A Difference Between Periodic Review and Continuous Review Inventory Systems Is

Abstruse

Nosotros analyze a continuous review lost sales inventory organization with two types of orders—regular and emergency. The regular social club has a stochastic pb fourth dimension and is placed with the cheapest acceptable supplier. The emergency order has a deterministic lead time is placed with a local supplier who has a higher toll. The emergency order is non always filled since the supplier may not have the ability to provide the order on an emergency basis at all times. This emergency order has a higher toll per detail and has a known probability of being 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 price optimal past comparing 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: 10.4236/ajor.2018.85020.

one. Introduction

In this newspaper 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 take chances by splitting replenishment requirements among multiple suppliers is a sourcing policy that has attracted the attending of academic researchers for more 20 years. The policy is theoretically appealing for several reasons. First, pooling lead-time doubtfulness amid several suppliers is a way to reduce the prophylactic stock needed to meet service targets or alternatively, the expected number of backorders for a prescribed level of rubber stock. 2nd, successive deliveries of smaller orders volition reduce cycle stock. Tertiary, the incremental ordering price of the 2d equally subsequent orders may be relatively pocket-sized in a variety of settings.

Since the tragic events of September xi, 2001, several security initiatives have been implemented at US international border checkpoints. These aim to minimize the take a chance of trans-border flows of merchandise being conduits for impairment to national security. For many companies delivering products by trucks from Canada to the US, these measures have increased both the mean and variance of border crossing times. Consequently, some U.s. companies (for example the automotive manufacturing industry) that traditionally obtained their raw material from Canada have switched to using simply American suppliers or using both a Canadian and an American supplier. In the latter example, companies split their requirements between the less expensive and less pb-fourth dimension reliable Canadian supplier and the more expensive just more than lead-time reliable American supplier. This has boosted interest in research on inventory systems with multiple suppliers and provides the motivation for our research.

In this paper we volition compare ii inventory policies. The policy we volition focus near 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 only when the inventory level is dangerously low and a stock out risk is imminent. We consider the case of a manufacturer facing demand that is anticipated 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 border crossing times. When inventory falls to dangerously low levels, an emergency society is placed with a local supplier or with the competition. The order is filled with a certain probability. If the gild is filled, it volition be filled in a certain fixed amount of time which is deterministic. There is a fixed cost of placing the order which is incurred whether the society is filled or not. The variable cost of the emergency order, which is proportional to the order size, is merely incurred if the order is filled. There are various reasons why the emergency order may not be filled. If the emergency club is placed with a local supplier with whom the company does not take a large amount of business, this company may reserve their stock for college priority customers and choose not to fill the gild of this manufacturer. If the guild is placed with the competition, and so for strategic business concern reasons, they may choose not to make full that particular order. We will compare the full long run cost charge per unit 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 order placements is price effective when compared to a single sourcing (Q, R) model. At that place are two main contributions of this paper―ane) Determine the long run expected total toll of a (Q, R) arrangement with emergency orders and lost sales, 2) Provide a method by which to compare a lost sales (Q, R) system to a lost sales (Q, R) organization with emergency orders and illustrate with numerical examples when the system with emergency orders is cost effective.

Rapid advancements in estimator and it forth with the crucial role of responsiveness as a winning supply concatenation strategy have additional interest in continuous review inventory policies. However, the bulk of the inventory models dealing with multiple suppliers are for the periodic review case. The focus on periodic review is mainly considering of the mathematical complexity in dealing with continuous review systems. In this newspaper we will deal with a continuous review (Q, R) inventory system with lost sales and two suppliers. Past identifying and establishing the equivalence of this model to some other mathematical model which is more than tractable, we will obtain analytic expressions for the full price of running the system. Nosotros will then compare the two policies numerically in a multifariousness of scenarios and identify situations where a policy with ii suppliers performs ameliorate than a system with a unmarried supplier.

The paper has half-dozen sections. The 2d section consists of literature review of inventory systems for multiple suppliers. Section 3 is the crux of the paper where the mathematical model and detailed analysis are presented. Section iii is likewise fundamental in demonstrating how nosotros address what has been heretofore seen equally a mathematical difficulty of continuous review (Q, R) systems. The steady state inventory level distribution is also derived in this section. In Section 4, the expression for the long-run expected cost rate is adult. In Section 5 some numerical examples are presented. Section six consists of conclusions and future inquiry.

2. Literature Survey

The literature on inventory systems with multiple suppliers can be broadly separated into 2 classes―one, where replenishment orders are split simultaneously amid many suppliers and two, where orders are placed at different times with different suppliers. In this paper we focus on the second instance and hence just provide a literature survey of papers of this kind. For data on models of the get-go 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 inquiry includes the early papers of Barankin [iii] and Neuts [4] , which studied periodic review inventory systems with regular and emergency replenishments, where the regular order atomic number 82-fourth dimension is ane period and the emergency replenishment is instantaneous. The more than recent studies on emergency ordering have all been in the periodic review realm. The latest study of a periodic review arrangement with emergency orders was by Johansen [5] . They study an inventory organization with compound Poisson demands and backorders using Markov determination processes. They do not accept whatever analytical comparisons of price, 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.

Nevertheless, since all these papers focus on periodic review models while we address 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 non include the literature on inventory systems where suppliers are either available or unavailable but no emergency ordering is done during the unavailable period of the suppliers nor does it include the example of transhipment of inventory between ii locations. Further we practice not present a literature review of ii echelon inventory models or simulation studies of inventory models.

Dohi, Kaio and Osaki, Southward. [6] and Giri and Dohi [vii] report continuous review inventory systems where later on 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 fourth dimension L2. The first paper derives the necessary and sufficient weather for t_o to exist which minimizes the long-run average price. The 2d paper they derive the optimal ordering time that minimizes average cost for a fixed ordering quantity model. Bradley [8] analyzed a production-inventory model in which in-business firm product and a sub-contractor are the inventory replenishment alternatives. Using Brownian move approximations, the author sought to determine the optimal policy parameters. Allon and Van Miegham [9] 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 but inexpensive supplier. Despite some similarities between their work and ours, at that place are several important differences. For instance, they deal with the instance of backorders while we deal with the case of lost sales. Also, in their paper the two orders are placed with the two suppliers simultaneously while in our paper, the guild with the expensive supplier is merely placed if the stock levels autumn dangerously low.

Allen and D'Esopo [10] were the first to consider a continuous review inventory system with emergency orders. They analyzed the standard (Q, R) inventory model with an additional parameter called the expediting level. Moinzadeh and Schmidt [11] considered the (Southward − 1, S) inventory system with emergency orders. Vocal and Zipkin [12] extend this model to include the case of multiple suppliers and develop operation evaluation tools for a variety of policies nether which the supply organization becomes a network of queues. Johansen and Thorstenson [thirteen] 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 order. They bargain with the case of consummate backordering and employ simulation to obtain the optimal values of the reorder levels and lodge quantities.

Moinzadeh and Nahmias [14] proposes a very general model with the (Q, R) organisation with back orders, ii 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 lead times for both orders are deterministic and that the demand tin follow a Poisson or normal distribution. Mohebbi and Posner [15] analyze the model by Moinzadeh and Nahmias [xiv] under the supposition of chemical compound Poisson demand and non-identical exponentially distributed atomic number 82 times using the level crossing approach and develop the total toll function. Duran, Gutierrez and Zequeira [16] analyze a system similar to the one past Moinzadeh and Nahmias [fourteen] . They analyze a continuous review (Q, R) system with backorders where the pb time has a stock-still component T. If the inventory level lies below a threshold level when T units of time has elapsed since order placement, the order is expedited and arrives later on a short but deterministic time and if the inventory level is non below the threshold level, there is a longer deterministic catamenia of time before the society 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 one past Axsater [17] which deals with a (Q, R) inventory arrangement with Poisson demands and emergency ordering. There are four important differences betwixt Axsater [17] and our model. Our newspaper is motivated past the fact the atomic number 82 time for the order placed with the cheaper (regular) supplier is highly variable, which necessitates the use of an alternating supplier. Hence in our case only the emergency lodge has a deterministic lead time and the regular order has a stochastic lead time whereas both orders take deterministic lead time in Axsater [17] . Further in our paper, which has applications to the auto industry, the need pattern is deterministic since the production lines at most motorcar companies run on a continuous footing while the need pattern in Axsater [17] is Poisson. Our paper deals with the situation where the availability of the emergency order is probabilistic whereas in Axsater [17] , the assumption is that the emergency gild will always get filled. The last difference is that we consider the lost sales example while Axsater [17] considers the backlogging case.

3. Problem Description and Analysis

We consider a continuous review (Q, R) inventory system for a US based manufacturer where the demand for the detail is anticipated and is one unit every T periods; i.due east., mean need per catamenia = one ÷ T. Note that because at each demand epoch, there is a need for just one detail, the (Q, R) policy is equivalent to the (due south, Q) policy and from here on, we will refer to the model as an (due south, Q) organisation. The maximum inventory level is Q + southward units. When the inventory level drops to the reorder indicate, southward, a regular order of size Q (Q > s) is placed with a Canadian supplier. The supposition that Q > due south ensures that at that place is at most i outstanding regular social club at any given time. Considering of the highly unpredictable border crossing times, the lead time for the regular order is causeless to exist exponentially distributed with rate μ. The organization we consider is a lost sales system. Hence if the inventory level drops to zero and there is a need for the item by a customer, the customer 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 time taken to procure items on an emergency ground 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 contest. This gild has a probability p of existence fulfilled. Note that an emergency order is placed but at the instant when the stock level drops to northward from n + one and not at times when the inventory level is northward. If the society is filled, it is for the unabridged south units. The local supplier may guarantee that the stock will be available 100p% of the time. In the case of the contest, obviously there volition be no guarantees and 100p% is the estimated percentage of time that the manufacturer tin can get the stock from the competition.

The organization described above is equivalent to a lost sales (s, Q) inventory system where demands occur once every T units with the following boosted conditions. The reorder policy with the Canadian supplier is the same equally in the model described in the previous paragraph. The emergency lodge described in that model is equivalent to an emergency guild placed when the inventory level drops to cipher which gets replenished with probability p. If the order is filled, information technology is filled instantaneously. In this newspaper we will model this system as a arrangement with instantaneous replenishment of the emergency order and follow through with the assay.

In order to determine the long-run expected cost rate, we need to make up one's mind the steady land inventory level distribution P(j), where P(j) denotes the steady country 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 state inventory level distribution P(j) is given by

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

$P\left(j\right)=\frac{{\left({\text{e}}^{\mu T}-1\right)}^{\mathrm{ii}}}{\mu T\left(1-p\right)}{\text{e}}^{\left(j-1\right)\mu T}\pi \left(0\right);\mathrm{i}\le j\le s$ (three.ii)

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

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

$P\left(j\right)=\frac{\left({\text{e}}^{\mu T}-\mathrm{i}\right)}{\left(1-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p{\left({\text{e}}^{\mu T}-1\right)}^2}{\mu T\left(1-p\right)}{\text{eastward}}^{\left(j-Q-\mathrm{ane}\right)\mu T}\pi \left(0\right);Q+\mathrm{i}\le j\le Q+\mathrm{south}$ (3.5)

where

$\pi \left(0\right)=\frac{1}{1+\frac{Q\left({\text{e}}^{\mu T}-1\right)}{1-p}{\text{e}}^{\mathrm{due\; south}\mu T}+\frac{\left(Q-s\right)p\left({\text{east}}^{\mu T}-\mathrm{i}\right)}{1-p}}$ (iii.6)

Proof: Allow I(t) denote the inventory level at time t. From our assumptions it is clear that the inventory level process {I(t); t ≥ 0} with country space E = {0, ane, 2, ..., Q + south} is a semi-regenerative process with the regeneration points beingness the demand epochs. Allow $\left\{{\tau }_0,{\tau }_1,{\tau }_{\mathrm{two}},\cdots \right\}=\left\{0,T,2T,\cdots \right\}$ exist the successive epochs at which demands occur. If $I_n=I\left({\tau }_n^{-}\right)$, then $\left(I,\tau \right)=\left\{I_n,{\tau }_n;\mathrm{north}\in N^0\right\}$ is a Markov renewal process with embedded Markov concatenation $\left\{I_n;\mathrm{northward}\in N^0\right\}$ . Allow united states of america 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:

$P\left(i,j,t\right)=K\left(i,j,t\right)+{\displaystyle {\sum }_{l=0}^{\mathrm{south}}{\displaystyle {\int }_0^t\theta \left(i,l,u\right)P\left(l,j,t-u\right)\text{d}u}}$ (three.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

$\theta \left(i,j,t\right)=\underset{\Delta \to 0}{\lim }P\left[I_{\mathrm{ane}}=j,t\le {\tau }_1\le t+\Delta |I_0=i\right]/\Delta$ (3.8)

and

$K\left(i,j,t\right)=P\left[I\left(t\right)=j,{\tau }_1>t|I_0=i\right]$ (3.9)

The various operating characteristics that are necessary to obtain the long-run expected cost charge per unit of the inventory system can exist obtained in terms of the steady land inventory level distribution if information technology exists.

From Markov renewal theory the steady state distribution of the inventory level exists as the country infinite is finite and the embedded Markov chain is irreducible. Allow P(j) be the steady state inventory level distribution. Then from Cinlar [18] , nosotros have

$P\left(j\right)=\frac{{\displaystyle \sum _{i=0}^{\mathrm{south}}\pi \left(i\right)K^{*}\left(i,j,0\right)}}{{\displaystyle \sum _{i=0}^{\mathrm{southward}}\pi \left(i\right)\mathrm{one\; thousand}\left(i\right)}}$ (3.ten)

where ${\mathrm{1000}}^{*}\left(i,j,0\right)$ is the Laplace transform of $M\left(i,j,t\right)$ evaluated at goose egg and m(i) is the hateful sojourn time in state i. The stationary distribution of the embedded Markov chain is given by π(i) and obtained by 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 ${\displaystyle {\sum }_j\pi \left(j\right)}=1$ .

Since time between ii consecutive (unit) demands is deterministic and equal to T,

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

In order to determine the steady land inventory level distribution, we first demand to determine ${\displaystyle {\int }_0^{\infty }\theta \left(i,j,t\right)\text{d}t}$ and ${\displaystyle {\int }_0^{\infty }\mathrm{Thou}\left(i,j,t\right)\text{d}t}$ .

To determine $\theta \left(i,j,t\right)$, we annotation that the transition points are either demand or replenishment epochs. For case let the inventory level simply before a demand occurs be 0 and hence afterwards the need occurs and earlier the side by side demand, the inventory level volition either remain at 0 if no replenishment occurs with probability e^−μt and the inventory level will attain Q if a replenishment occurs with probability 1 − e^−μt. In our case, the time between need epochs is deterministic. Hence $\theta \left(i,j,t\right)$ is non-zilch only when t = T. Denote past $p\left(i,j\right)$, ${\displaystyle {\int }_0^{\infty }\theta \left(i,j,t\right)\text{d}t}$ . Then $p\left(i,j\right)$ are the one pace transition probabilities of the Markov concatenation embedded in the Markov renewal process (I, τ) and are given by the function $\theta \left(i,j,T\right)$ .

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

The stationary distribution of the embedded Markoff 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 ${\displaystyle {\sum }_j\pi \left(j\right)}=1$ . Using the function $p\left(i,j\right)$ given above we obtain, on solving $\pi \left(j\right)={\displaystyle {\sum }_i\pi \left(i\right)p\left(i,j\right)}$,

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

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

$\pi \left(j\right)=\frac{\left({\text{e}}^{\mu T}-\mathrm{one}\right)}{\left(\mathrm{ane}-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p\left({\text{due east}}^{\mu T}-\mathrm{ane}\right)}{\left(1-p\right)}{\text{due east}}^{\left(j-Q-1\right)\mu T}\pi \left(0\right);Q+1\le j\le Q+s$ (3.15)

Using Equations (3.13) to (3.15) in ${\displaystyle {\sum }_j\pi \left(j\right)}=1$, we obtain

$\pi \left(0\right)=\frac{\mathrm{ane}}{1+\frac{Q\left({\text{e}}^{\mu T}-1\right)}{1-p}{\text{e}}^{s\mu T}+\frac{\left(Q-s\right)p\left({\text{e}}^{\mu T}-1\right)}{1-p}}$ (three.16)

In social club to determine the steady state inventory level distribution, we adjacent make up one's mind the function $M\left(i,j,t\right)$ given past $\mathrm{Chiliad}\left(i,j,t\right)=P\left[I\left(t\right)=j,{\tau }_{\mathrm{ane}}>t|I_0=i\right]$ . Hence for T > t, we have

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

and

$\begin{array}{l}{K}^{*}\left(i,j,0\right)\\ =\{\begin{array}{l}\left(1-{\text{due east}}^{-\mu T}\right)/\mu i=j=0;2\le i\le s+\mathrm{i},j=i-\mathrm{i}\hfill \\ T-\left(1-{\text{east}}^{-\mu T}\right)/\mu i=0,j=Q;2\le i\le s+\mathrm{one},j=i+Q-\mathrm{one}\hfill \\ \left(\mathrm{one}-p\right)\left(1-{\text{due east}}^{-\mu T}\right)/\mu i=1,j=0\hfill \\ \left(1-p\right)T-\left(1-p\right)\left(1-{\text{east}}^{-\mu T}\right)/\mu i=1,j=Q\hfill \\ p\left(1-{\text{due east}}^{-\mu T}\right)/\mu i=1,j=s\hfill \\ pT-p\left(1-{\text{due east}}^{-\mu T}\right)/\mu i=1,j=Q+s\hfill \\ Ts+2\le i\le Q+\mathrm{southward},j=i-1\hfill \\ 0\text{otherwise}\hfill \end{array}\end{array}$ (3.xviii)

Substituting for $M^{*}\left(i,j,0\right)$ and π(j) from (3.18) and (three.13) to (3.fifteen) in (iii.iv) we obtain

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

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

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

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

$P\left(j\right)=\frac{\left({\text{east}}^{\mu T}-1\right)}{\left(\mathrm{one}-p\right)}{\text{e}}^{s\mu T}\pi \left(0\right)-\frac{p{\left({\text{e}}^{\mu T}-1\right)}^2}{\mu T\left(1-p\right)}{\text{due east}}^{\left(j-Q-1\right)\mu T}\pi \left(0\right);Q+1\le j\le Q+\mathrm{south}$

Now that we have obtained the steady land inventory level distribution and the stationary probabilities of the embedded Markov concatenation, nosotros can determine the various operating characteristics required to derive the cost function.

4. Price Function Derivation

In this section we volition bargain with the problem of minimizing the full expected price rate. We will also decide conditions under which information technology is toll optimal to identify an emergency society with the local supplier rather than just wait for the order from your regular supplier. Nosotros use the following cost components

K_one: the set-upwards cost per order for the regular gild.

Yard₂: the set-up price per society for the emergency order. This price is incurred whether or non the social club is filled.

c₁: the cost per item for the regular gild.

c₂: the price per item for the emergency order. This toll is merely incurred if the gild is filled.

g: the shortage cost/unit brusk.

h: the inventory conveying cost/unit/unit fourth dimension.

Then the total expected cost rate is given past

$C\left(\mathrm{due\; south},Q\right)=\left({\mathrm{Chiliad}}_{\mathrm{ane}}+c_{\mathrm{i}}Q\right){\Gamma }_1+\left(G_2+c_2\mathrm{south}p\right){\Gamma }_2+h{\Gamma }_{\mathrm{iii}}+g{\Gamma }_4$ (iv.1)

where Γ_i is the reorder rate for the regular order, Γ_ii is the reorder rate for the emergency society Γ_iii is the average inventory level and Γ₄ is the shortage rate.

A regular guild is placed when the inventory level is south + 1 and a demand occurs which brings the level downwards to the reorder point s. This order has a atomic number 82 fourth dimension which is exponentially distributed with a mean of 1/μ. Hence

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

An emergency social club is placed when the inventory level is n+1 and a demand occurs or in the equivalent (instantaneous replenishment) organisation when the inventory level is one and a demand occurs. Hence

${\Gamma }_2=\pi \left(1\right)/T$ (4.3)

The average inventory level

${\Gamma }_3={\displaystyle {\sum }_{j}P\left(j\right)}$ (4.iv)

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

${\Gamma }_4=\pi \left(0\right)/T$ (4.5)

Substituting for Yard₁, G₂, G₃ and G₄ from (4.2) to (iv.5) in (4.ane) we obtain

$\begin{array}{c}C\left(s,Q\right)=\left(K_1+c_{1}Q\right)\pi \left(s+1\right)/T+\left(K_2+c_{2}s\left[\mathrm{i}-P\left(0\right)\right]\right)\pi \left(\mathrm{i}\right)/T\\ +h{\displaystyle {\sum }_{j}P\left(j\right)}+\mathrm{thou}\pi \left(0\right)/T\end{array}$ (4.6)

Notation that the fixed cost of placing an emergency order is incurred whether or not the order is satisfied while the variable cost is incurred only if the order is met. If the emergency lodge materializes, the inventory level does not touch naught. Hence probability that the variable toll of the emergency order is incurred is the probability the inventory level does not attain zero.

$C\left(s,Q\right)$ tin be obtained explicitly past substituting for π(j)s from Equations (3.13) to (3.15) and for P(j)s from (three.ane) to (iii.5).

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

The long-run expected cost charge per unit for the (s, Q) that organisation with no emergency orders is given by

$C\left(s,Q\right)=\left(K_{\mathrm{i}}+c_{1}Q\right){\Gamma }_5+h{\Gamma }_{\mathrm{half\; dozen}}+g{\Gamma }_7$ (4.7)

Annotation that the only difference betwixt Equation (4.7) and Equation (4.ane) is the absenteeism of the toll term for the emergency orders. The reorder charge per unit, Γ₅,tin be obtained by substituting p = 0 in Γ_one. The average inventory level, Γ₆, can be obtained by substituting p = 0 in Γ₃. The shortage rate, Γ₇, tin be obtained by substituting p = 0 in Γ_four.

5. Numerical Illustration

In this section we compare the long-run expected toll charge per unit of the system using emergency orders with that of the with standard lost sales (s, Q) policy and institute cases where the organization with emergency orders yields lower full costs or lower average inventory levels or both lower total costs and lower boilerplate inventory levels. For purposes of numerical illustration we causeless that the club size for the regular order Q is divisional above by the size of the container. We used 200 as the container size. In order to decide the optimal parameters s and Q, we do a complete enumeration and determine the pair (south, Q) that gives us the minimum toll. The values held fixed in our numerical analysis were Grand₁ = 140, Yard_two = seventy, c₁ = twenty, T = 0.02, μ = 2. We so considered six values of c₂, ten values of one thousand, ii values of h, and 13 values of p. Our numerical analysis covered all possible combination of those values; i.e., 1560 combinations. Tables one-half dozen tabulate the key results.

Table 1 and Table 2 address the question of how the inventory policy parameters (south, Q) and average inventory are affected by using the emergency ordering organization at the farthermost value combinations of c₂, g and h for the various values of p. The tables show the results only for factor combinations where the long-run expected price rate is lower for the emergency ordering than for the

Table 1. Illustrative impacts of emergency order system on policy parameters if belongings cost = one.

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

Table 2. Illustrative impacts of emergency order arrangement on policy parameters if holding cost = x.

Note: The shaded cells represent to scenarios where emergency sourcing is inferior to unmarried sourcing.

Tabular array three. Minimum emergency order receipt probabilities for dual sourcing to exist beneficial if belongings cost = 1.

Notation: Ten = single sourcing is ever cheaper than emergency sourcing for that combination of g and c_two.

Table 4. Minimum emergency gild receipt probabilities for dual sourcing to be benign if property cost = ten.

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

Tabular array 5. Percentage toll reduction achieved with dual sourcing for guaranteed emergency order receipt if holding cost = 1.

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

Table 6. Per centum toll reduction achieved with dual sourcing for guaranteed emergency lodge receipt if belongings cost = ten.

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

standard lost sales (s, Q) policy. Thus, the combination of (c_ii, g) = (fifty, 40) 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 tin can operate optimally with smaller inventories than the standard lost sales (southward, Q) policy. For case, at (c₂, yard, h) = (22, 180, 10), the buyer's average inventory level when the choice of emergency ordering is unavailable is lxx units just drops to less than 1-third of that corporeality (20 units) when emergency club fulfillment (receipt) is guaranteed; i.e. if p = one.

Table 3 and Table four focus on determining the minimum emergency social club receipt probability (p) that must be reached in lodge for dual sourcing to yield a lower long-run expected toll rate than the standard lost sales (s, Q) policy. Consider, as illustration, the column for c₂ = fifty in Table 1. This shows that unless the shortage toll exceeds a value somewhere in the range 120 £ yard < 150, dual sourcing does not make sense. Further, even if that condition is satisfied, the dual sourcing is advantageous only if the probability of order fulfilment past the emergency source is high. Case in point is that at 1000 = 150, the probability must attain 0.99. At the higher value of g = 180, the required probability threshold is less (0.90). The threshold order 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 belongings inventory (h).

These observations have important sourcing policy implications. In detail, as noted before, emergency lodge fulfilment by infrequently used domestic suppliers is unlikely to have a high probability, much less be guaranteed. As such, an emergency lodge policy might have to exist complemented by a policy of ordering more often from the domestic supplier; i.e., guild even in the absence of an emergency requirement. That style, the domestic supplier may accord the buyer'south lodge the high priority given to orders from its regular customers and thus exist more inclined to fill the emergency order. Clearly, this means higher long-run inventory toll rate and requires the buyer to decide the revised social club frequency from a domestic source would assure a sufficiently loftier gild fulfilment probability. Analysis of this emergency ordering tactic is beyond the scope of this newspaper and is considered as a matter for possible futurity enquiry.

Table 5 or Table 6 testify the percentage price reductions of using the emergency ordering policy instead of the standard lost sales (s, Q) policy, under the assumption of guaranteed order fulfillment (receipt). The reported percentages follow the expected directions: they are higher for 1) lower values of c_ii (lower detail price penalization of using the emergency source), 2) higher values of thou (avoidance of larger stockout penalties past using the emergency source), and 3) higher values of h (higher savings from the reduced inventory levels when the emergency source is used). For low holding cost (h = i% or v% of item price from regular supplier), the all-time improvement observed was a very moderate 4.37%. The increase of h to 10 (l% of the item price from regular supplier) yields more than hitting improvements. These improvements tin exist viewed every bit the premium a company would be willing to pay to accept guaranteed insurance against stockouts.

These observations may offer some insights into the strategies used by some companies to handle their United states of america-Canada trans-edge supply chain operations. To explicate these insights, we first notation that firms such as auto manufactures might face scenarios closer to the lower rows of Tabular array six than to any of the other scenarios in either Tabular array 5 or Table 6. Specifically, because these firms apply just-in-time delivery, failure of a just-in-time social club of parts to arrive as scheduled (a stockout) can crusade costly close down of a day's scheduled production. The fact that they utilize only-in-time every bit an inventory reduction strategy leads ane to deduce that their inventory conveying costs are loftier enough to exist warrant such a strategy. Indeed, holding cost data used for the models in Nozick and Turnquist (2001) support that deduction. These points arrive understandable that these firms have been willing to pay what is regarded equally a high premium for insurance against stockouts. However, the insurance that these firms have 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 across the machine manufacturing sector. Since, every bit we take noted, the values in Tabular array 5 and Table 6 provide a mensurate of what can be gained from having guaranteed inventory availability, they propose what may exist defensible levels of investment to have that guarantee. Thus, for a heir-apparent facing the scenario in the terminal row of Table 5 under the cavalcade for c₂ = 22 an investment limit of 4.37% of the expected cost rate for regular orders might be advisable since the total cost is lowered by 4.37% when emergency ordering is used (see Table 5). That sort of guideline might be useful for practical decisions of how much to invest in supply chain security initiatives that promise such guarantee; e.g., initiatives such as the Customs-Trade Partnership Against Terrorism (C-TPAT).

half-dozen. Conclusions

Using a lost sales context, we written report a dual sourcing policy that uses emergency ordering when a delay in the society from the regular source volition lead to an imminent stockout. We model the policy every bit a lost sales (s, Q) system. By demonstrating its equivalence to systems with instantaneous emergency guild delivery (with a known delivery probability) nosotros overcome some mathematical complexities of modeling continuous review systems. Through an extensive prepare of numerical examples, we notice that complementing regular orders with emergency orders does not ever reduce total costs (comprising ordering toll, inventory conveying cost, shortage cost, and item toll). We identify atmospheric condition under which this dual sourcing policy yields lower full costs than the standard lost sales (s, Q) policy without emergency. Farther, we quantify the magnitude of the price reductions if commitment of the emergency order is guaranteed; i.due east., guaranteed elimination of shortages.

A primal decision from our work is that the dual sourcing policy might have to exist supported by companion strategies to provide that guarantee. The results provided some insights into known visitor behaviour in the empirical context that motivated the study: US-Canada trans-border supply bondage. That is, we surmise that the reasons firms such as car manufacturers pursue strategies aimed at guaranteed elimination of shortages might be the resulting big total cost reductions. 2 items we see every bit potential future research goals are i) developing airtight-form solutions to readily produce the newspaper'due south results and insights without reliance on extensive numerical examples and 2) studying policies that tolerate some increase in inventory holding costs in club to have guarantees that the emergency source will deliver the emergency order.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of involvement.

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