Computers have had a dramatic impact on the management of industrial production systems and the fields of operations research and industrial engineering. The speed and data-handling capabilities of computers allow engineers and scientists to build larger, more realistic models of organized systems and to get meaningful solutions to those models through the use of simulation techniques.
Simulation consists of calculating the performance of a system by evaluating a model of it for randomly selected values of variables contained within it. Most simulation in operations research is concerned with “stochastic” variables; that is, variables whose values change randomly within some probability distribution over time. The random sampling employed in simulation requires either a supply of random numbers or a procedure for generating them. It also requires a way of converting these numbers into the distribution of the relevant variable, a way of sampling these values, and a way of evaluating the resulting performance.
A simulation in which decision making is performed by one or more real decision makers is called “operational gaming.” Such simulations are commonly used in the study of interactions of decision makers as in competitive situations. Military gaming has long been used as a training device, but only relatively recently has it been used for research purposes. There is still considerable difficulty, however, in drawing inferences from operational games to the real world.
Experimental optimization is a means of experimenting on a system so as to find the best solution to a problem within it. Such experiments, conducted either simultaneously or sequentially, may be designed in various ways, no one of which is best in all situations.
Decision analysis and support
Since their widespread introduction in business and government organizations in the 1950s, the primary applications of computers have been in the areas of record keeping, bookkeeping, and transaction processing. These applications, commonly called data processing, automate the flow of paperwork, account for business transactions (such as order processing and inventory and shipping activities), and maintain orderly and accurate records. Although data processing is vital to most organizations, most of the work involved in the design of such systems does not require the methods of operations research.
In the 1960s, when computers were applied to the routine decision-making problems of managers, management information systems (MIS) emerged. These systems use the raw (usually historical) data from data-processing systems to prepare management summaries, to chart information on trends and cycles, and to monitor actual performance against plans or budgets.
More recently, decision support systems (DSS) have been developed to project and predict the results of decisions before they are made. These projections permit managers and analysts to evaluate the possible consequences of decisions and to try several alternatives on paper before committing valuable resources to actual programs.
The development of management information systems and decision support systems brought operations researchers and industrial engineers to the forefront of business planning. These computer-based systems require knowledge of an organization and its activities in addition to technical skills in computer programming and data handling. The key issues in MIS or DSS include how a system will be modeled, how the model of the system will be handled by the computer, what data will be used, how far into the future trends will be extrapolated, and so on. In much of this work, as well as in more traditional operations research modeling, simulation techniques have proved invaluable.
New software tools for decision making
The explosive growth of personal computers in business organizations in the early 1980s spawned a parallel growth in software to assist in decision making. These tools include spreadsheet programs for analyzing complex problems with trails that have different sets of data, data base management programs that permit the orderly maintenance and manipulation of vast amounts of information, and graphics programs that quickly and easily prepare professional-looking displays of data. Business programs (software) like these once cost tens of thousands of dollars; now they are widely available, may be used on relatively inexpensive hardware, are easy to use without learning a programming language, and are powerful enough to handle sophisticated, practical business problems.
The availability of spreadsheet, data base, and graphics programs on personal computers has also greatly aided industrial engineers and operations researchers whose work involves the construction, solution, and testing of models. Easy-to-use software that does not require extensive programming knowledge permits faster, more cost-effective model building and is also helpful in communicating the results of analysis to management. Indeed, many managers now have a computer on their desk and work with spreadsheets and other programs as a routine part of their managerial duties.
Examples of operations research models and applications
As previously mentioned, many operational problems of organized systems have common structures. The most common types of structure have been identified as prototype problems, and extensive work has been done on modeling and solving them.
Though all the problems with similar structures do not have the same model, those that apply to them may have a common mathematical structure and hence may be solvable by one procedure. Some real problems consist of combinations of smaller problems, some or all of which fall into different prototypes. In general, prototype models are the largest that can be solved in one step. Hence, large problems that consist of combinations of prototype problems usually must be broken down into solvable units; the overall model used is an aggregation of prototype and possibly other models.
Allocation problems involve the distribution of resources among competing alternatives in order to minimize total costs or maximize total return. Such problems have the following components: a set of resources available in given amounts; a set of jobs to be done, each consuming a specified amount of resources; and a set of costs or returns for each job and resource. The problem is to determine how much of each resource to allocate to each job.
If more resources are available than needed, the solution should indicate which resources are not to be used, taking associated costs into account. Similarly, if there are more jobs than can be done with available resources, the solution should indicate which jobs are not to be done, again taking into account the associated costs.
If each job requires exactly one resource (e.g., one person) and each resource can be used on only one job, the resulting problem is one of assignment. If resources are divisible, and if both jobs and resources are expressed in units on the same scale, it is termed a transportation or distribution problem. If jobs and resources are not expressed in the same units, it is a general allocation problem.
An assignment problem may consist of assigning workers to offices or jobs, trucks to delivery routes, drivers to trucks, or classes to rooms. A typical transportation problem involves distribution of empty railroad freight cars where needed or the assignment of orders to factories for production. The general allocation problem may consist of determining which machines should be employed to make a given product or what set of products should be manufactured in a plant during a particular period.
In allocation problems the unit costs or returns may be either independent or interdependent; for example, the return from investing a dollar in selling effort may depend on the amount spent on advertising. If the allocations made in one period affect those in subsequent periods, the problem is said to be dynamic, and time must be considered in its solution.
Linear programming (LP) refers to a family of mathematical optimization techniques that have proved effective in solving resource allocation problems, particularly those found in industrial production systems. Linear programming methods are algebraic techniques based on a series of equations or inequalities that limit a problem and are used to optimize a mathematical expression called an objective function. The objective function and the constraints placed upon the problem must be deterministic and able to be expressed in linear form. These restrictions limit the number of problems that can be handled directly, but since the introduction of linear programming in the late 1940s, much progress has been made to adapt the method to more complex problems.
Since linear programming is probably the most widely used mathematical optimization technique, numerous computer programs are available for solving LP problems. For example, LP techniques are now used routinely for such problems as oil and chemical refinery blending, choosing vendors or suppliers for large, multiplant manufacturing corporations, determining shipping routes and schedules, and managing and maintaining truck fleets.
Inventories include raw materials, component parts, work in process, finished goods, packing and packaging materials, and general supplies. The control of inventories, vital to the financial strength of a firm, in general involves deciding at what points in the production system stocks shall be held and what their form and size are to be. As some unit costs increase with inventory size—including storage, obsolescence, deterioration, insurance, investment—and other unit costs decrease with inventory size—including setup or preparation costs, delays because of shortages, and so forth—a good part of inventory management consists of determining optimal purchase or production lot sizes and base stock levels that will balance the opposing cost influences. Another part of the general inventory problem is deciding the levels (reorder points) at which orders for replenishment of inventories are to be initiated.
Inventory control is concerned with two questions: when to replenish the store and by how much. There are two main control systems. The two-bin system (sometimes called the min-max system) involves the use of two bins, either physically or on paper. The first bin is intended for supplying current demand and the second for satisfying demand during the replenishment period. When the stock in the first bin is depleted, an order for a given quantity is generated. The reorder-cycle system, or cyclical-review system, consists of ordering at fixed regular intervals. Various combinations of these systems can be used in the construction of an inventory-control procedure. A pure two-bin system, for example, can be modified to require cyclical instead of continuous review of stock, with orders being generated only when the stock falls below a specific level. Similarly, a pure reorder-cycle system can be modified to allow orders to be generated if the stock falls below the reorder level between the cyclical reviews. In yet another variation, the reorder quantity in the reorder-cycle system is made to depend on the stock level at the review period or the need to order other products or materials at the same time or both.
The classic inventory problem involves determining how much of a resource to acquire, either by purchasing or producing it, and whether or when to acquire it to minimize the sum of the costs that increase with the size of inventory and those that decrease with increases in inventory. Costs of the first type include the cost of the capital invested in inventory, handling, storage, insurance, taxes, depreciation, deterioration, and obsolescence. Costs that decrease as inventory increases include shortage costs (arising from lost sales), production setup costs, and the purchase price or direct production costs. Setup costs include the cost of placing a purchase order or starting a production run. If large quantities are ordered, inventories increase but the frequency of ordering decreases, hence setup costs decrease. In general, the larger the quantity ordered the lower the unit purchase price because of quantity discounts and the lower production cost per unit resulting from the greater efficiency of long production runs. Other relevant variables include demand for the resource and the time between placing and filling orders.
Inventory problems arise in a wide variety of contexts; for example, determining quantities of goods to be purchased or produced, how many people to hire or train, how large a new production or retailing facility should be or how many should be provided, and how much fluid (operating) capital to keep available. Inventory models for single items are well developed and are normally solved with calculus. When the order quantities for many items are interdependent (as, for example, when there is limited storage space or production time) the problem is more difficult. Some of the larger problems can be solved by breaking them into interacting inventory and allocation problems. In very large problems simulation can be used to test various relevant decision rules.
In the 1970s several Japanese firms, led by the Toyota Motor Corporation, developed radically different approaches to the management of inventories. Coined the “just-in-time” approach, the basic element of the new systems was the dramatic reduction of inventories throughout the total production system. By relying on careful scheduling and the coordination of supplies, the Japanese ensured that parts and supplies were available in the right quantity, with proper quality, at the exact time they were needed in the manufacturing or assembly process.
Two things made just-in-time work—a dogged attention to quality at all levels of the total system obviated the need for parts inventories to cover defectives found in the manufacturing process, and a close coordination of information and plans with suppliers and vendors permitted them to align their schedules and shipments with the last-minute needs of the manufacturer. Elements of the just-in-time approach now have been adopted by numerous companies in the United States and Europe, although many cannot use the system to its fullest extent because their supplier networks are larger and more widely dispersed than in Japan.
A second Japanese technique, called kanban (“card”), also permits Japanese firms to schedule production and manage inventories more effectively. In the kanban system, cards or tickets are attached to batches, racks, or pallet loads of parts in the manufacturing process. When a batch is depleted in the assembly process, its kanban is returned to the manufacturing department and another batch is shipped immediately. Since the total number of parts or batches in the system is held constant, the coordination, scheduling, and control of the inventory is greatly simplified.
Replacement and maintenance
Replacement problems involve items that degenerate with use or with the passage of time and those that fail after a certain amount of use or time. Items that deteriorate are likely to be large and costly (e.g., machine tools, trucks, ships, and home appliances). Nondeteriorating items tend to be small and relatively inexpensive (e.g., light bulbs, vacuum tubes, ink cartridges). The longer a deteriorating item is operated the more maintenance it requires to maintain efficiency. Furthermore, the longer such an item is kept the less is its resale value and the more likely it is to be made obsolete by new equipment. If the item is replaced frequently, however, investment costs increase. Thus the problem is to determine when to replace such items and how much maintenance (particularly preventive) to perform so that the sum of the operating, maintenance, and investment costs is minimized.
In the case of nondeteriorating items the problem involves determining whether to replace them as a group or to replace individuals as they fail. Though group replacement is wasteful, labour cost of replacements is greater when done singly; for example, the light bulbs in a large subway system may be replaced in groups to save labour. Replacement problems that involve minimizing the costs of items, failures, and the replacement labour are solvable either by numerical analysis or simulation.
The “items” involved in replacement problems may be people. If so, maintenance can be interpreted as training or improvements in salary, status, or fringe benefits. Failure can be interpreted as departure, and investment as recruiting, hiring, and initial training costs. There are many additional complexities in such cases; for example, the effect of one person’s resigning or being promoted on the behaviour of others. Such controllable aspects of the environment as location of work and working hours can have a considerable effect on productivity and failure rates. In problems of this type, the inputs of the behavioral sciences are particularly useful.
A queue is a waiting line, and queuing involves dealing with items or people in sequence. Thus, a queuing problem consists either of determining what facilities to provide or scheduling the use of them. The cost of providing service and the waiting time of users are minimized. Examples of such problems include determining the number of checkout counters to provide at a supermarket, runways at an airport, parking spaces at a shopping centre, or tellers in a bank. Many maintenance problems can be treated as queuing problems; items requiring repair are like users of a service. Some inventory problems may also be formulated as queuing problems in which orders are like users and stocks are like service facilities.
Job shop sequencing
In queuing problems, the order in which users waiting for service are served is always specified. Selection of that order so as to minimize some function of the time to perform all the tasks is a sequencing problem. The performance measure may account for total elapsed time, total tardiness in meeting deadlines or due dates, and the cost of in-process inventories.
The most common context for sequencing problems is a batch, or job shop, production facility that processes many different products with many combinations of machines. In this context account may have to be taken of such factors as overlapping service (that is, if a customer consists of a number of items to be taken through several steps of a process, the first items completing the initial step may start on the second step before the last one finishes the first), transportation time between service facilities, correction of service breakdowns, facility breakdowns, and material shortages.
A simplified job shop sequencing problem, with two jobs and four machines, is shown in the figure. At the top of the figure is the operations sequence of the two jobs. Job A must go first to machine 1, then to 2, then to 3, and finally to 4, and the order of processing on the four machines cannot be changed. The processing time for the job is one hour at each machine, for a total of four hours of machining time. In this example, the job can only be on one machine at a time, as if the job consisted of a single product being processed through four machine tools.
Job B must follow a different sequence. It also starts on machine 1, but then it goes to machine 4, then to 2, and finally back to machine 4. Each machining operation on Job B also requires one hour.
Underneath the charts showing the required sequence of operations, two alternative schedules are shown for the two jobs. (In a bar chart, time is shown on the horizontal line, and the bars or blocks represent the time that each operation is scheduled on each of the four machines.) The first schedule assumes that Job A is run first. Once Job A is laid out on the schedule, Job B’s operations are placed on the chart as far to the left as possible, without violating the sequence constraints. In this case, the chart shows that both jobs (eight hours of work) can be completed in five hours. This is made possible by running both jobs at the same time (on separate machines) during the second, third, and fourth hours. The second schedule assumes that Job B is run first. This schedule requires a total of six hours, one more than the previous schedule. If the total elapsed time for completion of the two jobs is an important criterion, the first schedule would be superior to the second.
Although this problem is easily solved, solutions to actual job shop sequencing problems require the use of sophisticated models and the calculating power of computers. It is not unusual for job shops to have 5,000 customer orders in process at any given time, with each order requiring 50 or 60 distinct processing or machine operations. The number of combinations of feasible sequences is astronomical in such problems, and they provide many problems in modeling and systems development for operations researchers and industrial engineers.
Manufacturing progress function
Because of the enormous complexity of a typical mass production line and the almost infinite number of changes that can be made and alternatives that can be pursued, a body of quantitative theory of mass production manufacturing systems has not yet been developed. The volume of available observational data is, however, growing, and qualitative facts are emerging that may eventually serve as a basis for quantitative theory. An example is the “manufacturing progress function.” This was first recognized in the airframe industry. Early manufacturers of aircraft observed that as they produced increasing numbers of a given model of airplane, their manufacturing costs decreased in a predictable fashion, declining steeply at first, then continuing to decline at a lower rate. When an actual cost graph is drawn on double logarithmic paper plotting the logarithm of the cost per unit as a function of the logarithm of the total number of units produced results in data points that almost form a straight line. Over the years similar relationships have been found for many products manufactured by mass production techniques. The slope of the straight line varies from product to product. For a given class of products and a given type of production technology, however, the slope appears remarkably constant.
Manufacturing progress functions can be of great value to the manufacturer, serving as a useful tool in estimating future costs. Furthermore, the failure of costs to follow a well-established progress function may be a sign that more attention should be given to the operation in order to bring its cost performance in line with expectation.
Though manufacturing progress functions are sometimes called “learning curves,” they reflect much more than the improved training of the manufacturing operators. Improved operator skill is important in the start-up of production, but the major portion of the long-term cost improvement is contributed by improvements in product design, machinery, and the overall engineering planning of the production sequence.
A network may be defined by a set of points, or “nodes,” that are connected by lines, or “links.” A way of going from one node (the “origin”) to another (the “destination”) is called a “route” or “path.” Links, which may be one-way or two-way, are usually characterized by the time, cost, or distance required to traverse them. The time or cost of traveling in different directions on the same link may differ.
A network routing problem consists of finding an optimum route between two or more nodes in relation to total time, cost, or distance. Various constraints may exist, such as a prohibition on returning to a node already visited or a stipulation of passing through every node only once.
Network routing problems commonly arise in communication and transportation systems. Delays that occur at the nodes (e.g., railroad classification yards or telephone switchboards) may be a function of the loads placed on them and their capacities. Breakdowns may occur in either links or nodes. Much studied is the “traveling salesman problem,” which consists of starting a route from a designated node that goes through each node (e.g., city) only once and returns to the origin in the least time, cost, or distance.
This problem arises in selecting an order for processing a set of production jobs when the cost of setting up each job depends on which job has preceded it. In this case the jobs can be thought of as nodes, each of which is connected to all of the others, with setup costs as the analogue of distances between them. The order that yields the least total setup cost is therefore equivalent to a solution to the traveling salesman problem.
The complexity of the calculations is such that even with the use of computers it is very costly to handle more than 20 nodes. Less costly approximating procedures are available, however. More typical routing problems involve getting from one place to another in the least time, cost, or distance. Both graphic and analytic procedures are available for finding such routes.
Competitive problems deal with choice in interactive situations where the outcome of one decision maker’s choice depends on the choice, either helpful or harmful, of one or more others. Examples of these are war, marketing, and bidding for contracts. Competitive problems are classifiable as certain, risky, or uncertain, depending on the state of a decision maker’s knowledge of his opponent’s choices. Under conditions of certainty, it is easy to maximize gain or minimize loss. Competitive problems of the risk type require the use of statistical analysis for their solution; the most difficult aspect of solving such problems usually lies in estimating the probabilities of the competitor’s choices; for example, in bidding for a contract on which competitors and their bids are unknown.
The theory of games was developed to deal with a large class of competitive situations of the uncertainty type in which each participant knows what choices he and each other participant has. There is a well-defined “end state” that terminates the interaction (e.g., win, lose, or draw), and the payoffs associated with each end state are specified in advance and are known to each participant. In situations in which all the alternatives are open to competition, or some of their outcomes are not known in advance, operational gaming can sometimes be used. The military have long constructed operational games; their use by business is more recent.
Search problems involve finding the best way to obtain information needed for a decision. Though every problem contains a search problem in one sense, situations exist in which search itself is the essential process; for example, in auditing accounts, inspection and quality control procedures, in exploration for minerals, in the design of information systems, and in military problems involving the location of such threats as enemy ships, aircraft, mines, and missiles.
Two kinds of error are involved in search: those of observation and those of sampling. Observational errors, in turn, are of two general types: commission, seeing something that is not there; and omission, not seeing something that is there. In general, as the chance of making one of these errors is decreased, the chance of making the other is increased. Furthermore, if fixed resources are available for search, the larger the sample (and hence the smaller the sampling error), the less resources available per observation (and hence the larger the observational error).
The cost of search is composed of setup or design cost, cost of observations, cost of analyzing the data obtained, and cost of error. The objective is to minimize these costs by manipulating the sample size (amount of observation), the sample design (how the things or places to be observed are selected), and the way of analyzing the data (the inferential procedure).
Almost all branches of statistics provide useful techniques for solving search problems. In search problems that involve the location of physical objects, particularly those that move, physics and some fields of mathematics (e.g., geometry and trigonometry) are also applicable.
A “reversed-search” problem arises when the search procedure is not under control but the object of the search is. Most retailers, for example, cannot control the manner in which customers search for goods in their stores, but they can control the location of the goods. This type of problem also arises in the design of libraries and information systems, and in laying land and sea mines. These, too, are search problems, and solution techniques described above are applicable to them.
Frontiers of operations research
Operations research is a rapidly developing application of the scientific method to organizational problems. Its growth has consisted of both technical development and enlargement of the class of organized systems and the class of problems to which it is applied.
Tactics and strategy are relative concepts. The distinction between them depends on three considerations: (1) the longer the effect of a decision and the less reversible it is, the more strategic it is; (2) the larger the portion of a system that is affected by a decision, the more strategic it is; and (3) the more concerned a decision is with the selection of goals and objectives, as well as the means by which they are to be obtained, the more strategic it is.
Strategy and tactics are separable only in thought, not in action. Every tactical decision involves a strategic choice, no matter how implicit and unconscious it may be. Since the strategic aspects of decisions are usually suppressed, an organization’s strategy often emerges as an accidental consequence of its tactical decisions.
Operations research is becoming increasingly concerned with strategic decisions and the development of explicit strategies for organizations so as to improve the quality of their tactical decisions and make even the most immediate and urgent of these contribute to its long-run goals.
The system design problem
Operations research has traditionally been concerned with finding effective solutions to specific operational problems. It has developed better methods, techniques, and tools for doing so. But operations researchers have found that too many of their solutions are not implemented and, of those that are, too few survive the inclination of organizations to return to familiar ways of doing things. Therefore, operations researchers have gradually come to realize that their task should not only include solving specific problems but also designing problem-solving and implementation systems that predict and prevent future problems, identify and solve current ones, and implement and maintain these solutions under changing conditions.
The planning problem
Operations researchers have come to realize that most problems do not arise in isolation but are part of an interacting system. The process of seeking simultaneous interrelated solutions to a set of interdependent problems is planning. More and more operations research effort is being devoted to developing a rational methodology of such planning, particularly strategic planning.
Most organizations resist changes in their operations or management. The organizational need to find better ways of doing things is often not nearly as great as is the need to maximize use of what it already knows or has. This is apparent in many underdeveloped countries that, while complaining about the lack of required resources, use what resources they have with considerably less efficiency than do most developed countries. Operations research, therefore, has been addressing itself more and more to determining how to produce the willingness to change.
Types of organization
Operations researchers have become increasingly aware of the need to distinguish between different types of organization because their distinguishing features affect how one must go about solving their problems. Two important classifications exist, the first of which is homogeneous–heterogeneous. Homogeneous organizations are those in which membership involves serving the objectives of the whole (e.g., a corporation or military unit), while heterogeneous organizations are those whose principal objective it is to serve the objectives of its members (e.g., a university or city). The second classification is unimodal–multimodal. Unimodal organizations are hierarchical organizations with a single decision-making authority that can resolve differences between any lower level decision makers. Multimodal organizations have no such authority but have diffused decision making and hence require agreement among the several decision makers in order to reach conclusions.
Since current skills in operations research are largely restricted to homogeneous unimodal organizations, attempts are under way to develop methodologies adequate for improving the other three types of organization.
In order to solve any of the preceding problems more effectively, operations research requires a better understanding of human behaviour, individual and collective, than is currently available. Furthermore, what understanding the behavioral sciences claim to provide is seldom available in a form that lends itself to symbolic representation and hence to operations research methodology. Operations researchers, therefore, are increasingly working with behavioral scientists to develop behavioral theories that are expressible in a more usable form.
As the scope of problems to which operations research addresses itself increases, it becomes more apparent that the number of disciplines and interdisciplines that have an important contribution to make to their solution also increases. An attempt to provide such a higher order integration of scientific activity is being made in the management sciences.