Research On Inventory Management Of One Product And Many Merchants Under Uncertain Demand Environment

Under the background of economic globalization,enterprises can not easily satisfy the ever-changing demand fluctuation only by using one single supply channel.Multi-sourcing replenishment policy emerges as the times require.Multi-sourcing replenishment policy is a kind of supply strategy widely adopted by enterprises.It refers to that enterprises usually use multiple suppliers to jointly fulfill the replenishment,and different suppliers possess different lead times and ordering costs.A special case is dual-sourcing policy,under which companies adopt a slow(cheaper)channel to replenish its basic inventory while turn to a faster(more expensive)channel in emergencies.Multiple suppliers can help diversify the risks of supply disruptions,also reduce the supplying cost and improve the service level.Due to its practical relevance,inventory systems with multiple suppliers have been studied for several decades.However,the dynamic programming formulation of the problem is in general intractable due to the curse of dimensionality,which stems from the lead time differences of multiple suppliers,thus the optimal policy for this system is still unknown.Indeed,even for the problem with only two uncapacitated suppliers,if the lead times of the two suppliers differ more than one period,the dynamic programming formulation of the problem is hard to solve due to the curse of dimensionality.Its optimal policy is complicated and depends on the vector of the outstanding orders.Therefore,a stream of research has focused on analyzing the structural properties of the optimal inventory policy and designing simple heuristic policies for dual-soucing systems.However,for inventory systems with more than two capacitated suppliers and general lead times,little is known about the optimal policy,and to the best of my knowledge,there is no any heuristic policy proposed or studied in the literature.In the first part of my thesis,I consider a periodic-review,infinite-horizon inventory system with multiple capacitated suppliers.These suppliers provide different lead times and incur different ordering costs.A supplier with a shorter lead time incurs a higher unit cost.The order quantity from each supplier is subject to a capacity constraint.The system faces random customer demands and the unsatisfied demands are fully backlogged.My purpose is to design an easy-to-implement and cost effective inventory replenishment policy for the system.I focus on a natural extension of the dual-index base-stock(DI-VSW)policy of Veeraraghavan and Scheller-Wolf(2008),which I call the multi-index base-stock policy(MIBS).For a system with m suppliers,the MIBS is determined by m parameters——m base-stock levels.Under the MIBS,I track different inventory positions for different suppliers based on their lead times.In particular,if there are only two suppliers,the MIBS reduces to the DI-VSW of Veeraraghavan and Scheller-Wolf(2008).By analyzing the dynamic recursions of the system under the MIBS,I show separability properties of the policy parameters of the MIBS and prove that given other(m-1)parameters,the base-stock level of the fastest supplier can be computed by a newsboy fractile.Based on this result and following Veeraraghavan and Scheller-Wolf(2008),I can employ a simulation-based procedure to evaluate and optimize the policy parameters for the MIBS.However,this optimization procedure can merely reduce the optimization of the policy parameters to(m-1)-dimension(from m-dimension),the procedure still involves an exhaustive search over(m-1)parameters.In Veeraraghavan and Scheller-Wolf(2008),where there are only two suppliers,this procedure is computationally efficient because only one-dimensional search is required.However,as the number of suppliers increases,the computational complexity of the procedure will increase exponentially.In order to tackle this problem,I provide a simple heuristic for computing the near-optimal policy parameters.The construction of the heuristic is based on the optimal base-stock levels of m separate single-supplier systems and the newsboy fractile of the fastest supplier I established in my system.Our numerical results show that this heuristic policy is quite efficient.In the meantime,I numerically investigate the value of the MIBS.I find that with more suppliers,the MIBS can bring significant cost savings compared with the DI-VSW.I also conduct numerical experiments with some of the parameters calibrated by real data to investigate how the MIBS performs compared with the extended policies of the single-index policy and the TBS policy.The results show that the MIBS is outstanding.An even more challenging difficulty faced by managers is that,given the rapidly changing business environment,it is hard for managers to predict with precision the demand distributions of their products.Consequently,robust optimization method enters into my mind.Robust optimization will handle demand by focusing on the worst-case scenario,with no requirements on distribution.In the second part of our thesis,I study a periodic-review,infinite-horizon stochastic inventory system with multiple supply sources(delivery modes).The supply sources have different supply lead times and unit costs,and the shorter the lead time of a supplier,the higher its unit cost.Random demands can be correlated over time,and demands are constrained in an uncertainty set in which the first and second moments are known,whereas their underlying distributions are not known.Then I implement robust optimization to model multi-period inventory problem,I want to look a certain number of time periods ahead to aggregate knowledge before decision making.This reactive scheduling method is rolling horizon approach,which solves iteratively the deterministic problem by moving forward the optimization horizon in every iteration.In every iteration,I solve a robust rolling horizon inventory problem with uncertain demands belonging to the corresponding prediction horizon(time window).Assuming the decision maker have solved the problem,only the first period's decision will be actually applied.Then the multi-period inventory problem is updated,thus the decision maker employ new information and repeat the procedure.So I study a robust rolling horizon model for the problem.Sun and Van Mieghem(2019)show that the robust optimal policy for sourcing from two supply sources is a dual-index,dual-base-stock policy that caps the slow order.I prove that the structure of the robust optimal policy for inventory management with more than two supply sources MI-cap is not a natural extension of the optimal policy for the system with two supply sources.Rather,I show that,under an appropriate condition on supply sources' unit costs and lead times,the robust optimal policy for sourcing from three or more suppliers is a base-stock policy for the fastest supply source and a capped base-stock policy for the slowest supply source,but for the middle supply sources,the optimal policy is a sort of "gap-of-base-stock" policy.I further study a special uncertainty set——central limit theorem(CLT)uncertainty set,and derive closed-form expressions of the robust optimal policy RCLT.In computational results,I compare an extension of VSW policy derived in our first part with robust optimal multiple sourcing policy to see the efficacy under conjunct actions of a middle supplier's structure and a "cap" on the slowest source.The numerical results show that our policy performs favorably compared with an uncapped policy in most cases.Besides,modern inventory models are generally based on two important assumptions:1.a risk-neutral setting of optimizing expected cost or profit;2.complete knowledge of distribution function of random demand in the model.However,not all decision makers are risk-neutral in reality,some are risk-averse.In the third part of our thesis,I consider the dual-sourcing inventory model with demand distribution unknown,and the decision maker is risk averse.I generalize the traditional risk-neutral finite-periods' dual-sourcing inventory model.Coherent risk measures arose from an axiomatic approach for quantifying risk.Common examples include mean-absolute deviation and conditional-value-at-risk.There exists a one-to-one correspondence between the risk averse formulations involving coherent risk measures and min-max type formulations.Hence I introduce the coherent risk measures instead of expectation for a unifying treatment of both two models.Next,I consider the inventory model of which the objective function is coherent risk measures.Our main work is:Firstly,I list the multi-period,risk-aversion,recursive dynamic programming for the dual-sourcing inventory model with lead times of 2 and 3.It can be extended to any two consecutive lead times' situation using similar method.Then,I use the transfer technique to reorganize the minimum cost function of state variables.By redefining the inventory level and two order-up-to levels,I realize dimension reduction of recursive dynamic programming.Due to the coherent risk measure's advantage of convexity conservation,I can use Karush(1959)lemma to decompose the convex function in the above recursive formulation into three different convex functions.Then,starting from the last stage,I use backward induction to prove that the optimal value function is a convex function of inventory level,and it has the feature of convexity conservation such that I can obtain the optimal ordering policy for each period.Based on that,I consider the situation with fixed ordering cost.I also deal with the minimum cost function through dimension reduction and convex function's decomposition.Then,with properties of k-convex function and a related lemma,I find that the optimal value function in the last period is a k-convex function of its inventory level.Applying the theorem given by Porteus(2002),I can prove that the optimal value function of each period is still k-convex,and the optimal ordering policy can be obtained.In order to get the optimal order quantity for each period,I begin to solve two unconstrained convex functions and set their optimal solutions as two order-up-to levels.By classifying and discussing the relationship between the inventory level and two order-up-to levels,I finally obtain the optimal ordering policy for the risk-averse dual-sourcing model,which is a dual-index base stock policy.For the case of adding fixed ordering cost,I can also get two order-up-to levels by solving a convex function and a k-convex function without constraints.Then the optimal ordering policy is obtained by comparing the size of inventory level and three indexes,which is a generalization of(s,S)policy.The policies' concise structures in both cases are consistent with that under the circumstance of risk-neutral setting.My work is of great significance to replenish the theory of multi-sourcing inventory system,and the corresponding optimal ordering policies are of great reference value to decision makers.