On the value of warehouse sharing in an inventory system for medical consumables

Yu, Yugang; Guo, Dandan; Zheng, Shengming

doi:10.1108/MSCRA-02-2025-0012

Purpose

This paper investigates the value of warehouse sharing when a third-party service provider (3PS) manages medical consumables inventory instead of hospitals. With the advancement of information technology, the 3PS commonly establish central warehouses to facilitate horizontal collaboration, replacing separate client-specific warehouses.

Design/methodology/approach

We examine warehouse sharing in a decentralized supply chain with multiple suppliers, multiple hospitals and a 3PS, considering two scenarios: no sharing and warehouse sharing. Based on the economic order quantity (EOQ) model, cost functions for suppliers and the 3PS are built in a continuous-time, constant-demand setting, where delivery intervals are sequentially determined.

Findings

Warehouse sharing results in more frequent shipments from suppliers than no sharing when the delivery frequency required by hospitals is high, but causes more frequent 3PS shipments than no sharing under intermediate hospital delivery frequency requirements. When requesting a low delivery frequency, both suppliers and 3PS may make less frequent deliveries than without sharing. Numerical experiments show that warehouse sharing benefits suppliers but may harm 3PS profitability due to misaligned scale effects. Furthermore, collaborating with larger hospitals that require more suppliers is more profitable for the 3PS.

Originality/value

This research analyzes warehouse sharing in a decentralized healthcare supply chain and identifies how hospital delivery requirements non-monotonically affect warehouse sharing's value, revealing a counterintuitive cost trade-off: warehouse sharing improves supplier efficiency but reduces 3PS profitability due to misaligned scale effects.

1. Introduction

With global health spending continuing to rise (Reynolds, 2022), the efficient supply of medical consumables has gradually become even more of an integral part of hospital healthcare activities (Leppälä et al., 2023). A well-maintained inventory control system helps patients receive medication without delay and prevents stock shortages at the hospital. Inadequate inventory can cause critical problems, such as deterioration of the patient's condition or development of antimicrobial resistance due to delays in treatment procedures, which can even be life-threatening to the patient. However, excessive inventory can lead to wastage and inefficient management of medical consumables and create high costs. As medical consumables are directly related to human life, managing inventory efficiently while maintaining satisfactory service has become a challenging task in the competitive business environment.

In recent years, with the development of supply chain management, some researchers have applied supply chain concepts to the medical field, such as zero inventory and Just In Time (JIT) purchasing; they have proposed optimization methods focusing on hospitals' procurement, inventory and distribution (Bijvank and Vis, 2012). In the 1990s, Japanese researchers introduced the “JIT” production model to the medical field and proposed the Supply-Processing-Distribution (SPD) management model (Liu et al., 2016), which reflects the idea of realizing integration management of procurement, inventory, distribution and consumption with the support of informationization. The SPD model has been successfully applied in several hospitals (Ito et al., 2006), such as the Anhui Provincial Hospital and the First Affiliated Hospital of Anhui Medical University in China.

In the SPD business model, hospitals do not manage the inventory of medical consumables; instead, they designate a third-party service provider (3PS) to manage the inventory on their behalf. The SPD model resembles the Vendor-Managed Inventory (VMI) model in the ownership of inventory. Under VMI, the supplier takes over inventory planning and replenishment decisions for the buyer (Kim, 2005; Kadiyala et al., 2020). In this sense, both SPD and VMI shift inventory management away from hospitals and inventory ownership is typically transferred to hospitals upon usage. However, the key difference between the two models is that suppliers can independently select a 3PS in the VMI model, whereas the 3PS is designated by hospitals in the SPD model. Moreover, in the SPD model, suppliers ship to the 3PS's warehouse, and then the 3PS, rather than the suppliers, decides when to deliver to the departments within the hospitals.

In the traditional SPD model, the 3PS operates several single warehouses (including staffing) within each hospital, and each single warehouse serves only that hospital. Accordingly, each supplier shipment is destined for a single hospital and is delivered to the corresponding single warehouse. With the development of information technology, the 3PS can instead operate a regional central warehouse outside hospitals to manage inventory for multiple hospitals in the same region. We refer to this configuration as warehouse sharing, in which supplier shipments consolidate consumables for multiple hospitals and are delivered to the central warehouse before being distributed to each hospital. Warehouse sharing reshapes the logistics trade-offs for both parties. For suppliers, consolidation can reduce the number of hospital-specific delivery trips, although meeting the aggregated demand at the central warehouse may require different replenishment frequencies. For the 3PS, operating a central warehouse can generate economies of scale in inventory and operations (e.g. labor and facility-related costs), but it may also increase outbound transportation costs due to additional distribution from the central warehouse to hospitals.

The practice of warehouse sharing in inventory systems for medical consumables motivates the investigation of the following questions: (1) In the scenario with and without warehouse sharing, what optimal policies should suppliers and 3PS adopt, and how do these policies differ across the two scenarios? (2) What value does warehouse sharing create for suppliers and for the 3PS, respectively, in terms of cost implications and the distribution of benefits between the two parties? (3) What are the boundary conditions for the value of warehouse sharing?

We try to answer the questions in a stylized, continuous-time, constant-demand setting. The suppliers and 3PS are responsible for both fixed transportation costs and linear holding costs. First, each supplier decides the delivery interval, and then the 3PS reacts to these to minimize costs and satisfy the hospitals' demands. We examine the impact of warehouse sharing by comparing the optimal cost in two settings: with and without warehouse sharing. We summarize the main findings as follows.

First, whether in the scenario with warehouse sharing or without, the optimal delivery intervals of suppliers are longer than those of the 3PS when the delivery frequency required by hospitals is high; otherwise, the two are likely to set the same delivery interval. The delivery frequency required by a hospital is determined by the storage capacity and the daily demand of each department within the hospital. Hospitals communicate their demands and the available storage space to the 3PS. Consequently, the 3PS schedules deliveries according to these requirements, ensuring a continuous supply without exceeding storage capacities. Warehouse sharing makes suppliers deliver more frequently compared to the scenario without sharing, if hospitals require a high delivery frequency. In contrast, when the delivery frequency required by hospitals is medium, it leads to more frequent deliveries for the 3PS central warehouse compared to having individual deliveries to single warehouses. Additionally, compared to the non-sharing scenario, warehouse sharing induces less frequent deliveries of both suppliers and 3PS when hospitals request low delivery frequency.

Second, investing in a central warehouse can be profitable for both the 3PS and suppliers. Warehouse sharing can enable suppliers and the 3PS to deliver more frequently to provide a higher service quality while reducing total costs. Due to warehouse sharing, suppliers can transport consumables for all hospitals to a central warehouse in a single shipment. However, the capacity of single shipments is limited, forcing suppliers to deliver more frequently to satisfy demands. Switching to a central warehouse sharing can significantly reduce the number of shipments for suppliers, so warehouse sharing still benefits suppliers. However, warehouse sharing may hurt the 3PS even though it can achieve horizontal collaboration. When hospitals require a high delivery frequency, potentially due to constrained storage capacities or high consumption rates in departments, 3PS schedules shipments to hospital departments at intervals that closely align with these requirements. This synchronization satisfies the hospital's requirements promptly and refines the 3PS's delivery schedules, thereby reducing the overall number of shipments required. However, when the 3PS maintains alignment with hospital requirements in both the sharing and non-sharing scenarios, warehouse sharing would increase the 3PS's total costs due to the higher transportation costs and holding costs of the central warehouse.

Finally, based on numerical experiments, we show that it is necessary for the 3PS to collaborate with larger hospitals that require more suppliers. For suppliers, it is interesting that even if there is no competition between suppliers for consumable sales, the value of warehouse sharing still decreases with too many suppliers. Intuitively, considering economies of scale, the higher the suppliers' transportation and holding costs, the better the warehouse sharing should be for suppliers. However, due to the interaction between the suppliers and 3PS caused by hospitals' requirements for delivery frequency, cost savings from warehouse sharing for suppliers are nonmonotonic in the transportation and holding costs. Furthermore, changes in the suppliers' costs will also lead to a nonmonotonic influence on the value of warehouse sharing for the 3PS due to the first-mover advantage of suppliers.

The rest of the paper is organized as follows. Section 2 reviews the literature. Section 3 lays out the model setup. After Section 4 presents the model without warehouse sharing as a benchmark, Section 5 presents the model with warehouse sharing and studies the impact of warehouse sharing on the optimal delivery intervals. The effect of warehouse sharing on cost for different parameter changes is investigated in Section 6, based on numerical experiments. Finally, Section 7 concludes the paper. All proofs are given in Appendix.

2. Literature review

Our work relates to three streams of literature: central warehouses, inventory management systems and healthcare operations management. In this section, we present an overview of these research areas and highlight our contributions.

This paper is first related to the literature on central warehouses. Roundy (1985) considers a distribution system with one warehouse and multiple retailers (OWMR). Each retailer has a constant demand, and no backlogs or shortages are allowed. The warehouse orders unlimited supplies from outside suppliers and replenishes the retailers' inventories; this is called the OWMR problem. They introduce two simple policies and prove that for any data set, the worst-case effectiveness was 94% and 98%, respectively. However, Arkin et al. (1989) prove the OMWR problem is NP-hard. Cha et al. (2008) focus on a two-stage supply chain consisting of one warehouse and n retailers and suggest a more flexible policy for joint replenishment and delivery scheduling. Beyond replenishment and delivery scheduling, Lei et al. (2024) consider a seasonal system in which a retailer prepositions inventory at a central warehouse and then makes periodic joint replenishment and pricing decisions for multiple stores under stochastic demand.

Although centralized strategies are widely recognized for their potential to minimize total systems costs due to economies of scale and optimized resource allocation, they may not be achievable due to various practical considerations (both technically and in terms of the level of trust that would be necessary; Ding and Kaminsky, 2020). Thus, extensive research efforts have been undertaken in decentralized logistics systems. Abdul-Jalbar et al. (2003) consider the retailer-driven OWMR problem in the centralized and decentralized settings, truncating the retailer's economic order quantity (EOQ) delivery intervals to rational numbers and applying the algorithm of Wagelmans et al. (1992) to find the supplier's optimal replenishment schedule. By numerical example, they show that as the number of retailers increases, so does the number of instances where decentralization is better. Baboli et al. (2008) investigate the inventory costs and transportation costs for a single-supplier-single-retailer model and compare centralized with decentralized decision-making. They find centralization can lead to savings for the warehouse and the whole system; however, it induces losses for the retailer. Ding and Kaminsky (2020) analytically calculate the costs of centralized and decentralized strategies in a multi-supplier, multi-retailer supply chain with a central warehouse. They show that the benefits of centralization are limited, and the decentralized system is almost as effective as the centralized system.

This study contributes to the literature on central warehouse sharing by extending the OWMR problem to a multi-supplier setting in the healthcare supply chain. In the traditional OWMR problem, the supplier and retailer pay inventory costs and decide optimal order intervals. However, in the research setting of our work, the suppliers and a third-party service provider (rather than the demand side, i.e. hospitals) determine the optimal delivery intervals.

This study is also related to the literature on vendor managed inventory (VMI). In VMI systems, suppliers determine the retailers' replenishment schedule and distribution operations through information sharing of retailers (Kadiyala et al., 2020). This information sharing and collaboration not only reduces the total costs but also improves the overall operational efficiency of the supply chain. Archetti and Speranza (2016) empirically show that VMI leads to total cost savings of 24% on average compared to retailer managed inventory (RMI) systems. Cetinkaya and Lee (2000) consider shipment consolidation in VMI systems facing stochastic demands of retailers in a specific region, and they calculate the optimal replenishment quantity and dispatch frequency simultaneously using a theoretical model for coordinating inventory and transport decisions. Mishra and Raghunathan (2004) show that VMI can intensify the competition between manufacturers of competing brands, thus benefiting retailers who stock two brands from two competing manufacturers. Chen et al. (2010) deal with the problem of coordinating a vertically separated distribution system under vendor-managed inventory and consignment arrangements, revealing that the profit gained by adopting VMI and consignment is significant. Ru et al. (2018) use the classical infinite perspective periodic review inventory model in a supply chain consisting of one supplier and one retailer. The study shows that retailers are more likely to benefit from VMI when their inventory holding costs are low, while manufacturers are more likely to benefit from VMI when their inventory holding costs are high. Kadiyala et al. (2020) propose a dynamic inventory mechanism under a VMI agreement. They find that VMI can result in significant profit improvements for both firms when the retailers have high market power and when the supplier has less knowledge than retailers about the end customer and market. Cui et al. (2023) study an inventory-routing problem under demand ambiguity in which a single supplier jointly determines replenishment quantities, delivery timing and routes. From a service-level perspective, they propose an adaptive policy framework and quantify stockout risk via a service-violation index that captures both the frequency and severity of violations. However, some empirical and anecdotal evidence also suggests that VMI-type agreements have proven difficult to maintain over multiple planning horizons due to a decline of trust (Kouvelis et al., 2006; Brinkhoff et al., 2015).

Our SPD setting relates to the VMI literature because both frameworks delegate replenishment and delivery planning from the demand side to an external decision maker, thereby inducing a similar inventory–transportation trade-off. Unlike standard VMI, in which the supplier typically assumes this role, our setting features a hospital-designated 3PS. Specifically, suppliers ship consumables to the 3PS's warehouse, and the 3PS subsequently determines the delivery timing to point-of-use departments within hospitals.

Our work also contributes to the literature on healthcare operations management. In related papers on healthcare operations management, several scholars have discussed logistics outsourcing in healthcare systems. Kim (2005) stresses that VMI enables hospitals to improve their procurement processes and inventory control of medicines, resulting in a reduction of total inventory by more than 30%, so the total supply chain cost can be decreased significantly. Tsui et al. (2008) state that VMI can help the hospital reduce the number of staff required, reduce stock holding, and improve customer service. In the setting of a hospital pharmacy trusteeship (HPT), which outsources hospital pharmacy services to a major pharmaceutical company using a VMI system, Liu et al. (2017) develop two medicine logistics planning models by using a time-space network approach for deterministic and stochastic variables. Shang et al. (2022) are motivated by a real-world healthcare supply case of a medical implant company and study a location-inventory-routing problem in the healthcare supply chain with a VMI system. They develop a mixed integer linear programming formulation that can save up to 12.6% of the cost. Bendavid et al. (2012) evaluate a radio-frequency identification (RFID)-enabled traceability system for managing consignment products based on a case study in a hospital operating theater. The results show that this traceability system, combined with a redesigned replenishment process, can improve service levels and reduce inventory. Using patient-level data, Jiang et al. (2023) quantify how geographic virtual pooling and data-driven advance scheduling can substantially reduce the fraction of patients exceeding wait-time targets while limiting incremental travel time and show that a majority of hospitals are not worse off financially. Dorgham et al. (2022) model logistics pooling among hospitals as a network design problem with multiple suppliers, warehouses and commodities under demand uncertainty and show how collaborative warehousing and distribution can reduce total logistics costs while accounting for data imprecision via a chance-constrained fuzzy optimization approach.

Several scholars have investigated the management of inventory in healthcare systems. Ahmadi et al. (2022) consider a healthcare system where a central warehouse is responsible for purchasing from suppliers and distributing the perishable medicines among multiple regional hospitals. They develop intelligent inventory management (IIM) approaches to provide near-optimal order quantities and remaining life distributions for products ordered for hospitals. Chan et al. (2024) study a periodic-review model in which elective-surgery booking information provides imperfect advance demand signals and quantify sizable inventory reductions without compromising service levels when such information is incorporated into ordering decisions. Inventory sharing via transshipment has also been studied in hospital contexts, particularly for highly perishable items. Zhang et al. (2023) analyze a two-location hospital system for platelet inventory management and derive structural results showing that simple age-based rules can determine transshipment directions; they further develop a myopic policy with strong performance that was implemented in practice and reduced outdates.

In this stream of literature, we investigate the value of horizontal collaboration in inventory systems for medical consumables. In our model, a 3PS manages the inventory of medical consumables. The suppliers first decide on the delivery intervals of shipping to the 3PS, and subsequently, the 3PS determines the intervals of distributing the inventory to the hospital departments.

3. Model setup

In this work, we consider a supply chain with n suppliers, m hospitals and a third-party service provider. For convenience, we now use “3PS” as a name, referring to the hospital's chosen third-party service provider. Each supplier provides a unique medical consumable to all the hospitals. The set $S = \{1, \dots, n\}$ represents suppliers, and $H = \{1, \dots, m\}$ represents hospitals. For ease of exposition, we use i (j) as the index associated with suppliers (hospitals).

Based on the actual data of the First Affiliated Hospital of Anhui Medical University (see Figure 1), we find that the demands of medical consumables are relatively stable. In particular, this pattern is most consistent with high-volume, routine and standardized consumables that are used broadly across departments as part of daily clinical operations, such as disposable gloves, gauze, syringes and infusion-related supplies. Motivated by this observation, we assume that each hospital faces a continuous and fixed demand for each medical consumable, which means that the demand rate d_ij for consumables from supplier i to hospital j is an exogenous constant. This assumption is commonly used in the literature (e.g. Roundy, 1985; Chen and Chen, 2005; Ding and Kaminsky, 2020). We acknowledge that the demand for some items can be more volatile, including low-volume or procedure-specific products, as well as periods affected by seasonality or unexpected shocks. Our analysis focuses on settings where demand can be approximated by an average rate. Extending the model to time-varying or stochastic demand is a meaningful direction for future research. Supplier i produces the unique medical consumable at a fixed rate $Σ_{j = 1}^{m} d_{i j}$ ⁠.

Figure 1

Line chart of hospital consumption quantity for seven medical consumable codes from July to September 2022.

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The figure is a time-series line chart. The x-axis lists dates from 2022-07-01 to 2022-09-15 at roughly biweekly intervals, and the y-axis reports consumption quantity (approximately 0–65,000). Seven lines represent consumable codes 1–7 as indicated by the legend. Overall, the relative ranking across codes is stable over time. Code 1 is consistently the highest series, remaining around the mid-50,000s to low-60,000s and ending with a noticeable uptick above 60,000. Code 2 is the second-highest series and shows moderate fluctuations around the mid-20,000s, with a visible local peak in early August and a dip around mid-August. Code 7 forms a middle tier, varying within roughly 18,000–22,000 and peaking around early September before easing slightly toward mid-September. Codes 3, 4, and 5 constitute lower-to-mid tiers, remaining comparatively stable. Code 3 stays around 5,000–7,000, Code 4 around 6,000–8,000, and Code 5 around 8,000–12,000 with a small rise near the end of the period. Code 6 is the lowest series and remains near 1,000 throughout with minimal variation.

Medical consumables consumption at a hospital from July 2022 to September 2022. Source(s): Authors' analysis based on hospital operational data

To investigate the value of warehouse sharing, we initially consider the scenario without warehouse sharing. In the absence of warehouse sharing, 3PS builds individual warehouses within each hospital to stock the consumables required by that specific hospital. The exclusive warehouse for hospital j is called single warehouse j in our work. Each supplier i first sets its delivery interval $T_{i j}^{s}$ to ship the consumables directly to the corresponding single warehouse and 3PS then sets the delivery interval $T_{j}^{h}$ for delivering the consumables to the hospital j departments.

Next, we explore the warehouse sharing scenario, in which 3PS establishes a central warehouse outside the hospitals to stock consumables for all hospitals it serves, achieving sharing through unified inventory management and delivery. With warehouse sharing, the decision-making timing is as follows. Each supplier i first determines the delivery interval $Γ_{i}^{s}$ of replenishing its inventory at the central warehouse, consolidates the consumables needed for all hospitals and delivers them to this central warehouse. Subsequently, 3PS decides the delivery interval $Γ_{j}^{h}$ for shipping medical consumables to the departments of hospital j. Figure 2 presents a detailed comparison of the no warehouse sharing and warehouse sharing scenarios. The symbol H_i denotes all the departments within hospital i.

Figure 2

Two diagrams comparing supply chain structures without and with warehouse sharing.

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The figure contains two diagrams labeled (a) and (b). Panel (a), “Without warehouse sharing,” shows a decentralized structure in which multiple suppliers provide medical consumables to multiple third-party service providers, and each service provider ships directly to its corresponding hospital. Panel (b), “With warehouse sharing,” shows a centralized structure in which multiple suppliers deliver to a shared central warehouse operated by the third-party service provider, and the central warehouse then distributes to multiple hospitals. The key difference is that panel (b) consolidates inventory and outbound flows at a single shared warehouse, whereas panel (a) keeps inventory and fulfillment separate across service providers.

Decentralized healthcare supply chain. Source(s): Authors' own creation

Whenever supplier i ships to 3PS, it incurs a fixed transportation cost $k_{i}^{s}$ ⁠. Without warehouse sharing, 3PS needs to deliver medical consumables from the single warehouse (within the hospital) to the requesting departments of that hospital. With warehouse sharing, 3PS needs to deliver medical consumables from the central warehouse to various departments of hospitals. The hospitals' demands and storage space limitations define the minimum frequency of deliveries, but 3PS can flexibly choose a delivery interval that may align with or be more frequent than the hospitals' requirements to optimize overall costs. For 3PS, the fixed transportation cost for single warehouses to deliver to departments is y, while the fixed delivery cost from the central warehouse to departments is c. Because a single warehouse is located in each hospital, while the central warehouse is built outside all hospitals, the shorter travel distance for single warehouses means that c > y. Linear holding costs are incurred both at the suppliers and 3PS. The inventory holding cost at supplier i is $h_{i}^{s}$ ⁠. The holding cost rates at single warehouses and the central warehouse are denoted as hⁿ and h^w, respectively. Consistent with observations in practice, the holding cost at the central warehouse is higher than that at a single warehouse, that is, hⁿ < h^w.

In practice, the demand for medical consumables must be met without backlogging. Hence, whenever a supplier delivers to a 3PS warehouse, 3PS must minimize inventory held while ensuring enough inventory to cover the demands of all related hospitals until the next delivery. However, since deliveries might not align perfectly, 3PS may deliver during a specific supplier interval to meet demand in the subsequent supplier interval. This means that the demand from hospitals during a particular supplier interval might exceed the actual demand faced by the hospitals in that interval. As a result, the optimal delivery interval decisions can become complex and nonstationary, even if the demand for each medical consumable is known and stable. To simplify the analysis of our model and get analytical solutions, we assume that each supplier has the same fixed transportation cost and holding cost, i.e. $k_{1}^{s} = k_{2}^{s} = \dots = k_{n}^{s}$ and $h_{1}^{s} = h_{2}^{s} = \dots = h_{n}^{s}$ ⁠. The symmetry assumption also enables us to capture the effect of warehouse sharing without getting into the issue of heterogeneity among suppliers, ensuring that our findings are solely driven by warehouse sharing rather than other factors like asymmetries among suppliers.

Due to the storage capacity constraints of hospital departments, 3PS cannot ship a large amount of medical consumables per delivery from any warehouse to departments. We use an exogenous variable Q_j to describe the inventory capacity of the hospital’s j departments for storing medical consumables after they have been delivered from the single warehouse or the central warehouse.

To avoid excessive inventory stock at the hospital's departments, 3PS must schedule deliveries to them in both scenarios, whether from single warehouses within the hospital or from a central warehouse outside the hospital. This means that the delivery interval should not exceed ${\bar{T}}_{j}$ where ${\bar{T}}_{j} = \frac{Q_{j}}{\sum_{i \in S} d_{i j}}$ ⁠. Similarly, we assume the value of ${\bar{T}}_{j}$ is the same at each hospital, and we use $\bar{T}$ to denote ${\bar{T}}_{j}$ ⁠, i.e. $\bar{T} = {\bar{T}}_{1} = {\bar{T}}_{2} = \dots = {\bar{T}}_{m}$ ⁠. Due to the travel distance constraint, the delivery intervals required by the supplier cannot be very low. We introduce an exogenous variable $\underline{T}$ to describe the lower bound of suppliers' delivery intervals such that $\bar{T} \geq \underline{T}$ ⁠. Table 1 summarizes the notation used in this paper.

Table 1

Notation

set and parameters
$S = \{1, \dots, n\}$	Set of suppliers
$H = \{1, \dots, m\}$	Set of hospitals
d_ij	Demand for consumables from supplier i at hospital j
$k_{i}^{s}$	Supplier i's fixed delivery cost to the central (single) warehouse
y	3PS's fixed delivery cost from single warehouse j to departments of hospital j
c	3PS's fixed delivery cost from the central warehouse to departments of hospital j
Q_j	Inventory capacity of hospital j departments
$h_{i}^{s}$	Holding cost rate at supplier i
hⁿ	Holding cost rate of the consumables i at single warehouse j
h^w	Holding cost rate of the consumables i at the central warehouse
$\bar{T}$	Upper bound on 3PS's delivery interval
$\underline{T}$	Lower bound on suppliers' delivery interval
$T_{i j}^{s}$	Supplier i's delivery interval of replenishing inventory at a single warehouse j
$Γ_{i}^{s}$	Supplier i's delivery interval of replenishing inventory at the central warehouse
$T_{j}^{h}$	3PS's delivery interval from single warehouse j to hospital j departments
$Γ_{j}^{h}$	3PS's delivery interval from the central warehouse to hospital j departments
$L_{j} ({\tilde{L}}_{j})$	Suppliers who deliver less frequently than 3PS without(with)warehouse sharing
$G_{j} ({\tilde{G}}_{j})$	Suppliers who deliver more frequently than 3PS without(with)warehouse sharing
$E_{j} ({\tilde{E}}_{j})$	Suppliers who deliver as frequently as 3PS without (with) warehouse sharing
$C_{i}^{S N} (C_{i}^{S W})$	Supplier i's total costs without (with) warehouse sharing
C^PN(C^PW)	3PS's total costs without (with) warehouse sharing

set and parameters
$S = \{1, \dots, n\}$	Set of suppliers
$H = \{1, \dots, m\}$	Set of hospitals
d_ij	Demand for consumables from supplier i at hospital j
$k_{i}^{s}$	Supplier i's fixed delivery cost to the central (single) warehouse
y	3PS's fixed delivery cost from single warehouse j to departments of hospital j
c	3PS's fixed delivery cost from the central warehouse to departments of hospital j
Q_j	Inventory capacity of hospital j departments
$h_{i}^{s}$	Holding cost rate at supplier i
hⁿ	Holding cost rate of the consumables i at single warehouse j
h^w	Holding cost rate of the consumables i at the central warehouse
$\bar{T}$	Upper bound on 3PS's delivery interval
$\underline{T}$	Lower bound on suppliers' delivery interval
$T_{i j}^{s}$	Supplier i's delivery interval of replenishing inventory at a single warehouse j
$Γ_{i}^{s}$	Supplier i's delivery interval of replenishing inventory at the central warehouse
$T_{j}^{h}$	3PS's delivery interval from single warehouse j to hospital j departments
$Γ_{j}^{h}$	3PS's delivery interval from the central warehouse to hospital j departments
$L_{j} ({\tilde{L}}_{j})$	Suppliers who deliver less frequently than 3PS without(with)warehouse sharing
$G_{j} ({\tilde{G}}_{j})$	Suppliers who deliver more frequently than 3PS without(with)warehouse sharing
$E_{j} ({\tilde{E}}_{j})$	Suppliers who deliver as frequently as 3PS without (with) warehouse sharing
$C_{i}^{S N} (C_{i}^{S W})$	Supplier i's total costs without (with) warehouse sharing
C^PN(C^PW)	3PS's total costs without (with) warehouse sharing

Source(s): Authors' own creation

4. Benchmark: No warehouse sharing

This section considers the scenario without warehouse sharing as a benchmark. We represent the delivery intervals of suppliers as $T^{s} = (T_{11}^{s}, T_{12}^{s}, \dots, T_{n m}^{s})$ ⁠, where $T_{i j}^{s}$ is the delivery interval from supplier i intended to hospital j. Similarly, $T^{h} = (T_{1}^{h}, T_{2}^{h}, \dots, T_{m}^{h})$ captures delivery intervals of single warehouses.

To calculate the expected cost, we decompose inventory by supplier and intended hospital and divide suppliers into sets as follows.

L_{j} = \{i \in S : T_{i j}^{s} > T_{j}^{h}\}, E_{j} = \{i \in S : T_{i j}^{s} = T_{j}^{h}\}, G_{j} = \{i \in S : T_{i j}^{s} < T_{j}^{h}\} .

The cost function for supplier i can be formulated as follows:

\begin{aligned} \min C_{i}^{S N} = & \sum_{j \in H} [\frac{k_{i}^{s}}{\max (T_{i j}^{s}, T_{j}^{h})} + \sum_{i \in L_{j}} H_{i j}^{L S} (T_{i j}^{s}, T_{j}^{h}) + \sum_{i \in E_{j}} H_{i j}^{E S} (T_{i j}^{s}, T_{j}^{h}) + \sum_{i \in G_{j}} H_{i j}^{G S} (T_{i j}^{s}, T_{j}^{h})], \\ s.t. \bar{T} \geq T_{j}^{h} > 0, T_{i j}^{s} \geq \underline{T} > 0, \forall i \in S, j \in H \end{aligned}

(1)

We use the superscript “S” to denote suppliers and “N” to denote the case without warehouse sharing. In Equation 1, the first term is the fixed transportation costs from supplier i to single warehouse j. Note that suppliers only ship inventory to single warehouses for cycles when single warehouses of 3PS ship inventory to departments, so the actual number of deliveries from supplier i within a cycle depends on $\max (T_{i j}^{s}, T_{j}^{h})$ ⁠. For holding costs of suppliers, we consider three cases. We let $H_{i j}^{E S} (T_{i j}^{s}, T_{j}^{h})$ be the average holding cost per unit time for medical consumable i ultimately intended for single warehouse j if $T_{i j}^{s} = T_{j}^{h}$ ⁠; we let $H_{i j}^{G S} (T_{i j}^{s}, T_{j}^{h})$ be the average holding cost if $T_{i j}^{s} < T_{j}^{h}$ ⁠; and we let $H_{i j}^{L S} (T_{i j}^{s}, T_{j}^{h})$ be the average holding cost if $T_{i j}^{s} > T_{j}^{h}$ ⁠.

Similarly, the cost function for the 3PS is:

\begin{aligned} \min C^{P N} = & \sum_{j \in H} [\frac{y}{T_{j}^{h}} + \sum_{i \in L_{j}} H_{i j}^{L H} (T_{i j}^{s}, T_{j}^{h}) + \sum_{i \in E_{j}} H_{i j}^{E H} (T_{i j}^{s}, T_{j}^{h}) + \sum_{i \in G_{j}} H_{i j}^{G H} (T_{i j}^{s}, T_{j}^{h})], \\ s.t. \bar{T} \geq T_{j}^{h} > 0, T_{i j}^{s} \geq \underline{T} > 0, \forall i \in S, j \in H \end{aligned}

(2)

Similarly, we use the superscript “P” to denote 3PS and “N“ to denote the case without warehouse sharing. In Equation 2, the first term is the transportation costs from single warehouse j (within hospital j) to the departments of hospital j. Similarly, $\sum_{i \in L_{j}} H_{i j}^{L H} (T_{i j}^{s}, T_{j}^{h})$ ⁠, $\sum_{i \in E_{j}} H_{i j}^{E H} (T_{i j}^{s}, T_{j}^{h})$ and $\sum_{i \in G_{j}} H_{i j}^{G H} (T_{i j}^{s}, T_{j}^{h})$ denote the inventory holding cost at single warehouse j when $T_{i j}^{s} > T_{j}^{h}$ ⁠, $T_{i j}^{s} = T_{j}^{h}$ and $T_{i j}^{s} < T_{j}^{h}$ ⁠, respectively. Each single warehouse holds medical consumables from all suppliers.

We characterize the inventory holding cost for the suppliers and 3PS for three cases: $T_{i j}^{s} = T_{j}^{h}$ ⁠, $T_{i j}^{s} < T_{j}^{h}$ and $T_{i j}^{s} > T_{j}^{h}$ ⁠. Let $\frac{T_{j}^{h}}{T_{i j}^{s}} = {\tilde{b}}_{i j} + {\tilde{a}}_{i j}$ ⁠, where ${\tilde{b}}_{i j}$ is the integer part of $\frac{T_{j}^{h}}{T_{i j}^{s}}$ and ${\tilde{a}}_{i j}$ is the fractional part. ${\tilde{a}}_{i j}$ is rational, we further let ${\tilde{a}}_{i j} = \frac{{\tilde{p}}_{i j}}{{\tilde{q}}_{i j}}$ ⁠, where integers ${\tilde{p}}_{i j}$ and ${\tilde{q}}_{i j}$ are coprime. Similarly, let $\frac{T_{i j}^{s}}{T_{j}^{h}} = b_{i j} + a_{i j}$ and define $a_{i j} = \frac{p_{i j}}{q_{i j}}$ correspondingly if a_ij is rational.

We assume that for each supplier and each single warehouse, the first delivery occurs at the start of time 0. The average holding costs associated with the inventory of consumables from supplier i that is ultimately intended for hospital j depend on the relationship between $T_{i j}^{s}$ and $T_{j}^{h}$ ⁠. We consider the three possible cases:

Case 1) $T_{i j}^{s} = T_{j}^{h}, \forall j, i \in E_{j}$

Because the first delivery for both occurs at time 0, $T_{i j}^{s} = T_{j}^{h}$ implies that supplier i and single warehouse j always deliver at the same frequency, it follows that no inventory is held at single warehouse j, and 3PS has no inventory holding costs. The inventory levels at suppliers follow a standard sawtooth pattern with a delivery interval of $T_{i j}^{s}$ ⁠. That is,

H_{i j}^{E H} (T_{i j}^{s}, T_{j}^{h}) = 0,

(3)

H_{i j}^{E S} (T_{i j}^{s}, T_{j}^{h}) = \frac{1}{2} T_{i j}^{s} d_{i j} h_{i}^{s} .

(4)

Case 2) $T_{i j}^{s} < T_{j}^{h}, \forall j, i \in G_{j}$

Note that $T_{i j}^{s} < T_{j}^{h}$ implies that supplier i delivers more frequently than single warehouse j. As mentioned before, suppliers only transfer inventory during cycles when the single warehouse delivers inventory to the hospital departments. The inventory levels when $T_{i j}^{s} < T_{j}^{h}$ are illustrated in Figure 3. As for 3PS, the inventory level only increases to $d_{i j} T_{j}^{h}$ after supplier i has shipped the inventory to single warehouses. Then, the inventory level sustains until a single warehouse j delivers inventory to the hospital departments. The inventory level is zero at other times. To evaluate $H_{i j}^{G H} (T_{i j}^{s}, T_{j}^{h})$ and $H_{i j}^{G S} (T_{i j}^{s}, T_{j}^{h})$ ⁠, we introduce a technical lemma as follows.

Figure 3

Two side-by-side line charts illustrating inventory dynamics over time, with a sawtooth pattern at suppliers and a stepwise pattern at a single warehouse.

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The figure contains two line charts arranged horizontally with a common time axis. The left chart plots inventory level at suppliers over time and shows a repeating sawtooth pattern: inventory increases during replenishment and then decreases steadily as it is consumed, followed by an instantaneous replenishment that resets inventory to a higher level. The right chart plots inventory level at a single warehouse over time and shows a stepwise pattern: inventory remains at a high level, drops as it is depleted, stays near zero between replenishments, and then jumps back to the target level when replenished. A legend indicates the inventory level curve and highlights the supplier delivery interval and the third-party service provider delivery interval using distinct markers. Overall, the two panels contrast continuous consumption with periodic replenishment at suppliers versus batched replenishment and stockout periods at the warehouse.

Inventory levels when $T_{i j}^{s} < T_{j}^{h}$ ⁠. Source(s): Authors' own creation

Lemma 1.

Let Δ_ij(k) be the time between the kth delivery of single warehouse j and supplier i's last replenishment. If supplier i replenishes inventory simultaneously with the kth delivery from single warehouse j to the hospital's departments, we let Δ_ij(k) = 0. Then, the long-run average of Δ_ij(k) is

\frac{1}{N} \sum_{k = 1}^{N} Δ_{i j} (k) \overset{N \to \infty}{\to} \{\begin{cases} \frac{T_{i j}^{s} ({\tilde{q}}_{i j} - 1)}{2 {\tilde{q}}_{i j}} & if \frac{T_{j}^{h}}{T_{i j}^{s}} \in Q, \\ \frac{T_{i j}^{s}}{2} & otherwise . \end{cases}

Here, Δ_ij(k) denotes the average holding time of consumables from supplier i at single warehouse j, which is mainly affected by the relative relationship of both delivery intervals $(1 - \frac{1}{{\tilde{q}}_{i j}}) T_{i j}^{s}$ ⁠. Thus, we can calculate the average holding cost per unit time at supplier i and single warehouse j based on Lemma 1.

Theorem 1.

When $T_{i j}^{s} < T_{j}^{h}, \forall j, i \in G_{j}$ , the average holding costs per unit time for single warehouse j and supplier i are

H_{i j}^{G H} (T_{i j}^{s}, T_{j}^{h}) = \{\begin{cases} \frac{1}{2} (1 - \frac{1}{{\tilde{q}}_{i j}}) T_{i j}^{s} d_{i j} h^{n} & if \frac{T_{j}^{h}}{T_{i j}^{s}} \in Q, \\ \frac{1}{2} T_{i j}^{s} d_{i j} h^{n} & otherwise, \end{cases}

(5)

H_{i j}^{G S} (T_{i j}^{s}, T_{j}^{h}) = \{\begin{cases} \frac{1}{2} ((1 - \frac{1}{{\tilde{q}}_{i j}}) T_{i j}^{s} + T_{j}^{h}) d_{i j} h_{i j}^{s} & if \frac{T_{j}^{h}}{T_{i j}^{s}} \in Q, \\ \frac{1}{2} (T_{i j}^{s} + T_{j}^{h}) d_{i j} h_{i j}^{s} & otherwise . \end{cases}

(6)

Case 3) $T_{i j}^{s} > T_{j}^{h}, \forall j, i \in L_{j}$

When $T_{i j}^{s} > T_{j}^{h}, \forall j, i \in L_{j}$ ⁠, single warehouse j delivers more frequently than 3PS. As we see from Figure 4, supplier i faces b_ij or b_ij + 1 delivers from single warehouse j to the hospital within each delivery interval. Each time a supplier ships to a single warehouse, 3PS needs to ensure that there is enough inventory to cover demands for hospitals until the supplier's next delivery. That means that supplier i delivers amounts of $b_{i j} d_{i j} T_{j}^{h}$ or $(b_{i j} + 1) d_{i j} T_{j}^{h}$ to a single warehouse j per shipment. Then, the inventory level in single warehouse j falls in steps of $d_{i j} T_{j}^{h}$ ⁠. This observation allows us to exactly characterize $H_{i j}^{L H} (T_{i j}^{s}, T_{j}^{h})$ and $H_{i j}^{L S} (T_{i j}^{s}, T_{j}^{h})$ ⁠, which are shown in Theorem 2.

Figure 4

Two side-by-side line charts showing inventory dynamics over time with a sawtooth pattern at suppliers and a stepwise pattern at a single warehouse.

View large Download slide

The figure has two panels arranged horizontally with a common time axis. The caption indicates the case where the supplier delivery interval is longer than the warehouse replenishment interval. In the left panel for suppliers, the inventory level follows a repeating sawtooth pattern: inventory increases during replenishment to a high level and then declines steadily with consumption before the next replenishment. In the right panel for a single warehouse, the inventory level changes in a stepwise manner: it drops as inventory is depleted, may remain low or at zero between replenishments, and then jumps back upward when replenished. A legend distinguishes the inventory level curve and uses different markers to denote the supplier delivery interval and the third-party service provider delivery interval.

Inventory levels when $T_{i j}^{s} > T_{j}^{h}$ ⁠. Source(s): Authors' own creation

Theorem 2.

When $T_{i j}^{s} > T_{j}^{h}, \forall j, i \in L_{j}$ ⁠, the average holding costs per unit time for single warehouse j and supplier i are

H_{i j}^{L H} (T_{i j}^{s}, T_{j}^{h}) = \{\begin{cases} \frac{1}{2} (T_{i j}^{s} - \frac{1}{q_{i j}} T_{j}^{h}) d_{i j} h^{n} & if \frac{T_{i j}^{s}}{T_{j}^{h}} \in Q, \\ \frac{1}{2} T_{i j}^{s} d_{i j} h^{n} & otherwise, \end{cases}

(7)

H_{i j}^{L S} (T_{i j}^{s}, T_{j}^{h}) = \{\begin{cases} \frac{1}{2} (T_{i j}^{s} + (1 - \frac{1}{q_{i j}}) T_{j}^{h}) d_{i j} h_{i}^{s} & if \frac{T_{i j}^{s}}{T_{j}^{h}} \in Q, \\ \frac{1}{2} (T_{i j}^{s} + T_{j}^{h}) d_{i j} h_{i}^{s} & otherwise . \end{cases}

(8)

We use backward induction to obtain optimal solutions. Firstly, given the delivery interval $T_{i j}^{s}$ from supplier i to single warehouse j, 3PS chooses the delivery interval from the single warehouse to minimize total cost.

First, if $\bar{T} < T_{i j}^{s}$ ⁠, the upper bound of 3PS's delivery interval is shorter than the delivery interval of suppliers, which means the optimal delivery interval of single warehouses must be shorter than that of suppliers. In such a circumstance, the inventory levels are shown in Figure 4, and the corresponding inventory holding cost is expressed as Equation (8). Thus, we can substitute Equation (8) into Equation (2) to obtain the optimal delivery interval of single warehouse j and $T_{j}^{h *} = \bar{T}$ ⁠.

Then, if $\bar{T} \geq T_{i j}^{s}$ ⁠, the relationship of delivery intervals between single warehouses and suppliers has three possible cases. We can substitute Equations 3, 5 and 7 into Equation (2), respectively, and then compare the results to obtain the optimal delivery interval of single warehouses. We define $F (T_{i j}^{s}, \bar{T})$ as follows:

F (T_{i j}^{s}, \bar{T}) = \{\begin{cases} \frac{y T_{i j}^{s}}{y - \sum_{i \in S} \frac{1}{2} d_{i j} h^{n} (1 - \frac{1}{q_{i j}^{F N}}) {T_{i j}^{s}}^{2}} & if \frac{\bar{T}}{T_{i j}^{s}} \in Q, \\ \frac{y T_{i j}^{s}}{y - \sum_{i \in S} \frac{1}{2} d_{i j} h^{n} {T_{i j}^{s}}^{2}} & otherwise, \end{cases} where \frac{\bar{T}}{T_{i j}^{s}} = b_{i j}^{F N} + \frac{p_{i j}^{F N}}{q_{i j}^{F N}} .

When $F (T_{i j}^{s}, \bar{T}) \leq 0$ ⁠, the optimal delivery interval of single warehouse j is: $T_{j}^{h *} = T_{i j}^{s}$ ⁠. Otherwise, the optimal delivery interval of single warehouse j is:

T_{j}^{h *} = \{\begin{cases} \bar{T} & if F (T_{i j}^{s}, \bar{T}) \leq \bar{T}, \\ T_{i j}^{s} & if \bar{T} < F (T_{i j}^{s}, \bar{T}) . \end{cases}

Building upon single warehouse j's optimal delivery interval $T_{j}^{h *}$ ⁠, we move to examine the optimal delivery intervals of suppliers. We analyze the locally optimal delivery interval from supplier i intended for single warehouse j when $T_{i j}^{s} = T_{j}^{h}$ and define it as ${\hat{T}}_{i j}^{S} = \sqrt{\frac{2 k_{i}^{s}}{d_{i j} h_{i}^{s}}}$ ⁠. The optimal delivery intervals of supplier i and single warehouse j are characterized as follows:

Theorem 3.

Without warehouse sharing, there exists three thresholds α₀, α₁, and α₂ such that,

when $\underline{T} < \bar{T} \leq {\hat{T}}_{i j}^{S}$ , the optimal delivery intervals are

T_{i j}^{s *} = \{\begin{cases} {\hat{T}}_{i j}^{S} & if \bar{T} \leq α_{0}, \\ \bar{T} & if α_{0} < \bar{T}, \end{cases} and T_{j}^{h *} = \bar{T};

(9)

when $\underline{T} < {\hat{T}}_{i j}^{S} < \bar{T}$ , the optimal delivery intervals are

T_{i j}^{s *} = \{\begin{cases} \bar{T} & if {\hat{T}}_{i j}^{S} < α_{1} and F ({\hat{T}}_{i j}^{S}, \bar{T}) < \bar{T}, \\ {\hat{T}}_{i j}^{S} & otherwise, \end{cases} and T_{j}^{h *} = T_{i j}^{s *};

(10)

when ${\hat{T}}_{i j}^{S} \leq \underline{T} < \bar{T}$ , the optimal delivery intervals are

T_{i j}^{s *} = \{\begin{cases} \bar{T} & if \underline{T} < α_{2} and F (\underline{T}, \bar{T}) < \bar{T}, \\ \underline{T} & otherwise, \end{cases} and T_{j}^{h *} = T_{i j}^{s *} .

(11)

Theorem 3 shows that the supplier's deliveries are not more frequent than those of 3PS without warehouse sharing. Recall that the demand in our model is deterministic. Each time a supplier makes a shipment to 3PS's warehouse, it can accurately determine and ship the exact amount that 3PS will demand until the supplier's next shipment. Suppliers have the first-mover advantage and will ship less frequently to avoid high delivery costs.

5. Warehouse sharing

In this section, we proceed to examine the more involved scenario where 3PS establishes a central warehouse outside hospitals. Trucks carrying multiple hospitals' medical consumables from each supplier individually ship to the central warehouse, and then trucks containing consumables from multiple suppliers will be shipped by 3PS from the central warehouse to each hospital's departments. We use new notation to represent the supplier's delivery intervals as $Γ_{i}^{s} = (Γ_{1}^{s}, Γ_{2}^{s}, \dots Γ_{n}^{s})$ ⁠, and the central warehouse's delivery intervals as $Γ_{j}^{h} = (Γ_{1}^{h}, Γ_{2}^{h}, \dots Γ_{m}^{h})$ ⁠. Similar to the previous analysis in the scenario without warehouse sharing, we divide suppliers into sets to calculate the expected cost.

{\tilde{L}}_{j} = \{i \in S : Γ_{i}^{s} > Γ_{j}^{h}\}, {\tilde{E}}_{j} = \{i \in S : Γ_{i}^{s} = Γ_{j}^{h}\}, {\tilde{G}}_{j} = \{i \in S : Γ_{i}^{s} < Γ_{j}^{h}\} .

The cost function for supplier i is:

\begin{aligned} \min C_{i}^{S W} = & \frac{k_{i}^{s}}{\max (Γ_{i}^{s}, Γ_{j}^{h})} + \sum_{j \in H} [\sum_{i \in {\tilde{L}}_{j}} H_{i j}^{L S} (Γ_{i}^{s}, Γ_{j}^{h}) + \sum_{i \in {\tilde{E}}_{j}} H_{i j}^{E S} (Γ_{i}^{s}, Γ_{j}^{h}) + \sum_{i \in {\tilde{G}}_{j}} H_{i j}^{G S} (Γ_{i}^{s}, Γ_{j}^{h})], \\ s.t. \bar{T} \geq Γ_{j}^{h} > 0, Γ_{i}^{s} \geq \underline{T} > 0, \forall i \in S, j \in H \end{aligned}

(12)

The cost function for 3PS is:

\begin{aligned} \min C^{P W} = & \sum_{j \in H} [\frac{c}{Γ_{j}^{h}} + H_{j}^{L H} (Γ_{i}^{s}, Γ_{j}^{h}) + H_{j}^{E H} (Γ_{i}^{s}, Γ_{j}^{h}) + H_{j}^{G H} (Γ_{i}^{s}, Γ_{j}^{h})], \\ s.t. \bar{T} \geq Γ_{j}^{h} > 0, Γ_{i}^{s} \geq \underline{T} > 0, \forall i \in S, j \in H \end{aligned}

(13)

The cost function structure of suppliers and 3PS are similar to those in the case without sharing. The superscript “S“ and “P“ represent suppliers and 3PS, respectively, while the superscript “W“ represents the case of warehouse sharing. The first term reflects fixed transportation costs, and the other three terms represent the holding costs for the three cases. As before, we assume the first delivery occurs at the start of time 0. Then we can obtain the average holding costs of suppliers and 3PS for three cases: $Γ_{i}^{s} = Γ_{j}^{h}$ ⁠, $Γ_{i}^{s} < Γ_{j}^{h}$ and $Γ_{i}^{s} > Γ_{j}^{h}$ ⁠. These are stated in the following theorem.

Theorem 4.

With warehouse sharing, the average holding costs at the suppliers and 3PS are as follows.

When $Γ_{i}^{s} = Γ_{j}^{h}, \forall j, i \in {\tilde{E}}_{j}$ , the average holding costs per unit time are:

H_{j}^{E H} (Γ_{i}^{s}, Γ_{j}^{h}) = 0,

(14)

H_{i j}^{E S} (Γ_{i}^{s}, Γ_{j}^{h}) = \frac{1}{2} Γ_{i}^{s} d_{i j} h_{i}^{s} .

(15)

When $Γ_{i}^{s} < Γ_{j}^{h}, \forall j, i \in {\tilde{G}}_{j}$ , the average holding costs per unit time are:

H_{j}^{G H} (Γ_{i}^{s}, Γ_{j}^{h}) = \{\begin{cases} \sum_{i \in {\tilde{G}}_{j}} \frac{1}{2} (1 - \frac{1}{q_{i j}}) Γ_{i}^{s} d_{i j} h^{w} & if \frac{Γ_{j}^{h}}{Γ_{i}^{s}} \in Q, \\ \sum_{i \in {\tilde{G}}_{j}} \frac{1}{2} Γ_{i}^{s} d_{i j} h^{w} & otherwise, \end{cases}

(16)

H_{i j}^{G S} (Γ_{i}^{s}, Γ_{j}^{h}) = \{\begin{cases} \frac{1}{2} ((1 - \frac{1}{q_{i j}}) Γ_{i}^{s} + Γ_{j}^{h}) d_{i j} h_{i}^{s} & if \frac{Γ_{j}^{h}}{Γ_{i}^{s}} \in Q, \\ \frac{1}{2} (Γ_{i}^{s} + Γ_{j}^{h}) d_{i j} h_{i}^{s} & otherwise . \end{cases}

(17)

If $Γ_{i}^{s} > Γ_{j}^{h}, \forall j, i \in {\tilde{L}}_{j}$ , the average holding costs per unit time are:

H_{j}^{L H} (Γ_{i}^{s}, Γ_{j}^{h}) = \{\begin{cases} \sum_{i \in {\tilde{L}}_{j}} \frac{1}{2} (Γ_{i}^{s} - \frac{Γ_{j}^{h}}{q_{i j}}) d_{i j} h^{w} & if \frac{Γ_{i}^{s}}{Γ_{j}^{h}} \in Q, \\ \sum_{i \in {\tilde{L}}_{j}} \frac{1}{2} Γ_{i}^{s} d_{i j} h^{w} & otherwise, \end{cases}

(18)

H_{i j}^{L S} (Γ_{i}^{s}, Γ_{j}^{h}) = \{\begin{cases} \frac{1}{2} (Γ_{i}^{s} + (1 - \frac{1}{q_{i j}}) Γ_{j}^{h}) d_{i j} h_{i}^{s} & if \frac{Γ_{i}^{s}}{Γ_{j}^{h}} \in Q, \\ \frac{1}{2} (Γ_{i}^{s} + Γ_{j}^{h}) d_{i j} h_{i}^{s} & otherwise . \end{cases}

(19)

Similar to Section 4, we use backward induction to obtain the optimal solution. Given the delivery interval $Γ_{i}^{s}$ of supplier i, we can discuss $\bar{T}$ and $Γ_{i}^{s}$ ⁠.

The total costs of 3PS are decreasing in the delivery interval $Γ_{j}^{h}$ ⁠. Considering the delivery frequency required by hospitals, the optimal delivery interval of 3PS to hospital j could only be $Γ_{j}^{h *} = \bar{T}$ when $\bar{T} \leq Γ_{i}^{s}$ ⁠. Otherwise, it can be $Γ_{i}^{s}$ or $\bar{T}$ ⁠. The optimal interval critically depends on the threshold $F (Γ_{i}^{s}, \bar{T})$ ⁠:

F (Γ_{i}^{s}, \bar{T}) = \{\begin{cases} \frac{c Γ_{i}^{s}}{c - \sum_{i \in S} \frac{1}{2} d_{i j} h^{w} (1 - \frac{1}{q_{i j}^{F S}}) {Γ_{i}^{s}}^{2}} & if \frac{\bar{T}}{Γ_{i}^{s}} \in Q, \\ \frac{v_{j} Γ_{i}^{s}}{y - \sum_{i \in S} \frac{1}{2} d_{i j} h^{w} {Γ_{i}^{s}}^{2}} & otherwise, \end{cases} where \frac{\bar{T}}{Γ_{i}^{s}} = b_{i j}^{F S} + \frac{p_{i j}^{F S}}{q_{i j}^{F S}} .

Similarly, when $F (Γ_{i}^{s}, \bar{T}) \leq 0$ ⁠, the optimal delivery interval by 3PS from the central warehouse directly to departments of hospital j is: $Γ_{j}^{h *} = Γ_{i}^{s}$ ⁠. Otherwise, when $F (Γ_{i}^{s}, \bar{T}) > 0$ ⁠, the optimal delivery interval is:

Γ_{j}^{h *} = \{\begin{cases} \bar{T} & if F (Γ_{i}^{s}, \bar{T}) \leq \bar{T}, \\ Γ_{i}^{s} & if \bar{T} < F (Γ_{i}^{s}, \bar{T}) . \end{cases}

We also define ${\hat{Γ}}_{i}^{S} = \sqrt{\frac{2 k_{i}^{s}}{\sum_{j \in H} d_{i j} h_{i}^{s}}}$ as the locally optimal solutions for suppliers in the scenario with warehouse sharing. Then we can get the optimal delivery intervals of the suppliers and 3PS based on a discussion of ${\hat{Γ}}_{i}^{S}$ ⁠, $\underline{T}$ and $\bar{T}$ ⁠. The results are given in Theorem 5.

Theorem 5.

With warehouse sharing, there exist three thresholds β₀, β₁ and β₂ such that,

when $\underline{T} < \bar{T} \leq {\hat{Γ}}_{i}^{S}$ , the optimal delivery intervals are:

Γ_{i}^{s *} = \{\begin{cases} {\hat{Γ}}_{i}^{S} & if \bar{T} \leq β_{0}, \\ \bar{T} & if β_{0} < \bar{T}, \end{cases} and Γ_{j}^{h *} = \bar{T};

(20)

when $\underline{T} < {\hat{Γ}}_{i}^{S} < \bar{T}$ , the optimal delivery intervals are:

Γ_{i}^{s *} = \{\begin{cases} \bar{T} & if {\hat{Γ}}_{i}^{S} < β_{1} and F ({\hat{Γ}}_{i}^{S}, \bar{T}) < \bar{T}, \\ {\hat{Γ}}_{i}^{S} & otherwise, \end{cases} and Γ_{j}^{h *} = Γ_{i}^{s *};

(21)

when ${\hat{Γ}}_{i}^{S} \leq \underline{T} < \bar{T}$ , the optimal delivery intervals are:

Γ_{i}^{s *} = \{\begin{cases} \bar{T} & if \underline{T} < β_{2} and F (\underline{T}, \bar{T}) < \bar{T}, \\ \underline{T} & otherwise, \end{cases} and Γ_{j}^{h *} = Γ_{i}^{s *} .

(22)

Similar to the scenario without sharing, when the delivery frequency required by the hospital is high (for example, for some small hospitals with quite limited capacity for storing medical consumables, such that $\bar{T} \leq {\hat{Γ}}_{i}^{S}$ ⁠), the optimal delivery interval by 3PS from the central warehouse to hospital j departments can only be $\bar{T}$ ⁠. Otherwise, suppliers and 3PS always have the same delivery interval.

So far, we have analyzed the case without and with warehouse sharing. Next, we investigate the effect of warehouse sharing on optimal delivery intervals.

There are three thresholds, $\underline{T}$ ⁠, ${\hat{T}}_{i}^{S}$ and ${\hat{Γ}}_{i}^{S}$ on $\bar{T}$ combining Theorem 3 and Theorem 5. In our analysis, we can definitively establish that ${\hat{Γ}}_{i}^{S} < {\hat{T}}_{i j}^{S}$ always holds, nevertheless, the specific relationship involving $\underline{T}$ and $\bar{T}$ is not yet determined. Table 2 shows all the possible optimal solutions.

Table 2

Optimal solutions

Scenario	Value of $(T_{i j}^{s }, T_{j}^{h })$	Value of $(Γ_{i}^{s }, Γ_{j}^{h })$
(i) $\underline{T} < \bar{T} < {\hat{Γ}}_{i}^{S} < {\hat{T}}_{i j}^{S}$	$({\hat{T}}_{i j}^{S}, \bar{T})$ or $(\bar{T}, \bar{T})$	$({\hat{Γ}}_{i}^{S}, \bar{T})$ or $(\bar{T}, \bar{T})$
(ii) $\underline{T} < {\hat{Γ}}_{i}^{S} < \bar{T} < {\hat{T}}_{i j}^{S}$	$({\hat{T}}_{i j}^{S}, \bar{T})$ or $(\bar{T}, \bar{T})$	$({\hat{Γ}}_{i}^{S}, {\hat{Γ}}_{i}^{S})$ or $(\bar{T}, \bar{T})$
(iii) ${\hat{Γ}}_{i}^{S} < \underline{T} < \bar{T} < {\hat{T}}_{i j}^{S}$	$({\hat{T}}_{i j}^{S}, \bar{T})$ or $(\bar{T}, \bar{T})$	$(\underline{T}, \underline{T})$ or $(\bar{T}, \bar{T})$
(iv) $\underline{T} < {\hat{Γ}}_{i}^{S} < {\hat{T}}_{i j}^{S} < \bar{T}$	$({\hat{T}}_{i j}^{S}, {\hat{T}}_{i j}^{S})$ or $(\bar{T}, \bar{T})$	$({\hat{Γ}}_{i}^{S}, {\hat{Γ}}_{i}^{S})$ or $(\bar{T}, \bar{T})$
(v) ${\hat{Γ}}_{i}^{S} < \underline{T} < {\hat{T}}_{i j}^{S} < \bar{T}$	$({\hat{T}}_{i j}^{S}, {\hat{T}}_{i j}^{S})$ or $(\bar{T}, \bar{T})$	$(\underline{T}, \underline{T})$ or $(\bar{T}, \bar{T})$
(vi) ${\hat{Γ}}_{i}^{S} < {\hat{T}}_{i j}^{S} < \underline{T} < \bar{T}$	$(\underline{T}, \underline{T})$ or $(\bar{T}, \bar{T})$	$(\underline{T}, \underline{T})$ or $(\bar{T}, \bar{T})$

Source(s): Authors' own creation

In summary, when the delivery frequency required by hospitals is relatively high, warehouse sharing has no effect on 3PS. 3PS can only ship at $\bar{T}$ intervals even if there is another delivery interval that could result in a lower cost. Furthermore, there are more choices for 3PS when hospitals request a lower delivery frequency. 3PS can deliver at $\bar{T}$ intervals to minimize the average transportation cost or ship at the same frequency as suppliers to lower the holding costs.

With a lower delivery frequency required by hospitals, the suppliers and 3PS can achieve Pareto optimality by having the same delivery frequency. Moreover, under lower requirements of hospitals, indicated by a larger $\bar{T}$ ⁠, warehouse sharing can decrease the delivery frequency of both suppliers and 3PS with a larger ${\hat{T}}_{i j}^{S}$ $({\hat{T}}_{i j}^{S} > α_{1})$ and a smaller ${\hat{Γ}}_{i}^{S}$ $({\hat{Γ}}_{i}^{S} < β_{1})$ ⁠. Specifically, when supplier i's locally optimal delivery interval without warehouse sharing ${\hat{T}}_{i j}^{S}$ is a larger value, delivering at the same frequency of ${\hat{T}}_{i j}^{S}$ as supplier i can minimize the holding cost at single warehouse j for 3PS, without increasing the transportation cost too much. This is possible because the fixed transportation costs from single warehouses to departments are low. When supplier i's locally optimal delivery interval with warehouse sharing ${\hat{Γ}}_{i}^{S}$ is a smaller value, delivering at the same frequency of ${\hat{Γ}}_{i}^{S}$ as supplier i will significantly increase the transportation cost of 3PS, enough that the reduction of the holding cost is not enough to offset it. Therefore, the suppliers and 3PS will both deliver at intervals of $\bar{T}$ ⁠. Thus, warehouse sharing increases the delivery intervals of both suppliers and 3PS.

In other cases, warehouse sharing is more likely to reduce the optimal delivery intervals of suppliers and 3PS. For suppliers, each shipment only contains consumables demanded by an individual hospital without sharing, yet each supplier truck can contain consumables demanded by multiple hospitals under sharing. However, given the limited capacity of each shipment, the volume that can be transported per trip is constrained. Less volume allocated to each individual hospital can be transported per trip with warehouse sharing compared to deliveries made to a single hospital's warehouse. To prevent stockouts, suppliers must, therefore, increase the frequency of deliveries, ensuring that the cumulative demand of all hospitals is adequately met despite the lower volume per shipment. Thus, warehouse sharing leads to more frequent deliveries for suppliers. Even so, the number of shipments from suppliers to the central warehouse is significantly reduced compared to the number of shipments that would be required to ship to all single warehouses. As for 3PS, delivering at the same frequency as supplies arrive can avoid inventory backlogs and minimize the total costs, even when shorter intervals lead to much higher transportation costs.

6. Case study

Due to the complexity of such problems, it is hard to analytically examine the impact of warehouse sharing on the costs of different parties. To better understand the potential value of warehouse sharing, we consider a computational study in this section to explore the following questions: (1) How do changes in the number of suppliers or hospitals affect the value of warehouse sharing? (2) How do changes in transportation costs and inventory holding costs affect the value of warehouse sharing? We initially select parameters for our computational study that are intended to roughly capture “order-of-magnitude“ parameters of the medical consumables industry. The parameter ranges are selected to align with EOQ-based modeling conventions and with commonly observed cost relationships in medical consumables logistics, such as higher holding costs at a warehouse than within hospitals. We use simple distributions to generate heterogeneous instances while avoiding reliance on any single calibrated parameter set.

Specifically, we generate $d_{i j} \sim \exp (1), k_{i}^{s} \sim U n i f (20,30), y \sim U n i f (10,20), c \sim U n i f (20,40), h_{i}^{s} \sim U n i f (1,2), h^{n} \sim U n i f (1,4), h^{w} \sim U n i f (4,16)$ ⁠. For the purpose of analysis, we assume that demand is aggregated so that consumables with similar demand patterns from the same supplier can be viewed as a single consumable (much as is typically done in logistics network design problems; see Simchi-Levi et al., 2004). Thus, the exponential distribution is a reasonable approximation of this type of variability. We generate $d_{i j} \sim \exp (1)$ to approximate the demand of medical consumables. For each of the following subsections, we randomly generate 30 sets of parameters and record the average value of warehouse sharing by comparing two scenarios: without and with warehouse sharing.

6.1 Number of Suppliers

In this subsection, we analyze the value of warehouse sharing by varying the number of suppliers. All suppliers and hospitals are assumed to be identical. We initially let m = 8, $n \in \{4,5, \dots, 50\}$ and explore how the number of suppliers affects the average value or cost reduction due to warehouse sharing.

We define the cost change because of warehouse sharing as $Δ_{i}^{S} = \frac{C_{i}^{S N} - C_{i}^{S W}}{C_{i}^{S N}} \times 100 %$ for supplier i and $Δ^{H} = \frac{C^{P N} - C^{P W}}{C^{P N}} \times 100 %$ for 3PS. The changes are shown in Figure 5. As the number of suppliers increases, the average value of warehouse sharing(i.e. the decrease in total costs) initially increases and then decreases for suppliers. Although there is no direct competition among suppliers, the delivery frequency required by hospitals is higher as the number of suppliers increases due to limited space within hospitals (i.e. $\bar{T}$ decreases).

Figure 5

Two line charts showing how average cost reduction changes with the number of suppliers, for suppliers (panel a) and the 3 P S (panel b).

View large Download slide

The figure contains two line charts arranged side by side. In both panels, the x-axis is the number of suppliers (from 3 to 48) and the y-axis is average cost reduction (percent). Panel (a), “Impact on Suppliers,” shows cost reduction starting around the mid-60% range, increasing to a peak a little above 70% at around the mid-teens in supplier count, and then gradually declining and leveling off in the low-60% range as the number of suppliers increases further. Panel (b), “Impact on 3PS,” includes a dashed horizontal reference line at 0%. The curve begins negative (around −40%) for small numbers of suppliers, increases toward zero, becomes positive around the mid-teens, and then rises sharply to a stable positive level (around 45–50%) for larger numbers of suppliers.

Impact of warehouse sharing with different numbers of suppliers. Source(s): Authors' own creation

When the number of suppliers is few, the delivery frequencies required by hospitals are relatively low, implying a larger $\bar{T}$ $(\bar{T} > {\hat{T}}_{i j}^{S}, \forall i, j)$ ⁠. Whether sharing or not, suppliers and 3PS always deliver at the same frequency. As a result, warehouse sharing creates a positive value for suppliers due to greatly reducing the number of shipments. However, the increase in transportation costs of 3PS is higher than the decrease in holding costs, resulting in a negative value.

As the number of suppliers increases, the delivery frequency required by hospitals would be lower for the case with warehouse sharing firstly $({\hat{Γ}}_{i}^{S} < \bar{T} < {\hat{T}}_{i j}^{S}, \forall i, j)$ ⁠. Thus, 3PS can only deliver at the interval of $\bar{T}$ without warehouse sharing. However, suppliers are likely to choose the locally optimal delivery ${\hat{T}}_{i j}^{S}$ intervals due to the decreasing $\bar{T}$ ⁠. In contrast, with warehouse sharing, both suppliers and 3PS prefer to have the same delivery intervals to avoid inventory backlogs. In this way, in addition to reducing the number of supplier shipments, warehouse sharing can also lower holding costs for both suppliers and 3PS. Thus, the average value of warehouse sharing increases with the number of suppliers.

Finally, the delivery frequency required by hospitals would be much higher with a large number of suppliers $(\bar{T} < {\hat{Γ}}_{i}^{S}, \forall i)$ ⁠. The delivery intervals of 3PS in this case are always $\bar{T}$ ⁠. As the value of $\bar{T}$ decreases, suppliers are more likely to choose locally optimal delivery intervals in the case of warehouse sharing, which results in more inventory backlogs and the value of warehouse sharing decreases for suppliers. After that, the decisions for both are not affected by the number of suppliers, and hence the value of warehouse sharing remains stable.

6.2 Number of hospitals

In this subsection, we analyze the average value of warehouse sharing by varying the number of hospitals. We let n = 8, $m \in \{4,5, \dots, 50\}$ to study how the number of hospitals impacts the average value or cost reduction due to warehouse sharing.

Figure 6 shows that as the number of hospitals increases, the value of sharing increases with a decreasing growth rate for suppliers and decreases in the number of hospitals for 3PS. Changes in the hospital count have no impact on the optimal delivery intervals of both suppliers and 3PS in the case without warehouse sharing. Also, the total costs (transportation cost plus holding cost) of the suppliers and 3PS increase linearly with the number of hospitals.

Figure 6

Two line charts showing how average cost reduction changes with the number of hospitals, for suppliers (panel a) and the 3 P S (panel b).

View large Download slide

The figure contains two line charts arranged side by side. In both panels, the x-axis is the number of hospitals (about 10 to 60) and the y-axis is average cost reduction (percent). Panel (a), “Impact on Suppliers,” shows a monotonic increase with diminishing returns: average cost reduction starts in the mid-50% range at low hospital counts, rises quickly to around 70% by roughly 10 hospitals, then increases more gradually, approaching the high-80% range as the number of hospitals reaches 60. Panel (b), “Impact on 3PS,” includes a dashed horizontal reference line at 0%. The curve begins positive (around 50%) for small numbers of hospitals, declines steadily as hospitals increase, approaches 0% around the mid-range (roughly 30 hospitals), and becomes negative thereafter, reaching roughly −30% by about 60 hospitals.

Impact of warehouse sharing with different numbers of hospitals. Source(s): Authors' own creation

The limited capacity per shipment constrains the volume of consumables delivered per trip. Consequently, in the warehouse-sharing scenario, as the number of hospitals increases, the delivery quantity allocated to each hospital decreases, necessitating more frequent shipments from suppliers to satisfy the cumulative demand across all hospitals. Additionally, suppliers also produce more consumables to satisfy the demands of more hospitals, resulting in holding costs linearly increasing with the number of hospitals, although the shorter optimal delivery intervals from suppliers to the central warehouse could mitigate that increase. As a result, in the case of warehouse sharing, the total costs of suppliers increase at a less than linear rate, which increases the value of warehouse sharing at a decreasing rate. Moreover, the frequent deliveries of suppliers make the delivery frequency required by hospitals much lower. Hence, 3PS delivers at the same frequency as suppliers to minimize the holding costs. However, the shorter optimal delivery intervals lead to much higher transportation costs for 3PS from the central warehouse to the hospitals as the number of hospitals increases. Thus, the value of warehouse sharing for 3PS is decreasing, and a negative value could even result.

6.3 Variation in transportation cost of Suppliers

In this subsection, we analyze how suppliers' transportation cost variations affect the performance of warehouse sharing. We let m = 10 and n = 10 denote 10 suppliers and 10 hospitals, respectively. We keep other costs as before but differentiate suppliers by letting $k_{i}^{s} \in \{1,2,3, \dots, 140\}$ ⁠.

As seen in Figure 7, the value of warehouse sharing for suppliers and 3PS first increases and then decreases in the suppliers' transportation costs.

Figure 7

Two line charts showing average cost reduction versus supplier transportation cost, for suppliers (panel a) and the 3 P S (panel b).

View large Download slide

The figure contains two line charts arranged side by side. In both panels, the x-axis is supplier transportation cost (approximately 0–140) and the y-axis is average cost reduction (percent). Panel (a), “Impact on Suppliers,” starts at roughly 20% cost reduction when transportation cost is 0, rises steeply to about 60% by around 10 units, and continues increasing to a peak in the low-70% range (approximately 72–73%) around 30–40 units. After the peak, the curve declines gradually: it is around 70% near 60 units, about 68% near 80 units, and then slowly decreases further, ending near the mid-60% range (approximately 65%) by 140 units. Panel (b), “Impact on 3PS,” includes a dashed horizontal reference line at 0%. The curve begins strongly negative (around −50%) at transportation cost 0 and increases steadily, reaching around −20% near 30 units. It crosses the 0% line at roughly 40 units, continues upward to a maximum near +20% around 70 units, and then declines. After the peak, it approaches 0% again and crosses below 0% at roughly 110 units, continuing downward to end around −15% by 140 units.

Impact of warehouse sharing with changing supplier transportation costs. Source(s): Authors' own creation

Warehouse sharing creates positive value for suppliers but negative value for 3PS with low transportation costs. When the transportation cost is relatively low, regardless of warehouse sharing, the locally optimal delivery intervals of suppliers ${\hat{T}}_{i j}^{S}, \forall i, j$ and ${\hat{Γ}}_{i}^{S}, \forall i$ are smaller. Responsively, the delivery frequency required by hospitals is much lower. Both suppliers and 3PS have the same delivery frequency (see Theorem 3 and 5). For suppliers, warehouse sharing can reduce the number of shipments to decrease total transportation costs. However, for 3PS, shipments in the same delivery interval as suppliers significantly increase in transportation cost with zero holding costs.

As transportation costs increase, the delivery frequency required by hospitals becomes higher in the case without warehouse sharing. This means that suppliers and 3PS deliver at different intervals, and then inventory backlogs exist for both, whereas they deliver at the same frequency in the case of warehouse sharing. As supplier transportation costs increase, the value of warehouse sharing for both suppliers and 3PS increases. Initially, with low transportation costs, warehouse sharing provides a positive value for suppliers but may result in a negative value for 3PS. However, as transportation costs rise, the negative impact on 3PS diminishes and can potentially become positive.

Finally, when suppliers' transportation costs are relatively high, the locally optimal delivery intervals ${\hat{T}}_{i j}^{S}$ and ${\hat{Γ}}_{i}^{S}, \forall i, j$ are longer than the hospitals' requirements. That makes the delivery frequency required by hospitals much higher. The optimal delivery intervals of 3PS can only be $\bar{T}$ while suppliers deliver at the locally optimal delivery interval. Furthermore, there are more inventory backlogs at the suppliers and 3PS due to the increase in the optimal delivery intervals of suppliers. The value of warehouse sharing decreases and can even be negative for 3PS because of the higher transportation costs and unit holding costs of the central warehouse.

6.4 Variation in holding cost of suppliers

We next explore how holding cost variations affect sharing performance. To explore the effect of holding cost on suppliers, we let $h_{i}^{s} \in \{1,2,3, \dots, 20\}$ ⁠.

In Figure 8, as a whole, the value of warehouse sharing experiences a slight decrease for suppliers, but there is a significant decrease for 3PS in the supplier holding cost. Intuitively, as suppliers' holding costs increase, they are likely to deliver more frequently to reduce inventory holding levels, and the frequency of supplier shipments improves faster with warehouse sharing. However, the results are nonmonotonic.

Figure 8

Two line charts showing average cost reduction versus supplier holding cost, for suppliers (panel a) and the 3 P S (panel b).

View large Download slide

The figure contains two line charts arranged side by side. In both panels, the x-axis is supplier holding cost (approximately 2.5–20.0) and the y-axis is average cost reduction (percent). Panel (a), “Impact on Suppliers,” starts at about 58% when holding cost is at the low end, increases to a peak in the low-to-mid 70% range (around 73–74%) at holding costs of roughly 4–5, and then declines gradually as holding cost increases. The curve falls to around 70% near holding cost 8, drops further to about 66% around 12, and continues decreasing steadily, reaching roughly 58% by holding cost 20. Panel (b), “Impact on 3PS,” includes a dashed horizontal reference line at 0%. The curve begins strongly positive (around 80%) at low holding cost, decreases steadily, drops sharply between holding costs of roughly 4 and 7, and crosses the 0% line at about 6. It continues downward to approximately −55% by around holding cost 10, after which it remains relatively flat, fluctuating in the range of roughly −55% to −58% from about 10 to 20.

Impact of warehouse sharing with changing supplier holding costs. Source(s): Authors' own creation

When the holding costs are relatively low, regardless of sharing, suppliers' locally optimal delivery intervals are longer, and the delivery frequency required by hospitals is much higher. The increase in supplier holding costs can lead to smaller locally optimal delivery intervals, which, in turn, lead to gradually lower requirements of delivery frequency to hospitals. Since ${\hat{Γ}}_{i}^{S} < {\hat{T}}_{i j}^{S}, \forall i, j$ ⁠, the lower requirements are more likely to exist with warehouse sharing and result in the same delivery interval for both. In this case, warehouse sharing can reduce inventory backlogs while reducing the number of shipments for suppliers. The value of warehouse sharing for suppliers increases with holding costs at the beginning. In contrast, supplier i will choose the locally optimal delivery interval without warehouse sharing, which decreases the holding costs. The shorter delivery intervals of suppliers can also reduce backlogs at single warehouses. The gap between the sharing and no-sharing cases is mitigated, and the value of warehouse sharing decreases for 3PS.

In contrast, when the holding costs are relatively high, and the delivery frequency required by hospitals is much lower for both cases, the suppliers and 3PS are more likely to have the same delivery intervals $\bar{T}$ ⁠. Higher transportation costs for the central warehouse will stabilize the value of warehouse sharing at a negative value. As for suppliers, the average holding cost increases linearly, but the gap between sharing and no-sharing gradually stabilizes. Finally, the value of warehouse sharing decreases linearly for suppliers.

7. Conclusions

This study examines the value of warehouse sharing in the supply chains of the healthcare industry. We consider a supply chain where multiple suppliers provide multiple consumables for multiple hospitals, and a third-party service provider manages inventory for hospitals. We extend the traditional EOQ model to analyze the optimal delivery intervals of suppliers and the 3PS in two scenarios: without and with warehouse sharing. In the former, the 3PS builds single warehouses in each hospital. In the latter, the 3PS has a central warehouse built outside hospitals to manage the consumables of several hospitals in a region centrally. There are several major findings from this research.

First, we characterize how warehouse sharing reshapes the optimal delivery decisions of suppliers and the 3PS and show that these adjustments are governed by hospitals' required delivery frequency. When hospitals require very frequent deliveries, suppliers optimally operate on a longer replenishment cycle than the 3PS; otherwise, their delivery cycles tend to coincide. Warehouse sharing further induces suppliers to replenish the shared warehouse more frequently under high-frequency hospital requirements, while prompting the 3PS to increase outbound delivery frequency when hospitals' requirements are at an intermediate level. In contrast, when hospitals' delivery-frequency requirements are low, sharing mainly relaxes operational pressure and can lead to less frequent deliveries. Managerially, this implies that hospitals' service requirements are the primary lever that determines whether sharing increases operational intensity for suppliers, for the 3PS, or for neither.

Second, warehouse sharing can create efficiency gains by consolidating inbound shipments and pooling inventory, thereby reducing supplier shipment counts and lowering total inventory held across the network. However, these gains are not symmetric. When hospitals impose very high delivery-frequency requirements, the 3PS may have little space to smooth outbound deliveries and may be forced to operate at the requested rate. Under such binding service constraints, the central warehouse faces higher transportation and holding burdens, so the 3PS's total cost may increase even if the system benefits overall.

Finally, we identify important network-level drivers of the value of warehouse sharing. For the 3PS, collaborating with larger hospitals and with hospitals sourcing from more suppliers tends to strengthen consolidation and pooling benefits, making sharing more attractive. In particular, our results imply a threshold effect in supplier breadth. The 3PS's incremental profit from sharing becomes positive only when the number of upstream suppliers is sufficiently large, because the gains from inbound consolidation and inventory pooling outweigh the induced increase in outbound delivery effort. This threshold is not universal, but can be computed for a given application from the local cost environment, including transportation and holding costs, together with the hospital's delivery frequency requirement. Moreover, because suppliers and the 3PS interact strategically under hospitals' delivery-frequency requirements, the value of warehouse sharing can vary nonmonotonically with transportation and holding costs. Changes in suppliers' costs may also affect the 3PS's gains in a nonmonotone manner due to suppliers' first-mover advantage in replenishment decisions. Practically, these patterns suggest that warehouse sharing should be evaluated and priced based on the specific cost environment and network composition rather than treated as a universally beneficial scale initiative.

We conclude the paper by suggesting possible future research directions. First, considering the effect of seasonal changes or unexpected public health events, demand uncertainty can be introduced to explore more interesting insights. Second, a challenging direction is to incorporate competition among suppliers and deeply discuss how strategic interactions among suppliers impact the supply chain. Lastly, one could introduce more horizontal collaboration, such as joint purchasing alliances, to improve the resource utilization efficiency and risk resilience of the overall supply chain. To summarize, future research could conduct an in-depth exploration into demand uncertainty, supplier competition and horizontal collaboration to make up for the shortcomings of existing research and promote the further development of healthcare supply chain management theory and practice.

The supplementary material for this article can be found online

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On the value of warehouse sharing in an inventory system for medical consumables

1. Introduction

2. Literature review

3. Model setup

4. Benchmark: No warehouse sharing

5. Warehouse sharing

6. Case study

6.1 Number of Suppliers

6.2 Number of hospitals

6.3 Variation in transportation cost of Suppliers

6.4 Variation in holding cost of suppliers

7. Conclusions

References

Further reading

Supplementary data

Email Alerts

Cited By

On the value of warehouse sharing in an inventory system for medical consumables Open Access

1. Introduction

2. Literature review

3. Model setup

4. Benchmark: No warehouse sharing

5. Warehouse sharing

6. Case study

6.1 Number of Suppliers

6.2 Number of hospitals

6.3 Variation in transportation cost of Suppliers

6.4 Variation in holding cost of suppliers

7. Conclusions

References

Further reading

Supplementary data

Email Alerts

Suggested Reading

Related Chapters

Recommended for you

Cited By

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On the value of warehouse sharing in an inventory system for medical consumables