Performance Analysis of a Manufacturing Line Operated under Optimal Surplus-Based Production Control

Abstract

We examine optimality and performance of a tandem manufacturing line driven by a surplus-based decentralized production control strategy. The main objective of this type of production strategies is to guarantee that the cumulative number of produced products follows the cumulative production demand on the output of the given network. The basic idea of surplus-based control strategy is presented for the case of one manufacturing machine. We prove that this strategy is optimal. Then, a flow model of a line composed of arbitrarily many machines with bounded buffers is analyzed. We prove that the surplus-based production control enables this network to efficiently follow the product demand and establish the relation between the efficiency in the production tracking error and the intermediate inventory levels of a line. Performance and robustness of the flow model of the closed-loop manufacturing line are illustrated by computer simulations.

1. Introduction

Nowadays, the highly dynamic market environment requires that production control policies implemented in manufacturing industries should be capable of providing quick and accurate responses to constant and rapid changes in the production demand. This strongly shifts the interests of manufacturers to the need of theoretical analysis of the currently existing policies, that is, the study of conformity between the production output and the demand of their product, as well as the relation between stock level (buffer content) and the production surplus of the manufacturing network.

Currently, there is a substantial literature on manufacturing control policies and their performance. A number of classifications of these policies were introduced by various authors. In this paper, we follow the classification introduced in [1], which puts the control policies into three categories: token-based, time-based, and surplus-based, respectively. In token-based approaches, the so-called tokens are generated and utilized to trigger certain events in the manufacturing system. The famous examples of such a policy are Kanban [2, 3], Conwip [4], and Basestock [5]. The time-based approaches offer to make the control decisions on the basis of the time when a certain operation should take place, they are exemplified by Material Resource Planning, Least Stack, and Earliest Due Date strategies (see, e.g., [6]). In the surplus approach, control decisions are based on the production tracking error, (here the term “production tracking” refers to the action of following or keeping track of the production demand trajectory on the output of a given network) which is the difference between the cumulative demand and output of the system. For extensive surveys and further details concerning, in particular, the production line control mechanisms, we refer the reader to [1, 7–9].

In the afore-mentioned literature, considerable research effort has been invested into the issue of optimality. In [10], the authors showed that for unreliable manufacturing systems with parameters from a certain domain, the zero-inventory policies are optimal even if there is an uncertainty in the manufacturing capacity. This was demonstrated under the assumption that the demand rate is constant and the production rate can be adjusted proceeding from the deviation from the optimal inventory level. In [11], the optimality of pull controlled flow shop was established in the case where the performance index encompasses the buffer holding costs and system shortfall/inventory costs. This research treated the production demand and processing rates as deterministic.

In [12], a broad class of dynamic scheduling problems associated with single-server, multiclass, continuous-flow, flexible manufacturing systems was considered. The objective was to minimize the integral of an instantaneous cost function defined on the inventory/backlog state of the system. The authors provide sufficient conditions under which the optimal solution comes to implementation of the myopic scheduling policy. The paper also presents examples and counterexamples that explicitly illustrate the behavior and limitations of the myopic scheduling policies.

In [13], a model based on stochastic discrete-time controlled dynamical systems was developed in order to derive optimal policies for controlling the material flow in supply networks. Contrary to most studies in the area, which typically assume a given (parameterized) control strategy and analyze how the dynamics depends on the parameters, the authors do not assume a certain family of strategies a priori, thus allowing any control law in the form of a function of the current state of the system. The individual nodes are controlling their inflows in a decentralized fashion by placing orders to their immediate suppliers. An explicit optimal state-feedback control is derived with respect to the cost functional that typically takes into account both inventory holding costs and ordering costs.

Extended reviews on optimal control in production networks can be found in, for example, [8, 9, 14].

This paper offers the extension of our previous results (see [15]) on an analytical analysis for pull-type policies that concern a single machine and a line of machines with both being driven by the cumulative production demand. In addition to our previous work (see [15]), where we also examined the performance of one machine and the line, here we present some novel results. In this paper, in case of one machine, we find the optimal control policy that minimizes the mismatch between the cumulative production output and cumulative demand. Then, we examine a tandem line, where each machine is driven by the obtained optimal policy. For this network, we obtain the bounds of the production surplus for all machines, as well as in [15], but now assuming that the manufacturing line has a limited inventory (buffer content) for storing its intermediate production. Also in this paper we provide the analytical results obtained on the tradeoff relationship between the production tracking error bounds and the inventory level (buffer capacity) of a manufacturing line. In our simulation results, we show that this relation have an important meaning and can be used as a tool in practice.

Thus, we first tackle the problem of optimal control for one manufacturing machine that nominally produces products in lots of a given size and is controlled by carrying out only two commands (“on” and “off”), unlike [13], where the size of the lot was controllable and the set of feasible controls was a polygon. Another difference is that we deal with a deterministic model of the system. Similarly to [13] and contrary to the bulk of the literature in the area, we do not limit the class of the strategies a priori and take into account all control laws that are fed by the currently available data, which concerns not only the current system state but also the past states. The system is influenced by uncertain disturbances that are due to both market fluctuations in demand rate and random fluctuations in the production rate of the machine. By assuming known bounds on these disturbances and by applying the min-max dynamical programming, we prove that surplus-based pull control policy is optimal.

Further, we apply classical tools from the control theory in order to evaluate the performance of this technique for a unidirectional production line of 𝑁 machines with limited capacity intermediate buffers.

The production flow process is described by means of difference equations and in order to analyze performance, Lyapunov theory approach is employed (see, e.g., [16, 17] and references therein). Each machine in the line coordinates its individual production with those of the rest of the system. Its primary objective may be viewed as manufacturing of sufficient quantity of parts to satisfy the demand of its immediate downstream machine and some desired amount as backup material storage in its downstream buffer. The production strategy itself is intuitive and we will show that it can be associated with existing techniques.

To the best of our knowledge, concerning the previous results on performance analysis of surplus-based approaches (see, e.g., [9, 10, 18–23], as well as the references from our review), the novelty of our result can be summarized as follows. The studied tandem production model is considered in discrete time, where the production speed of each machine is defined as deterministic with bounded perturbations. The future production demand is assumed to be unknown and with bounded fluctuations. As the result, for a line of 𝑁 machines with limited capacity intermediate buffers, strict “worst case” bounds on production tracking errors were obtained as well as their relation with the system inventory levels. In our simulation results, it is shown that the obtained relation can be a very important tool to be used in the decision-making process by a production line manager.

The paper is organized as follows. In Section 2, a discrete-time flow model of a single manufacturing machine is presented and the optimal control strategy is obtained. Section 3 is devoted to study of a flow model of a unidirectional manufacturing line. Performance of the closed-loop production line is illustrated by computer simulations in Section 4. Finally, Section 5 presents conclusions and discussions of future work.

2. Analysis of One Manufacturing Machine

2.1. Flow Model

At discrete time 𝑘 and for one manufacturing machine, the cumulative number of produced products can be described as the sum of its production rates at each time step till time 𝑘. So, the flow model of one manufacturing machine in discrete time is defined as𝑦(𝑘+1)=𝑦(𝑘)+𝑢(𝑘)+𝑓(𝑘),∀𝑘∈𝑁, (2.1) where 𝑦(𝑘)∈ℝ is the cumulative output of the machine in time 𝑘, 𝑢(𝑘)∈ℝ is the control signal, 𝑓(𝑘)∈ℝ is an unknown external disturbance that affects on the production rate of the machine and 𝑁 is a set of naturals.

Under the assumption that there is always sufficient quantity of the raw material to feed the machine, the control aim is to track the nondecreasing cumulative production demand. We define the cumulative production demand by using 𝑦_𝑑(𝑘)∈ℝ given by𝑦_𝑑(𝑘)=𝑦_𝑑0+𝑣_𝑑𝑘+𝜑(𝑘), (2.2) where 𝑦_𝑑0 is a positive constant that represents the initial production demand, 𝑣_𝑑 is a positive constant that defines the average desired demand rate, and 𝜑(𝑘)∈ℝ is the bounded fluctuation that is imposed on the linear demand 𝑣_𝑑𝑘, for example, market fluctuation.

At each time step 𝑘, the control input 𝑢 can take only two values: 1 or 0. With regard to (2.1), this means that the nominal size of the lot produced by the machine for the unit time is assumed to be 1. (The unity is taken for the definiteness. The case of arbitrary production rate can be reduced to that at hand by proper scaling of all variables involved.)

To efficiently fulfill the control aim, we are going to minimize the production error 𝜀(𝑘)=𝑦_𝑑(𝑘)−𝑦(𝑘) over the class of all control strategies fed by the available data:𝑢(𝑘)=𝑈_𝑘𝑦(0),…,𝑦(𝑘),𝑦_𝑑(0),…,𝑦_𝑑(𝑘)∈{0;1}. (2.3)

Due to (2.1) and (2.2), the evolution of the production error 𝜀(𝑘) obeys the following equation:𝜀(𝑘+1)=𝜀(𝑘)−𝑢(𝑘)+𝜉(𝑘),where𝜉(𝑘)∶=𝑣_𝑑+Δ𝜑(𝑘)−𝑓(𝑘), (2.4) and Δ𝜑(𝑘)=𝜑(𝑘+1)−𝜑(𝑘) is the deviation of the demand rate Δ𝜑(𝑘)+𝑣_𝑑 from its average desired value 𝑣_𝑑. We assume that despite of the disturbances, the demand rate is realistic, that is, it lies within the nominal bounds 𝑢(𝑘)+𝑓(𝑘)|_𝑢(𝑘)=0=𝑓(𝑘) and 𝑢(𝑘)+𝑓(𝑘)|_𝑢(𝑘)=1=𝑓(𝑘)+1 of the actual production rate 𝑢(𝑘)+𝑓(𝑘). As a result, the signed deviation 𝜀(𝑘)=𝑦_𝑑(𝑘)−𝑦(𝑘) of the cumulative output from the cumulative demand is controllable: it can be both decreased 𝜀(𝑘+1)≤𝜀(𝑘) and increased 𝜀(𝑘+1)≥𝜀(𝑘) by applying the proper feasible control 𝑢(𝑘)=0,1.

This implies the following condition (also known as capacity condition):0<𝜉(𝑘)<1,∀𝑘∈ℕ. (2.5)

2.2. Optimal Control Strategy

In this section, we examine the following two optimization problems:𝐽_𝑇=sup_𝜉(0)()^𝑇_{,…,𝜉𝑇−1𝑘=0}|𝜀(𝑘)|^𝑝⟶min_𝑈𝐽_∞,=limsup_𝑘→∞sup_𝜉(⋅)|𝜀(𝑘)|^𝑝⟶min_𝑈. (2.6)(2.7) Here, 𝑝≥1 is a given constant, sup is taken over all 𝜉(⋅) satisfying (2.5), and 𝑈={𝑈_𝑘(⋅)}^∞_𝑘=0 is the set of all possible control strategies given by (2.3).

The problem (2.6) is to optimize the worst-case summary error, where the sum is over the given and finite time horizon 𝑇 and the worst case is considered with respect to feasible perturbations 𝜑(𝑘),𝑓(𝑘), and demand rate 𝑣_𝑑. This optimization provides the best performance guarantees in the face of uncertainties in 𝑣_𝑑,𝜑(𝑘), and 𝑓(𝑘). The performance criterion (2.7) is aimed at minimization of the worst-case production error at large enough times via, maybe, sacrificing the exactness of tracking the production demand during an initial time interval. Though the control objectives in (2.6) and (2.7) are different, the main result of this section (Theorem 2.1) shows that they are achieved by a common control strategy.

In (2.6) and (2.7), the larger values of the parameter 𝑝 are used to penalize large production errors more severely. In the literature, the most popular values are 𝑝=1,2. We consider arbitrary 𝑝∈[1,+∞) partly in order to show that the solution of the optimization problem is not influenced by the value of 𝑝. This is presumably due to the use of discrete controls 𝑢=0,1; if the set of feasible controls is not finite, the solution typically depends on 𝑝.

The following theorem is the main result of the section.

Theorem 2.1. The following control strategy 𝑢(𝑘)=sign₊(𝜀(𝑘)) (2.8) is optimal with respect to the performance index (2.6) for any given 𝑇, as well as with respect to the performance criterion (2.7). This is true irrespective of the choice of 𝑝∈[1,+∞).

Here, the control action is given by𝑢=sign₊(𝜀)∶=1if𝜀>00if𝜀<00,1if𝜀=0. (2.9) Basically, this controller works as a pull control that authorizes the machine to produce if the product surplus is negative, stops the machine if the surplus is positive, and randomly selects between the two previous decisions if the surplus is on the boundary.

The proof of Theorem 2.1 is given in Appendix A.

Now that for one machine the optimal tracking controller is derived, we can extend our analysis of this strategy by applying it to a line of 𝑁 manufacturing machines with bounded intermediate buffers.

3. A Line of Machines with Bounded Buffers

3.1. Flow Model

The flow model of a manufacturing line is presented in this section. Figure 1 shows a schematics of a line of 𝑁 manufacturing machines with machines 𝑀_𝑗, buffers 𝐵_𝑗, and infinite product supply. The optimal control strategy from the previous section is modified with respect to the buffers and the machines present in the line. New limitations such as desired buffer content and buffer capacity restriction are considered in the model.

Figure 1 

Schematics of a line of 𝑁 manufacturing machines.

The flow model of the manufacturing line is defined as𝑦₁(𝑘+1)=𝑦₁(𝑘)+𝛽₁(𝑘)sign₋𝑤₂(𝑘)−𝛾₂,𝑦_𝑗(𝑘+1)=𝑦_𝑗(𝑘)+𝛽_𝑗(𝑘)sign_Buff𝑤_𝑗(𝑘)−𝛽_𝑗(𝑘)sign₋𝑤_𝑗+1(𝑘)−𝛾_𝑗+1𝑦,𝑗=2,…,𝑁−1,_𝑁(𝑘+1)=𝑦_𝑁(𝑘)+𝛽_𝑁(𝑘)sign_Buff𝑤_𝑁(𝑘)−𝛽_𝑁(𝑘), (3.1)(3.2)(3.3) where 𝑦_𝑗(𝑘) is the cumulative output of machine 𝑀_𝑗 in time 𝑘,𝑤_𝑗(𝑘)=𝑦_𝑗−1(𝑘)−𝑦_𝑗(𝑘) is the content of buffer 𝐵_𝑗, 𝛽_𝑗(𝑘)=𝑢_𝑗(𝑘)+𝑓_𝑗(𝑘),forall𝑗=1,…,𝑁, 𝑓_𝑗 is the external disturbance affecting machine 𝑀_𝑗 (e.g., production speed variations, undesired delay or setup time), 𝑢_𝑗 is the control input of machine 𝑀_𝑗, sign_Buff(𝑥)=(1,if𝑥≥0|0, otherwise), sign₋(𝑥)=(1,if𝑥≤0|0, otherwise), and 𝛾_𝑗+1 is the threshold value of the buffer content 𝑤_𝑗+1.

In order to give a solution to the demand tracking problem, we propose the following control inputs:𝑢_𝑗(𝑘)=𝜇_𝑗sign₊𝜀_𝑗+1(𝑘)+𝑤_𝑑𝑗+1−𝑤_𝑗+1(𝑘𝑢),∀𝑗=1,…,𝑁−1,_𝑁(𝑘)=𝜇_𝑁sign₊𝑦_𝑑(𝑘)−𝑦_𝑁(𝑘), (3.4) where 𝜇_𝑗 is the constant processing speed of machine 𝑗, 𝑤_𝑑𝑗+1 is the constant that represents the desired inventory level of buffer 𝐵_𝑗+1, and 𝜀_𝑗+1 is the production error of machine 𝑀_𝑗+1. Here, for simplicity, we restrict the value of sign₊ function, which was defined in the previous section, to sign₊(𝑥)=(1,if𝑥>0|0, otherwise). The tracking error of each machine is given by𝜀_𝑗(𝑘)=𝜀_𝑗+1(𝑘)+𝑤_𝑑𝑗+1−𝑤_𝑗+1(𝑘𝜀),∀𝑗=1,…,𝑁−2,_𝑁−1(𝑘)=𝜀_𝑁(𝑘)+𝑤_𝑑𝑁−𝑤_𝑁(𝑘)𝜀_𝑁,(𝑘)=𝑦_𝑑(𝑘)−𝑦_𝑁(𝑘). (3.5)(3.6)(3.7) It follows from (3.7) that the error of machine 𝑀_𝑁 is defined exactly as for the single machine case. The buffer restriction, as seen from (3.3), is the only difference in the flow model of machine 𝑀_𝑁 with the flow model of (2.1). For (3.5), (3.6) new considerations are applied for the production error of each machine 𝑀_𝑗, where 𝑗=1,…,𝑁−1. Here, 𝜀_𝑗(𝑘) depends on number of produced products 𝑦_𝑗(𝑘) with respect to current demand 𝑦_𝑑(𝑘) and desired buffer content (buffer inventory level) 𝑤_𝑑𝑗+1 of each downstream buffer. This means that every upstream machine needs to produce 𝑤_𝑑𝑗+1 lots more than the downstream one. Constant parameter 𝑤_𝑑 is introduced in order to prevent downstream machines from starvation, for example, in case of a sudden growth of the product demand.

Figure 2 shows a schematic of the information flow throughout the manufacturing line of 𝑁 machines. The squares represent the manufacturing machines each with inside label 𝑀_𝑗 and outside short thick black arrows denoting the external perturbations (𝑓_𝑖), which are affecting its production rates. Each machine (exept for 𝑀₁) has a buffer connected to it, each one denoted by 3 joined squares. The product flow directions are denoted by a thick white arrows with a thin black frame. The transferring of the production tracking error (𝜀_𝑗) information is shown by arched thin black arrows, going from one machine to another in upstream manner. For each machine, the upstream and downstream inventory level (𝑤_𝑗) information transfer is depicted by a curved thin black arrow coming from each buffer and the desired downstream buffer inventory level is shown by a short thin black arrow pointing to each machine.

Figure 2 

Flow model diagram for a line of 𝑁 manufacturing machines.

The substantial difference in the model for 𝑁 machines compared to the model of one machine can be appreciated through the flow model (3.2) as well as graphically through the Figure 2. For each machine, we introduce an extra restriction on production that is based on the buffer content of its upstream and downstream buffer. Functions sign_Buff(𝑤_𝑗(𝑘)−𝛽_𝑗(𝑘)) and sign₋(𝑤_𝑗+1(𝑘)−𝛾_𝑗+1) together with the control input 𝑢_𝑗(𝑘) are acting as the restrictions that are imposed on production of 𝑀_𝑗. Thus, any machine 𝑀_𝑗, with 𝑗=2,…,𝑁−1, is activated only if three authorizations are given. The first authorization comes from control input of 𝑀_𝑗, which is based on the current production error value of this machine (𝜀_𝑗(𝑘)). The second authorization comes from the restriction on the upstream buffer content that is granted if the buffer contains at least the minimal number of products required (𝛽_𝑗(𝑘)) in order for the machine 𝑀_𝑗 to start its work. Finally, the third authorization comes from the downstream buffer of given machine. This authorization is possible only if the downstream buffer have sufficient storage in order to accept incoming production.

Note that we could easily associate this control algorithm with Basestock policy (see [7] and references there in) as well as with Hedging Point policy (see [1] and references there in). From (3.5) the production error of each intermediate machine can be interpreted as 𝜀_𝑗(𝑘)=𝑦_𝑑(𝑘)−𝑦_𝑗(𝑘)+𝑤_𝑑𝑗+1+⋯+𝑤_𝑑𝑁,𝑗=1,…,𝑁−2.

This means that each machine is keeping track of the current demand as well as of its Hedging point or its Basestock level, which in this case is the sum of the desired buffer contents of all the downstream buffers of 𝑀_𝑗. Also, due to our intermediate buffer content limitation (𝛾_𝑗), this policy could be associated to a Kanban or a combination of local Conwip controllers (one for each intermediate machine) and a surplus-based pull control (for the output machine 𝑀_𝑁).

It is also important to take into account that the control actions are decentralized throughout the network. In other words, the control action of each machine in the line depends only on the production error of its neighboring downstream machine (except for machine 𝑀_𝑁, which control action depends directly on cumulative demand input) and the current buffer content of its upstream and downstream buffer (see Figure 2). This gives our flow model an extra robustness with respect to the undesired events such as temporal machine setup or breakdown.

For further analysis, let us rewrite flow model (3.1), (3.2), and (3.3) in feedback with (3.4) asΔ𝜀₁(𝑘)=𝑣_𝑑+Δ𝜑(𝑘)−𝑓₁(𝑘)−𝜇₁sign₊𝜀₁(𝑘)sign₋𝑤₂(𝑘)−𝛾₂,Δ𝜀_𝑗(𝑘)=𝑣_𝑑+Δ𝜑(𝑘)−𝑓_𝑗(𝑘)−𝜇_𝑗sign₊𝜀_𝑗(𝑘)sign_Buff𝑤_𝑗(𝑘)−𝛽_𝑗(𝑘)sign₋𝑤_𝑗+1(𝑘)−𝛾_𝑗+1,Δ𝜀_𝑁(𝑘)=𝑣_𝑑+Δ𝜑(𝑘)−𝑓_𝑁(𝑘)−𝜇_𝑁sign₊𝜀_𝑁(𝑘)sign_Buff𝑤_𝑁(𝑘)−𝛽_𝑁(𝑘), (3.8)(3.9)(3.10) where Δ𝜀_𝑗(𝑘)=𝜀_𝑗(𝑘+1)−𝜀_𝑗(𝑘).

For system (3.8), (3.9), and (3.10), let us introduce the following assumptions.

Assumption 3.1 (Boundedness of perturbations). There are constants 𝛼₁, 𝛼₂, and 𝛼₃ such that 𝑊_𝑗(𝑘)=Δ𝜑(𝑘)−𝑓_𝑗(𝑘),forall𝑗=1,…,𝑁, satisfies 𝛼₁<𝑊_𝑗(𝑘)<𝛼₂,∀𝑘∈ℕ, (3.11) and 𝑓_𝑗(𝑘) satisfies 𝑓_𝑗(𝑘)≤𝛼₃,∀𝑘∈ℕ. (3.12)

Assumption 3.2 (Capacity condition). Constants 𝛼₁, 𝛼₂ satisfy the following inequalities 𝛼₂<𝜇_𝑗−𝑣_𝑑,𝛼₁>−𝑣_𝑑. (3.13)(3.14) Thus, from (3.11), (3.13), and (3.14), the following condition holds: 0<𝑣_𝑑+𝑊_𝑗(𝑘)<𝜇_𝑗,∀𝑗=1,…,𝑁. (3.15)

It is important to notice that each machine 𝑀_𝑗 in the line has a processing speed of 𝜇_𝑗 lots per time unit, which can differ from the rest of the machines.

The buffer content condition is considered as𝛽_𝑗(𝑘)≤𝑤_𝑗(𝑘)<𝛾_𝑗,∀𝑗=2,…,𝑁. (3.16) Note that the physical restriction on buffer content is given as0≤𝑤_𝑗(𝑘)≤𝛾_𝑗+𝜇_𝑗−1+𝛼₃,∀𝑗=2,…,𝑁. (3.17) Here, 𝛾_𝑗=𝜇_𝑗+𝛼₂−𝛼₁+𝑤_𝑑𝑗. Thus, from (3.5), (3.6) and (3.16), the following tracking error condition holds𝜀_𝑗(𝑘)≥𝛽_𝑗(𝑘)−𝑤_𝑑𝑗+𝜀_𝑗−1(𝑘),∀𝑗=2,…,𝑁, (3.18) where 𝑤_𝑑𝑗 satisfies the following assumption.

Assumption 3.3 (Desired buffer content condition). The constants 𝑤_𝑑𝑗 comply with the following inequality: 𝑤_𝑑𝑗≥𝜇_𝑗+𝜇_𝑗−1+𝛼₃+𝛼₂−𝛼₁, (3.19) from where it follows that 𝑤_𝑑𝑗≥𝛽_𝑗(𝑘)+𝜇_𝑗−1+𝛼₂−𝛼₁,∀𝑘∈ℕ,𝑗=2,…,𝑁. (3.20)

One can note the following relation between the buffer content and the tracking errors of its neighboring machines. That is, if the first part of inequality (3.16) is not satisfied, that is, the buffer content is lower than the minimum, then𝜀_𝑗−1(𝑘)>^(),()3.63.17𝜇_𝑗−1+𝛼₂−𝛼₁+𝜀_𝑗(𝑘). (3.21) In case the second part of (3.16) is unsatisfied, that is, the buffer is full, then𝜀_𝑗(𝑘)≥^()3.6𝜇_𝑗+𝛼₂−𝛼₁+𝜀_𝑗−1(𝑘). (3.22)

3.2. Results on Performance

In this section, we present obtained results on the production tracking error trajectories behavior of flow model (3.8).

Theorem 3.4. Assume that the discrete time system defined by (3.8), (3.9), and (3.10) satisfies Assumptions 3.1, 3.2, and 3.3. Then, all solutions of (3.8), (3.9), and (3.10) are uniformly ultimately bounded by limsup_𝑘→∞𝜀_𝑗(𝑘)≤𝑣_𝑑+𝛼₂,liminf_𝑘→∞𝜀_𝑗(𝑘)≥𝑣_𝑑+𝛼₁−𝜇_𝑗. (3.23)

Proof. The proof of Theorem 3.4 is given in Appendix B.

Theorem 3.5. Assume that the discrete time system defined by (3.8), (3.9), and (3.10) satisfies Assumptions 3.1 and 3.2. Then, all solutions of (3.8), (3.9), and (3.10) are uniformly ultimately bounded by limsup_𝑘→∞𝜀_𝑗(𝑘)≤𝑣_𝑑+𝛼₂+𝑥_𝑗,liminf_𝑘→∞𝜀_𝑗(𝑘)≥𝑣_𝑑+𝛼₁−𝜇_𝑗, (3.24)(3.25) where 𝑥₁=0 and 𝑥_𝑗=∑^𝑗_𝑖=2max((𝜇_𝑖−1−𝛼₁+𝛼₂−𝑤_𝑑𝑖+(𝜇_𝑖+𝛼_3𝛽𝑖)),0) for all 𝑗=2,…,𝑁.

Proof. The proof of Theorem 3.5 is given in Appendix C.

Now, in order to support the proposed development, let us supplement our analysis by a simulation example.

4. Simulation Examples

Consider the following example of a production line of 4 manufacturing machines (see Figure 1) operating under surplus-based regulators (3.4). The processing speed for each machine is set to 𝜇_𝑗=6 lots per time unit forall𝑗=1,3 and 𝜇_𝑗=4 lots per time unit forall𝑗=2,4, the desired inventory level of each buffer is selected considering (3.20) as 𝑤_𝑑𝑗=12 lots, with 𝑗=2,…,4, and the mean demand rate 𝑣_𝑑=3.5 lots per time unit with fluctuation rate of Δ𝜑(𝑘)=0.2sin(5𝑘). The tracking error of each machine in the line is depicted in Figure 3. Here, the initial conditions (𝑦_𝑑0, 𝑦₁(0), 𝑦₂(0), 𝑦₃(0), 𝑦₄(0)) were set to the zero value. After the first 40 time steps, as it is shown in Figures 4 and 3, the system reaches its steady state. Tracking errors (see the dashed lines of Figure 3) are maintained inside [−2.7, 3.7] lots for machine 𝑀₁, [−0.7, 3.7] lots for machine 𝑀₂, [−2.7, 3.7] lots for machine 𝑀₃, and [−0.7, 3.7] lots for machine 𝑀₄, which satisfy the bounds (3.23). From Figure 5 it can be observed that the inventory level of each buffer satisfies the buffer limit given by the second part of inequality (3.17) and the capacity condition (3.16) is sometimes violated due to the discrete nature of the model. Here, 𝛾₂=14.8 (lots), 𝛾₃=16.8 (lots), 𝛾₄=14.8 (lots).

(a)

(b)

(c)

(d)

Figure 3 

Tracking error 𝜀𝑗(𝑘).

Figure 4 

Outputs 𝑦𝑗(𝑘) versus demand 𝑦𝑑(𝑘).

(a)

(b)

(c)

Figure 5 

Buffer content 𝑤𝑗(𝑘).

Now, let us show the effectiveness of Theorem 3.5 by means of a following example. Consider a similar production line of 4 manufacturing machines (see Figure 1) operating under surplus-based regulators (3.4). The nominal speed for each machine is 𝜇₁+𝑓₁(𝑘)=10.56+0.5sin(180𝑘), 𝜇₂+𝑓₂(𝑘)=10.7+0.5sin(20𝑘), 𝜇₃+𝑓₃(𝑘)=15.5+0.5sin(45𝑘) and 𝜇₄+𝑓₄(𝑘)=20.5+0.5sin(90𝑘) lots per time unit. The desired inventory level of each buffer is selected as 𝑤_𝑑𝑗=𝜇_𝑗𝜌 lots, with 𝑗=2,…,4 and 𝜌 as a constant. The experiment was executed 37 times. Thus, the value of constant 𝜌 was modified 37 times as well, starting from 𝜌=1 and with increments of 0.25 units till 𝜌=10. The demand rate was selected as 𝑣_𝑑+Δ𝜑(𝑘)=10+0.05cos(45𝑘) lots per time unit. Here, each 𝛾_𝑗 was selected according to (3.16). For this experiment, the initial conditions (𝑦_𝑑0, 𝑦₁(0), 𝑦₂(0), 𝑦₃(0), 𝑦₄(0)) were set to the zero value.

The relation between the maximal value of steady-state (SS) production tracking errors of machines 𝑀₂, 𝑀₃, 𝑀₄ (see Figure 1) and the desired inventory levels of there upstream buffers is depicted in Figure 6. Each graphic of Figure 6 shows the maximal value of each 𝜀_𝑗 in SS with its values from simulation and from analytical result given by (3.24). It could be appreciated that our analytical result describes in rather accurate manner the tradeoff between the desired inventory level and the accuracy of the production demand satisfaction. It is important to notice that the production tracking error precision is limited. Thus, for each of 3 machines, the production surplus can be decreased by incrementing the desired inventory level till some threshold value, which is given by (3.20), after which the bound on the surplus value remains constant. Doing so improves the service level of the network but increases the inventory costs. At the same time, lowering the inventory level further than the threshold decreases the service level of the network. This trade-off relationship plays an important role in the decision-making process of a production line manager. By using the relation provided beforehand in Theorem 3.5, a number of scheduling related decisions can be significantly reduced, which in consequence can decrease a computational time spent on planning for an efficient distribution of resources in a production line. In conclusion, presented simulation results reflect the desired flow model behavior, that is, all technical conditions proposed in this section correspond to analytical results described in Section 3. Also, the result shown on Figure 6 underlines the practical importance of the obtained theoretical results.

Figure 6 

Production tracking errors versus desired inventory levels.

5. Conclusion

In the case of a single manufacturing machine in isolation, a surplus-based pull controller was proven to be optimal for the cumulative production demand tracking problem. This control strategy was extended to a tandem production line with variable processing speed of each machine and restrictions on the sizes of intermediate buffers. The performance of the closed-loop system was addressed in the form of bounds on the production tracking errors that occur for each machine in the line, respectively. The analytical results describing the tradeoff relationship between the production tracking errors and the inventory level in the manufacturing line were obtained. All theoretical results were illustrated and confirmed by computer simulation. In simulation results, it was shown that our analytical results on performance can be used as an important tool in the decision-making process of the production line manager.

Our future work includes study of performance of re-entrant networks, and multiple part type production systems. A special attention will be given to detailed production analysis of these networks in the presence of unexpected delays and setup times.

Appendices

A. Proof of Theorem 2.1

Based on (2.4), it is easy to see that without any loss of generality, the class of admissible control strategies (2.3) can be reduced to those processing only the tracking errors:𝑢(𝑘)=𝑈_𝑘[𝜀(0),…,𝜀(𝑘)]∈{0;1}. (A.1)

We start with the problem (2.6). The proof is based on the min-max dynamic programming. So, we first introduce the cost-to-go:𝑉_𝜏(𝑎)=min_{𝑈𝜏(⋅),…,𝑈𝑇−1(⋅)}sup_𝜉(⋅)^𝑇_𝑘=𝜏|𝜀(𝑘)|^𝑝,𝑉_𝑇[𝑎]∶=|𝑎|^𝑝, (A.2) where the minimum is over all functions 𝑈_𝑘(𝜀_𝑘,…,𝜀_𝑇−1)∈{0;1}, and 𝜀(𝑘) is obtained from (2.4), where 𝑘=𝜏,…,𝑇−1 and 𝜀(𝜏)=𝑎. This function satisfies the Bellman equation ([24]):𝑉_𝜏−1(𝑎)=min_𝑢=0;1sup_()𝜉∈0;1|𝑎|^𝑝+𝑉_𝜏𝑎−𝑢+𝜉, (A.3) and the optimal strategy is given by 𝑢(𝜏−1)=𝑈⁰_𝜏−1[𝜀(𝜏−1)], where 𝑈⁰_𝜏−1[𝑎] is the point 𝑢 furnishing the minimum in (A.3).

Lemma A.1. The cost-to-go (A.2) is the piece-wise smooth even function depicted in Figures 7 and 8, and𝑈⁰_𝜏(𝑎)=sign₊(𝑎)for𝜏=0,…,𝑇−1. (A.4)

(a)

(b)

Figure 7 

(a) The graph of 𝑉𝑇; (b) the graph of 𝑉𝑇−1.

(a)

(b)

Figure 8 

(a) The graph of 𝑉𝑇−2; (b) the graph of 𝑉𝑇−𝑛 with 𝑛≥3.

Proof. We first note that (A.3) can be shaped into 𝑉_𝜏−1(𝑎)=minsup_𝜉∈(0;1)𝑉_𝜏[𝑎^{𝑆00𝑥0𝑓𝑒37}+𝜉];sup_𝜉∈(0;1)𝑉_𝜏[𝑎^{𝑆10𝑥0𝑓𝑒37}−𝜉]+|𝑎|^𝑝. (A.5) Here, 𝑆₀ and 𝑆₁ correspond to 𝑢=0 and 𝑢=1, respectively. So, 𝑈⁰_𝜏−1(𝑎)=𝜎_min, where 𝜎_min=0,1 is the index of the term 𝑆_𝜎 furnishing the minimum in (A.5). We also note that since the function 𝑎↦|𝑎|^𝑝 is even, simple induction on 𝜏=𝑇,…,0 and the last equation from (A.2) show that 𝑉_𝜏[⋅] is even for any 𝜏. With this in mind, it becomes clear that firstly, 𝜎_max=0,1 for 𝑎=0 and secondly, substitution 𝑎∶=−𝑎 in (A.5) switches 𝜎_min to the alternative value. This permits us to focus on 𝑎>0 in the subsequent proof. For 𝑎>0, formula (A.4) (to be justified) takes the form 𝑈⁰_𝜏(𝑎)=1.
We proceed with immediate proof of the lemma, arguing by induction on 𝜏=𝑇−𝑛,𝑛=0,1,….
(i)𝑛=0. The claim is immediate from the last equation in (A.2).(ii)𝑛=1. (a)𝑎≥1/2. Then evidently, 𝑆₁=|𝑎|^𝑝, and 𝑆₀=|𝑎+1|^𝑝>𝑆₁. So, due to (A.5), 𝑉_𝑇−1(𝑎)=2|𝑎|^𝑝, as is depicted in Figure 7(b), and 𝑈⁰_𝜏(𝑎)=1.(b)0<𝑎<1/2. Since𝑉_𝑇(⋅) is even,𝑆₁=|𝑎−1|^𝑝<|𝑎+1|^𝑝=𝑆₀. So𝑉_𝑇−1=|𝑎−1|^𝑝+|𝑎|^𝑝, as is depicted in Figure 7(b), and𝑈⁰_𝜏(𝑎)=1.(iii)𝑛=2. (a)𝑎≥1.Similarly, in (A.5), the supremum 𝑆₀ is equal to 2|𝑎+1|^𝑝, whereas 𝑆₁=2|𝑎|^𝑝<𝑆₀. (b)1/2≤𝑎<1: 𝑆₁=2|𝑎|^𝑝𝑎>^𝑝1/21𝑎<^𝑝||1/2<2𝑎+1^𝑝=𝑆₀, (A.6)(c)0≤𝑎<1/2: 𝑆₁=2|𝑎−1|^𝑝𝑎<1−^𝑝1/21𝑎>1−^𝑝||1/2<2𝑎+1^𝑝=𝑆₀. (A.7)Thus 𝑉_𝑇−2(𝑎)=3|𝑎|^𝑝𝑎≥^𝑝|𝑎|1/21+^𝑝1−^𝑝1/2≤𝑎<^𝑝2||1/2𝑎−1^𝑝+|𝑎|^𝑝𝑎<1−^𝑝1/2, (A.8) as is depicted in Figure 8(a), so 𝑈⁰_𝜏(𝑎)=1.
Figure 8(a) is a particular case of Figure 8(b). So to complete the proof, it suffices to show that:
(D) Figure 8(b) is correct and 𝑈⁰_𝑇−𝑛(𝑎)=1 for 𝑛=2,3,…, arguing by induction on 𝑛.
Suppose that (D) is true for some 𝑛≥2. To compute 𝑉_{𝑇−𝑛−1}(𝑎), we consider separately several cases.(i)𝑎≥^𝑝𝑛/(𝑛+1). Here ^𝑝𝑛/(𝑛+1)>^𝑝(𝑛−1)/𝑛. It follows that in (A.5) the supremum 𝑆₁ is attained at 𝜉=0 and thus equals (𝑛+1)|𝑎|^𝑝, whereas 𝑆₀=(𝑛+1)|𝑎+1|^𝑝>𝑆₁. Thus, (D) does hold for 𝑛∶=𝑛+1.(ii)^𝑝(𝑛−1)/𝑛≤𝑎≤^𝑝𝑛/(𝑛+1). Then evidently 𝑆₁=𝑛, whereas 𝑆₀=(𝑛+1)|𝑎+1|^𝑝>𝑆₁. Thus (D) does hold for 𝑛∶=𝑛+1.(iii)1−^𝑝(𝑛−1)/𝑛≤𝑎≤^𝑝(𝑛−1)/𝑛. Since the left end 𝑎−1 of the interval [𝑎−1,𝑎] is still to the right of the first fracture point of the graph from Figure 3(b), the situation replicates the previous one.(iv)1−^𝑝𝑛/(𝑛+1)≤𝑎≤1−^𝑝(𝑛−1)/𝑛. That end is to the left of the first fracture point. So either 𝑆₁=𝑛 (and is attained at the third fracture point) or 𝑆₁=(𝑛+1)|𝑎−1|^𝑝 (and is attained at 𝜉=1). Elementary comparison shows that in fact 𝑆₁=𝑛, and so the situation still replicates the previous two ones.(v)0≤𝑎≤1−^𝑝𝑛/(𝑛+1). Then, conversely, 𝑆₁=(𝑛+1)|𝑎−1|^𝑝, whereas 𝑆₀=(𝑛+1)|𝑎+1|^𝑝>𝑆₁. Thus, (D) does hold for 𝑛∶=𝑛+1, which completes the proof.

For the performance index (2.6), Theorem 2.1 is straightforward from Lemma A.1 and the dynamic programming principle ([24]).

To deal with (2.7), we introduce the following intermediate performance criterion:𝐽_av=limsup_𝑇→∞sup_{()𝜉(0),…,𝜉𝑇−1}1𝑇^𝑇_𝑘=0|𝜀(𝑘)|^𝑝. (A.9) It is clear thatinf_𝑈𝐽_av≥limsup_𝑇→∞1𝑇min_𝑈𝐽==_𝑇^(A).2=limsup_𝑇→∞𝑉^𝑇₀(𝑎)𝑇, (A.10) where the upper index 𝑇 in 𝑉^𝑇_𝜏 underscores that the cost-to-go is computed for the time horizon [0∶𝑇]. As a result, Lemma A.1 and the evident inequality 𝐽_∞≥𝐽_av imply the following lower estimates:inf_𝑈𝐽_∞≥inf_𝑈𝐽_av≥|𝑎|^𝑝if|𝑎|≥11otherwise. (A.11) Now, we are going to show that this lower estimate of 𝐽_∞ is attained at the control strategy (2.8), which will complete the proof.

Let the system (2.4) be driven by the control law (2.8). By invoking (2.5), we conclude that𝜀[]∈(𝜀[𝑘][𝑘(𝑘)(𝜀[𝑘][𝑘])(𝑘)(𝜀[𝑘][𝑘])(𝑘)𝑘+1−1,𝜀])if𝜀>0,𝜀+1if𝜀<0−1,𝜀+1if𝜀=0. (A.12) Hence,𝑓₋[𝜀(𝑘)]≤𝜀(𝑘+1)≤𝑓₊[𝜀(𝑘)],where𝑓₋(𝜀)∶=min{𝜀;−1},𝑓₊(𝜀)∶=max{𝜀;1}. (A.13) It follows that𝜀₋(𝑘)≤𝜀(𝑘)≤𝜀₊(𝑘)∀𝑘, (A.14) where 𝜀₋(𝑘) and 𝜀₊(𝑘) are the solutions of the following recursions:𝜀_±(𝑘+1)=𝑓_±𝜀_±(𝑘),𝜀_±(0)=𝑎. (A.15) It is evident that𝜀_±(𝑘)∈[min{−|𝑎|,−1};max{|𝑎|;1}], (A.16) which completes the proof.

B. Proof of Theorem 3.4

Proof. Let us prove that Theorem 3.4 holds for a line of 2 manufacturing machines (𝑗=1,2) defined by (3.8) and (3.10). With this goal, let us introduce the following Lyapunov function: 𝑉^2𝑀𝜀₁;𝜀₂=max𝑉₁𝜀₁,𝑉₂𝜀₂, (B.1) where 𝑉_𝑗𝜀_𝑗=max−𝜀_𝑗−𝜇_𝑗+𝑣_𝑑+𝛼₁,𝜀_𝑗−𝑣_𝑑−𝛼₂,0>0,∀𝜀_𝑗∉𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂,𝑗=1,2. (B.2) Here, for the sake of brevity 𝑉^2𝑀(𝜀₁(𝑘),𝜀₂(𝑘))=𝑉_𝑘^2𝑀, 𝑉_𝑗(𝜀_𝑗(𝑘))=𝑉_𝑗,𝑘, with 𝑉^2𝑀=0forall𝜀_𝑗∈[𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂].
Thus, Δ𝑉_𝑘^2𝑀 along the solutions of 𝜀₁(𝑘) and 𝜀₂(𝑘) is given by Δ𝑉_𝑘^2𝑀=𝑉^2𝑀_𝑘+1−𝑉_𝑘^2𝑀=max𝑉_1,𝑘+1,𝑉_2,𝑘+1+min−𝑉_1,𝑘,−𝑉_2,𝑘, (B.3) where 𝑉_𝑗,𝑘+1=max−𝜀_𝑗(𝑘)−𝑊_𝑗(𝑘)+𝛼₁−𝜇_𝑗+𝜇_𝑗𝜂_𝑗,𝑘,𝜀_𝑗(𝑘)+𝑊_𝑗(𝑘)−𝛼₂−𝜇_𝑗𝜂_𝑗,𝑘,0,𝑗=1,2. (B.4) Here, for the sake of brevity we introduce 𝜂_𝑗,𝑘 as 𝜂_1,𝑘=sign₊𝜀₁(𝑘)sign₋𝑤₂(𝑘)−𝛾₂,𝜂_2,𝑘=sign₊𝜀₂(𝑘)sign_Buff𝑤₂(𝑘)−𝛽₂(𝑘). (B.5) In order to perform a more detailed analysis on Δ𝑉_𝑘^2𝑀, let us divide this proof into 3 cases.
Case 1 (Sufficient buffer content). Suppose that 𝑤₂(𝑘) satisfies the following inequality: 𝛽₂(𝑘)≤𝑤₂(𝑘)<𝛾₂, (B.6) which means that machine 𝑀₂ has sufficient material in its buffer 𝐵₂ in order to start working and machine 𝑀₁ always has an access to the infinite raw material supply. Thus, these machines have an independent behavior and it will be sufficient to analyse the increment of only one of the functions 𝑉_𝑗,𝑘 in order to determine the behavior of Δ𝑉_𝑘^2𝑀.
Let us assume that 𝜀_𝑗(𝑘) satisfies the following condition: 𝜀_𝑗(𝑘)>0, (B.7) and in consequence from (B.5) it follows that 𝜂_𝑗,𝑘=1.
Then, Δ𝑉_𝑗,𝑘 along the solutions of 𝜀_𝑗(𝑘) is given by Δ𝑉_𝑗,𝑘=max−𝜀_𝑗(𝑘)−𝑊_𝑗(𝑘)+𝛼₁,𝜀_𝑗(𝑘)+𝑊_𝑗(𝑘)−𝛼₂−𝜇_𝑗,_{𝑉𝑗,𝑘+1}0𝜀+min^𝑗(𝑘)+𝜇_𝑗−𝑣_𝑑−𝛼₁,−𝜀_𝑗(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑗,𝑘}0,𝑗=1,2. (B.8) From where with help of Assumptions 3.1 and 3.2, it can be easily deduced that Δ𝑉_𝑗,𝑘=0if𝜀_𝑗(𝑘)≤𝑣_𝑑+𝛼₂,−𝜀_𝑗(𝑘)+𝑣_𝑑+𝛼₂<0if𝑣_𝑑+𝛼₂<𝜀_𝑗(𝑘)≤𝜇_𝑗+𝛼₂−𝑊_𝑗(𝑘),−𝜇_𝑗+𝑣_𝑑+𝑊_𝑗(𝑘)<0if𝜀_𝑗(𝑘)>𝜇_𝑗+𝛼₂−𝑊_𝑗(𝑘). (B.9) Now, suppose that for 𝜀_𝑗(𝑘) the following condition holds: 𝜀_𝑗(𝑘)≤0, (B.10) and in consequence from (B.5) it yields that 𝜂_𝑗,𝑘=0. Then, Δ𝑉_𝑗,𝑘 along the solutions of 𝜀_𝑗(𝑘) is given by Δ𝑉_𝑗,𝑘=max−𝜀_𝑗(𝑘)−𝑊_𝑗(𝑘)+𝛼₁−𝜇_𝑗,𝜀_𝑗(𝑘)+𝑊_𝑗(𝑘)−𝛼₂,_{𝑉𝑗,𝑘+1}0𝜀+min^𝑗(𝑘)+𝜇_𝑗−𝑣_𝑑−𝛼₁,−𝜀_𝑗(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑗,𝑘}0,𝑗=1,2. (B.11) Here, with help of Assumptions 3.1 and 3.2 it can be easily deduced that Δ𝑉_𝑗,𝑘=0if𝑣_𝑑+𝛼₁−𝜇_𝑗≤𝜀_𝑗(𝑘)≤0,−𝑣_𝑑−𝑊_𝑗(𝑘)<0if𝜀_𝑗(𝑘)<−𝜇_𝑗+𝛼₁−𝑊_𝑗(𝑘),𝜀_𝑗(𝑘)+𝜇_𝑗−𝑣_𝑑−𝛼₁<0if−𝜇_𝑗+𝛼₁+𝑊_𝑗(𝑘)≤𝜀_𝑗(𝑘)<𝑣_𝑑+𝛼₁−𝜇_𝑗. (B.12)
Summarizing, for conditions (B.6), (B.7), and (B.10) from (B.9) and (B.12) it holds that if 𝑉_𝑗,𝑘>0, its increment Δ𝑉_𝑗,𝑘<0. From the definition of min, it yields that for Δ𝑉_𝑘^2𝑀 given by (B.3) the following inequality is satisfied: Δ𝑉_𝑘^2𝑀≤𝑉_𝑖,𝑘+1−𝑉_𝑖,𝑘≤0, (B.13) where 𝑖=argmax_𝑗=1,2{𝑉_𝑗,𝑘+1}. Note that Δ𝑉_𝑗,𝑘=𝑉_𝑗,𝑘+1−𝑉_𝑗,𝑘=0 only if either first condition of (B.9) or first condition of (B.12) is satisfied and for all 𝜀_𝑗(𝑘)∉[𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂] it follows that Δ𝑉_𝑘^2𝑀<0. Thus, in this case it holds that for 𝑉_𝑘^2𝑀>0 given by (B.1) its increment Δ𝑉_𝑘^2𝑀<0.Case 2 (Insufficient buffer content). Let us assume that 𝑤₂(𝑘) satisfies the following inequality: 𝑤₂(𝑘)<𝛽₂(𝑘),∀𝑘∈ℕ, (B.14) and 𝜀₂(𝑘) satisfies 𝜀₂(𝑘)≤0,∀𝑘∈ℕ. (B.15) Then, from (3.21) it holds that 𝜀₁(𝑘)>𝜇₁+𝛼₂−𝛼₁+𝜀₂(𝑘). (B.16) Here, similarly to Case 1, the behavior of these two machines can be considered independently. Thus, for 𝜀₁(𝑘) satisfying (B.16), it holds that Δ𝑉_1,𝑘 is given by (B.9) or (B.12) if 𝜀₂(𝑘)<−𝜇₁−𝛼₂+𝛼₁ or Δ𝑉_1,𝑘 is given by (B.9) if −𝜇₁−𝛼₂+𝛼₁≤𝜀₂(𝑘)≤0. For 𝜀₂(𝑘) satisfying (B.15) the increment Δ𝑉_2,𝑘 is given by (B.12). In consequence for Δ𝑉_𝑘^2𝑀 given by (B.3), the inequality (B.13) in this case is also satisfied.
Now, let us assume that 𝜀₂(𝑘) satisfies 𝜀₂(𝑘)>0,∀𝑘∈ℕ. (B.17) Then, from (3.21), it holds that 𝜀₁(𝑘) is given by (B.16). In this case, 𝑀₂ has a positive tracking error, but its buffer 𝐵₂ has insufficient raw material content (B.14) in order to start working (𝜂_2,𝑘=0). Machine 𝑀₁ has a positive error as well, but due to its infinite raw material supply access, it can immediately initiate its production process (𝜂_1,𝑘=1). Thus, for (B.17) and (B.16), let us rewrite Δ𝑉_𝑘^2𝑀 from (B.3) as Δ𝑉_𝑘^2𝑀=max−𝜀₁(𝑘)−𝑊₁(𝑘)+𝛼₁,𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇₁,−𝜀₂(𝑘)−𝑊₂(𝑘)+𝛼₁−𝜇₂,𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂,_{𝑉2𝑀𝑘+1}0𝜀+min¹(𝑘)+𝜇₁−𝑣_𝑑−𝛼₁,−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂,𝜀₂(𝑘)+𝜇₂−𝑣_𝑑−𝛼₁,−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑘2𝑀}0. (B.18) It follows from (3.11), (B.17), and (B.16) that Δ𝑉_𝑘^2𝑀 from (B.18) can be reduced to Δ𝑉_𝑘^2𝑀𝜀=max¹(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇₁,𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂,_{𝑉2𝑀𝑘+1}0+min−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂,−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑘2𝑀}0. (B.19) Now, let us prove that, for 𝜀₁(𝑘) given by (B.16), inequality 𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇₁>𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂ (B.20) is satisfied.
Indeed, from condition (B.16), it yields that 𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇₁>𝜀₂(𝑘)+𝑊₁(𝑘)−𝛼>₁^(3.11)𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂. (B.21) Thus, inequality (B.20) is satisfied. Also, from (B.21), it holds that 𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇>₁^{(),(3.11B).17}0. (B.22) Now, considering (B.20) and (B.22), we can rewrite 𝑉^2𝑀_𝑘+1 given by the first term of (B.19) as 𝑉^2𝑀_𝑘+1=𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂−𝜇₁. (B.23) Let us prove that, for 𝜀₁(𝑘) given by (B.16), inequality −𝜀₂(𝑘)+𝑣_𝑑+𝛼₂>−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂ (B.24) is satisfied. Here, from condition (B.16), it yields that −𝜀₂(𝑘)+𝑣_𝑑+𝛼₂>−𝜀₁(𝑘)+𝜇₁+𝛼₂−𝛼₁+𝑣_𝑑+𝛼>₂^{(3.11),(3.14)}−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂. (B.25) Thus, inequality (B.24) is satisfied. From inequalities (B.16), (3.13), it follows that −𝜀₁(𝑘)+𝑣_𝑑+𝛼₂<0. (B.26) From (B.24), (B.26), we can rewrite 𝑉_𝑘^2𝑀 given by the second term of (B.19) as 𝑉_𝑘^2𝑀=𝜀₁(𝑘)−𝑣_𝑑−𝛼>₂^(B.26)0. (B.27) Having 𝑉^2𝑀_𝑘+1 given by (B.23) and 𝑉_𝑘^2𝑀 given by (B.27), we can finally reduce Δ𝑉_𝑘^2𝑀 from (B.19) to 𝑉_𝑘^2𝑀=−𝜇₁+𝑣_𝑑+𝑊₁(𝑘)<^(3.15)0. (B.28) Thus, for this case, it holds that for 𝑉_𝑘^2𝑀>0 given by (B.1) its increment Δ𝑉_𝑘^2𝑀<0.
Case 3 (Limited buffer content). Suppose that 𝑤₂(𝑘) satisfies the following inequality: 𝑤₂(𝑘)≥𝛾₂,∀𝑘∈ℕ, (B.29) and let us first assume that 𝜀₂(𝑘) satisfies 𝜀₂(𝑘)≤0,∀𝑘∈ℕ. (B.30) Then, from (3.22), it holds that 𝜀₁(𝑘)≤𝜀₂(𝑘)−𝜇₂−𝛼₂+𝛼₁, (B.31) where −𝜇₂−𝛼₂+𝛼<₁^(3.11)0.
In this case, machines 𝑀₁ and 𝑀₂ are not working (𝜂_𝑗,𝑘=0) and their behavior can be considered similar to the first part of Case 2. It follows that for (B.30) and (B.31) the increments Δ𝑉_1,𝑘 and Δ𝑉_2,𝑘 are given by (B.12), respectively. Thus, for Δ𝑉_𝑘^2𝑀 given by (B.3) the inequality (B.13) in this case is also satisfied.
Now, let us assume that 𝜀₂(𝑘) satisfies 𝜀₂(𝑘)>0,∀𝑘∈ℕ. (B.32) In this case, 𝑀₂ has sufficient material to start working (𝜂_2,𝑘=1) and 𝑀₁ is stopped (𝜂_1,𝑘=0) due to the limited capacity of its downstream buffer 𝐵₂. Thus, two situations may occur. Firs, consider that 𝑀₁ is stopped, but its tracking error 𝜀₁(𝑘)≤0. This may occur if 𝜀₂(𝑘) satisfies 0<𝜀₂(𝑘)≤^(B.31)𝜇₂+𝛼₂−𝛼₁. The behavior of these 2 machines can be considered independently and by following the procedure from Case 1, we arrive to the conclusion that for 𝑉_𝑘^2𝑀>0 given by (B.1) its increment Δ𝑉_𝑘^2𝑀<0. In the second situation, consider that 𝜀₁(𝑘) satisfies 𝜀₁(𝑘)>0,∀𝑘∈ℕ, (B.33) which by (B.31) implies that 𝜀₂(𝑘)>𝜇₂+𝛼₂−𝛼₁. (B.34) Then, for (B.33) and (B.34), let us rewrite Δ𝑉_𝑘^2𝑀 from (B.3) as Δ𝑉_𝑘^2𝑀=max−𝜀₁(𝑘)−𝜇₁−𝑊₁(𝑘)+𝛼₁,𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂,−𝜀₂(𝑘)−𝑊₂(𝑘)+𝛼₁,𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂,_{𝑉2𝑀𝑘+1}0𝜀+min¹(𝑘)+𝜇₁−𝑣_𝑑−𝛼₁,−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂,𝜀₂(𝑘)+𝜇₂−𝑣_𝑑−𝛼₁,−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑘2𝑀}0. (B.35)
It follows from (3.11) and (3.15) that Δ𝑉_𝑘^2𝑀 from (B.35) can be reduced to Δ𝑉_𝑘^2𝑀𝜀=max¹(𝑘)+𝑊₁(𝑘)−𝛼₂,𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂,_{𝑉2𝑀𝑘+1}0+min−𝜀₁(𝑘)+𝑣_𝑑+𝛼₂,−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂,_{−𝑉𝑘2𝑀}0. (B.36) Now, let us derive from (B.31) that the following inequality 𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂>𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂ (B.37) is satisfied. Indeed, from (B.31), it holds that 𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂≥𝜀₁(𝑘)+𝜇₂+𝛼₂−𝛼₁+𝑊₂(𝑘)−𝛼₂−𝜇_2(3.11)>𝜀₁(𝑘)+𝑊₁(𝑘)−𝛼₂. (B.38) Thus, inequality (B.37) is satisfied. From (B.34) and (3.11), it also holds that 𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂>0. (B.39) Considering (B.37) and (B.39), we can rewrite 𝑉^2𝑀_𝑘+1 from the first part of (B.36) as 𝑉^2𝑀_𝑘+1=𝜀₂(𝑘)+𝑊₂(𝑘)−𝛼₂−𝜇₂. (B.40) Let us show that from (B.31) the following inequality: −𝜀₁(𝑘)+𝑣_𝑑+𝛼₂>−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂ (B.41) is satisfied. Here, from condition (B.31), it yields that −𝜀₁(𝑘)+𝑣_𝑑+𝛼₂>−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂+𝜇₂+𝛼₂−𝛼>₁^{(3.11),(3.14)}−𝜀₂(𝑘)+𝑣_𝑑+𝛼₂. (B.42) Thus, inequality (B.41) is satisfied. From inequalities (B.31), (3.13), it follows that −𝜀₂(𝑘)+𝑣_𝑑+𝛼₂<0. (B.43) From (B.41), (B.43), we can rewrite 𝑉_𝑘^2𝑀 given by the second part of (B.36) as 𝑉_𝑘^2𝑀=𝜀₂(𝑘)−𝑣_𝑑−𝛼>₂^(B.43)0. (B.44) Having 𝑉^2𝑀_𝑘+1 given by (B.40) and 𝑉_𝑘^2𝑀 given by (B.44), we can finally reduce Δ𝑉_𝑘^2𝑀 from (B.36) to Δ𝑉_𝑘^2𝑀=−𝜇₂+𝑣_𝑑+𝑊₂(𝑘)<^(3.15)0. (B.45) Thus, for this case it holds that for 𝑉_𝑘^2𝑀>0 given by (B.1) its increment Δ𝑉_𝑘^2𝑀<0. Summarizing for 3 cases, we have shown that for 𝑉_𝑘^2𝑀>0 given by (B.1) its increment Δ𝑉_𝑘^2𝑀<0 for all 𝜀_𝑗(𝑘)∉[𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂] and Δ𝑉_𝑘^2𝑀=0 for all 𝜀_𝑗(𝑘)∈[𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂]. Thus, limsup_𝑘→∞𝑉_𝑘^2𝑀=0 which completes our proof.
In this proof, we have analyzed the increment of the proposed Lyapunov function by means of 3 cases. Now, for a line of 𝑁 manufacturing machines (𝑗=1,…,𝑁) defined by (3.8), (3.9), and (3.10), the Lyapunov function (B.1) is extended to 𝑉_𝑘^𝑁𝑀=max𝑉₁𝜀₁,…,𝑉_𝑁𝜀_𝑁. (B.46) Here, the same reasoning is followed as for the proof for 2 machines.

C. Proof of Theorem 3.5

For a different value of the desired buffer inventory level constant, the Lyapunov function is modified as follows. The function for 𝑗=1 remains as in (B.2), and for 𝑗=2,…,𝑁, it is now given by𝑉_𝑗𝜀_𝑗=max−𝜀_𝑗−𝜇_𝑗+𝑣_𝑑+𝛼₁,𝜀_𝑗−𝑣_𝑑−𝛼₂−^𝑗_𝑖=2max𝜇_𝑖−1−𝛼₁+𝛼₂−𝑤_𝑑𝑖+𝜇_𝑖+𝛼_3𝛽𝑖,0_𝑥𝑗,0>0,∀𝜀_𝑗∉𝑣_𝑑+𝛼₁−𝜇_𝑗,𝑣_𝑑+𝛼₂+𝑥_𝑗,𝑗=2,…,𝑁. (C.1) Thus, Δ𝑉_𝑘^𝑁𝑀 along the solutions of 𝜀_𝑗(𝑘) is given byΔ𝑉_𝑘^𝑁𝑀=𝑉^𝑁𝑀_𝑘+1−𝑉_𝑘^𝑁𝑀=max𝑉_1,𝑘+1,…,𝑉_𝑁,𝑘+1+min−𝑉_1,𝑘,…,−𝑉_𝑁,𝑘, (C.2) where for 𝑗=2,…,𝑁,𝑉_𝑗,𝑘+1=max−𝜀_𝑗(𝑘)−𝑊_𝑗(𝑘)+𝛼₁−𝜇_𝑗+𝜇_𝑗𝜂_𝑗,𝑘,𝜀_𝑗(𝑘)+𝑊_𝑗(𝑘)−𝛼₂−𝜇_𝑗𝜂_𝑗,𝑘−𝑥_𝑗,0. (C.3) Here, 𝜂_𝑗,𝑘 is given by𝜂_1,𝑘=sign₊𝜀₁(𝑘)sign₋𝑤₂(𝑘)−𝛾₂,𝜂_𝑗,𝑘=sign₊𝜀_𝑗(𝑘)sign_Buff𝑤_𝑗(𝑘)−𝛽_𝑗(𝑘)sign₋𝑤_𝑗+1(𝑘)−𝛾_𝑗+1,𝜂_𝑁,𝑘=sign₊𝜀_𝑁(𝑘)sign_Buff𝑤_𝑁(𝑘)−𝛽_𝑁(𝑘). (C.4)

The analysis of (C.2) is subdivided into the same 3 cases. Case 1 (Sufficient buffer content) and the first part of Case 2 (𝑤_𝑗(𝑘)<𝛽_𝑗(𝑘) and 𝜀_𝑗(𝑘)≤0,forall𝑗=2,…,𝑁) are solved identically to the proof for the line of 2 machines. For the second part of Case 2 and for the Case 3 (just as in proof of Theorem 3.4), the proof relies on the condition (3.16) in combination with (3.5) and (3.6). Due to the extensive technical details and the similarity of the procedure with the proof of Theorem 3.5, we omit the complete analysis for a line of 𝑁 machines and restrict ourselves by only giving this general idea of the procedure.

Acknowledgments

The research leading to these results has received partial funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. INFSO-ICT-223844. This work was supported by the Russian Foundation for Basic Research (Grant 09-08-00803) and the Russian Federal Programm (Grants N 2010-1.1-111-128-033 and 16.740.11.0042).