Example of a Company That Uses the Continuous Review Inventory Systems Is

Abstruse

We analyze a continuous review lost sales inventory organization with ii types of orders—regular and emergency. The regular order has a stochastic lead time and is placed with the cheapest adequate supplier. The emergency social club has a deterministic lead time is placed with a local supplier who has a higher cost. The emergency social club is not always filled since the supplier may not have the ability to provide the order on an emergency footing at all times. This emergency order has a higher toll per item and has a known probability of existence filled. The total costs for this system are compared to a system without emergency placement of orders. This paper provides managers with a tool to assess when dual sourcing is cost optimal by comparison the single sourcing and dual sourcing models.

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Haughton, M. and Isotupa, K. (2018) A Continuous Review Inventory System with Lost Sales and Emergency Orders. American Periodical of Operations Research, 8, 343-359. doi: x.4236/ajor.2018.85020.

i. Introduction

In this paper we analyze an inventory system with two types of orders, a regular order and an emergency order under the lost sales framework. Reducing stock out risk by splitting replenishment requirements amid multiple suppliers is a sourcing policy that has attracted the attention of academic researchers for more than 20 years. The policy is theoretically appealing for several reasons. First, pooling lead-time uncertainty among several suppliers is a way to reduce the safety stock needed to run across service targets or alternatively, the expected number of backorders for a prescribed level of prophylactic stock. Second, successive deliveries of smaller orders will reduce cycle stock. Third, the incremental ordering cost of the 2d equally subsequent orders may be relatively pocket-sized in a diversity of settings.

Since the tragic events of September xi, 2001, several security initiatives have been implemented at US international edge checkpoints. These aim to minimize the hazard of trans-edge flows of merchandise existence conduits for damage to national security. For many companies delivering products by trucks from Canada to the US, these measures accept increased both the hateful and variance of edge crossing times. Consequently, some US companies (for example the automotive manufacturing industry) that traditionally obtained their raw material from Canada have switched to using just American suppliers or using both a Canadian and an American supplier. In the latter case, companies divide their requirements betwixt the less expensive and less atomic number 82-time reliable Canadian supplier and the more than expensive but more lead-time reliable American supplier. This has boosted interest in research on inventory systems with multiple suppliers and provides the motivation for our inquiry.

In this paper we volition compare two inventory policies. The policy we will focus most of our attention on is a (Q, R) inventory policy with two suppliers―a regular supplier and an emergency supplier. The emergency supplier is used but when the inventory level is dangerously low and a stock out take a chance is imminent. We consider the instance of a manufacturer facing demand that is predictable and occurring at equally spaced fourth dimension intervals. The lead time for the regular order is probabilistic and highly variable with a high variance due to the unpredictability in edge crossing times. When inventory falls to dangerously low levels, an emergency lodge is placed with a local supplier or with the competition. The order is filled with a certain probability. If the social club is filled, information technology will be filled in a sure stock-still amount of time which is deterministic. In that location is a fixed cost of placing the lodge which is incurred whether the order is filled or not. The variable cost of the emergency social club, which is proportional to the order size, is only incurred if the order is filled. There are various reasons why the emergency order may non be filled. If the emergency guild is placed with a local supplier with whom the visitor does non have a large amount of business, this company may reserve their stock for higher priority customers and choose not to fill the club of this manufacturer. If the gild is placed with the competition, and then for strategic business reasons, they may choose non to make full that particular order. We volition compare the total long run cost rate of this policy to the traditional (Q, R) inventory policy with anticipated demand and lost sales. With the help of numerical examples nosotros provide some situations where dual sourcing with emergency gild placements is price effective when compared to a single sourcing (Q, R) model. There are two main contributions of this newspaper―i) Decide the long run expected total cost of a (Q, R) system with emergency orders and lost sales, 2) Provide a method by which to compare a lost sales (Q, R) arrangement to a lost sales (Q, R) system with emergency orders and illustrate with numerical examples when the system with emergency orders is cost constructive.

Rapid advancements in estimator and information applied science along with the crucial role of responsiveness as a winning supply chain strategy have additional interest in continuous review inventory policies. Withal, the bulk of the inventory models dealing with multiple suppliers are for the periodic review case. The focus on periodic review is mainly because of the mathematical complexity in dealing with continuous review systems. In this paper we will deal with a continuous review (Q, R) inventory system with lost sales and two suppliers. By identifying and establishing the equivalence of this model to another mathematical model which is more than tractable, we will obtain analytic expressions for the total toll of running the organisation. We volition then compare the two policies numerically in a variety of scenarios and place situations where a policy with 2 suppliers performs meliorate than a arrangement with a single supplier.

The paper has 6 sections. The second section consists of literature review of inventory systems for multiple suppliers. Section iii is the crux of the newspaper where the mathematical model and detailed assay are presented. Section 3 is also central in demonstrating how we address what has been heretofore seen as a mathematical difficulty of continuous review (Q, R) systems. The steady land inventory level distribution is as well derived in this section. In Department 4, the expression for the long-run expected toll charge per unit is developed. In Section 5 some numerical examples are presented. Section six consists of conclusions and time to come research.

2. Literature Survey

The literature on inventory systems with multiple suppliers can be broadly separated into two classes―i, where replenishment orders are divide simultaneously among many suppliers and two, where orders are placed at different times with unlike suppliers. In this paper nosotros focus on the second case and hence only provide a literature survey of papers of this kind. For information on models of the first blazon, useful sources are Thomas and Tyworth [1] and the review paper by Minner [2] on multiple supplier inventory models in both the periodic and continuous review cases.

Past related research includes the early papers of Barankin [3] and Neuts [iv] , which studied periodic review inventory systems with regular and emergency replenishments, where the regular lodge pb-fourth dimension is one flow and the emergency replenishment is instantaneous. The more recent studies on emergency ordering take all been in the periodic review realm. The latest written report of a periodic review system with emergency orders was by Johansen [5] . They study an inventory system with compound Poisson demands and backorders using Markov conclusion processes. They practice not have any belittling comparisons of toll, just based on numerical examples they show that a combination of normal orders and emergency orders yield lower system costs than having no emergency orders.

However, since all these papers focus on periodic review models while we accost the continuous review case, we exclude the periodic review models from further consideration in our literature survey. Our literature survey focuses only on continuous review inventory systems with both regular and emergency orders. Our literature survey does not include the literature on inventory systems where suppliers are either available or unavailable simply no emergency ordering is done during the unavailable menstruum of the suppliers nor does it include the case of transhipment of inventory between two locations. Farther we do not present a literature review of two echelon inventory models or simulation studies of inventory models.

Dohi, Kaio and Osaki, South. [half dozen] and Giri and Dohi [7] study continuous review inventory systems where after a fixed amount of time, t_o, if stock is depleted, an emergency order is placed which arrives after a pb time L1 and if stock is not depleted, a regular order is placed which arrives after a lead time L2. The beginning paper derives the necessary and sufficient conditions for t_o to exist which minimizes the long-run boilerplate cost. The 2d paper they derive the optimal ordering time that minimizes average cost for a fixed ordering quantity model. Bradley [8] analyzed a product-inventory model in which in-house production and a sub-contractor are the inventory replenishment alternatives. Using Brownian movement approximations, the author sought to determine the optimal policy parameters. Allon and Van Miegham [ix] studied a continuous review inventory model with dual sourcing. They considered the problem of splitting orders between a responsive and expensive supplier versus a slow simply inexpensive supplier. Despite some similarities between their work and ours, there are several of import differences. For example, they deal with the case of backorders while we bargain with the case of lost sales. Also, in their newspaper the ii orders are placed with the two suppliers simultaneously while in our paper, the lodge with the expensive supplier is only placed if the stock levels autumn dangerously low.

Allen and D'Esopo [10] were the showtime to consider a continuous review inventory arrangement with emergency orders. They analyzed the standard (Q, R) inventory model with an boosted parameter called the expediting level. Moinzadeh and Schmidt [11] considered the (S − 1, South) inventory organization with emergency orders. Vocal and Zipkin [12] extend this model to include the case of multiple suppliers and develop performance evaluation tools for a diversity of policies under which the supply system becomes a network of queues. Johansen and Thorstenson [xiii] adopted the standard (Q, R) policy for regular orders and an (s, S) type policy for emergency orders, where s and S depend ingeneral on the time remaining until the receipt of a regular lodge. They deal with the example of complete backordering and use simulation to obtain the optimal values of the reorder levels and order quantities.

Moinzadeh and Nahmias [14] proposes a very full general model with the (Q, R) arrangement with back orders, two reorder points and two reorder quantities and proposes a heuristic control policy for the case where lead times are deterministic. Their paper assumes that the atomic number 82 times for both orders are deterministic and that the demand can follow a Poisson or normal distribution. Mohebbi and Posner [15] analyze the model past Moinzadeh and Nahmias [14] under the assumption of compound Poisson need and non-identical exponentially distributed pb times using the level crossing approach and develop the total price function. Duran, Gutierrez and Zequeira [16] analyze a system similar to the 1 past Moinzadeh and Nahmias [fourteen] . They analyze a continuous review (Q, R) system with backorders where the lead time has a fixed component T. If the inventory level lies below a threshold level when T units of time has elapsed since gild placement, the order is expedited and arrives later on a short but deterministic time and if the inventory level is not below the threshold level, there is a longer deterministic period of time earlier the order arrives. They present an algorithm to determine the policy parameters that minimize the total cost. The newspaper most closely related to our paper is the 1 by Axsater [17] which deals with a (Q, R) inventory system with Poisson demands and emergency ordering. At that place are iv important differences betwixt Axsater [17] and our model. Our paper is motivated by the fact the lead time for the order placed with the cheaper (regular) supplier is highly variable, which necessitates the use of an alternate supplier. Hence in our case only the emergency order has a deterministic pb fourth dimension and the regular order has a stochastic pb time whereas both orders have deterministic lead time in Axsater [17] . Farther in our newspaper, which has applications to the auto industry, the need pattern is deterministic since the product lines at almost auto companies run on a continuous basis while the demand pattern in Axsater [17] is Poisson. Our newspaper deals with the situation where the availability of the emergency order is probabilistic whereas in Axsater [17] , the assumption is that the emergency club volition always get filled. The terminal departure is that we consider the lost sales case while Axsater [17] considers the backlogging example.

3. Trouble Description and Assay

We consider a continuous review (Q, R) inventory organization for a Usa based manufacturer where the need for the particular is predictable and is 1 unit every T periods; i.e., mean demand per flow = 1 ÷ T. Notation that because at each demand epoch, at that place is a demand for only one particular, the (Q, R) policy is equivalent to the (due south, Q) policy and from hither on, we will refer to the model as an (due south, Q) organisation. The maximum inventory level is Q + s units. When the inventory level drops to the reorder signal, s, a regular order of size Q (Q > s) is placed with a Canadian supplier. The assumption that Q > southward ensures that there is at most 1 outstanding regular society at any given time. Considering of the highly unpredictable border crossing times, the lead time for the regular social club is assumed to be exponentially distributed with rate μ. The system we consider is a lost sales system. Hence if the inventory level drops to zero and there is a need for the detail by a customer, the client is sent away with his demand unsatisfied. The business is investigating the possibility of procuring stock on an emergency basis from a local supplier or from the competition if the reorder doesn't arrive when the stock level drops to n.

The fourth dimension taken to procure items on an emergency basis from the local supplier or the competition is deterministic and takes nT units of time where n is less than the reorder level, s, for obvious reasons. Hence when the stock level drops to n, an emergency order of size s is placed with the local supplier or competition. This order has a probability p of beingness fulfilled. Note that an emergency gild is placed only at the instant when the stock level drops to n from due north + ane and not at times when the inventory level is due north. If the society is filled, it is for the entire s units. The local supplier may guarantee that the stock will exist available 100p% of the time. In the example of the competition, obviously there will exist no guarantees and 100p% is the estimated percentage of time that the manufacturer can go the stock from the contest.

The system described above is equivalent to a lost sales (s, Q) inventory system where demands occur in one case every T units with the following additional atmospheric condition. The reorder policy with the Canadian supplier is the aforementioned as in the model described in the previous paragraph. The emergency order described in that model is equivalent to an emergency order placed when the inventory level drops to zero which gets replenished with probability p. If the order is filled, it is filled instantaneously. In this paper we will model this system as a system with instantaneous replenishment of the emergency order and follow through with the analysis.

In order to determine the long-run expected cost rate, we need to determine the steady state inventory level distribution P(j), where P(j) denotes the steady land probability that inventory level is j. In the analysis that follows, π(j) denotes the stationary distribution of the embedded Markov chain.

Theorem 1: The steady country inventory level distribution P(j) is given by

$P\left(0\right)=\frac{\left({\text{east}}^{\mu T}-1\right)\pi \left(0\right)}{\mu T}$ (three.1)

$P\left(j\right)=\frac{{\left({\text{east}}^{\mu T}-1\right)}^2}{\mu T\left(\mathrm{ane}-p\right)}{\text{eastward}}^{\left(j-1\right)\mu T}\pi \left(0\right);1\le j\le s$ (3.2)

$P\left(j\right)=\frac{\left({\text{e}}^{\mu T}-1\right)}{\left(1-p\right)}{\text{due east}}^{\mathrm{southward}\mu T}\pi \left(0\right)-\frac{p\left({\text{e}}^{\mu T}-\mathrm{ane}\right)}{\left(1-p\right)}\pi \left(0\right);\mathrm{southward}+1\le j\le Q-1$ (iii.3)

$P\left(Q\right)=\frac{\left({\text{eastward}}^{\mu T}-1\right)}{\left(1-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p\left({\text{e}}^{\mu T}-\mathrm{i}\right)}{\left(1-p\right)}\pi \left(0\right)+\pi \left(0\right)-\frac{\left({\text{e}}^{\mu T}-\mathrm{i}\right)\pi \left(0\right)}{\mu T}$ (3.4)

$P\left(j\right)=\frac{\left({\text{east}}^{\mu T}-\mathrm{ane}\right)}{\left(1-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p{\left({\text{east}}^{\mu T}-\mathrm{one}\right)}^{\mathrm{ii}}}{\mu T\left(\mathrm{ane}-p\right)}{\text{e}}^{\left(j-Q-1\right)\mu T}\pi \left(0\right);Q+1\le j\le Q+s$ (3.five)

where

$\pi \left(0\right)=\frac{1}{1+\frac{Q\left({\text{east}}^{\mu T}-\mathrm{i}\right)}{1-p}{\text{eastward}}^{s\mu T}+\frac{\left(Q-\mathrm{due\; south}\right)p\left({\text{eastward}}^{\mu T}-1\right)}{1-p}}$ (3.vi)

Proof: Allow I(t) denote the inventory level at time t. From our assumptions it is clear that the inventory level procedure {I(t); t ≥ 0} with state space E = {0, 1, 2, ..., Q + s} is a semi-regenerative procedure with the regeneration points being the demand epochs. Allow $\left\{{\tau }_0,{\tau }_1,{\tau }_2,\cdots \right\}=\left\{0,T,2T,\cdots \right\}$ be the successive epochs at which demands occur. If $I_{\mathrm{due\; north}}=I\left({\tau }_n^{-}\right)$, so $\left(I,\tau \right)=\left\{I_{\mathrm{due\; north}},{\tau }_n;n\in N^0\right\}$ is a Markov renewal process with embedded Markov concatenation $\left\{I_n;n\in N^0\right\}$ . Let us ascertain the inventory level distribution as follows:

$P\left(i,j,t\right)=\mathrm{Pr}\left[I\left(t\right)=j|I_0=i\right]$

And then from Markov renewal theory (refer Cinlar (1975)), $P\left(i,j,t\right)$ satisfies the post-obit Markov renewal equation:

$P\left(i,j,t\right)=\mathrm{Chiliad}\left(i,j,t\right)+{\displaystyle {\sum }_{l=0}^s{\displaystyle {\int }_0^t\theta \left(i,\mathrm{50},u\right)P\left(l,j,t-u\right)\text{d}u}}$ (3.vii)

where $\theta \left(i,j,t\right)$ is the derivative of the semi-Markov kernel of the Markov renewal process (I, τ) and is given by

$\theta \left(i,j,t\right)=\underset{\Delta \to 0}{\lim }P\left[I_1=j,t\le {\tau }_{\mathrm{ane}}\le t+\Delta |I_0=i\right]/\Delta$ (3.8)

and

$\mathrm{Grand}\left(i,j,t\right)=P\left[I\left(t\right)=j,{\tau }_1>t|I_0=i\right]$ (three.9)

The various operating characteristics that are necessary to obtain the long-run expected cost rate of the inventory system can be obtained in terms of the steady country inventory level distribution if it exists.

From Markov renewal theory the steady state distribution of the inventory level exists equally the land space is finite and the embedded Markov concatenation is irreducible. Permit P(j) be the steady state inventory level distribution. Then from Cinlar [18] , we have

$P\left(j\right)=\frac{{\displaystyle \sum _{i=0}^s\pi \left(i\right){\mathrm{Chiliad}}^{*}\left(i,j,0\right)}}{{\displaystyle \sum _{i=0}^s\pi \left(i\right)\mathrm{chiliad}\left(i\right)}}$ (3.x)

where $M^{*}\left(i,j,0\right)$ is the Laplace transform of $M\left(i,j,t\right)$ evaluated at zero and m(i) is the hateful sojourn time in country i. The stationary distribution of the embedded Markov chain is given by π(i) and obtained past solving the equations

$\pi \left(j\right)={\displaystyle \sum _i\pi \left(i\right){\displaystyle \underset{0}{\overset{\infty }{\int }}\theta \left(i,j,t\right)\text{d}t}}$ (3.11)

and the normalizing status ${\displaystyle {\sum }_j\pi \left(j\right)}=\mathrm{i}$ .

Since time between two consecutive (unit of measurement) demands is deterministic and equal to T,

${\displaystyle \sum _{i=0}^s\pi \left(i\right)\mathrm{1000}\left(i\right)}=T$ (3.12)

In order to determine the steady state inventory level distribution, nosotros first need to determine ${\displaystyle {\int }_0^{\infty }\theta \left(i,j,t\right)\text{d}t}$ and ${\displaystyle {\int }_0^{\infty }K\left(i,j,t\right)\text{d}t}$ .

To determine $\theta \left(i,j,t\right)$, we note that the transition points are either demand or replenishment epochs. For example allow the inventory level simply before a demand occurs be 0 and hence after the demand occurs and before the side by side demand, the inventory level will either remain at 0 if no replenishment occurs with probability e^−μt and the inventory level volition achieve Q if a replenishment occurs with probability 1 − e^−μt. In our case, the time between demand epochs is deterministic. Hence $\theta \left(i,j,t\right)$ is not-zero only when t = T. Denote by $p\left(i,j\right)$, ${\displaystyle {\int }_0^{\infty }\theta \left(i,j,t\right)\text{d}t}$ . Then $p\left(i,j\right)$ are the one step transition probabilities of the Markov chain embedded in the Markov renewal process (I, τ) and are given by the role $\theta \left(i,j,T\right)$ .

$p\left(i,j\right)=\{\begin{array}{l}{\text{east}}^{-\mu T}i=j=0;\mathrm{two}\le i\le s+1,j=i-1\hfill \\ 1-{\text{e}}^{-\mu T}i=0,j=Q;2\le i\le \mathrm{south}+1,j=i+Q-1\hfill \\ \left(\mathrm{one}-p\right){\text{e}}^{-\mu T}i=1,j=0\hfill \\ \left(1-p\right)\left(1-{\text{due east}}^{-\mu T}\right)i=1,j=Q\hfill \\ p{\text{e}}^{-\mu T}i=1,j=\mathrm{due\; south}\hfill \\ p\left(1-{\text{due east}}^{-\mu T}\right)i=\mathrm{ane},j=Q+s\hfill \\ 1s+2\le i\le Q+s,j=i-\mathrm{ane}\hfill \\ 0\text{otherwise}\hfill \end{array}$

The stationary distribution of the embedded Markov chain is obtained by solving the equations $\pi \left(j\right)={\displaystyle {\sum }_i\pi \left(i\right)p\left(i,j\right)}$ and the normalizing condition ${\displaystyle {\sum }_j\pi \left(j\right)}=\mathrm{one}$ . Using the function $p\left(i,j\right)$ given above we obtain, on solving $\pi \left(j\right)={\displaystyle {\sum }_i\pi \left(i\right)p\left(i,j\right)}$,

$\pi \left(j\right)=\frac{\left({\text{east}}^{\mu T}-1\right)}{\left(\mathrm{ane}-p\right)}{\text{eastward}}^{\left(j-1\right)\mu T}\pi \left(0\right);1\le j\le s$ (3.13)

$\pi \left(j\right)=\frac{\left({\text{due east}}^{\mu T}-\mathrm{ane}\right)}{\left(\mathrm{ane}-p\right)}{\text{e}}^{\mathrm{due\; south}\mu T}\pi \left(0\right)-\frac{p\left({\text{due east}}^{\mu T}-1\right)}{\left(1-p\right)}\pi \left(0\right);\mathrm{due\; south}+1\le j\le Q$ (3.14)

$\pi \left(j\right)=\frac{\left({\text{east}}^{\mu T}-1\right)}{\left(1-p\right)}{\text{due east}}^{s\mu T}\pi \left(0\right)-\frac{p\left({\text{e}}^{\mu T}-\mathrm{one}\right)}{\left(1-p\right)}{\text{e}}^{\left(j-Q-\mathrm{ane}\right)\mu T}\pi \left(0\right);Q+1\le j\le Q+s$ (three.xv)

Using Equations (three.13) to (3.fifteen) in ${\displaystyle {\sum }_j\pi \left(j\right)}=1$, we obtain

$\pi \left(0\right)=\frac{1}{\mathrm{one}+\frac{Q\left({\text{e}}^{\mu T}-1\right)}{\mathrm{one}-p}{\text{eastward}}^{\mathrm{southward}\mu T}+\frac{\left(Q-\mathrm{south}\right)p\left({\text{due east}}^{\mu T}-\mathrm{i}\right)}{\mathrm{one}-p}}$ (3.16)

In order to decide the steady state inventory level distribution, we next determine the function $\mathrm{Thousand}\left(i,j,t\right)$ given by $K\left(i,j,t\right)=P\left[I\left(t\right)=j,{\tau }_1>t|I_0=i\right]$ . Hence for T > t, we have

$G\left(i,j,t\right)=\{\begin{array}{l}{\text{e}}^{-\mu t}i=j=0;2\le i\le s+\mathrm{ane},j=i-\mathrm{ane}\hfill \\ 1-{\text{due east}}^{-\mu t}i=0,j=Q;\mathrm{two}\le i\le s+\mathrm{i},j=i+Q-\mathrm{ane}\hfill \\ \left(1-p\right){\text{e}}^{-\mu t}i=\mathrm{ane},j=0\hfill \\ \left(\mathrm{one}-p\right)\left(\mathrm{ane}-{\text{due east}}^{-\mu t}\right)i=1,j=Q\hfill \\ p{\text{e}}^{-\mu t}i=\mathrm{ane},j=s\hfill \\ p\left(\mathrm{i}-{\text{east}}^{-\mu t}\right)i=1,j=Q+s\hfill \\ 1\mathrm{southward}+2\le i\le Q+\mathrm{south},j=i-\mathrm{i}\hfill \\ 0\text{otherwise}\hfill \end{array}$ (3.17)

and

$\begin{array}{l}{\mathrm{Grand}}^{*}\left(i,j,0\right)\\ =\{\begin{array}{l}\left(\mathrm{i}-{\text{e}}^{-\mu T}\right)/\mu i=j=0;\mathrm{ii}\le i\le \mathrm{south}+1,j=i-1\hfill \\ T-\left(\mathrm{i}-{\text{e}}^{-\mu T}\right)/\mu i=0,j=Q;2\le i\le s+\mathrm{ane},j=i+Q-1\hfill \\ \left(\mathrm{i}-p\right)\left(1-{\text{eastward}}^{-\mu T}\right)/\mu i=1,j=0\hfill \\ \left(\mathrm{one}-p\right)T-\left(\mathrm{ane}-p\right)\left(1-{\text{due east}}^{-\mu T}\right)/\mu i=1,j=Q\hfill \\ p\left(\mathrm{ane}-{\text{e}}^{-\mu T}\right)/\mu i=\mathrm{i},j=s\hfill \\ pT-p\left(\mathrm{i}-{\text{e}}^{-\mu T}\right)/\mu i=\mathrm{one},j=Q+s\hfill \\ T\mathrm{south}+2\le i\le Q+s,j=i-\mathrm{i}\hfill \\ 0\text{otherwise}\hfill \end{array}\end{array}$ (three.18)

Substituting for ${\mathrm{1000}}^{*}\left(i,j,0\right)$ and π(j) from (iii.eighteen) and (3.13) to (3.fifteen) in (three.four) we obtain

$P\left(0\right)=\frac{\left({\text{due east}}^{\mu T}-\mathrm{one}\right)\pi \left(0\right)}{\mu T}$

$P\left(j\right)=\frac{{\left({\text{eastward}}^{\mu T}-1\right)}^2}{\mu T\left(\mathrm{ane}-p\right)}{\text{eastward}}^{\left(j-1\right)\mu T}\pi \left(0\right);\mathrm{ane}\le j\le s$

$P\left(j\right)=\frac{\left({\text{e}}^{\mu T}-\mathrm{i}\right)}{\left(1-p\right)}{\text{e}}^{\mathrm{southward}\mu T}\pi \left(0\right)-\frac{p\left({\text{eastward}}^{\mu T}-1\right)}{\left(\mathrm{one}-p\right)}\pi \left(0\right);\mathrm{southward}+\mathrm{one}\le j\le Q-1$

$P\left(Q\right)=\frac{\left({\text{e}}^{\mu T}-1\right)}{\left(1-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p\left({\text{e}}^{\mu T}-1\right)}{\left(\mathrm{ane}-p\right)}\pi \left(0\right)+\pi \left(0\right)-\frac{\left({\text{e}}^{\mu T}-1\right)\pi \left(0\right)}{\mu T}$

$P\left(j\right)=\frac{\left({\text{due east}}^{\mu T}-1\right)}{\left(1-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p{\left({\text{e}}^{\mu T}-\mathrm{one}\right)}^{\mathrm{ii}}}{\mu T\left(\mathrm{one}-p\right)}{\text{e}}^{\left(j-Q-1\right)\mu T}\pi \left(0\right);Q+1\le j\le Q+\mathrm{south}$

Now that we accept obtained the steady country inventory level distribution and the stationary probabilities of the embedded Markov chain, we tin can make up one's mind the various operating characteristics required to derive the price function.

4. Cost Role Derivation

In this section we will deal with the trouble of minimizing the total expected cost rate. We will also determine conditions nether which it is cost optimal to identify an emergency gild with the local supplier rather than just look for the order from your regular supplier. We utilize the following cost components

Yard_i: the gear up-up cost per guild for the regular social club.

K₂: the set-up cost per order for the emergency gild. This cost is incurred whether or not the order is filled.

c_i: the cost per item for the regular society.

c₂: the cost per detail for the emergency gild. This toll is only incurred if the order is filled.

one thousand: the shortage cost/unit curt.

h: the inventory carrying cost/unit/unit time.

Then the total expected cost rate is given by

$C\left(s,Q\right)=\left(M_1+c_{1}Q\right){\Gamma }_1+\left(K_{\mathrm{ii}}+c_2\mathrm{south}p\right){\Gamma }_{\mathrm{ii}}+h{\Gamma }_3+g{\Gamma }_{\mathrm{four}}$ (four.ane)

where Γ₁ is the reorder rate for the regular order, Γ_two is the reorder charge per unit for the emergency club Γ₃ is the boilerplate inventory level and Γ_four is the shortage rate.

A regular order is placed when the inventory level is s + 1 and a demand occurs which brings the level down to the reorder point south. This order has a lead time which is exponentially distributed with a mean of 1/μ. Hence

${\Gamma }_1=\pi \left(s+1\right)/T$ (4.2)

An emergency order is placed when the inventory level is due north+1 and a demand occurs or in the equivalent (instantaneous replenishment) organization when the inventory level is one and a demand occurs. Hence

${\Gamma }_2=\pi \left(1\right)/T$ (4.three)

The average inventory level

${\Gamma }_3={\displaystyle {\sum }_{j}P\left(j\right)}$ (4.4)

A shortage occurs when inventory level is aught and a need occurs. Hence

${\Gamma }_4=\pi \left(0\right)/T$ (iv.5)

Substituting for Thousand_ane, Grand₂, G₃ and G₄ from (4.two) to (four.5) in (4.1) we obtain

$\begin{array}{c}C\left(s,Q\right)=\left({\mathrm{Thousand}}_1+c_{1}Q\right)\pi \left(\mathrm{due\; south}+1\right)/T+\left(K_2+c_{\mathrm{two}}s\left[\mathrm{i}-P\left(0\right)\right]\right)\pi \left(1\right)/T\\ +h{\displaystyle {\sum }_{j}P\left(j\right)}+g\pi \left(0\right)/T\end{array}$ (four.vi)

Note that the fixed cost of placing an emergency lodge is incurred whether or not the order is satisfied while the variable cost is incurred simply if the social club is met. If the emergency order materializes, the inventory level does not touch zero. Hence probability that the variable toll of the emergency order is incurred is the probability the inventory level does not reach goose egg.

$C\left(\mathrm{due\; south},Q\right)$ can be obtained explicitly by substituting for π(j)s from Equations (3.13) to (three.xv) and for P(j)s from (iii.1) to (3.5).

For a fixed south, the cost function is convex in Q and for a fixed Q, it is convex in s. Although nosotros were unable to show the convexity of the toll role in two variables, our experience with diverse numerical examples indicates that the price function is convex.

The long-run expected toll rate for the (southward, Q) that organization with no emergency orders is given by

$C\left(s,Q\right)=\left(K_1+c_{\mathrm{i}}Q\right){\Gamma }_5+h{\Gamma }_6+g{\Gamma }_{\mathrm{vii}}$ (four.7)

Notation that the simply difference between Equation (4.vii) and Equation (4.1) is the absence of the price term for the emergency orders. The reorder charge per unit, Γ_five,tin be obtained by substituting p = 0 in Γ_one. The average inventory level, Γ₆, can be obtained past substituting p = 0 in Γ₃. The shortage rate, Γ₇, can be obtained by substituting p = 0 in Γ_iv.

5. Numerical Illustration

In this section nosotros compare the long-run expected toll rate of the organisation using emergency orders with that of the with standard lost sales (s, Q) policy and establish cases where the system with emergency orders yields lower total costs or lower average inventory levels or both lower full costs and lower average inventory levels. For purposes of numerical analogy nosotros assumed that the guild size for the regular order Q is bounded above by the size of the container. We used 200 as the container size. In order to determine the optimal parameters s and Q, we do a complete enumeration and determine the pair (due south, Q) that gives us the minimum cost. The values held stock-still in our numerical assay were K₁ = 140, K₂ = 70, c₁ = twenty, T = 0.02, μ = 2. We then considered six values of c₂, 10 values of g, 2 values of h, and xiii values of p. Our numerical assay covered all possible combination of those values; i.east., 1560 combinations. Tables ane-6 tabulate the key results.

Table one and Tabular array two address the question of how the inventory policy parameters (s, Q) and average inventory are affected by using the emergency ordering system at the extreme value combinations of c₂, k and h for the various values of p. The tables show the results just for factor combinations where the long-run expected cost rate is lower for the emergency ordering than for the

Tabular array one. Illustrative impacts of emergency order arrangement on policy parameters if holding cost = 1.

Annotation: The shaded cells represent to scenarios where emergency sourcing is inferior to single sourcing.

Table ii. Illustrative impacts of emergency lodge organization on policy parameters if holding toll = 10.

Annotation: The shaded cells correspond to scenarios where emergency sourcing is junior to single sourcing.

Table three. Minimum emergency order receipt probabilities for dual sourcing to be benign if holding price = i.

Note: X = single sourcing is always cheaper than emergency sourcing for that combination of g and c₂.

Table 4. Minimum emergency order receipt probabilities for dual sourcing to be beneficial if holding cost = 10.

Annotation: X = single sourcing is always cheaper than emergency sourcing for that combination of g and c₂.

Table 5. Percentage cost reduction achieved with dual sourcing for guaranteed emergency social club receipt if holding cost = 1.

Note: X = unmarried sourcing is ever cheaper than emergency sourcing for that combination of yard and c₂.

Table 6. Percent cost reduction achieved with dual sourcing for guaranteed emergency society receipt if holding toll = ten.

Annotation: 10 = single sourcing is ever cheaper than emergency sourcing for that combination of g and c_ii.

standard lost sales (southward, Q) policy. Thus, the combination of (c₂, g) = (50, forty) is excluded considering the standard lost sales (s, Q) policy for that combination was superior to the emergency order policy at every value of p. The tables ostend the expected outcome that the emergency order policy can operate optimally with smaller inventories than the standard lost sales (south, Q) policy. For example, at (c₂, thou, h) = (22, 180, 10), the buyer's average inventory level when the option of emergency ordering is unavailable is 70 units only drops to less than ane-third of that amount (20 units) when emergency club fulfillment (receipt) is guaranteed; i.e. if p = 1.

Table iii and Table iv focus on determining the minimum emergency order receipt probability (p) that must be reached in order for dual sourcing to yield a lower long-run expected price rate than the standard lost sales (s, Q) policy. Consider, as illustration, the column for c₂ = 50 in Tabular array 1. This shows that unless the shortage cost exceeds a value somewhere in the range 120 £ g < 150, dual sourcing does not brand sense. Further, even if that condition is satisfied, the dual sourcing is advantageous merely if the probability of gild fulfilment by the emergency source is high. Case in betoken is that at one thousand = 150, the probability must reach 0.99. At the higher value of g = 180, the required probability threshold is less (0.90). The threshold guild fulfilment probability requirements are lower (or at least, non-increasing) for lower unit prices of emergency ordered items (c_ii) and for college unit costs of holding inventory (h).

These observations have important sourcing policy implications. In particular, as noted before, emergency club fulfilment past infrequently used domestic suppliers is unlikely to have a high probability, much less exist guaranteed. As such, an emergency order policy might accept to be complemented by a policy of ordering more oft from the domestic supplier; i.eastward., order even in the absence of an emergency requirement. That way, the domestic supplier may accord the buyer'south order the loftier priority given to orders from its regular customers and thus be more than inclined to fill up the emergency order. Conspicuously, this means higher long-run inventory cost rate and requires the buyer to determine the revised order frequency from a domestic source would assure a sufficiently high social club fulfilment probability. Analysis of this emergency ordering tactic is across the scope of this paper and is considered as a affair for possible future research.

Tabular array 5 or Table half-dozen evidence the percent toll reductions of using the emergency ordering policy instead of the standard lost sales (s, Q) policy, nether the assumption of guaranteed order fulfillment (receipt). The reported percentages follow the expected directions: they are college for 1) lower values of c₂ (lower item cost punishment of using the emergency source), 2) college values of grand (avoidance of larger stockout penalties by using the emergency source), and three) higher values of h (higher savings from the reduced inventory levels when the emergency source is used). For depression property cost (h = 1% or v% of item toll from regular supplier), the best improvement observed was a very moderate 4.37%. The increase of h to 10 (50% of the detail price from regular supplier) yields more than striking improvements. These improvements tin can exist viewed as the premium a company would be willing to pay to have guaranteed insurance confronting stockouts.

These observations may offer some insights into the strategies used by some companies to handle their United states-Canada trans-border supply chain operations. To explain these insights, we first notation that firms such as machine manufactures might face scenarios closer to the lower rows of Table vi than to whatever of the other scenarios in either Table five or Table 6. Specifically, because these firms utilise simply-in-fourth dimension delivery, failure of a just-in-fourth dimension social club of parts to go far equally scheduled (a stockout) can cause costly shut downwardly of a solar day's scheduled production. The fact that they use merely-in-time as an inventory reduction strategy leads ane to deduce that their inventory carrying costs are high plenty to exist warrant such a strategy. Indeed, holding price data used for the models in Nozick and Turnquist (2001) support that deduction. These points make it understandable that these firms have been willing to pay what is regarded as a high premium for insurance against stockouts. All the same, the insurance that these firms accept procured is more preventive rather than remedial (as is emergency sourcing). That insurance involves investing in supply concatenation security initiatives that are necessary for firms to receive expedited border checkpoint processing of their trans-border shipments.

The derivable insights from these results apply beyond the automobile manufacturing sector. Since, as we have noted, the values in Table 5 and Table six provide a measure out of what tin can be gained from having guaranteed inventory availability, they suggest what may be defensible levels of investment to have that guarantee. Thus, for a buyer facing the scenario in the last row of Table 5 under the column for c_two = 22 an investment limit of iv.37% of the expected toll rate for regular orders might exist advisable since the total cost is lowered past iv.37% when emergency ordering is used (encounter Tabular array 5). That sort of guideline might exist useful for practical decisions of how much to invest in supply chain security initiatives that hope such guarantee; e.k., initiatives such equally the Community-Trade Partnership Against Terrorism (C-TPAT).

half-dozen. Conclusions

Using a lost sales context, we study a dual sourcing policy that uses emergency ordering when a delay in the social club from the regular source will lead to an imminent stockout. We model the policy as a lost sales (south, Q) system. By demonstrating its equivalence to systems with instantaneous emergency society delivery (with a known delivery probability) nosotros overcome some mathematical complexities of modeling continuous review systems. Through an extensive gear up of numerical examples, nosotros find that complementing regular orders with emergency orders does not e'er reduce full costs (comprising ordering price, inventory carrying cost, shortage cost, and item toll). Nosotros identify conditions under which this dual sourcing policy yields lower total costs than the standard lost sales (s, Q) policy without emergency. Further, we quantify the magnitude of the cost reductions if delivery of the emergency order is guaranteed; i.east., guaranteed elimination of shortages.

A key determination from our work is that the dual sourcing policy might accept to be supported by companion strategies to provide that guarantee. The results provided some insights into known company behaviour in the empirical context that motivated the study: United states of america-Canada trans-border supply chains. That is, we surmise that the reasons firms such every bit automobile manufacturers pursue strategies aimed at guaranteed elimination of shortages might be the resulting big full cost reductions. 2 items nosotros see every bit potential time to come research goals are one) developing closed-grade solutions to readily produce the newspaper's results and insights without reliance on all-encompassing numerical examples and 2) studying policies that tolerate some increase in inventory property costs in order to have guarantees that the emergency source volition deliver the emergency society.

Acknowledgements

This piece of work was funded by the SSHRC grant of the authors. The authors would like to thank the referees for their feedback which significantly improved the presentation of the paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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