Addressing Industrial Problems Using Various Mathematical Models

Vol. 5, No. 3, May 2005, pp. 8–18 issn 1532-0545
05
0503
0008

informs

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doi 10.1287/ited.5.3.8 © 2005 INFORMS

I N F O R M S Transactions on Education

Addressing Industrial Problems Using Various Mathematical Models Kathryn E. Stecke

School of Management, University of Texas at Dallas, Richardson, Texas 75083, [email protected]

M

athematics has been called the language of science. Mathematics is used to solve many real-world problems in industry, the physical sciences, economics, social and human sciences, engineering, and technology, for example. We overview the many industrial problems that have been solved using fuzzy logic technology, multiobjective metaheuristics, neural networks, tabu search, genetic algorithms, simulated annealing, decision analysis, Petri nets, and queueing models. This overview could be useful for Ph.D. students in Industrial Engineering, Operations Research, and Operations Management who are looking to solve some real problems. Future industrial applications are also be described.

1.

Introduction

approach to solve a real-world problem type. This paper helps this quest by overviewing the variety of industrial problems that have been solved successfully using a variety of mathematical models or methods. View this as food for thought for your research.

The origin of the word mathematics is in the Greek word “manthanein,” to learn. Mathematics is a tool that helps us learn how to design and operate our systems in better ways. This paper is focused on the Ph.D. audience. A Ph.D. is a world expert on some combination of topic/model/method. One may be an expert on a model, but looking for possible application areas. Another may have solved a particular problem using some mathematical approach. This paper aims to fill gaps by overviewing what types of mathematical models have been used to address which industrial applications. It was recently suggested to me that a specific contribution of this type of classification would be to specify what types of models are the best to use to solve which industrial problems. I do not believe that this is possible. For any type of industrial problem, there are peculiarities specific to every application. For a specific industrial problem, the particular best solution approach has to be searched for and tested. As shown in this paper, many different models and approaches can and have been used to solve a particular industrial problem. For most models, parameters have to be fine-tuned for a specific application. For some specific industrial applications, a particular model may be shown to provide the best solutions. For another industrial application of the same type, a very different model type may prove to be best. This is one focus of some Ph.D. research: to examine the many methods or models to discover the best

2.

Fuzzy Logic Technology

Many decision-making and problem-solving tasks are too complex to be understood quantitatively. However, people succeed by using knowledge that is imprecise rather than precise. Fuzzy set theory, originally introduced by Zadeh (1965), resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit. Since knowledge can be expressed in a more natural way by using fuzzy sets, many engineering and decision problems can be greatly simplified. See Dubois and Prade (1980). Figure 1 provides an example of the simplification that can be provided by fuzzy set theory when making various decisions. In a situation such as in Figure 1, various traditional computing tools would use “precise information,” while fuzzy set theory may use a simple but “significant information.” Fuzzy set theory implements classes of groupings of data with boundaries that are not sharply defined (i.e., fuzzy). Any methodology or theory implementing 8

Stecke: Addressing Industrial Problems Using Various Mathematical Models INFORMS Transactions on Education 5(3), pp. 8–18, © 2005 INFORMS

Figure 1

Precision and Significance in the Real World

“crisp” definitions such as classical set theory, arithmetic, and programming, may be “fuzzified” by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries. The benefit of extending crisp theory and analysis methods to fuzzy techniques is the strength in solving real-world problems, which inevitably entail some degree of imprecision and noise in the variables and parameters measured and processed for the application. Accordingly, linguistic variables are a critical aspect of some fuzzy logic applications, where general terms such as large, medium, and small are used to capture a range of numerical values. While similar to conventional quantification, fuzzy logic allows these stratified sets to overlap (e.g., a 85 kilogram man may be classified in both the “large” and “medium” categories, with varying degrees of belonging or membership to each group). Fuzzy set theory encompasses fuzzy logic, fuzzy arithmetic, fuzzy mathematical programming, fuzzy topology, fuzzy graph theory, and fuzzy data analysis, although the term fuzzy logic is often used to describe all of these. Fuzzy logic is an artificial intelligence-like technology that allows computer systems to capture and represent inexact knowledge and relationships. Unlike binary logic used in many computer-based decisions, fuzzy logic relies on the use of “shades of grey” or fuzzy sets, which represent answers and possible answers in probabilistic terms. This allows computers to use heuristics to evaluate imperfect or incomplete sets of data and still arrive at reasonable answers. Fuzzy logic emerged into the mainstream of information technology in the late 1980’s and early 1990’s. Fuzzy logic is a departure from classical two-valued sets and Boolean logic in that it implements soft linguistic variables (i.e., large, tall, cold) on a continuous range of truth values which allows intermediate values to be defined between conventional binary (true or false). It can often be considered a superset of Boolean or crisp logic, in the way fuzzy set theory is a superset of conventional set theory. Formally, fuzzy logic is a structured, model-free estimation that approximates a function through linguistic

9

input/output associations. Since fuzzy logic can handle approximate information in a systematic way, it is ideal for controlling nonlinear systems and for modeling complex systems where an inexact model exists or systems where ambiguity or vagueness is common. Fuzzy rule-based systems apply these methods to solve many types of real-world and industrial problems, especially where a system is difficult to model, is controlled by a human operator or expert, or where ambiguity or vagueness is common. A typical fuzzy system can consist of a rule base, membership functions, and an inference procedure. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement. Unfortunately, U.S. manufacturers have not been so quick to embrace this technology while the Europeans and Japanese have been aggressively building real products around it. Fuzzy logic is found in a variety of control applications including chemical process control, manufacturing, and in such consumer products as washing machines, video cameras, and automobiles. Other control applications include robotics, automation, tracking, and consumer electronics. Fuzzy logic is used by both Matsushita and Hitachi in laundry washing machines that can automatically determine the fabric and colors of the clothes that are put in and appropriately choose the correct wash cycle, to result in one-touch washing. Fuzzy logic is used by Hitachi in the Japanese bullet trains, allowing them to stop and start so smoothly that the people inside feel no movement. This also allows efficient electricity and fuel usages. Fuzzy logic has also been used for intelligent cruise control, anti-lock-brake systems, automatic transmission control, adaptive traffic signal control, mobile robots, and baggage handling at the Denver airport. Some other industrial applications of fuzzy logic are: automatic control of dam gates for the hydroelectric power plants of Tokio Electric Power; robot control for Hirota, Fuji Electric, Toshiba, and Omron; preventing unwanted temperature fluctuations in the air conditioning systems of Mitsubishi and Sharp; efficient and stable control of the car engines of Nissan; the cruise control for the automobiles of Nissan and Subaru; the improved efficiency of the industrial control applications of Aptronix, Omron, Meiden, Sha, Micom, Mitsubishi, NisshinDenki, and Oku-Electronics; positioning of wafer steppers in the production of the semiconductors of Canon; the optimized planning of bus time tables by Toshiba, Nippon-System, and Keihan-Express; automatic motor-control for vacuum cleaners while recognizing the surface conditions and degree of soil

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by Matsushita; the back light control for Sanyo’s camcorders; software design for industrial processes by Aptronix, Harima, and Ishikawajima-OC Engineering; controlling machine speed and temperature for the steel works of Kawasaki Steel, New-Nippon Steel, and NKK; improved fuel consumption for automobiles by NOK and Nippon Denki Tools; improved sensitivity and efficiency for the elevator controls of Fujitec, Hitachi, and Toshiba; and improved safety of nuclear reactors of Hitachi, Bernard, and Nuclear Fuel Division.

3.

Neural Networks

The study of the human brain is thousands of years old. The first step toward artificial neural networks came in 1943 when McCulloch and Pitts described the model of a simple neural network with electrical circuits. Artificial neural networks are crude electronic networks of neurons based on the neural structure of the brain. A neural network can be viewed as a massive distributed processor that has a natural propensity for storing experimental knowledge and making it available for use. A basic element of a neural network is the perceptron (Rosenblatt 1958). The perceptron, built in hardware, is the oldest neural network still in use today. A single-layer perceptron was found to be useful in classifying a continuous-valued set of inputs into one of two possible classes. The perceptron computes a weighted sum of the inputs, subtracts a threshold, and passes one of two possible values as the result. Neural nets are emulations of biological neurons, the most sophisticated collection of which is the human brain. Digital computers are built around Boolean (true-false) operations. Biological neurons can process a continuous range of intermediate values (i.e., not quite, almost, maybe). Also biological neurons can learn from experience, detect subtle relationships between various inputs, and adapt to changing and uncertain circumstances. Although there is a good understanding of how an individual or a very small group of biological neurons might work, it is currently impossible to understand how a human brain works. The human brain has around one hundred billion neurons. Each neuron may be connected to as many as a thousand other neurons. To date it is impossible to simulate such a large system. A neural net with a few thousand neurons, each with a few hundred connections, is considered very complex. An artificial neuron is a device with many inputs and one output. The neuron has two modes of operation; the training mode and the using mode. In the training mode, the neuron can be trained to fire

Figure 2

A Simple Neuron Model

(or not), for particular input patterns. In the using mode, when a taught input pattern is detected at the input, its associated output becomes the current output. See Figure 2. If the input pattern does not belong in the taught list of input patterns, the firing rule is used to determine whether to fire or not. The firing rule is an important concept in neural networks and accounts for their high flexibility. A firing rule determines how one calculates whether a neuron should fire for any input pattern. It relates to all of the input patterns, not only the ones on which the node was trained. What has been attained is the development of simple systems that exhibit the kind of generalized processing and adaptive properties inherent in biological neural networks, in order to improve the capabilities of computational tools. An advantage of neural networks is their resilience against distortions in input data and their capability of learning through training. A use of neural networks comes from their ability to derive meaning from complicated or imprecise data. They are used to extract patterns and detect trends that are too complex to be noticed by either humans or other computational devices. Neural networks process data sets and learn the patterns that exist within these data sets. Since they learn the patterns, they are in a sense programming themselves to react in a particular way. Neural networks process data records one at a time and learn by comparing their classification of the new data record with a known actual classification of the data. Figure 2 shows a model of a neuron, with its various inputs, an output, and the use of learning and teaching by a single neuron. Neural nets have been designed into a variety of industrial applications, such as navigation and vision or pattern recognition in robotics, expert systems, decision analysis, control systems, signal processing, character recognition, speech recognition, and financial analysis. For example, Siemens uses neural networks for process automation in industry. Neural networks have been used to control robot arm tracking movements. Neural networks have been used to classify loan default data. They have also been used to find patterns in data. For example, law

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enforcement agents have looked for travel patterns that might indicate drug smuggling activities.

4.

Genetic Algorithm Flowchart

Tabu Search

Tabu search is a meta-heuristic that guides a local heuristic search procedure to explore a solution space beyond local optimality (Glover and Laguna 1998, 2002). Adaptive memory is used to provide a more flexible search behavior. Tabu search is an iterative improvement search technique. It avoids getting trapped in local optima by using a limited memory of past moves, which helps subsequent iterations to diversify the search. As iterations are performed, memory of past learning is stored in lists that record how recently or how frequently specific attributes were encountered in solutions previously examined. Some lists are devoted to only the highest quality solutions. Old memory is updated by new learning as the iterations proceed. The memory of past iterations helps tabu search to continue exploration without becoming confounded by the absence of an improving move, and without falling back into a local optimum from which it had previously emerged. A simple form of tabu search methodology constrains a search by classifying certain moves as forbidden, and freeing the search with a short term memory function. There have been many industrial applications using this technique. Some described in Glover and Laguna (1998) include job shop, flow shop, flexible flow line, and audit scheduling; resource allocation of a single plant (and multiple plants) having capacity-based economies and diseconomies of scope; production planning with workforce learning; process plan optimization for machined parts in minimum cycle time using multiple-spindle, multi-turret CNC turning centers; determining the location of hub facilities in the design of communication networks (applications include traffic networks (airline passengers flow and parcel delivery networks) and telecommunication networks (location of digital switching offices for digital data service networks and location of base stations for wireless networks)); transportation, routing, and network design; vehicle routing; routing and distribution; automated guided vehicle system flowpath design; VLSI systems with learning; task assignment for balancing assembly lines; and facility layout in manufacturing. Brief descriptions of how tabu search was used to address these problems are found in Glover and Laguna (1998). Also, the complete references are provided, where each example can be studied in detail.

5.

Figure 3

Genetic Algorithms

Genetic algorithms are adaptive heuristic search algorithms using the evolutionary ideas of natural selection and genetics (Holland 1992). GAs use random

directed search to seek optimal solutions to problems. Similar to tabu search, GAs are artificial intelligence, meta-heuristic techniques. In an ecological setting, natural selection and random variations determine the attributes of an individual and species, which in turn determines the fitness of an individual to the environment. The variation in generations is brought about by the crossover and mutation of chromosomes. Various steps in modeling a problem using a GA are shown in Figure 3. In a similar manner, a GA views solutions as chromosomes, which are members of a population. The fitness of a chromosome determines its chances of procreating progenies. A fitness function is the objective function to be optimized. A big advantage of a GA (in common with tabu search) is its ability to deal with many types of constraints and objective functions. GAs not only solve problems, but have been implemented for machine learning. These can be thought of as living programs that learn from their environment and evolve over time, imitating a living creature. Genetic algorithms have been used by aircraft seat manufacturers to deal with the problem of getting their seats, restraints, and related interiors certified according to dynamic test standards. These tests are required to verify structural strength and the ability of these systems to protect an occupant from injuries in a crash environment, i.e., in an aircraft emergency landing. Analysis shows how these numerical techniques are a powerful tool for aircraft seat designers, to be used in the search for an optimum regarding structural strength and injury risk concerns. There are also applications of genetic algorithms to a variety of industrial scheduling problems, automated design, composite material design, multiobjective design of automotive components for crashworthiness, weight savings, and other desirable characteristics, mobile communication infrastructure optimization, plant flow layout problems, aircraft design, robot trajectory generation, and many others.

6.

Simulated Annealing

Annealing is the process of heating a solid and cooling it slowly so as to remove strain and crystal

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imperfections. During this process, the free energy of the solid is minimized. The initial heating is necessary to avoid becoming trapped in a local minimum. Functions can be viewed as the free energy of some systems. So imitating how nature reaches a minimum during the annealing process could yield optimization algorithms. Simulated annealing is mathematically imitating the actual annealing process (van Laarhoven and Aarts 1987). It is a powerful stochastic search algorithm applicable to a wide range of problems for which little prior knowledge is available. It asymptotically and probabilistically converges to a global optimum. A major advantage of simulated annealing is its ability to avoid becoming trapped in local minima. Early applications of simulated annealing solved problems such as computer-aided design of integrated circuits, layout, code design, neural network theory, optimization of cutting patterns (i.e., in the garment industry), and vehicle routing (with capacity and timewindow constraints). Descriptions of these industrial applications, as well as complete references for each, can be found in van Laarhoven and Aarts (1987).

7.

Decision Analysis

Decision analysis (DA) is generally applied to two broad classes of problems. One class consists of problems that involve sequential decisions, where uncertainties and probabilistic dependencies play a critical role. The second class includes one-time decisions, where a set of alternatives are compared on the basis of multiple, often competing and conflicting, goals or objectives. It is important to identify the comparison criterion for each alternative, which may differ in many ways. For multi-objective decisions, a corresponding utility function is needed, which allows the alternative’s overall desirability to be identified, based on evaluation measures. Decision trees and influence diagrams are often used in modeling a DA problem. These allow a decision analyst to explicitly understand the sequence of decisions and the uncertainties related to the decision. Decision analysis is implemented by breaking complicated decisions into small pieces. The crux of DA is in the distinction of possible choices (the alternatives), the identification of measures that define the goal, and the relative desirability of different sets of characteristics (preferences). DA is normative, rather than descriptive, as it provides a systematic quantitative approach to make decisions, rather than providing a description of how an unaided decision is made. The benefits of DA include a formal representation of decisions that clearly lays out goals and alternatives, captures a chronological sequence of choices and possible events, and provides well-informed decisions.

There are many applications of decision analysis in industry. Keeney and Raiffa (1976) provide many examples of decision-making problems under uncertainty. There are often preferences for possible consequences or outcomes. Details as well as complete references concerning these examples can be found in Keeney and Raiffa (1976). Also, Smith and von Winterfeldt (2004) overview many of the decision analysis papers that have been published in Management Science over the last 50 years. Many of these are industrial applications. Some examples are as follows. Bodin and Gass (2004) provide some detailed, worked-out examples showcasing the Analytic Hierarchy Process. Bordley (2003) demonstrates a nice, intuitive, visual representation of decision trees. Ulvila and Gaffney (2004) use decision analysis to evaluate computer intrusion detection systems. Their method can be used to decide the optimal operating point on an intrusion detector, choose the best intrusion detection system, compare the value of one intrusion detection system to another, determine the value of an intrusion detection system over no detector, and determine how to adjust the operation of an intrusion detector to respond to changes in its environment.

8.

Petri Nets

Petri nets are a graphical and modeling tool that are applicable to and have been used for many problems in industry. Petri nets have been used to describe and study information processing systems that can be characterized as being concurrent, asynchronous, distributed, parallel, and nondeterministic and/or stochastic (Peterson 1981). As a graphical tool, Petri nets have been used as a visual communication aid similar to networks and flow charts. Tokens are used to simulate the dynamic and concurrent activities of systems. As a mathematical tool, state equations and algebraic equations can be set up to describe the behavior of systems. For some Petri net models, simulation is the best way to analyze the described system’s behavior. Decision-free timed Petri nets are equivalent to linear state equations in a
max +-based algebra. Efficient techniques can solve such equations for a variety of system performance measures and characteristics. Petri nets have been used to address problems in flexible manufacturing, performance evaluation, industrial control systems, discrete event dynamic systems, fault tolerant systems, programmable logic and VLSI arrays, local-area networks, neural networks, and decision models. Figure 4 shows a schematic of a manufacturing and assembly process and its Petri net model. Various

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Figure 4

Two Independent Manufacturing Processes and Their Petri Net Models

10.

Using Queueing Theory to Solve Machine Interference Problems

We now demonstrate using queueing theory to solve a machine interference problem (Stecke 1992). Suppose that there are N machines and O operators. As an example, consider a computer system consisting of N independent processors (CPUs) and O independent memory modules such that each CPU can access each memory module. In such a system, independent programs may simultaneously request access to the same memory module and interference will occur. The assumptions of the model are: 1. The time between breakdowns (or production time) of any of the machines is a sample from a negative exponential distribution with mean  (or mean rate ). A breakdown is random and is independent of the operating behavior of the other machines. Then, when there are n machines not working at time t, non-deterministic processes, such as machine tool failures and product quality, can be modeled using the above Petri net model. A Petri net allows the flexibility of modeling an operation that can have a random time delay. In the above model, parts are tokens. There are 3 parts in the input buffer for molding and 4 parts waiting in the input buffer for welding. After each operation, the parts join unique queues for inspection, before waiting again for assembly.

9.

Queueing Models

Queueing theory is the study of waiting times, busy times, queue lengths, and other properties of queues. Queues are a familiar phenomenon in everyday life. Often a competition for limited resources results in queues, which generates a possibility of using queueing models. Queueing theory can be used to understand the properties of these real world queues. More abstract queues, such as in machine shops, communication channels, and processing jobs in a computer system can be modeled using queueing. Other examples include grocery checkout counters, bank tellers, parts of different types waiting to be processed before a machine, email messages at a server, and cars at a traffic signal. The aim of queueing theory is to understand queueing phenomenon and their operating characteristics so as to optimize or evaluate the performance of a system. Predicting and controlling the behavior of queues is another objective of queueing theory. Suri (1998) suggests using queueing theory to provide quick solutions to industrial problems. Many queueing models, used to address a variety of industrial situations, can be found in Buzacott and Shanthikumar (1993) and Tempelmeier and Kuhn (1993). A detailed example is now provided.

Prob(one of the M − n machines goes down in the interval (t t + t = (N − n) t + O t, where t is a small increment of time. 2. Any one of the n down machines requires only one of the O operators to fix it. The service time distribution is negative exponential with mean 1/ for each machine and each operator. The service times are mutually independent and also independent of the number of down machines. Then Prob(one of the n down machines is fixed in an interval t)

=

  n t + O t

for 1 ≤ n ≤ 0

 O t + O t

for 0 ≤ n ≤ N 

3. The machines are randomly tended. The system can be pictured as shown in Figure 5. Note that the model is that of a finite source, closed queueing model in which the arrival (breakdown) rate increases as the number in the system (number of down machine, n) increases. The queue contains n − O down machines not being serviced. Let the average queueing (waiting, interference) time be denoted by I. Let Figure 5

M-Machine Service Queueing Systems with O Operators

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Nq = expected (or average) number of down machines in the queue E n = expected (or average) number of down machines M t = the number of down machines at time t. Define Pn t = Prob M t = n
M 0 = i. A transition from state Pn t to Pn+1 t + t is caused by a breakdown of one of the N − n working machines; a transition from Pn t to Pn−1 t + t means that one of the down machines has been fixed. The state P0 t occurs when all of the machines are up. Then the stochastic process, M t, can be modeled as a birth and death process, with rates   N − n n = 0 1     N  n =  0 n > N  n n = 1 2     0 n =  O n = 0 + 1     N  The expected (average) number of down machines is En =

N  n=0

nPn =

N 

n − 1Pn−1 =

n=1

N  n=1

nPn − 1 − P0 

It can be seen that there is no closed-form expression for E n in general, but for a particular problem (system), E n is easily computed. There is a closedform expression for single-server (only one operator) systems. In this case En = N +

+ 1 − P0  

Some system performance measures are 1. Machine efficiency is Me =

E n N

or percentage of average production obtained (of the fraction of total production time on all machines). 2. Average operator utilization is Oe =

0  nPn n=0

O

+

N  n=0+1

O − nPn 

n=0

N  n=0+1

Pn

n − OPn 

By dividing measure 3 by the number of operators, O, and measure 4 by the number of machines, N , some related measures are 3’. Coefficient of loss for operators is 0 n=0 O − nPn O or percentage of idle operators. 4’. Coefficient of loss for machines is N n=0+1 n − OPn N or percentage of interference time. The following example shows the advantages obtained in system performance and productivity from the pooling of operators. In this case, several operators have the same assignment of machines. Table 1 has values for operator utilization for pairs of N  O parameters that have the same machine per operator ratio (N /O = 4 and then 15). Notice that the operator utilization is increasing for a given  even though the ratio of the number of machines per operator stays the same. This is an indication that it is better, when feasible, to pool operators rather than to assign a particular number of machines to each operator individually. A good example can be found in Feller (1968). The example considers two cases: (1) 6 machines serviced by one operator and (2) 20 machines serviced by three operators. The results show that, even though the workload by one operator increases from system 1 (6 machines/operator) to system 2 (6 23 machines/operator), the machines were serviced more efficiently in system 2. The advantages of pooling are well known. Notice in Table 1 that, for a given number of machines per operator (assuming N /O is integer), as O (and therefore N ) increases, the operator utilization slowly increases. Likewise, under the same conditions, the machine efficiency will slowly increase. Table 1

of the fraction of time an operator would be working. 3. Average number of idle operators is 0 

4. Average number of machines waiting is

Operator Utilizations for Proportional Parameters



N

O

Operator utilization

0.45

4 8 16

1 2 4

0.881 0.934 0.994

0.05

15 30 60

1 2 4

0.656 0.682 0.705

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Some of the theory and results regarding machine interference from queueing theory are presented in some of the standard operations research textbooks (see Hillier and Lieberman 2005). In particular, a closed queueing network problem is solved under different assumptions: 1. The service time per machine by one operator, S, is constant. 2. The production time per machine, P , is constant. The limiting equations have been solved recursively to prepare tables useful in deciding the economically optimal number of operators for a given number N  of machines. In reviewing some old notation and introducing some new notation, recall that I= S= P= Me = X= D= NS = NP = NI = Then

average interference time per machine average service time per machine average production time per machine machine efficiency = fraction of machine running = S + P / S + P + I (usually service time is not included) S/ S + P  Prob(delay) = Prob(a down machine incurs interference time) average number of machines being serviced average number of up machines (producing) average number of machines waiting for service. 

S +P Ns = Me NX = N S +P +S



I S +P





I =N S +P +S

Table 2 O

36 S = = 036 S +P 36 + 64

NS = Me NX = Me 14 036 = 504Me NI = N 1 − Me  = 14 1 − Me  NP = 1 − xNMe = 064 14Me = 896Me  Each machine causes a profit of $10.00 for each hour producing (longer than an hour of real time). The

7

6

5

4

3

483 0574 8593 3000 8593 5593

451 147 802 2500 8020 5720

3886 3206 691 2000 6910 4910

299 5698 5313 1500 5313 3813

NP = 1 − XNMe = 064 14Me From Table 4, we see that, for X = S/ S +P  = 036 and O = 7, Me = 0987. Then NP = 896, Me = 896 0987 = 8844. The other entries are similarly calculated. The maximum profit per hour is $57.20, which is attained when five operators are tending the 14 machines. Suppose that a policy of preventive maintenance for the 14 machines is introduced, costing $10.00/hour. Because of this extra care, the service time required decreases from 40 to 20 hours, and the production time increases from 60 to 80 hours. Now X=

S 20 = = 02 S +P 100

NS = Me NX = 14 02Me = 28Me NI = N 1 − Me  = 14 1 − Me 

NI = N 1 − Me 

X=

8

cost to service the machines is $5.00/hr/operator. The operators get paid whether or not they are working. Using Table 4 we construct our own table, as shown in Table 2. To see how entries from Table 3 were calculated, look at the NP row and the column for seven operators.

NP = 1 − XNMe

N = 14

9

NS 5035 502 4975 0014 0056 0182 NI NP 895 8924 8844 Labor cost: $5 × O 4500 4000 3500 NP 8950 8924 8844 Net Profit ($/hr) 4450 4924 5344



Before demonstrating the use of the tables with an example, it should be pointed out that they are very extensive. The number of machines tabulated ranges from 4 to 250, and Xe 0001 0950 in increasing increments. (See Table 4.) Suppose that the Diedinthewool textile firm has 14 looms. From time studies it was found that on average the machines are running for 64 minutes and require 36 minutes of service. Then

Fourteen Machines, Three to Nine Operators

NP = 1 − XNMe = 08 14Me = 112Me  The profit per hour producing remains the same ($10.00). The service cost is now Labor cost = $500 × O + $1000 We construct a new table (Table 3) using Table 4. As before, the maximum profit per hour is attained when the 14 machines are tended by five operators. Note that the cost for three operators and maintenance is the same as the cost for five (the optimum) in Table 3 O

Fourteen Machines, Two Operators 6

5

4

3

NS 2797 2078 2724 2526 NI 0014 0098 0378 1372 1119 13902 10898 10102 NP LaborCost: $5 O + $10 4000 3500 3000 2500 Profit: $10 × NP 11190 13902 10898 10102 Net Profit ($/hr) 7190 10402 7898 7602

2 1952 4242 7806 2000 7806 5806

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Illustration of Peck and Hazelwood’s Tables for N = 14, Given Me and I as a Function of , N, and O

s/ s + p

O

I

Me

0.06 0.062

1 3 2 1 3 2 1

0714 0046 0219 0732 005 0231 075

0902 0999 099 0894 0999 0989 0885

3 2 1 3 2 1 3

0054 0244 0766 0058 0256 0782 0062

0998 0988 0876 0998 0987 0866 0998

2 1 3 2 1 3 2

0269 0798 0074 0301 0833 0088 0333

0985 0856 0998 0982 083 0997 0977

1 4 3 2 1 4 3

0863 0021 0102 0367 089 0026 0117

0803 0999 0996 0973 0775 0999 0995

2 1 4 3 4 3 2 7

0401 0912 0031 0133 0374 0708 0963 0016

0967 0746 0999 0994 096 0867 0642 0999

6 5 4 3 2 7 6

006 0178 0415 0748 0973 002 0073

0997 0986 0952 0848 0616 0999 0996

028

5 4 3 2 7 6 5

0205 0457 0785 0981 0025 0087 0234

0983 0943 0828 0592 0999 0995 098

029

4 3 2

0499 0818 0986

0932 0807 0569

03

0.064

0.066

0.07

0.075 0.08

0.085

0.09

0.095

0.23

0.24

0.25

s/ s + p

01

0105

011

0115

012

0125

130

O

I

Me

s/ s + p

O

I

Me

2 1 4

0435 0931 0036

0961 0718 0999

0135

3 2 1 4 3 2 1

0151 0469 0946 0943 0169 0502 0958

0992 0954 069 0999 0991 0947 0663

5 4 3 2 1 5

0024 0069 0297 0691 0992 0028

0999 0995 0976 0885 0527 0999

4 3 2 1 4 3 2

005 0189 0536 0967 0058 0209 0569

0998 0989 0938 0637 0998 0987 0929

4 3 2 1 5 4 3

0107 0321 0719 0994 0032 0119 0345

0994 0973 0873 0509 0999 0994 0969

1 4 3 2 1 5 4

0975 0066 023 0601 0981 0017 0075

0613 0997 0985 0919 0589 0999 0997

2 1 5 4 3 2 1

0745 0995 0036 0132 037 077 0997

0959 0492 0999 0992 0965 0846 0476

3 2 1 5 4 3 2

0252 0632 0986 002 0085 0274 0662

0982 0909 0567 0999 0996 098 0987

5 4 3 2 1 5 4

0041 0145 0395 0793 0997 0047 0159

0998 0991 0961 0831 046 0998 099

1 3 4

0989 0582 0874

042 0815 0998 0053 0174 0445

0956 0817 0446 0998 0988 095

2 8 7 6 5 4 3

0993 0012 0047 0139 0331 0622 0896

0547 0909 0764  0528 0999 0997 099 0965 0896 0743

3 2 1 5 4 3 6 5 4 3

0234 0476 0765 0957

0978 0935 0837 0663

04

2 8 7 6 5 4 3

0995 0015 0056 016 0366 0661 0916

051 0999 0997 0987 0958 0882 0723

2 8 7 6 5 4 3

0999 0034 0106 0262 0513 0795 0966

0446 0998 0992 0974 0926 082 0645

042

2 8 7 6

0996 0019 0067 0183

0492 0999 0996 0985

2 8 7 6 5 4 3

0999 0041 0123 0291 055 0822 0973

0433 0998 0991 097 0916 0804 0627

014

0145

015

0155

016

0165

033

034

s/ s + p

017

018

019

02

021

022

044

O

I

Me

2

0835

0802

1 6 5 4 3 2 1

0999 0015 0059 0189 047 0854 0999

0433 0999 0997 0987 0945 0787 042

6 5 4 3 2 1 6

0019 0074 0222 0521 0886 0999 0025

0999 0996 0983 0932 0757 0397 0999

5 4 3 2 6 5 4

009 0257 057 0913 0032 0109 0295

0995 0978 0918 0727 0999 0993 0973

3 2 6 5 4 3 2

0619 0934 004 013 0333 0665 0951

0902 0697 0998 0991 0967 0885 0669

7 6 5 6 5 4

0013 0049 0153 0419 069 0907

0999 0997 0969 0947 0871 0738

3 9 8 7 6 5 4

0991 0033 0014 0252 0486 0751 0936

0563 0998 0992 0975 0932 0845 0706

3 10 9 8 7 6 5

0995 0012 0046 0135 0307 0555 0806

0535 0999 0997 0989 0967 0915 0818

4 3 10

0957 0997 0017

0675 051 0999 continued

Stecke: Addressing Industrial Problems Using Various Mathematical Models

17

INFORMS Transactions on Education 5(3), pp. 8–18, © 2005 INFORMS

Table 4

Continued

s/ s + p

O

I

Me

0.26

7 6 5 4

0032 0103 0265 0541

0998 0993 0975 0921

3 2 7 6 5

0848 099 0039 012 0297

0786 0548 0998 0992 097

0.27

s/ s + p

031

032

O

I

Me

s/ s + p

5 4 3

0402 0698 0932

0951 0868 0702

036

2 8 7 6 5 4 3 2 8 7

0998 0023 0078 0208 0439 0732 0945 0998 0028 0092

0476 0999 0995 0982 0944 0852 0682 0461 0998 0994

the previous example, yet the net profit is more than 30% higher because the machines are utilized more. These tables can be used for many types of problems. One such application is in determining the number of operators required in order to meet some demand or production requirement. As in our examples, the cost of additional labor can be calculated and compared with the cost of lost production. Additional costs might be added if the required production is not met. Trade-offs between assigning N machines to one operator versus, say, an assignment of 2N machines to two operators can be examined. All things being equal, the second assignment must be better. But all things are not equal. For example, the patrolling time or the average time to walk from machine to machine could be larger for a larger group of machines. This increase can be absorbed into the average service time.

11.

Future Industrial Applications

There is a large project within the University of Michigan’s College of Engineering on developing and operating reconfigurable manufacturing systems. These are manufacturing systems of the future, which can change or add on to their functionality to respond to future needs. For example, current needs may require only 3 axes of motion from a set of machine tools. If future needs call for the additional capability of 2 additional axes of motion, “reconfigurability” allows the quick addition of these new capabilities. Such machines do not yet exist. Mathematics of many types, within the expertise of industrial, mechanical, and control engineers, is required to allow for this future manufacturing system capability. Another future application falls in the realm of homeland security. Many supply chains are particularly vulnerable to disruptions because of their design characteristics and operating philosophies. For example, global manufacturing, single-source

038

O

I

Me

2 9 8 7 6 5 4

0999 0016 0057 016 0353 0622 067

042 0999 0996 0987 0959 0895 0771

3 9 8 7

0984 0023 0078 0203

0593 0999 0995 0981

s/ s + p

046

O

I

Me

9 8 7 6

0063 0172 0366 0622

0996 0985 0956 0896

5 4 3 10 9 8 7

0852 0971 0998 0025 0084 0215 0429

079 0647 0487 0999 0994 098 0945

suppliers, JIT manufacture, excessive outsourcing, complexity, and changing legislation all unfortunately can contribute to making a supply chain vulnerable. Disruption effects on direct targets could be substantial. Long lasting and rippling effects could be felt throughout multiple business sectors because of the increase in business interrelations. A current complex JIT environment creates supply chains that integrate many private and public entities with unclear contingency plans and roles in a disaster. JIT also provides insufficient buffers to absorb unusual system disturbances in supply networks. Since most companies’ operations are not flexible enough for essential quick responses, disruptions can create bullwhip and queueing effects, increasing negative economic effects further upstream and downstream (Schmitt et al. 2004). There is a need to understand these systems to probe for weaknesses, predict outcomes, and test policies. The scale and complexities involved pose significant problems. Schmitt and Stecke (2003) have applied novel methods of simulating and modeling to synthesize the huge, dynamic mass of information that flows in supply networks. They are working with Sandia National Laboratories in the development of an “agent-based” model of supply networks of 14,000 manufacturing firms in the Pacific Northwest to assess the economic impact of various disruptions in critical infrastructure. The overall mathematical modeling effort should contribute to the U.S.’s effort and ability to prepare for possible attacks on critical infrastructure such as electric power, telecommunications, and transportation, and improve the effectiveness and efficiency of responses should such attacks occur. The following examples are future problems that neural networks and other models could address. They could help identify hackers and others who are unauthorized to access a company or a government security system or computer system.

18

Stecke: Addressing Industrial Problems Using Various Mathematical Models

Neural-network-based predictive software could be developed that does a real-time threat analysis of airline passengers at the time they purchase a ticket. If the system rates someone as a threat, then they would be stopped at check-in. Neural network pixel/pattern analysis/recognition systems can be developed to determine facial features. This can help in identifying people. Facial tones and patterns are good candidates for future neutral network applications, particularly where very large groups of individual faces must be processed quickly. Such identification would be useful for security checks to let employees in or to keep undesirable people out. Drinking water systems can be a potential target for future terrorist attacks. Conventional chemical testing methods can take days to identify harmful contaminants. Neural networks could be used for the real-time detection of significant disturbances in a drinking water distribution system. Using neural networks would require establishing baseline water quality conditions, which can be used to train an artificial neural network on what is considered “normal.” Then various contaminants could be introduced into a pilot drinking water distribution system, one at a time, at different concentrations. A trained neural network should then be able to note an anomaly. Acknowledgments

The author would like to thank Sanjay Kumar at the University of Texas at Dallas for his help in locating some material for this piece. The author would also like to thank the referees for their clarification suggestions.

References Bodin, L., S. Gass. 2004. Exercises for teaching the analytic hierarchy process. INFORMS Trans. Ed. 4(2), http://ite.pubs.informs. org/Vol4No2/BodinGass/index.php. Bordley, R. F. 2002. Decision rings: Making decision trees visual & non-mathematical. INFORMS Trans. Ed. 2(3), http://ite.pubs. informs.org/Vol2No3/Vol2No3toc.php, 64–74.

INFORMS Transactions on Education 5(3), pp. 8–18, © 2005 INFORMS

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