In Vitro Evaluation of an Immediate Response Starling‐Like Controller for Dual Rotary Blood Pumps

Rotary ventricular assist devices (VADs) are used to provide mechanical circulatory support. However, their lack of preload sensitivity in constant speed control mode (CSC) may result in ventricular suction or venous congestion. This is particularly true of biventricular support, where the native flow‐balancing Starling response of both ventricles is diminished. It is possible to model the Starling response of the ventricles using cardiac output and venous return curves. With this model, we can create a Starling‐like physiological controller (SLC) for VADs which can automatically balance cardiac output in the presence of perturbations to the circulation. The comparison between CSC and SLC of dual HeartWare HVADs using a mock circulation loop to simulate biventricular heart failure has been reported. Four changes in cardiovascular state were simulated to test the controller, including a 700 reduction in circulating fluid volume, a total loss of left and right ventricular contractility, reduction in systemic vascular resistance ( ) from 1300 to 600 , and an elevation in pulmonary vascular resistance ( ) from 100 to 300 . SLC maintained the left and right ventricular volumes between 69–214 and 29–182 respectively, for all tests, preventing ventricular suction (ventricular volume = 0 ) and venous congestion (atrial pressures > 20 ). Cardiac output was maintained at sufficient levels by the SLC, with systemic and pulmonary flow rates maintained above 3.14 for all tests. With the CSC, left ventricular suction occurred during reductions in SVR, elevations in PVR, and reduction in circulating fluid simulations. These results demonstrate a need for a physiological control system and provide adequate in vitro validation of the immediate response of a SLC for biventricular support.     

Stress and Exposure Time on von Willebrand Factor Degradation

Despite the prevailing use of the continuous flow left ventricular assist devices (cf‐LVAD), acquired von Willebrand syndrome (AvWS) associated with cf‐LVAD still remains a major complication. As AvWS is known to be dependent on shear stress (τ) and exposure time (t_exp), this study examined the degradation of high molecular weight multimers (HMWM) of von Willebrand factor (vWF) in terms of τ and t_exp. Two custom apparatus, i.e., capillary‐tubing‐type degrader (CTD) and Taylor‐Couette‐type degrader (TCD) were developed for short‐term (0.033 sec ≤ t_exp ≤ 1.05 s) and long‐term (10 s ≤ t_exp ≤ 10 min) shear exposures of vWF, respectively. Flow conditions indexed by Reynolds number (Re) for CTD were 14 ≤ Re ≤ 288 with corresponding laminar stress level of 52 ≤  ≤ 1042 dyne/cm². Flow conditions for TCD were 100 ≤ Re ≤ 2500 with corresponding rotor speed of 180 ≤ _o ≤ 4000 RPM and laminar stress level of 50 ≤  ≤ 1114 dyne/cm². Due to transitional and turbulent flows in TCD at Re > 1117, total stress (i.e., = laminar + turbulent) was also calculated using a computational fluid dynamics (CFD) solver, Converge CFD (Converge Science Inc., Madison, WI, USA). Inhibition of ADAMTS13 with different concentration of EDTA (5 mM and 10 mM) was also performed to investigate the mechanism of cleavage in terms of mechanical and enzymatic aspects. Degradation of HMWM with CTD was negligible at all given testing conditions. Although no degradation of HMWM was observed with TCD at Re < 1117 ( = 1012 dyne/cm²), increase in degradation of HMWM was observed beyond Re of 1117 for all given exposure times. At Re ~ 2500 ( = 3070 dyne/cm²) with t_exp = 60 s, a severe degradation of HMWM (90.7 ± 3.8%, abnormal) was observed, and almost complete degradation of HMWM (96.1 ± 1.9%, abnormal) was observed with t_exp = 600 s. The inhibition studies with 5 mM EDTA at Re ~ 2500 showed that loss of HMWM was negligible (<10%, normal) for all given exposure times except for t_exp = 10 min (39.5 ± 22.3%, borderline‐abnormal). With 10 mM EDTA, no degradation of HMWM was observed (11.1 ± 4.4%, normal) even for t_exp = 10 min. This study investigated the effect of shear stress and exposure time on the HMWM of vWF in laminar and turbulent flows. The inhibition study by EDTA confirms that degradation of HMWM is initiated by shear‐induced unfolding followed by enzymatic cleavage at given conditions. Determination of magnitude of each mechanism needs further investigation. It is also important to note that the degradation of vWF is highly dependent on turbulence regardless of the time exposed within our testing conditions.     

In vitro evaluation of an adaptive Starling‐like controller for dual rotary ventricular assist devices

Rotary ventricular assist devices (VADs) operated clinically under constant speed control (CSC) cannot respond adequately to changes in patient cardiac demand, resulting in sub‐optimal VAD flow regulation. Starling‐like control (SLC) of VADs mimics the healthy ventricular flow regulation and automatically adjusts VAD speed to meet varying patient cardiac demand. The use of a fixed control line (CL – the relationship between ventricular preload and VAD flow) limits the flow regulating capability of the controller, especially in the case of exercise. Adaptive SLC (ASLC) overcomes this limitation by allowing the controller to adapt the CL to meet a diverse range of circulatory conditions. This study evaluated ASLC, SLC and CSC in a biventricular supported mock circulation loop under the simulated conditions of exercise, sleep, fluid loading and systemic hypertension. Each controller was evaluated on its ability to remain within predefined limits of VAD flow, preload, and afterload. The ASLC produced superior cardiac output (CO) during exercise (10.1 L/min) compared to SLC (7.3 L/min) and CSC (6.3 L/min). The ASLC produced favourable haemodynamics during sleep, fluid loading and systemic hypertension and could remain within a predefined haemodynamic range in three out of four simulations, suggesting improved haemodynamic performance over SLC and CSC.     

In Vivo Evaluation of Active and Passive Physiological Control Systems for Rotary Left and Right Ventricular Assist Devices

Preventing ventricular suction and venous congestion through balancing flow rates and circulatory volumes with dual rotary ventricular assist devices (VADs) configured for biventricular support is clinically challenging due to their low preload and high afterload sensitivities relative to the natural heart. This study presents the in vivo evaluation of several physiological control systems, which aim to prevent ventricular suction and venous congestion. The control systems included a sensor‐based, master/slave (MS) controller that altered left and right VAD speed based on pressure and flow; a sensor‐less compliant inflow cannula (IC), which altered inlet resistance and, therefore, pump flow based on preload; a sensor‐less compliant outflow cannula (OC) on the right VAD, which altered outlet resistance and thus pump flow based on afterload; and a combined controller, which incorporated the MS controller, compliant IC, and compliant OC. Each control system was evaluated in vivo under step increases in systemic (SVR ～1400–2400 dyne/s/cm⁵) and pulmonary (PVR ～200–1000 dyne/s/cm⁵) vascular resistances in four sheep supported by dual rotary VADs in a biventricular assist configuration. Constant speed support was also evaluated for comparison and resulted in suction events during all resistance increases and pulmonary congestion during SVR increases. The MS controller reduced suction events and prevented congestion through an initial sharp reduction in pump flow followed by a gradual return to baseline (5.0 L/min). The compliant IC prevented suction events; however, reduced pump flows and pulmonary congestion were noted during the SVR increase. The compliant OC maintained pump flow close to baseline (5.0 L/min) and prevented suction and congestion during PVR increases. The combined controller responded similarly to the MS controller to prevent suction and congestion events in all cases while providing a backup system in the event of single controller failure.     

Hemodialysis using a low bicarbonate dialysis bath: Implications for acid-base homeostasis

The low bath bicarbonate concentration ([]) used by a nephrology group in Japan (25.5 mEq/L), coupled with a bath [acetate] of 8 mEq/L, provided an opportunity to study the acid-base events occurring during hemodialysis when flux is from the patient to the bath. We used an analytic tool that allows calculation of delivery during hemodialysis and the physiological response to it in 17 Japanese outpatients with an average pre-dialysis blood [] of 25 mEq/L. Our analysis demonstrates that addition is markedly reduced and that all of it comes from acetate metabolism. The added to the extracellular fluid during treatment (19.5 mEq) was completely consumed by H+ mobilization from body buffers. In contrast to patients dialyzing with higher bath [] values in the US and Europe, organic acid production was suppressed rather than stimulated. Dietary analysis indicates that these patients are in acid balance due to the alkaline nature of their diet. In a larger group of patients using the same bath solution, pre-dialysis blood [] was lower, 22.2 mEq/L, but still in an acceptable range. Our studies indicate that a low bath [] is well tolerated and can prevent stimulation of organic acid production.     

Numerical Analyses for Low Reynolds Flow in a Ventricular Assist Device

Scientific and technological advances in blood pump developments have been driven by their importance in cardiac patient treatments and in the expansion of life quality in assisted people. To improve and optimize the design and development, numerical tools were incorporated into the analyses of these mechanisms and have become indispensable in their advances. This study analyzes the flow behavior with low impeller Reynolds number, for which there is no consensus on the full development of turbulence in ventricular assist devices (VAD). For supporting analyses, computational numerical simulations were carried out in different scenarios with the same rotation speed. Two modeling approaches were applied: laminar flow and turbulent flow with the standard, RNG and realizable κ ? ε; the standard and SST κ ? ω models; and Spalart–Allmaras models. The results agree with the literature for VAD and the range for transient flows in stirred tanks with an impeller Reynolds number around 2800 for the tested scenarios. The turbulent models were compared, and it is suggested, based on the expected physical behavior, the use of RNG, standard and SST , and Spalart–Allmaras models to numerical analyses for low impeller Reynolds numbers according to the tested flow scenarios.     

Optimal hybrid control of interarea oscillation in power system using energy storage systems

Unmanaged renewables integration into the power system will raise the possibility of small-signal instability due to higher load deviations in the transmission lines. In power grids with a higher penetration level of the renewables, the load deviation can cause interarea oscillation in the grid. Meanwhile, large-scale battery energy storage systems are promising solutions to enhance power system stability by smoothening the load profile and their fast response. To damp the interarea oscillations without compromising the voltage stability, we need to consider the battery's dynamic model in axis. The battery's dynamic model in axis allows us to integrate the battery into the power system and simulate both the battery's active and reactive power injection/absorption. In this paper, we model the battery energy storage on axis. Then, the battery model is augmented into the two-area four-machine power system. An optimum hybrid controller using linear quadratic regulator techniques is designed to damp generators' frequency deviations. The results show that the interarea oscillations are damped without losing the voltage stability of the system.     

Renewable hydrogen and ammonia for combined heat and power systems in remote locations: Optimal design and scheduling

Using hydrogen (H) and ammonia (NH) for renewable energy storage has the potential to enable economical power and heat supply with high renewable penetrations, especially in remote locations which are characterized by high energy costs. In this work we assess the economic competitiveness of renewable combined heat and power (CHP) systems in Mahaka HI, Nantucket MA, and Northwest Arctic Borough (NWAB) AK by optimally designing these systems for scenarios in which power and heat can be purchased over a range of historical energy prices as well as when 100% renewable supply is required. We use a combined optimal design and scheduling model which minimizes annualized net present cost by determining optimal technology selection and size simultaneously with optimal schedules for each period of a system operating horizon aggregated from full year hourly resolution data via a consecutive temporal clustering algorithm. We find that renewable generation meets at least 85% of power demands and 75% of heat demands under the lowest energy prices investigated. Higher conventional energy prices lead to increased renewable penetration which is facilitated by renewable NH as a seasonal energy storage medium, as are 100% renewable CHP systems. NH is used for power generation with heat cogeneration in all three locations, as well as directly for heating in NWAB. On an annual cost basis, NH-enabled 100% renewable CHP is only 3% more expensive in Mahaka and NWAB than systems which can purchase energy at the lowest prices, while it is 15% more expensive in Nantucket.     

Direct synthesis of fixed-order multi-objective controllers

This paper introduces a new methodology for the design of fixed-order multi-objective output feedback controllers. The problem comprises a set of linear matrix inequalities and an additional rank constraint. The primary idea is to classify convex subsets of the set of rank constrained matrices in such formulations, based on which two noniterative and relatively fast methods are developed. The proposed methods require solving a convex optimization problem at each step and can be applied with any weighted summation of design objectives such as performance, performance, passivity, and regional pole assignment. Several benchmark systems with performance criteria including H_∞, mixed H₂/H_∞, and pole placement are used to demonstrate a marked improvement in comparison to the existing methods.     

Effect of Inflow Cannula Tip Design on Potential Parameters of Blood Compatibility and Thrombosis

During ventricular assist device support, a cannula acts as a bridge between the native cardiovascular system and a foreign mechanical device. Cannula tip design strongly affects the function of the cannula and its potential for blood trauma. In this study, the flow fields of five different tip geometries within the ventricle were evaluated using stereo particle image velocimetry. Inflow cannulae with conventional tip geometries (blunt, blunt with four side ports, beveled with three side ports, and cage) and a custom‐designed crown tip were interposed between a mixed‐flow rotary blood pump and a compressible, translucent silicone left ventricle. The contractile function of the failing ventricle and hemodynamics were reproduced in a mock circulation loop. The rotary blood pump was interfaced with the ventricle and aorta and used to fully support the failing ventricle. Among these five tip geometries, high‐shear volume (, potential parameter of platelet activation) was found to be the greatest in the blunt tip. The cage tip was observed to have the highest low‐shear volume and recirculation volume ( and V_z > 0, respectively; potential parameters of thrombus formation). The crown tip, together with conventional tip geometries with side ports (blunt with four side ports and beveled with three side ports) showed no significant difference in either high‐shear volume or low‐shear volume. However, recirculation volume was reduced significantly in the crown tip. Despite limited generalizability to clinical situations, these transient‐state measurements supported the potential mitigation of complications by changing the design of conventional cannula tip geometries.     

Sliding mode control of time‐varying delay Markov jump with quantized output

This paper investigates the quantized sliding mode control of Markov jump systems with time‐varying delay. A dynamical adjustment law is explored to quantize the system output. By constructing an observer‐based integral sliding surface, a sliding mode controller is designed to take over the dynamical motion of state estimation and ensure the reachability of sliding surface. A new scaling manner is developed to build the bound between the system output and quantized error. With the help of separation strategies for controller synthesis and general transition probabilities and a lower bound theorem for nonlinear integral terms, a new synthesis method to ensure the required stability and meet the required performance is proposed in the form of linear matrix inequalities. The validity of the proposed control method is illustrated by a numerical example.     

Hemodynamic Response to Exercise and Head‐Up Tilt of Patients Implanted With a Rotary Blood Pump: A Computational Modeling Study

The present study investigates the response of implantable rotary blood pump (IRBP)‐assisted patients to exercise and head‐up tilt (HUT), as well as the effect of alterations in the model parameter values on this response, using validated numerical models. Furthermore, we comparatively evaluate the performance of a number of previously proposed physiologically responsive controllers, including constant speed, constant flow pulsatility index (PI), constant average pressure difference between the aorta and the left atrium, constant average differential pump pressure, constant ratio between mean pump flow and pump flow pulsatility (ratio_PI or linear Starling‐like control), as well as constant left atrial pressure control, with regard to their ability to increase cardiac output during exercise while maintaining circulatory stability upon HUT. Although native cardiac output increases automatically during exercise, increasing pump speed was able to further improve total cardiac output and reduce elevated filling pressures. At the same time, reduced venous return associated with upright posture was not shown to induce left ventricular (LV) suction. Although control outperformed other control modes in its ability to increase cardiac output during exercise, it caused a fall in the mean arterial pressure upon HUT, which may cause postural hypotension or patient discomfort. To the contrary, maintaining constant average pressure difference between the aorta and the left atrium demonstrated superior performance in both exercise and HUT scenarios. Due to their strong dependence on the pump operating point, PI and ratio_PI control performed poorly during exercise and HUT. Our simulation results also highlighted the importance of the baroreflex mechanism in determining the response of the IRBP‐assisted patients to exercise and postural changes, where desensitized reflex response attenuated the percentage increase in cardiac output during exercise and substantially reduced the arterial pressure upon HUT.     

Comparison of preload-sensitive pressure and flow controller strategies for a dual device biventricular support system

The use of rotary left ventricular assist devices (LVADs) has extended to destination and recovery therapy for end‐stage heart failure. Incidence of right ventricular failure while on LVAD support requires a second device be implanted to support the failing right ventricle. Without a commercially available implantable rotary right ventricular assist device, rotary LVADs are cannulated into the right heart and operation modified to provide suitable support for the pulmonary system. While this approach can alleviate the demand for transplant through long‐term biventricular support, it uncovers a new challenge with respect to controller strategies for these dual device support systems. This study compares the preload sensitivity of rotary, dual device biventricular assistance controllers in light of their ability to adjust the flow rate according to physiological demand. A Frank–Starling‐like flow controller which requires both inlet pressure and flow sensors is compared to pressure controllers which maintain atrial or inlet cannula pressures through the use of a single pressure sensor. It was found that cannula selection and the location of a pressure controller's single pressure sensor can be tailored to adjust the preload sensitivity. When located within the atria, this sensitivity is effectively infinite. Moving the sensor to the base of a 450‐mm cannula, however, decreased the sensitivity to 0.22 (L/min)/mm Hg. This indicates the potential for simple and reliable VAD controllers with increased preload sensitivity without the need for complex controllers requiring an array of hemodynamic sensors.     

Circulatory loop design and components introduce artifacts impacting in vitro evaluation of ventricular assist device thrombogenicity: A call for caution

Mechanical circulatory support (MCS) devices continue to be hampered by thrombotic adverse events (AEs), a consequence of device-imparted supraphysiologic shear stresses, leading to shear-mediated platelet activation (SMPA). In advancing MCS devices from design to clinical use, in vitro circulatory loops containing the device under development and testing are utilized as a means of assessing device thrombogenicity. Physical characteristics of these test circulatory loops may also contribute to inadvertent platelet activation through imparted shear stress, adding inadvertent error in evaluating MCS device thrombogenicity. While investigators normally control for the effect of a loop, inadvertent addition of what are considered innocuous connectors may impact test results. Here, we tested the effect of common, additive components of in vitro circulatory test loops, that is, connectors and loop geometry, as to their additive contribution to shear stress via both in silico and in vitro models. A series of test circulatory loops containing a ventricular assist device (VAD) with differing constituent components, were established in silico including: loops with 0~5 Luer connectors, a loop with a T-connector creating 90° angulation, and a loop with 90° angulation. Computational fluid dynamics (CFD) simulations were performed using a shear stress transport turbulence model to platelet activation index (PAI) based on a power law model. VAD-operated loops replicating in silico designs were assembled in vitro and gel-filtered human platelets were recirculated within (1 hour) and SMPA was determined. CFD simulations demonstrated high shear being introduced at non-smooth regions such as edge-connector boundaries, tubing, and at Luer holes. Noticeable peaks’ shifts of scalar shear stress (sss) distributions toward high shear-region existed with increasing loop complexity. Platelet activation also increased with increasing shear exposure time, being statistically higher when platelets were exposed to connector-employed loop designs. The extent of platelet activation in vitro could be successfully predicted by CFD simulations. Loops employing additional components (non-physiological flow pattern connectors) resulted in higher PAI. Loops with more components (5-connector loop and 90° T-connector) showed 63% and 128% higher platelet activation levels, respectively, versus those with fewer (0-connector (P = .023) and a 90° heat-bend loop (P = .0041). Our results underscore the importance of careful consideration of all component elements, and suggest the need for standardization in designing in vitro circulatory loops for MCS device evaluation to avoid inadvertent additive SMPA during device evaluation, confounding overall results. Specifically, we caution on the use and inadvertent introduction of additional connectors, ports, and other shear-generating elements which introduce artifact, clouding primary device evaluation via introduction of additive SMPA.     

Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite-dimensional control space

In this work, we derive an a priori error estimate of order for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box-constrains on the control are considered and finitely many pointwise state-constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.     

In Vivo Kinematics of Functional Ankle Instability Patients and Lateral Ankle Sprain Copers During Stair Descent

Patients with mechanic ankle instability experience increased tibiotalar and subtalar joint laxity. However, in vivo joint kinematics in functional ankle instability (FAI) patients and lateral ankle sprain (LAS) copers, especially during dynamic activities, are poorly understood. Ten FAI patients, 10 LAS copers, and 10 healthy controls were included in this study. A dual fluoroscopic imaging system was used to analyze the tibiotalar and subtalar joint kinematics during stair descent. Five key poses of stair descent were analyzed. Kinematic data from six degrees of freedom were calculated utilizing a solid modeling software. The range of motion and joint positions in each degree of freedom were compared among the three groups. The tibiotalar joints of FAI patients and LAS copers were significantly more inverted than those of healthy controls during the foot strike (p = 0.016, = 0.264). The subtalar joints of FAI patients were significantly more anteriorly translated (pose 2, p = 0.003, = 0.352; pose 3, p < 0.001, = 0.454; pose 4, p = 0.004, = 0.334), inverted (pose 4, p = 0.027, = 0.234; pose 5,p = 0.034, = 0.221), and externally rotated (pose 4, p = 0.037, = 0.217; pose 5; p = 0.004, = 0.331) than those of healthy controls during the mid‐stance and the heel off. The FAI patients showed excessive tibiotalar inversion and subtalar joint hypermobility during stair descent. Meanwhile, the LAS copers maintained subtalar joint stability, and only showed excessive tibiotalar inversion in foot strike. These data provide insight into the mechanisms behind the development of FAI after initial LAS. © 2019 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 37:1860–1867, 2019     

A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data

We analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L²‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L²‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.     

PRESAS: Block-structured preconditioning of iterative solvers within a primal active-set method for fast model predictive control

Model predictive control (MPC) for linear dynamical systems requires solving an optimal control structured quadratic program (QP) at each sampling instant. This article proposes a primal active-set strategy, called PRESAS , for the efficient solution of such block-sparse QPs, based on a preconditioned iterative solver to compute the search direction in each iteration. Rank-one factorization updates of the preconditioner result in a per-iteration computational complexity of , where m denotes the number of state and control variables and N the number of control intervals. Three different block-structured preconditioning techniques are presented and their numerical properties are studied further. In addition, an augmented Lagrangian based implementation is proposed to avoid a costly initialization procedure to find a primal feasible starting point. Based on a standalone C code implementation, we illustrate the computational performance of PRESAS against current state of the art QP solvers for multiple linear and nonlinear MPC case studies. We also show that the solver is real-time feasible on a dSPACE MicroAutoBox-II rapid prototyping unit for vehicle control applications, and numerical reliability is illustrated based on experimental results from a testbench of small-scale autonomous vehicles.     

Optimal parameter selection problem with application to the synchronization of delivery cycles in a supply chain with Wiener process

In this paper, we use optimal parameter selection technique to develop two models involving single‐vendor–multiple‐buyer supply chain, which are called the dynamic independent optimization (DIO) model and the dynamic synchronized cycles (DSC) model, respectively. These models are, respectively, similar to the traditional static independent policy model and the traditional static synchronized cycle model, except that the deterministic demands of the buyers in the above two static models are now being replaced by the stochastic demands satisfying a Wiener process, which have more real‐life applications. Similar to the above static synchronized cycles model, the synchronization of the supply chain in our DSC model is also achieved by scheduling the delivery days of the buyers and coordinating them with the vendor's production cycle. Finding the optimal expected system costs of the DIO model and the DSC model involves solving optimal parameter selection problems governed by ordinary differential equations, whose final times are continuous decision variables and discrete decision variables, respectively. Computational methods have been developed for solving these problems. Numerical results show that the coordinated policy is better than the independent optimization policy, in terms of minimizing the expected system cost of the entire supply chain. Sensitivity analysis is performed to test the effect of changing the cost coefficients and the value on the performances of these models, where is the ratio of the total mean demand rate of all the buyers over the vendor's production rate.