رویکرد توصیف سیاست مبتنی بر فرایند تصمیم گیری مارکوف برای یک مشکل کنترل موجودی تصادفی با منابع غیرقابل اعتماد

We consider a single-product periodic-review inventory system for a retailer who has adopted a dual sourcing strategy to cope with potential supply process interruptions. Orders are placed to a perfectly reliable supplier and/or to a less reliable supplier that offers a better price. The success of an order placed to the unreliable supplier depends on his supply status that has a Markovian nature. The inventory control problem for this unreliable supply chain is modeled as a discrete-time Markov decision process (MDP) in order to find the optimal ordering decisions. Through numerical experimentation, the structure of the optimal ordering policy under several cost scenarios and different supplier reliability levels is determined. Four basic policy structures are found and are referred as case 1: order only from the unreliable supplier; case 2: order simultaneously from both suppliers or only from the unreliable supplier depending on the inventory level; case 3: order from one or the other but not both suppliers simultaneously; and case 4: order only from the reliable supplier. For all cases, (s, S)-like policies characterize perfectly the optimal ordering decisions due to the existence of the fixed ordering cost. Further experimentation is done to study the effects of changes in several system parameters (cost parameters such as fixed ordering cost, unit purchasing cost, backorder cost as well as the supplier reliability level) on the ordering policy and cost of the system.

Uncertainties in the supply process should be taken into consideration when making ordering decisions in order to manage inventories in an effective way and assure a desired customer service level. Uncertainty might be present in the quantity, quality or timing of orders delivered due to several factors. Possible internal causes include random machine/equipment breakdowns, process adjustments, a heavier-than-normal workload, shortages in the capacity of the supplier or material availability, labor strikes, etc. External causes include transportation disruptions due to accidents or bad weather conditions, natural disasters, or market-related causes such as price fluctuations, price subject to inflationary increases, scarcity of goods, etc. For instance, exposure to heat might affect the quality of perishable food, a traffic accident can delay the delivery of an order, and fire could destroy product partially or entirely. It is possible that the market price is so high that some companies find it prohibitively expensive to purchase the product (Parlar and Berkin, 1991). Or in case of a scarcity of goods, a supplier may prefer to satisfy the demand of major players at the expense of minor players (Arreola-Risa and Decroix, 1998). Such factors can result in changing the status of a source of supply from ‘available’ to ‘unavailable’ (or in other terms, from ‘on’ to ‘off’, or ‘up’ to ‘down’), hence interrupt the supply process (Mohebbi, 2004). Proactive firms adopt several supply-side tactics to cope with potential supply chain disruptions such as sourcing from multiple suppliers or holding more inventory (Tomlin, 2006). Due to the risks arising from the dependency on a single supplier such as uncertainty in the quality and quantity of the supply, supplier diversification (i.e. using multiple suppliers for the same product) is increasingly used by the buyers (Swaminathan and Shanthikumar, 1999). In addition to reducing the risk of no or partial delivery of the order, using multiple suppliers has other advantages such as creating competition that can yield better quality and/or better price (Anupindi and Akella, 1993). Even when there is no risk for supply disruptions, many firms prefer to use dual sourcing where one supplier offers faster product delivery (i.e. better responsiveness) at higher sourcing cost compared to the other supplier. In this case, the firms usually get the greater part of their products from the cheaper supplier with longer lead time, but turn to the more expensive supplier with expedited service when needed (see e.g. Veeraraghavan and Scheller-Wolf, 2008, Allon and Van Mieghem, 2010, Sheopuri et al., 2010 and Arts et al., 2011). We consider an infinite-horizon, single-product, periodic-review inventory system for a retailer who has adopted a dual sourcing strategy to cope with potential supply interruptions. Amongst the two suppliers, one is perfectly reliable while the other is not but offers a lower price. The unreliable supplier can be in “up” or “down” states modeled as a two-state Markov process. Supplier availability at the beginning of period does not guarantee the successful delivery of the product by the end of period because the lead time is not zero and so there is a risk that the supplier becomes unavailable when the order is still in progress. An order in progress is canceled if the supplier ends up being unavailable by the end of period. When the unreliable supplier is down, the retailer's only option is to source from the reliable supplier. Fixed costs of ordering are considered along with the unit purchasing, holding and backordering costs. We formulate the problem as a Markov decision process (MDP) in order to determine the optimal policy which is a list of the optimal action to follow (i.e. the optimal order quantities from both suppliers) in each possible state of the system. While optimal, the MDP policy does not provide managerial insight into the structure of the policy. Therefore in order to develop this insight, we use the process of policy characterization to define a generalizable ordering policy in a structured way using a few control parameters. This process consists of the following steps. First, we find the optimal policy using the MDP for a set of problem instances in an extensive numerical experimentation over a range of cost configurations of the system and different supply unreliability levels. Then, through careful observation of MDP solutions, we identify candidate policy structures that mimic the optimal decisions in each state. The accuracy of the characterization can be found by comparing the cost found using the characterization with that of the MDP optimal policy. Characterization of the policy is important, because it makes it easier to interpret the structure of the policy. The effects of changes in system parameters on the optimal ordering decisions can be easily seen through the characterized policies by observing how their structure and/or the values of the control parameters change. To the best of our knowledge, none of the published work provides all the different optimal policy structures derived in this paper for similar inventory systems under supply disruptions. The contents of the paper are organized as follows: Section 2 provides a review of work published on inventory models under supply interruptions. Section 3 describes the unreliable supply problem considered for a retailer with two suppliers, and provides the MDP formulations of the inventory control and sourcing problem for the retailer. Section 4 introduces the MDP-based characterization procedure that is employed to convert the optimal ordering decisions into a policy that can be defined using a few control parameters. In Section 5, first, through an empirical analysis, various types of optimal ordering policy structures that exist for such an unreliable supply chain are derived. Then, further computational study is done to show the impacts of changes in several system parameters on the optimal ordering policy structure and/or optimal cost. Section 6 summarizes our findings and suggests directions for further research.

We consider a single-product periodic-review inventory system for a retailer who has adopted a dual sourcing strategy in order to cope with potential supply process interruptions. The retailer places his orders to a reliable supplier and/or to an unreliable supplier offering a better price. The unreliable supplier can be in either of two states: ‘up’, available to accept orders; and ‘down’, unavailable to accept orders. The process of transitioning between states is assumed to be a simple Markov process defined by two parameters: α, the probability of staying in the ‘up’ state; and β, the probability of transitioning from the ‘down’ state to the ‘up’ state. The aim of the study is to determine the structure of the optimal ordering policies for this retail system under different cost scenarios as well as different unreliability levels of supply through a comprehensive numerical experimentation. The optimal ordering decisions found for different scenarios are characterized into ordering policies with a practical structure (i.e. policies that are defined in a structured way using a few parameters) using an MDP-based characterization method, which finds the optimal policy characterization for a scenario within seconds. It is important to note that this policy structure is based upon characterization of the optimal policy rather than some predetermined fixed policy structure as is typically done in the literature. The policies found by the MDP-based characterization approach are guaranteed to be optimal if the optimal policy structure for a given scenario is one of the four structures described in this paper. In the extensive numerical experimentation we conducted, we did not observe any another policy structure, so the characterized policies found were all optimal. However, this characterization approach can be easily extended to consider more policy structures if new policy structures are discovered for the same problem under different parameter settings. The effects of changes in cost parameters and the reliability of the unreliable supplier on the optimal ordering policies are also investigated. Results indicate that as the values of the cost parameters of the system and/or the unreliability levels of the supply chain change, four different optimal policy structures can be observed, which are referred to as case 1: order only from the unreliable supplier; case 2: order simultaneously from both suppliers or order only from the unreliable one based on what the inventory level is; case 3: order from one or the other but not both suppliers simultaneously; and case 4: order only from the reliable supplier. For all cases, the order policy from each supplier is an (s, S)-like policy, where s represents the reorder level and S, order-up-to level. As the reliability level of the unreliable supplier decreases from 1 (i.e. totally reliable) to 0 (i.e. totally unreliable), the optimal policy switches from case 1 to case 2 or case 3, and then finally to case 4. The four cases of this policy structure are somewhat similar to that of Swaminathan and Shanthikumar (1999). The policy found by Anupindi and Akella (1993) is also similar although theirs does not include the case of ordering from only the more expensive supplier. However, for both these papers it is assumed that there is no fixed cost of ordering and that if the order is not delivered in the current period, it is delivered with certainty in the next. This work gives insight into the different policy types that can be encountered as optimal for the retail system considered. For large-scale problems that might not be computationally tractable using MDP, development of exact or heuristic methods that find the values of the control parameters for given policy types is an open research problem.