Introduction

Figure 1.1: Serial production line

Figure 1.2: Assembly system

If the machines never breakdown, the situation would be quite simple: to ensure smooth part flow, balance the capacity of the machines and speeds of the material handling devices so that the desired throughput is achieved. (The notions of machine capacity, throughput, etc., are defined precisely in Chapter 3; at this point, just intuitive understanding of these terms is sufficient.)

In reality, however, the machines always experience random breakdowns. This leads to a complex phenomenon of perturbation propagation. Indeed, when, say, machine m₂ in the serial line fails, machine m₃ may eventually become starved for parts and machine m₁ blocked by full buffer b₁. If m₂ continues staying down, the starvations will propagate downstream reaching eventually the last machine, m_M, and the throughput is lost. Similarly, the blockage will propagate upstream, causing all machines to stop and, hence, wasting their production capacity. Thus, failure of one machine may affect all other machines in the system – up- and downstream. This is what makes the investigation of production systems difficult (and necessary too – in order to develop methods for alleviating this perturbation propagation).

Similar phenomena take place in assembly systems as well: failures of m₁₂ (see Figure 1.2) may lead to the starvation of m₁₃,..., m_1M1 , m₀₁,..., m_0M0 and to blockage of m₁₁ and m₂₁,..., m_2M2 . (Note that it is typically assumed that the first machines m₁ and (m₁₁, m₂₁) are never starved and the last machines m_M and m_0M0 are never blocked.)

To alleviate this mutual interference, the material handling devices are used not only as means for moving parts but also as buffers – to protect against blockages and starvations. If the buffers are infinite, only starvations are possible, and the parts flow is improved. However, infinite buffers are impractical, economically detrimental, and, as it turns out, unnecessary to ensure the desired system performance.

This textbook presents rigorous engineering methods for reducing mutual machine interference by selecting buffer capacities, which are "just right" (rather than just-in-time) for guaranteeing a sufficiently smooth flow of parts and, thus, resulting in an acceptable system behavior. To accomplish this, three main problems of production systems engineering are addressed. These problems are described next.

1.2.2 Analysis, continuous improvement, and design problems

Analysis: The problem of analysis is addressed in this book in two formulations:

Problem A1: Given a production system (i.e., the machine and buffer characteristics), calculate its performance measures, e.g., its throughput, work-in-process, and the probabilities of blockages and starvations.

The second formulation is concerned with systems having a Finished Goods Buffer (FGB), as shown in Figure 1.3 for the case of a serial line; similarly, assembly systems may have FGBs. The purpose of FGBs is to filter out production and demand randomness and, thus, improve the level of customer demand satisfaction.

Figure 1.3: Serial production line with a finished goods buffer

Problem A2: Given a production system and customer demand specifications (i.e., the shipment size, D, and the shipping period, T), calculate the probability of demand satisfaction.

Continuous improvement: The problem of continuous improvement is also considered in two formulations – constrained and unconstrained improvability.

Problem CI1 (Constrained improvability): Given a production system, determine if its performance can be improved by re-allocating its limited resources (such as buffer capacity and/or workforce) and suggest an improved or even optimal allocation.

Problem CI2 (Unconstrained improvability): Given a production system, determine the machine and the buffer, which impede system performance in the strongest manner; such a machine and a buffer are referred to as bottleneck machine (BN-m) and bottleneck buffer (BN-b), respectively.

Design: Design issues are formalized as three problems:

Problem D1: Given the machine characteristics and the desired system throughput, determine the smallest (i.e., lean or "just-right") buffers capacity, which ensures this throughput.

Figure 1.4: Queuing model

Unfortunately, production systems typically have a qualitatively different structure. Indeed, instead of servers operating in parallel, they have serial connection as shown in Figures 1.1 and 1.2. In some cases, however, methods of Queuing Theory may be modified to be applicable to production systems as well.

This approach is not pursued here. Instead, methods, which are directly applicable to serial lines and assembly systems, are developed. In most cases, they are based on a three-step approach: First, the simplest systems, consisting of two machines, are investigated using the methods of Markov processes. Second, systems with more than two machines are analyzed using the results of the first step and recursive aggregation procedures; the convergence of these procedures and their accuracy are investigated. Third, these results are extended to non-Markovian cases by discrete event simulations and subsequent analytical approximations. In this framework, the recursive procedures mentioned above are viewed as governing equations of production systems. Their analyses, described throughout this book, lead to formulation of the fundamental facts of Production Systems Engineering.