s   L2 gðks  þ Q  ktÞ þ PfL1 E½Iðtjs  þ kðsu  s Þ  ktÞ þ Pfs   L2 < L1 > tgðks  þ kðsu  s Þ þ Q  ktÞ; 6 tgðks which results in

Our model can be analyzed as a renewal reward process (see Appendix F), and therefore, the average inventory per unit time, E½IðtÞ E½I as t ! 1, is equal to E½T . Using (2.13) and (2.14) we have that t the average inventory per unit time is: 2 2 Þ2 þ 2ðsu  s ÞL2 þ ðs   sl Þ2  þ k2 ½ðsu  s  Þ2 þ ðs u  s ÞL2  bI ¼ Q ðsu  sl Þ þ kðsu  sl ÞQ ½ðsu  s 2 2   2½Qðsu  sl Þ þ kðsu  sl Þ½ðsu  sÞ þ ðsu  sÞL2 

2

The expected total cost per cycle is composed of the expected fixed order cost, expected holding cost and expected purchasing cost. Using A1 as the cost of placing an order via mode 1 and A as the incremental cost of placing a second order via mode 2, the expected fixed order cost is:

  L2 g þ ðA þ A1 ÞPfL1 > s   L2 g: A1 PfL1 6 s Substituting the probability values, we obtain an expected fixed order cost of

  su  s þ L2 : A1 þ A su  sl The expected total inventory holding cost per cycle is

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9

 2 ½sl þ L2 ; su Þ). Fig. 9. Expected inventory as a function of time (Case 2: s

( h

h i k½ðs  s Þ2 þ ðsu  s ÞL2  Q2 Q u Þ2 þ 2ðsu  s ÞL2 þ ðs   sl Þ 2 þ ðsu  s þ 2k 2ðsu  sl Þ 2ðsu  sl Þ2

2)

The expected purchasing cost in a cycle is given by

  L2 g þ ðc1 Q þ c2 kðsu  s ÞÞPfL1 > s   L2 g: c1 QPfL1 6 s Substituting the probability values, we obtain an expected variable purchase cost per cycle of

Þc2 Qc1 þ kðsu  s





su  s þ L2 : su  sl

Adding the expected fixed order cost, expected holding cost and expected purchase cost, we obtain the following expression for the total cost per cycle:     su  s þ L2 s  s þ L2 Þc2 u TC II ¼ A1 þ A þ Qc1 þ kðsu  s su  sl su  sl " 2# Þ2 þ ðsu  s ÞL2  Q2 Q k½ðsu  s Þ2 þ 2ðsu  s ÞL2 þ ðs   sl Þ2  þ þh ½ðsu  s þ : 2k 2ðsu  sl Þ 2ðsu  sl Þ2

The expected cost per unit time, which we denote by GII , is equal to the expected total cost per cycle divided by the expected cycle length, and we therefore have:

GII ðQ; bÞ ¼

2kA1 Ds2 þ 2kADsðb þ L2 Þ þ 2kDs2 Qc1 2½Q Ds2 þ kDsðb2 þ bL2 Þ 2

2

þ

2k2 Dsbðb þ L2 Þc2 þ hQ Ds2 þ k2 hðb2 þ bL2 Þ 2

2½Q Ds2 þ kDsðb þ bL2 Þ hkDsQ ½b þ 2bL2 þ ðDs  bÞ2 

:

GbII ðQ Þ for any given b, where b 2 ð0; Ds  L2  (using, e.g., gradient search techniques). Then, in order to find the optimal values of Q    and b we define GII ¼ minQ >0 fGII Q b ; b 8b 2 ð0; Ds  L2 g. Although b is a continuous variable that corresponds to the length of time during which the order shipped via mode 2 should cover demand, we will assume that it is sufficient to consider a discrete set of candidate values for b (typically in practice the value of b will be measured in some discrete measure of time, such as days). We can see that the convexity condition for GII ðQ Þ is not restrictive by observing that whenever x 6 0 (from Eq. (2.5)), i.e., when mode 1 is the preferred single mode, then (2.16) will always hold. We also note that the left-hand side of (2.16) is the expected order cost in a cycle divided by the expected extra time in the cycle if an order is placed via mode 2; the left-hand side is therefore an upper bound on the fixed order cost per unit time. The right-hand side of the inequality is less than x, the amount by which the cost of buffering the uncertainty using mode 1 exceeds the cost of buffering the uncertainty using mode 2. Therefore, even if mode 1 is not the preferred single mode of operation, it is not unlikely for condition (2.16) to hold for a broad range of practical cases. Since we assume that su is a step function of Q such that su ¼ siu if qi 6 Q < qiþ1 , for i ¼ 1; 2; . . . ; n, where qi < qiþ1 ; siu < siþ1 u , and n is the number of intervals for Q, we follow the same procedure to find the minimum average cost that we developed in the previous section. In Algorithm 2 (see Appendix E) we present the procedure to find the minimum average inventory cost for the dual-mode model  2 ½sl þ L2 ; su . when s

2

þ

2½Q Ds2 þ kDsðb2 þ bL2 Þ

ð2:15Þ

;

; Ds ¼ su  sl , and Dc ¼ c2  c1 . Note that kb is the where b ¼ su  s quantity ordered via mode 2 when this order is placed. The decision variables in the above expected cost equation are therefore Q and b. In order to show the convexity of GII as a function of Q, we consider its second derivative, which is given by

@ 2 GII @Q

2

¼

k2 hDsðb2 þ bL2 Þð2bDs  Ds2 Þ þ 2k2 Ds2 ðb2 þ bL2 ÞDc 3

½Q Ds þ kðb2 þ bL2 Þ þ

2kDs2 ðDsA1 þ Aðb þ L2 ÞÞ 3

½Q Ds þ kðb2 þ bL2 Þ

: 2

It is straightforward to show that the denominator of @@QG2II is positive

8Q > 0 and b > 0, and the numerator is non-negative if: A1 þ Ap P x  khb; bp

ð2:16Þ

where p ¼ su DssþL2 is the probability that an order is placed via mode 2. Note that (2.16) provides a convexity condition for GII as a function of Q for any b, and therefore we need to find the value Q b for minimizing 

3. Numerical analysis This section reports results of a set of numerical tests that characterize the benefits of the dual-mode model. While we show numerous cases in which the dual-mode model outperforms the better of the two single mode solutions, we also analyze how changes in model parameters affect the average inventory cost of the dual-mode model (recall that the minimum dual-mode average

inventory cost equals G ¼ min GI ; GII ). Since the average purchasing cost per year is at least kc1 (whether we use the single or dual-mode model), to compare the results of the dual-mode model against the single-mode model, we subtract kc1 from the average inventory cost for both models. Therefore we will compare the percentage cost reduction over the inventory costs that can be modified when using an alternative ordering policy. Hence, the percentage cost reduction is given by:

%CR ¼

minfG1 ; G2 g  G  100%; minfG1 ; G2 g  kc1

where Gi is the minimum average inventory cost per unit time when the order is shipped by mode i exclusively, i ¼ f1; 2g and G is the minimum average inventory cost for the dual-mode model.

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

Table 1 Base parameter values for numerical analysis.

sl (days)

k (units/year)

L2 (days)

h ($/unit year)

c1 ($/unit)

A1 ($/order)

A ($/order)

A2 ($/order)

10,000

14

5

1.5

10

100

70

170

s1u (days)

s2u (days)

s3u (days)

s4u (days)

50

55

60

65

q1 (units) 0

q2 (units) 1000

q3 (units) 2000

q4 (units) 3000

We are of course particularly interested in characterizing conditions under which %CR > 0. For our numerical analysis, we created problem instances with parameter values similar to the cost structures that can be observed in practice, and where we can demonstrate the different effects that changes in a particular parameter may have for the dualmode model. Table 1 shows the base parameter values used for the numerical analysis of our proposed model. For the fixed order cost, we assume that the fixed cost for mode 1, A1 , is less than or equal to the fixed order cost of mode 2, A2 . Also, assuming that orders can be placed with the same supplier, but using different shipping modes, we have that A1 þ A 6 A1 þ A2 , where A is the additional order cost of placing a second expedited order, since a portion of the fixed order cost was already incurred when the first order was placed. Before analyzing the effects of key parameters on the dualmode model, we discuss properties of optimal solutions for our proposed model. As we can see, based on the results in Tables 2–  is 8, we observe that for increasing values of Dc, the value of s non-decreasing (and the order size with supply mode 2 is non ¼ su , at which increasing), until it reaches its maximum value, s point the order size with supply mode 2 is zero. This means that in cases where the additional unit purchasing cost for using the more reliable mode is sufficiently small, it is optimal to always utilize both suppliers during each replenishment cycle, and conse 2 ½L2 ; sl þ L2 Þ. As Dc increases, the use of shipping quently s mode 2 becomes more expensive, and therefore, the model will in (decreasing the mode 2 order size), such that crease the value of s the probability of placing an order with delivery mode 2 is strictly  2 ðsl þ L2 ; su . less than one, and therefore, s However, even when supply mode 2 is the preferred single supplier mode, i.e., G2 < G1 , and Dc ¼ 0, we find some cases where it is not optimal to use supply mode 2 with probability one in every  2 ½sl þ L2 ; su Þ, as shown in Tables 2 replenishment cycle, i.e., s and 3. In other words, even though mode 2 is the cheaper single mode option, it is not optimal to maximize the size of the order with supplier 2 in our dual-mode model. This occurs because when Ds is sufficiently large, placing a large order with supplier 2 implies  (since the order with supply mode 2 is equal a smaller value of s

Table 2 Effect of

 Þ), and therefore an earlier decision time, s   L2 , on to kðsu  s whether to place the second order. As a result, the probability of placing an order with supply mode 2 increases, as well as the expected cost of holding this order in inventory. Since we do not have closed-form expressions for the optimal values of Q  and b for the dual-mode model, we perform numerical tests to evaluate how changes in key model parameters for modes 1 and 2 may affect the optimal solution of the dual-mode model and its performance relative the single-mode model. Sections 3.1–3.8 show how changes in a single parameter of interest influence the relative performance of the dual-mode model, all else being equal. Following this, in Section 3.9 we compare the results of the dual-mode model under various combinations of mode delivery lead time and variable cost. 3.1. Impact of increasing

sl

In order to analyze the effects of changes in the value of the lower limit on the lead time for delivery mode 1, sl , we performed numerical tests where we increased the value of sl while holding all other parameters equal. When sl increases (for a fixed su ), the mode 1 lead time uncertainty decreases, and therefore, the need for an emergency supplier also decreases. As a result, for increasing sl we expect to observe a decrease in the order size with delivery  . Since an increase in mode 2, kb, and therefore, an increase in s sl represents a decrease in Ds, we also expect to see a decrease in average inventory cost. Table 2 shows the optimal solutions for the dual-mode model and single-mode models and %CR, for increasing values of sl . As expected, we see a decrease in the order placed with delivery mode 2, except in cases when Dc ¼ 0. In this example, when Dc ¼ 0 it is beneficial to increase the use of mode 2, since we can benefit from using a more reliable mode without paying a premium in purchasing cost, and we thus observe increasing values for the size of the second order, kb. We also see a decrease in the average inventory cost per unit time, although the effect on %CR will depend on which supply mode has the lower average inventory cost. The average cost of mode 1 decreases as we increase sl , since there is a reduction in the uncertainty in the order arrival time, and

sl on the dual-mode model.

Dual-mode model

Single-mode model

%CR

sl (days)

s (days)

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

%

Dc ¼ 0 7 14 20

22 5 5

904 1233 1233

1315 986 839

102050.0 101937.3 101875.2

1507 1507 1507

102780.1 102636.2 102513.0

1506 1506 1506

102258.3 102258.3 102258.3

9.22 14.22 16.97

Dc ¼ 0:5 7 14 20

49 50 52

164 137 82

1343 1370 1425

102730.0 102630.0 102540.0

1507 1507 1507

102780.1 102636.2 102513.0

1506 1506 1506

107258.3 107258.3 107258.3

1.80 0.24 1.08

Dc ¼ 1 7 14 20

53 53 54

55 55 27

1452 1452 1480

102800.0 102680.0 102580.0

1507 1507 1507

102780.1 102636.2 102513.0

1506 1506 1506

112258.3 112258.3 112258.3

0.72 1.66 2.67

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

11

C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx Table 3 Effect of

su on the dual-mode model. q1 ¼ 0, q2 ¼ 1000; q3 ¼ 2000; q4 ¼ 3000.

Dual mode-model

Single-mode model 

%CR

s (days)

s (days)

s (days)

s (days)

s (days)

Dc ¼ 0 30 50 60

35 55 65

40 60 70

45 65 75

5 5 31

685 1233 932

1000 986 1397

101858.0 101937.3 102080.0

1000 1507 1781

102078.8 102636.2 102945.1

1506 1506 1506

102258.3 102258.3 102258.3

10.62 14.22 7.90

Dc ¼ 0:5 30 50 60

35 55 65

40 60 70

45 65 75

35 50 56

0 137 247

1247 1370 1534

102300.0 102630.0 102810.0

1000 1507 1781

102078.8 102636.2 102945.1

1506 1506 1506

107258.3 107258.3 107258.3

10.64 0.24 4.59

Dc ¼ 1 30 50 60

35 55 65

40 60 70

45 65 75

35 53 61

0 55 110

1247 1452 1671

102300.0 102680.0 102920.0

1000 1507 1781

102078.8 102636.2 102945.1

1506 1506 1506

112258.3 112258.3 112258.3

10.64 1.66 0.85

1 u

2 u

3 u

4 u

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

%

Table 4 Effect of L2 on the dual-mode model. Dual-mode model

Single-mode model 

%CR

L2 (days)

s (days)

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

Dc ¼ 0 5 14 30

5 14 30

1233 986 685

986 986 1123

101937.3 101971.4 102181.4

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

102258.3 102258.3 102258.3

14.22 12.71 3.40

Dc ¼ 0:5 5 14 30

50 54 55

137 27 0

1370 1480 1507

102630.0 102790.0 102980.0

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

107258.3 107258.3 107258.3

0.24 5.83 13.04

Dc ¼ 1 5 14 30

50 54 55

137 27 0

1452 1507 1507

102680.0 102790.0 102980.0

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

112258.3 112258.3 112258.3

1.66 5.83 13.04

therefore, in the safety stock required. The reduction in G1 is greater than the reduction in average inventory cost of the dual-mode model, G , and therefore, when G1 < G2 , the value of %CR decreases as we reduce the uncertainty in mode 1 lead time, making the dual-mode model relatively less attractive for increasing values of sl . Because an increase in sl does not affect the cost of mode 2, when G2 < G1 , an increasing value of sl leads to an increase in %CR, since G decreases.

3.2. Impact of increasing

su

When we increase su while holding all other parameters equal, we increase the uncertainty of mode 1. As a result we expect to see an increase in the amount shipped via mode 2, since the dualmode model will attempt to mitigate the impact of the increased uncertainty in L1 by using mode 2. Note also that an increase in the uncertainty in the arrival time of an order placed with supply mode 1 will increase the average inventory cost of mode 1 exclusively, as well as the average inventory cost of the dual-mode model, although it will not affect the inventory cost associated with using mode 2 exclusively. Table 3 shows the effect that increasing su has on the dualmode model and the single-mode model, for a specific instance. As expected, for increasing values of su we observe an increase in the order placed with mode 2 and an increase in the average inventory cost of the dual-mode model, G . Note that when Dc ¼ 0, the increase in su (and therefore, in Ds) has a different effect on the

%

quantity ordered by modes 1 and 2. As mentioned previously, when Dc ¼ 0 and for large values of Ds when G2 < G1 , the dualmode model will try to reduce the probability of using mode 2  2 ½sl þ L2 ; su Þ, and therefore, reducing kb. Also, for by choosing s increasing values of Ds, the lower limit on the order placed with mode 1 in order to avoid order crossing, Q P Ds (see Appendix A) increases, and therefore, this minimum order size constraint is satisfied at equality. We observe that the effect of increasing su on %CR will depend on which supplier has the lower average inventory cost. Since the average inventory cost of using mode 1 exclusively increases at a faster rate than the average inventory cost of the dual-mode model, G , when G1 < G2 ; %CR will increase. However, when G2 < G1 , the percentage cost reduction will decrease, since G2 is independent of su .

3.3. Impact of increasing mode 2 lead time, L2 We next discuss the effect of increasing L2 (while keeping the other parameters equal) on the dual-mode model. As L2 increases, the time at which we must decide on whether to place the second   L2 , will be earlier in the cycle, and therefore, the probaorder, s bility of placing the second order will increase. We can view the increase in L2 as a decrease in the responsiveness of delivery mode 2, and as a result, the average inventory cost of the dual-mode model will increase, and the model will be less beneficial. Table 4 shows the effect of the increase in L2 on the performance of the dualmode model. As expected, the average inventory cost of our

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

Table 5 Effect of demand rate on the dual-mode model. q1 ¼ 0; q2 ¼ 500; q3 ¼ 1500; q4 ¼ 3000. Dual-mode model

Single-mode model 

%CR

k (units/year)

s (days)

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

Dc ¼ 0 500 3000 7000 9000

50 5 5 5

0 411 959 959

270 527 786 1011

5442.7 31042.9 71608.4 91873.1

258 500 1055 1356

5424.3 31196.9 72044.5 92439.0

337 825 1260 1428

5505.0 31237.0 71889.0 92142.4

4.34 12.87 14.85 12.57

Dc ¼ 0:5 500 3000 7000 9000

50 51 51 51

0 33 77 77

270 681 1053 1258

5442.7 31236.0 72079.0 92447.0

258 500 1055 1356

5424.3 31196.9 72044.5 92439.0

337 825 1260 1428

5755.0 32737.0 75389.0 96642.0

4.34 3.27 1.69 0.33

Dc ¼ 1 500 3000 7000 9000

50 55 54 54

0 0 19 19

270 659 1036 1332

5442.7 31241.0 72100.0 92488.0

258 500 1055 1356

5424.3 31196.9 72044.5 92439.0

337 825 1260 1428

6005.0 34237.0 78889.0 101140.0

4.34 3.68 2.72 2.01

G2 ($/year)

%

Table 6 Effect of holding cost on the dual-mode model. Dual-mode model

Single-mode model

%CR

h ($/unit year)

s (days)

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

%

Dc ¼ 0 0.8 1.2 2.5

5 5 26

1370 1233 658

1173 986 986

101387.4 101703.0 102660.0

1507 1507 1507

101715.7 102241.7 103951.3

2062 1683 1166

101649.2 102019.9 102915.5

15.88 15.69 8.76

Dc ¼ 0:5 0.8 1.2 2.5

54 52 45

27 82 274

1670 1425 1233

101770.0 102280.0 103650.0

1507 1507 1507

101715.7 102241.7 103951.3

2062 1683 1166

106649.2 107019.9 107915.5

3.17 1.71 7.63

Dc ¼ 1 0.8 1.2 2.5

55 55 50

0 0 137

1647 1507 1370

101770.0 102300.0 103860.0

1507 1507 1507

101715.7 102241.7 103951.3

2062 1683 1166

111649.2 112019.9 112915.5

3.17 2.60 2.31

Table 7 Effect of holding cost as a percentage of purchasing cost on the dual-mode model. Holding cost as a percentage of c1

Holding cost as a percentage of c1 and c2

D

kb (units)

Q (units)

G ($/year)

s (days)

kb (units)

Q (units)

G ($/year)

5 6 5

1370 1205 1233

1000 894 867

100925.8 101239.2 102189.7

5 6 12

1370 1205 1041

1000 894 641

100923.3 101235.8 102162.1

0.00 0.00 0.03

Dc ¼ 0:5 8 12 25

54 50 46

27 137 247

1606 1382 1263

101576.3 102128.4 103499.9

55 50 44

0 137 301

1692 1382 1332

101581.4 102124.2 103517.2

0.01 0.00 0.02

Dc ¼ 1 8 12 25

54 54 52

27 27 82

1627 1546 1457

101587.8 102115.2 103716.7

55 54 49

0 27 164

1641 1591 1378

101564.4 102119.7 103678.1

0.02 0.00 0.04

h (%)

Dc ¼ 0 8 12 25

s (days)

proposed model increases for increasing values of L2 , and since the average inventory cost of the single-mode model is independent of L2 , we observe a decrease in %CR. 3.4. Change in demand rate: k In order to evaluate how the demand rate affects the performance of the dual-mode model we performed numerical tests

%

where we increased the value of k, while holding other parameters equal. As the demand rate increases we expect to observe an increase in the total quantity ordered through modes 1 and 2, i.e., Q þ kb, and an increase in the average inventory cost for the dual-mode model as well as for the single-mode model. Table 5 shows the optimal solution and average inventory cost for the dual-mode model and single-mode model for

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx Table 8 Effect of A on the dual-mode model. Dual mode model

Single mode model

s (days)



%CR

kb (units)

Q (units)

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

5 5 5

1233 1233 1233

986 986 1000

101937.3 102072.5 102297.4

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

102258.3 102258.3 102258.3

14.22 8.23 1.73

Dc ¼ 0:5 70 100 150

50 51 53

137 110 55

1370 1397 1452

102630.0 102680.0 102740.0

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

107258.3 107258.3 107258.3

0.24 1.66 3.94

Dc ¼ 1 70 100 150

53 54 55

55 27 0

1452 1480 1507

102680.0 102710.0 102760.0

1507 1507 1507

102636.2 102636.2 102636.2

1506 1506 1506

112258.3 112258.3 112258.3

1.66 2.80 4.69

A ($/order)

Dc ¼ 0 70 100 150

increasing values of demand rate. Note that for these instances, because the demand rates used were less than the base value of 10,000, we modified the break points at which su increases for increasing values of Q (see the caption of Table 5). This permitted analyzing solutions with different values of su , and showing how  in the optimal solution. the value of k affects the value of s We observed that for increasing values of k, when G1 < G2 , the value of %CR increases. Although we have not been able to prove it analytically, the reason is that for increasing values of k, the average inventory cost for supply mode 1 increases at a faster rate than the average inventory cost for the dual-mode model, G . The opposite happens when we increase k and G2 < G1 . Since G increases at a faster rate than the average inventory cost for supply mode 2, the value of %CR will decrease.

%

is independent of the supply mode has little impact on the results of the model. 3.6. Change in mode 2 incremental fixed order cost: A When the incremental fixed order cost for using shipping mode 2 in the dual-mode model increases, while holding all other parameters fixed, we expect to observe an increase in the average inventory cost of the dual-mode model, since it is more expensive to place a second order. Since the average inventory cost for the single-mode model is independent of the extra fixed order cost of placing a second order, as we increase A the value of %CR will decrease, making the dualmode model less beneficial. Table 8 shows the results of our numerical tests when we increase the value of A.

3.5. Change in holding cost parameter: h

3.7. Change in mode 2 purchasing cost: c2

To evaluate the effect of the holding cost per unit per unit time in the dual-mode model, we performed numerical tests where we increased the value of h and held the rest of the parameters fixed. Table 6 shows the order quantities and average inventory cost for the single-mode model and the dual-mode model. We observe that when h increases, the total quantity ordered in the dual-mode model, Q þ kb, decreases and the average inventory cost increases. Through our numerical tests, we cannot predict the effects of the increase of h on %CR, however, as this will depend on the value of h relative to the other parameters. Next, we wish to understand how computing the holding cost as a percentage of the item’s variable purchase cost might affect our results. To do this, we first calculated the holding cost per unit time as a percentage of the purchase cost of supply mode 1 exclusively. We then compared this to the case in which the holding cost is computed as a percentage of the value of each item, which depends on whether the item was purchased using supply mode 1 or supply mode 2, assuming a first-in, first-out (FIFO) inventory policy. In order to estimate these costs, we used a simulation-based optimization model to estimate the average inventory costs per unit time for our model. Table 7 shows the results for these two cases and the difference in their average inventory costs as a percentage of the average inventory cost for our dual-mode model, where the holding cost is calculated as a percentage of the purchasing cost of mode 1. As expected, the difference increases for increasing values of c2 . For every instance, however, this difference is less than 0.05%, and therefore, we conclude that our assumption of a holding cost that

When we increase the purchasing cost per unit for supply mode 2, and keep all other parameters fixed, we expect to observe an increase in the average inventory cost of the dual-mode model. Tables 2–6 and Table 8 show the optimal solution for the dualmode model and single-mode model for different values of c2 . We observe that although it may be intuitive to conclude that as we increase c2 the use of the dual-mode model will become less attractive, this is not always the case. The impact of the change in c2 depends on which shipping mode has the minimum cost when they are used independently. When shipping mode 2 is the preferred supply mode, as we increase c2 , the cost of using it exclusively also increases, but at a faster rate than the increase in the average inventory cost of the dual-mode model, G , and therefore the percentage cost reduction obtained by using the dual-mode model increases with increasing values of c2 . This increasing trend in %CR stops when c2 is large enough to make mode 1 the new preferred shipping mode. When mode 1 is the preferred single supply mode, as we increase c2 , the value of G1 does not change, and %CR decreases. Therefore the maximum %CR for increasing values of c2 is reached at the maximum value of c2 such that G2 6 G1 . 3.8. Change in mode 1 lead-time distribution For the development of our model we assumed that the lead time for supply mode 1 is Uniformly distributed. When the mode 1 lead time follows a Uniform Distribution, L1  Uðsl ; su Þ, the probability density of an order arrival at time t is equal for any time

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

Table 9 Dual-mode model for L1  fsl þ DsBetaða; bÞg. q1 ¼ 0; q2 ¼ 500; q3 ¼ 1500; q4 ¼ 3000. Dual-Mode Model

Single-Mode Model

k (units/year)

s (days)

kb (units)

Dc ¼ 0 500 3000 7000

45 5 7

7 411 921

Dc ¼ 0:5 500 3000 7000

43 43 42

Dc ¼ 1 500 3000 7000

43 41 42



%CR

G ($/year)

Q 1 (units)

G1 ($/year)

Q 2 (units)

G2 ($/year)

% L1  fsl þ DsBetaða; bÞg

% L1  Uðsl ; su Þ

241 585 604

5433.6 30987.9 71450.6

258 500 1055

5440.1 31292.0 72297.2

337 825 1260

5505.0 31237.0 71889.0

1.51 20.14 23.23

4.34 12.87 14.85

10 99 249

238 669 197

5428.3 31196.1 71980.2

258 500 1055

5440.1 31292.0 72297.2

337 825 1260

5755.0 32737.0 75389.0

2.69 7.42 13.80

4.34 3.27 1.69

10 74 74

241 500 1013

5430.9 31181.7 72000.9

258 500 1055

5440.1 31292.0 72297.2

337 825 1260

6005.0 34237.0 78889.0

2.10 8.54 12.90

4.34 3.68 2.72

Q (units)

Fig. 10. Percentage cost reduction for different values of L2 and Dc. A2 ¼ $150= order, A ¼ $50= order.

t 2 ½sl ; su , and the expected arrival time is E½L1  ¼ sl þ2su , which is equal to the midpoint of the interval. We are interested in the effects that using a non-Uniform distribution for the mode 1 lead time has on the dual-mode model. In particular, we assume that L1  fsl þ DsBetaða; bÞg1 where a ¼ 2 and b > 2, and therefore the distribution is unimodal and positively skewed. Under this assumption the probability of having an early arrival (before the middle point of the interval) is higher than when a Uniformly distributed lead time is assumed. Since we can no longer use the expressions found in Section 2, in order to find the optimal solution for the dual-mode model when we use a Beta distribution for L1 , we need to use a simulation-based optimization approach; in particular, we used the commercial package OptQuest from Arena. Table 9 shows the optimal values and average inventory costs for the single-mode model and dual-mode model when we assume that L1  fsl þ DsBetaða; bÞg. The last two columns of Table 9 show the %CR values for the different assumptions for the distribution of L1 . We observe that we obtain higher %CR values when we assume that L1 follows a Beta distribution, and the reasons for this are twofold: first, the average inventory cost of using supply mode 1 exclusively is higher when we use a positive skewed Beta distribution for L1 , compared with the average inventory cost of assuming a Uniformly distributed lead time. This is because the expected arrival time when L1  fsl þ DsBetaða; bÞg is less than sl þ2su , and 1 We define L1 as a scaled Beta variable, since the Beta distribution is defined on the interval ½0; 1 and we are interested in general cases where L1 2 ½sl ; su .

therefore, the holding cost due to safety stock (ksu  kE½L1 ) is increased under the Beta distribution (when compared to the Uniform); second, since the probability of an early arrival is higher when we assume that L1 follows a Beta distribution, compared with the use of the Uniform distribution, the probability of placing the second order with supply mode 2 decreases, and therefore, the average total cost of using the dual-mode model will decrease. In practice we expect that L1 follows a probability distribution with some positive skew, since the unusually long cases typically lead to a skew in the distribution. Hence, based on the previous analysis, we can expect to have higher %CR values than the ones observed under the assumption of a Uniformly distributed lead time, and therefore, the dual-mode model may be more beneficial in practice.

3.9. Comparing supply modes with different cost and lead-time parameters The analysis discussed in the previous sections shows how certain parameter changes affect the benefits of the dual-mode model. This section considers how a buyer might evaluate multiple supply mode options, where a given mode implies certain costs and leadtime distribution characteristics. That is, all else being equal, we would like to know how to choose from different supply alternatives. For example, using the instances in Fig. 10, we can see how the buyer might compare different deterministic supply modes that can be used as secondary sourcing, with different combinations of purchasing costs and lead times.

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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15

Fig. 11. Percentage cost reduction for different values of L2 , Dc and Ds. A2 ¼ $150= order, A ¼ $50= order.

Fig. 12. Percentage cost reduction for different values of L2 , Dc and E½L1 . A2 ¼ $150= order, A ¼ $50= order.

In Fig. 10 we observe that choosing the supply mode with the shortest lead time, while keeping the other parameters equal, will provide the biggest %CR value. However, in practice, a reduction in the lead time corresponds to an increase in the unit purchasing cost. As a result it may be more beneficial to choose a shipping mode with a longer lead time but with a reduced unit purchasing cost. Using our model and with this type of analysis, a buyer can negotiate purchasing costs based on delivery times. Next we would like to compare different supply modes with different uncertainty levels. The curves in Fig. 11 represent different shipping modes with Uniformly distributed lead times, each with the same mean, but with different values of variance (where Ds ¼ 41 days for the base case). This means that we will compare shipping modes with equal values for the expected arrival time, but with different values of uncertainty, Ds. Although intuition suggests that the dual-mode model will be more beneficial as the uncertainty in the lead time of shipping mode 1 increases, this is not always the case. As we can observe in Fig. 11, when Dc ¼ 0 we obtain the highest %CR value for the instances with the smallest values of uncertainty (Ds ¼ 29 days). The reason for this is that, for these instances, when Dc ¼ 0, mode 2 is the preferred supplier (G2 < G1 ), and therefore, as we decrease Ds the average inventory cost for supply mode 2 does not change, while the average inventory cost of the dual-mode model decreases, and consequently %CR increases. Please refer to Section 3.1 for a more detailed explanation.

Next we consider the dual-mode model for different modes with Uniformly distributed lead times and equal uncertainty levels, but increasing values of the expected delivery time. This means that we are comparing shipping modes with equal values of Ds, but increasing values of E½L1 , while holding all other parameters equal. Fig. 12 shows the results, where E½L1  ¼ 34:5 days for the base case. Note that the average inventory cost of using mode 2 exclusively is independent of E½L1 ; and therefore for instances with different expected lead times for supply mode 1, the value of G2 will not change. For the single-mode model when we use mode 1 exclusively, in order to avoid stockouts we need an order size at least as great as the reorder point, i.e., Q P ksu ; therefore for increasing values of E½L1 , the right hand side of the inequality increases and we will have increasing values of G1 . As expected, we have increasing values of %CR for increasing E½L1 . Note that when Dc ¼ 0 we have some cases when the %CR values are equal for different E½L1 . There are two reasons for this: first, when Dc ¼ 0 supply mode 2 is chosen as the preferred single supplier (G2 < G1 ), and therefore, we will compare the dual-mode model against the same value of average inventory cost for the single-mode model, for different values of E½L1 ; and second, for small values of L2 the constraint on order size to avoid order crossing, , and therefore for differQ P kDs, is more restrictive than Q P ks ent values of E½L1  and equal values of Ds, the values of Q and kb are

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

equal for different instances, which results in the same average inventory cost for the dual-mode model. As our results have shown, the proposed model presents saving opportunities for the buyer who uses a single source, and it can be used in a joint manner with the supplier to achieve inventory cost reductions for the buyer and better resource allocation for the supplier. 4. Conclusions When allocating orders to suppliers, there is often a tradeoff between cost and delivery performance. The use of a cheaper supply mode often implies dealing with higher uncertainty levels when compared with a more expensive mode. Assuming that shortages are highly undesirable, if a cheap and less reliable delivery mode is used, it may be beneficial to consider supplementing this with a secondary supply mode that is more reliable (and therefore more expensive). This paper focused on the benefits of this dual-mode ordering strategy by modeling a buyer’s ordering decisions with two delivery mode options. Our model allowed us to characterize the benefits of the dual-mode supply strategy and compare the optimal dual-mode policy with the better of the two single-supplier solutions. Using numerical analysis, we characterized situations under which the use of the dual-mode model is preferred over the single-mode model. We observed increasing benefits in our model for increasing values of the mean and variance of the lead time of the less reliable supply mode (assuming that supply mode 1 is used as the preferred single-mode supplier). We also observed that when the supply mode 1 lead time follows a positively skewed probability distribution, which is typically a more realistic assumption than a Uniform lead time, the percentage cost reduction due to the dualmode operation is greater compared with the results when we assume that supply mode 1 lead time is Uniformly distributed. Our model can be used by a manager who wants to analyze different combinations of shipping modes in order to minimize inventory costs while guaranteeing zero stockouts. Our work shows that a supplier that offers different shipping mode options with different delivery costs and reliability levels may provide additional value to potential buyers. Further research may consider different supplier pricing and incentives. For example, in order to increase the utilization of shipping modes with available capacity, a supplier might offer quantity discounts for the use of those modes, making shipping diversification more attractive for the buyer. Finally, we note that one dimension our model does not account for is the risk preference of the decision maker, effectively assuming that the buyer is risk neutral. An interesting extension of our work may include accounting for the way in which different buyer risk profiles influence the allocation of order quantities to different supply modes. Acknowledgements

Suppose we place an order at time t ¼ 0. In the worst case, this order may arrive at time su . Note that the inventory position at any time t P 0 is equal to Q þ ksu  kt. When the inventory position reaches the reorder point r ¼ ksu , another order is placed. That is, when Q þ ksu  kt ¼ ksu , an order is placed. This implies that an order is placed at time t ¼ Qk and by the assumption that Q < kðsu  sl Þ, we have that:

t¼

Q < su  sl : k

An order placed at time t will arrive at time t þ L1 , and with probability strictly greater than zero, this order will arrive at time t þ sl ; since t < su  sl , we have that:

t þ sl < su  sl þ sl ¼ su Therefore, with probability strictly greater than zero the order is received before su , which implies a positive probability of receiving the second order before the first, i.e., order crossing. Thus an absence of order crossing implies Q P kðsu  sl Þ. Appendix B. Single-mode solution with quantity-based upper bound This section presents the solution procedure to find an optimal order quantity from mode 1, Q 1 , and minimum average inventory cost, G1 , for the single-mode model when the lead-time upper bound for supply mode 1, su , is a step function of the order size Q such that, su ¼ siu if qi 6 Q < qiþi for i ¼ 1; 2; . . . ; n, where qi < qiþ1 , siu < siþ1 and n is the number of intervals. u 1. Find Q 1 using (2.2); 2. Find Gk1 ðQ 1 Þ, for

su ¼ sku such that qk < Q 1 6 qkþ1 , using

(2.3). Set G1 ¼ Gk1 ðQ  Þ; 3. while k > 1 do 4.

k1 Find Gk1 1 ðqk ; su Þ using (2.1), which is the average inventory cost per unit time using the right break point for st

Q in the ðk  1Þ interval; G1 Q 1

Gk1 1

6 then is the optimal order quantity and the minimum average inventory cost is G1 . Set k ¼ 1; 7. else 5. 6.

if

8. Set Q 1 ¼ qk ; G1 ¼ Gk1 1 ; k ¼ k  1; 9. end if 10. end while

Appendix C. Convexity Conditions for GI ðQ ; bÞ This section presents convexity conditions for the average inventory cost function for the dual-mode model when s 2 ½L2 ; L2 þ sl Þ, represented by: 2

This research was supported by the Center for Multimodal Solutions (CMS) at the University of Florida (CMS Project #2011–023). The authors wish to thank the anonymous reviewers for their useful comments and suggestions. Appendix A. Order crossing condition Assuming that we use a supply mode with a stochastic lead time, L1  Uðsl ; su Þ and a constant demand rate k, if we use a continuous review ðQ ; rÞ policy, where r ¼ ksu , we need Q P kðsu  sl Þ to prevent order crossing. In order to prove this assertion, we assume for a contradiction that Q < kðsu  sl Þ.

GI ðQ ; bÞ ¼

k2 b2 h þ 2c2 k2 b þ hQ þ 2c1 kQ þ hkQ ðsu  sl Þ þ 2kA 2ðQ þ kbÞ

First, we analyze the average inventory cost as a function of Q and b independently, and then we will consider the convexity condition for GI as a joint function of Q and b. To analyze the average inventory cost as a function of Q we need to find the first and second derivatives of GI : 2

@GI hQ þ 2khQb þ hk2 bDs  2k2 bDc  k2 hb2  2kA ¼ @Q 2ðQ þ kbÞ2 @ 2 GI @Q

2

¼

2k2 hb2  k2 hbDs þ 2k2 bDc þ 2kA ðQ þ kbÞ3

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

2

2

Since Q and b are strictly positive, the denominator of @@QG2I is strictly positive as well. The numerator can be analyzed as a qua-

H¼4

2

dratic function of b of the form hðbÞ ¼ ab þ bb þ c, where a ¼ 2k2 h, b ¼ 2k2 Dc  k2 hDs and c ¼ 2kA. Since a > 0; hðbÞ is convex, and the values of hðbÞ will depend on 2

its discriminant d ¼ b  4ac. In particular, we are interested in the conditions under which hðbÞ P 0, since we need

2

@ GI @Q 2

P 0 for

GI ðQ ; bÞ to be convex in Q. Note, that when d 6 0; hðbÞ has at most one real root, and therefore, hðbÞ P 0 8b, and when d > 0; hðbÞ will have two real roots, b1 and b2 , and hðbÞ will be positive for b 2 ð1; b1  [ ½b2 ; 1Þ, and negative otherwise. In order to have d 6 0 we need:

@ 2 GI @Q 2

@ 2 GI @Q @b

@ 2 GI @b@Q

@ 2 GI @b2

3 5;

where

@ 2 GI kxðQ  kbÞ  2k2 hQb þ 2k2 A ¼ @b@Q ðQ þ kbÞ3 2

2

Since @@bG2I and @@QG2I are non-negative when condition (C.1) holds, we need to transform H to the form:

Hnew ¼



h11

h12

0

new h22

2

2

x 6 4khA;

ðC:1Þ

where x ¼ h kD2s  kDc. This condition requires that the square of the upper bound on the difference between the average inventory cost of using shipping mode 1 and the average inventory cost of using shipping mode 2 must be less than or equal to twice the square of the ordering and holding cost of the EOQ model with holding cost per unit per unit time h and ordering cost A ¼ A1 þ A. Therefore, when condition (C.1) holds, we have that @ 2 GI =@Q 2 is non-negative for Q P 0 and b P 0, and hence, GI is convex for Q P 0 and b P 0. If condition (C.1) does not hold, we have that d > 0; therefore pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ðxÞ2 4khA and hðbÞ will have two real roots: b1 ¼ 2kh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xþ ðxÞ2 4khA , and hðbÞ will be nonnegative for b2 ¼ 2kh b 2 ð1; b1  [ ½b2 ; 1Þ. We can see that the sign of b1 and b2 depends on the value of x. When x < 0, we have b1 < b2 < 0, which means that the two real roots of hðbÞ are negative, and therefore hðbÞ > 0 8b P 0, and consequently fore,

2

@ GI @b2

@ 2 GI @b2

> 0. If x > 0, we have that b2 > b1 > 0, and there-

P 0 when b 2 ½0; b1  [ ½b2 ; 1Þ.

Finally we can conclude that GI ðQ ; bÞ is convex in Q

2

kbÞ2k hQ bþ2k A In order to do the transformation we use F ¼  kxðQ , 2k2 hb2 2kxbþ2kA and we have that,

new

h22 ¼ Fh12 þ h22 ¼

k2 ð4kAh  x2 Þ 2

2k hb2  2kxb þ 2kA

Substituting hðbÞ ¼ 2khb2  2xb þ 2A, we have: new

h22 ¼

k2 ð4kAh  x2 Þ : hðbÞ new

Because we assume that (C.1) holds, h22 P 0, and this implies that H is positive semidefinite, and therefore GI ðQ ; bÞ is jointly convex in Q and b, 8Q > 0 and b > 0. Appendix D. Algorithm for dual-mode model when s‰½L2 ; sl þ L2 Þ Algorithm 1 shows a detailed description of the solution procedure to find the optimal order quantities from modes 1 and 2, and minimum average inventory cost for the dual-mode model when the lead-time upper bound of mode 1, su , is a step function of Q  2 ½L2 ; sl þ L2 Þ. Note that Algorithm 1 uses the expressions deand s scribed in Section 2.2.1.

8Q > 0 and b > 0 when (C.1) holds or when x < 0. If (C.1) does not

hold

and

x > 0,

then

GI ðQ ; bÞ

is

convex

in

Q

8Q > 0 and b 2 ½0; b1  [ ½b2 ; 1Þ. We next analyze the average inventory cost as a function of b using its first and second derivatives: 2

@GI 2k2 hbQ þ 2k2 Q Dc þ k3 b2 h  khQ  k2 hQ Ds  2k2 A ¼ @b 2ðQ þ kbÞ2 @ 2 GI 2

@b

2

¼

2k2 hQ  2k3 Q Dc þ k3 hQ Ds þ 2k3 A

2

Q

of 3

the 3

form

kðQ Þ ¼ aQ 2 þ bQ þ c, 3

1. i 1 2. while i 6 n do 3. Find bi and Q i for siu 4. if Q i 2 ðqi ; qiþ1  then 5. Find the minimum average inventory cost for the ith interval GiI 6. else if Q i < qi then

ðQ þ kbÞ3

We observe that the denominator of @@bG2I is positive 8Q > 0 and b > 0 and that the numerator is a quadratic function of

Algorithm 1. Minimum average inventory cost for the dual-mode  2 ½L2 ; sl þ L2 Þ model when s

where

a ¼ 2k2 h;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x2 4kAh 2h

b ¼ hk Ds  2k Dc and c ¼ 2k A with roots: Q 1 ¼ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4kAh Q 2 ¼ xþ x . 2h Following the same analysis used for the convexity of GI ðQ ; bÞ in Q for any b, we have that GI ðQ ; bÞ is convex in b 8Q > 0 and b > 0 if condition (C.1) holds or if x > 0. If condition (C.1) does not hold and x < 0; GI ðQ ; bÞ is convex in b for Q 2 ½0; Q 1  [ ½Q 2 ; 1Þ and b > 0. Note that (C.1) it is not restrictive, since the order cost is typically greater than Dc and greater than the holding cost per unit during Ds. For the rest of this section we will assume that (C.1) holds, and therefore GI ðQ ; bÞ is convex in Q for any b and convex in b for any Q ; 8b > 0 and Q > 0. To establish convexity conditions for GI as a joint function of Q and b we need to analyze its Hessian:

7. 8.

if GiI ðqi ; bðqi ÞÞ 6 Gi1 I ðqi ; bðqi ÞÞ then The minimum average inventory cost for the ith

interval is GiI ðqi ; bðqi ÞÞ 9. else 10. The solution is not in the ith interval 11. end if 12. else Q i > qiþ1 13. 14.

if GiI ðqiþ1 ; bðqiþ1 ÞÞ 6 Giþ1 I ðqiþ1 ; bðqiþ1 ÞÞ then The minimum average inventory cost for the ith

interval is GiI ðqiþ1 ; bðqiþ1 ÞÞ 15. else 16. The solution is not in the ith interval 17. end if 18. end if 19. end while 20. GI ¼ minfGiI 8i 2 ½1; ng

Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049

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C. Pérez, J. Geunes / European Journal of Operational Research xxx (2014) xxx–xxx

Appendix E. Algorithm for dual-mode model when s‰½sl þ L2 ; su Þ Algorithm 2 shows a detailed description of the solution procedure to find the optimal order quantities from modes 1 and 2, and minimum average inventory cost for the dual-mode model when the lead-time upper bound of mode 1, su , is a step function of Q  2 ½sl þ L2 ; su . Note that Algorithm 2 uses the expressions deand s scribed in Section 2.2.2. Algorithm 2. Minimum average inventory cost for the dual-mode  2 ½sl þ L2 ; su  model when s 1. i 1 2. while i 6 n 3. Find bi and Q i for siu 4. if Q i 2 ðqi ; qiþ1  then 5. Find the minimum average inventory cost for the ith interval GiII 6. else if Q i < qi then 7. 8.

if GiII ðqi ; bðqi ÞÞ 6 Gi1 II ðqi ; bðqi ÞÞ then The minimum average inventory cost for the ith

interval is GiII ðqi ; bðqi ÞÞ 9. else 10. The solution is not in the ith interval 11. end if 12. else Q i > qiþ1 13. 14.

if GiII ðqiþ1 ; bðqiþ1 ÞÞ 6 Giþ1 II ðqiþ1 ; bðqiþ1 ÞÞ then The minimum average inventory cost for the ith

interval is GiII ðqiþ1 ; bðqiþ1 ÞÞ 15. else 16. The solution is not in the ith interval 17. end if 18. end if 19. end while 20. GII ¼ minfGiII 8i 2 ½1; ng

Appendix F. Validity of average inventory expression for dualmode model when s‰½sl þ L2 ; su  We consider our inventory model as a Renewal Reward Process, where the interarrival times T n ; n P 1, are equal to the length of the replenishment cycle, and each time a renewal occurs we receive a reward In ; n P 1, equal to the average inventory in the cycle. We observe that the pair ðT n ; In Þ; n P 1 are independent and identically distributed and that E½T n  ¼ E½T from (2.13), and

E½In  ¼ E½I from (2.14). Using the result showed by Ross [16], we have a Renewal Reward Process where E½T < 1 and E½I < 1, and therefore, with probability 1,

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Please cite this article in press as: Pérez, C., & Geunes, J. A (Q, R) inventory replenishment model with two delivery modes. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.049