پایداری دینامیک موجودی در زنجیره های تامین با سه تاخیر

Distinctive delays accounting for lead time in manufacturing, transportation of products and decision-making are considered in a single link supply chain that is modeled by continuous-time differential equations via system thinking. As seen in the literature, behavior of inventory variations in the presence of delays can become undesirably oscillatory. The novelty of this work is the analytical characterization of these oscillations with respect to two intrinsic parameters of the supply chain and the three delays considered separately. Particularly, inventory dynamics is characterized on the so-called stability maps drawn in the space of three delays, displaying which combinations of each delay lead to (un)desirable inventory behavior. Arising from the characterization is also a novel ordering policy design with which the inventory variations can be rendered insensitive to detrimental effects of delays. Case studies are provided along with managerial interpretations.

Inventory dynamics exhibit quite complex behavior in supply chains (SC) since inventory level variations are the end results of combined decision making, manufacturing and product shipment activities which are dynamically adapted against unpredictable and sometimes artificial consumer demand. While surfeit of inventories (overshoot) cause increased stocking costs, deficit of inventory levels (undershoot) may increase freight costs and the risk of depletion of inventories, all of which indicate inefficiency. Consequently, cost effective supply chain management naturally requires thorough understanding of decision making, manufacturing and product shipment dynamics that directly affect the underlying mechanisms of inventory behavior. One of the most critical parameters in supply chain management (SCM) is the delay (Sarimveis et al., 2008, Riddalls and Bennett, 2002b and Sterman, 2000). Delay is inevitable in SC due to physical constraints related to lead times (in manufacturing), transportation and delivery times (shipments), decision making durations (human behavior) and information availability (communication delays, data collection delays). In the presence of delays, what is known to the SC manager is not what is happening in the chain, but it represents the information regarding the SCs behavior in the time history. Moreover, there are multiple sources of delays in the SC and these delays are quantitatively different (An and Ramachandran, 2005). Therefore, available information pertaining to SC carries multiple delay signatures. What is detrimental to SCM is that delays mislead decision makers. This consequently prevents achieving successful SCM. Although it is known that delays bring detrimental effects, in some cases it is preferable that managers wait (adding delay) in order to observe the trends in the SC and in the market before making critical decisions (Sterman, 2000). Clearly, it is not straightforward to comment on the effects of delays to SCM. These two counter-intuitive arguments justify the need to study delay effects to dynamic behavior of the SC (Croson et al., 2004, Riddalls et al., 2000, Beamon, 1998 and Hafeez et al., 1996). We quest if there are ways to uncover the effects of delays to inventories and to SCM. If these effects can be understood with respect to intrinsic parameters defining the SC, then it would be possible to come up with new management strategies that can combat against undesirable effects of delays. This is exactly what forms the main objective here and it is aligned with the earlier work in Sarimveis et al. (2008), Warburton (2004), Ge et al. (2004), Riddalls and Bennett (2002b), Sterman, 2000 and Sterman, 1989, and Simchi-Levi et al. (2000). By performing stability analysis of the SC, we wish to reveal various dynamical behaviors of the SC and inventory levels with respect to delays and the parameters pertaining to management strategies. The stability/instability definitions used in this paper are along the lines of for instance Riddalls and Bennett (2002a) and Naim et al. (2004). For various combinations of management strategies, we are particularly interested in finding the delay values with which the inventories behave in a desirable way where inventory perturbations damp out (which we call as “stability”) rather than exhibiting oscillatory behavior (which we call as “instability”). It may be true that SC dynamics may eventually stabilize itself with the presence of bounds such as capacity limits, however, the long durations of inventory oscillations, which are known to have large periods, may put the SC into large financial losses before such bounds and extremis may take over and stabilize the SC. In this sense, the contribution of this paper can be seen as the characterization of delay effects to such persistent and undesirable transient behaviors observed in the inventory levels. As a result of our analysis, the SC manager has a decision making tool with which the SC can be operated in a stable regime based on various strategies and delays. With the tools we provide, it is also possible in some cases to dictate desirable inventory behavior by scheduling some of the activities with appropriate delays similar to the work in Lee and Feitzinger (1995), and to choose appropriate ordering policy with which the inventory levels are rendered insensitive to undesirable effects of delays. The results of this paper bridge the gap between surfacing undesirable effects of delays in SC and how to make proper decisions to avoid these effects in SCM. The mathematical framework of the study is constructed on Laplace domain, which is known to have been used first time in 1952 (Simon, 1952) for studying the stability of supply chains by Nobel Prize winner Herbert Alexander Simon. Forrester (1961) also derived differential equations for the same reason. Furthermore, Towill (1982) deployed Laplace transform for studying inventory and order based production control system. In Table 4 of Disney et al. (2006), it was shown that continuous time domain studies are more preferable due to various reasons except one, that is, the pure delays. The work presented here removes this concern, making continuous time domain analysis and its connection with Laplace transform a perfect platform to analyze SC and SCM. The particular SC problem studied in this paper is along the lines of Towill (1982), John et al. (1994), Riddalls and Bennett, 2002b and Riddalls and Bennett, 2003, where we consider an Automatic Pipeline Inventory and Order Based Production Control System (APIOBPCS) with two intrinsic deterministic parameters regulating a single inventory of a single product shipped via a single link transportation path. This model is also used in simplified forms in Hafeez et al. (1996) and Lewis et al. (1995). Interestingly APIOBPCS is similar to the heuristic stock acquisition strategy of Sterman which Sterman (1989) obtained from experiments involving multiple users playing a beer distribution game. What is different in this paper is that delay originates from three dissimilar physical sources hence we consider three different delays. These delays emerge from (i) decision making, (ii) production and (iii) transportation time. Hitherto, effects of each one of the three delays together were not investigated within a unified model, despite the fact that these delays are known to exist, (Ge et al., 2004, Riddalls and Bennett, 2002b and Sterman, 2000). With the analysis performed in this paper, we wish to present a broader picture as to how each delay governs the stability mechanisms of the SC. The paper is organized as follows. In Section 2, problem formulation is presented including the details of the mathematical model and the consideration of delays. Section 3 develops the system thinking and the authors’ earlier work (Sipahi and Delice, 2009) towards revealing the delay effects to inventory regulation and designing ordering policy in the presence of delays. Section 4 presents case studies and managerial decision making strategies extracted by exploiting the tools developed in Section 3. Discussions, limitations and future research directions conclude the paper in Section 5. Notations are standard. We use s for Laplace variable, CC for complex plane, j for complex number, View the MathML sourcej=-1. R(s)R(s) denotes real part of s , I(s)I(s) corresponds to imaginary part of s . Complex variable s lies in the left half complex plane C-C- when R(s)<0R(s)<0, and in the right half complex plane C+C+ when R(s)>0R(s)>0. RR, R+,-R+,-, ZZ, View the MathML sourceZ+∞ stands for the set of real numbers, positive real numbers, integer numbers and all positive integer numbers, respectively. CkCk denotes a circle with unity radius located on the origin of xk–ykxk–yk plane. II is the identity matrix with appropriate dimensions, and infinf denotes the greatest lower bound of a set.

Stability of inventory behavior controlled by a widely studied Automatic Pipeline Inventory Order Based Production Control System (APIOBPCS) is investigated with respect to delays originating from different physical reasons; lead time, transportation and decision-making. Analytical procedures are developed to tackle the stability problem by following control theory. End results of the stability analysis are the stability maps with respect to the delays, where on these maps stable and unstable inventory behavior are classified. Stability maps are supportive for managerial decision-making as they lay out which combinations of delays give rise to desirable inventory behavior. With the tools we develop, it also becomes possible to extract generalizing rules in designing the ordering policy in a way that undesirable effects of delays are mitigated, and the supply chain becomes insensitive to delays. Notice that additional constraints in the SC may exist and they will further narrow down the admissible stability regions found in the stability maps; for instance, the lead time can be less than 4 weeks and greater than a week. These constraints should be carefully superposed on the stability maps before judging on stability. Upon having considered these constraints, the arising stability tableau may assist the SC manager with the contractual agreements. Without availability of such a tableau, the SC manager would not realize that seemingly innocuous combination of transportation times and lead time delays could eventually put the inventory behavior into extremis. The limitations in our work are in parallel to the existing literature studying similar types of problems. The analysis performed in this paper is constructed on linear system theory. In this sense, the analysis uncovers how a supply chain that is perfectly in equilibrium gets perturbed by increased consumer demand or any other perturbation causing deviations from that equilibrium. The equilibrium is defined as ideal conditions where the demand product transportation, production and ordering rates are equal to each other, and the rate at which the products leave the inventory is equal to the rate at which the products arrive to the inventory. Even if there are delays in the pipelines, the pipelines are filled by precisely the same amount of products produced at the manufacturer at precisely the same amount at all times. This is obviously an ideal scenario which cannot occur in the presence of perturbations and variable consumer demand. Therefore, the inventory behavior will always oscillate around or deviate from this equilibrium. Arising question, upon these perturbations, is whether the ordering policy is robust enough to drive the inventory deviation back to steadiness or the ordering policy is poorly designed which will prevent the recovery of steadiness. This is exactly what we address in this paper. Considering the discussions above, our analysis is limited to revealing the stability mechanisms of deviations from an equilibrium under perturbations or changing consumer demand. The work is presented with three delays, and the absence of analytical techniques for handling arbitrary number of delays is still a limitation in the field of control theory. We also note that the ordering policy gain αiαi and ββ are constant parameters. Therefore, stability maps are extracted for a given (αiαi, β,λβ,λ) triplet and they are drawn with respect to delays, which are assumed to be time-invariant quantities. Regarding the computational speed of our stability analysis procedure, we state that the procedure only sweeps one parameter within a bounded range, and therefore the execution to extract the stability switching curves takes no more than 0.05 s. This is the strength of the approach developed and it can prove to be quite efficient in revealing multiple scenarios with different parameter settings. Since supply chain dynamics have large time constants, i.e. their response characteristics are in the order of days or weeks, rapid computations of stability maps within “seconds” can offer almost instantaneous information regarding the dynamics of supply chains. Future research is focused on several topics encompassing cost optimization considering delays, depleted inventories and inventory oscillations as costs; policy writing demonstrating how stability maps can assist in contractual agreements; SCM in early stage companies which are more prone to detrimental effects of delays due to lack of experience, elegant software utilities and large financial resources; analyzing stability and bullwhip in multi-echelon supply chains by expanding on the mathematical models studied; consideration of time-varying nature of multiple number of delays to account for various activities causing delays in the SC.