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Chapter 5. Network Models. Introduction. Many important optimization models have a natural graphical network representation. In this chapter, we discuss some specific examples of network models. There are several reasons for distinguishing network models from other LP models:
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Chapter 5 Network Models
Introduction • Many important optimization models have a natural graphical network representation. In this chapter, we discuss some specific examples of network models. There are several reasons for distinguishing network models from other LP models: • The network structure of these models allows them to be represented graphically in a way that is intuitive to users. This graphical representation can then be used as an aid in the spreadsheet model development. In fact, for a book at this level, the best argument for singling out network problems for special consideration is the fact that they can be represented graphically.
Introduction continued • Many companies have real problems, often extremely large, that can be represented as network models. In fact, many of the best management science success stories have involved large network models. • Specialized solution techniques have been developed specifically for network models. Although we do not discuss the details of these solution techniques - and they are not implemented in Excel’s Solver - they are important in real-world applications because they allow companies to solve huge problems that could not be solved by the usual LP algorithms.
Transportation models • In many situations, a company produces products at locations called origins and ships these products to customer locations called destinations. • Typically, each origin has a limited amount that it can ship, and each customer destination must receive a required quantity of the product. • Spreadsheet optimization models can be used to determine the minimum-cost shipping plan for satisfying customer demands.
Transportation models continued • For now, we assume that the only possible shipments are those directly from an origin to a destination. • That is, no shipments between origins or between destinations are possible. • This problem - generally called the transportation problem - has been studied extensively in management science. • In fact, it was one of the first management science models developed, more than half a century ago. • Example 5.1 is a typical example of a small transportation problem.
A typical transportation problem • A typical transportation problem requires three sets of numbers: capacities (or supplies), demands (or requirements), and unit shipping (and possibly production) costs. • The capacities indicate the most each plant can supply in a given amount of time under current operating conditions. In some cases it might be possible to increase the “base” capacities, by using overtime, for example. In such cases the model could be modified to determine the amounts of additional capacity to use (and pay for).
A typical transportation problem continued • The customer demands are typically estimated from some type of forecasting model. The forecasts are often based on historical customer demand data. • The unit shipping costs come from a transportation cost analysis - what does it really cost to send a single automobile from any plant to any region? • The unit “shipping” cost can also include the unit production cost at each plant.
Representing transportation as a network model • A network diagram of this model is shown here. This diagram is typical of network models.
Representing transportation as a network model continued • A node, indicated by a circle, generally represents a geographical location. • An arc, indicated by an arrow, generally represents a route for getting a product from one node to another. • The decision variables are usually called flows. They represent the amounts shipped on the various arcs. • Upper limits are called arc capacities, and they can also be shown on the model.
An alternative model • The transportation model already shown is very natural. • From its graphical representation one can see that all arcs go from left to right, that is, from plants to regions. • Therefore, the rectangular range of shipments allows us to calculate shipments out of plants as row sums and shipments into regions as column sums. • In anticipation of later models in this chapter, however, where the graphical network can be more complex, we present an alternative model of the transportation problem.
An alternative model • First, it is useful to introduce some additional network terminology. • An arc pointed into a node is called an inflow. An arrow pointed out of a node is called an outflow. • General networks can have both inflows and outflows for any given node. • Typical network models have one changing cell per arc. • It is useful to model network problems by listing all of the arcs and their corresponding flows in one long list. Constraints are placed in a separate section. • For each node in the network, there is a flow balance constraint.
Transportation models:Modeling issues • Some modeling issues to note include: • How the demand constraints are expressed ( “>=” or “<=” or “=”) depends on the context of the problem. • If all supplies and demands are integers it is not necessary to add explicit integer constraints. This allows us to use the “fast” simplex method. • Shipping costs are often nonlinear due to quantity discounts. • There is a streamlined version of the simplex method designed for transportation problems, called the transportation simplex method. • LeBlanc and Galbreth (2007a, 2007b) discuss a large network model they developed for a client. They recommend writing a macro in VBA to sum the appropriate flows in and out of nodes.
Extending the transportation model • Extending the basic Grand Prix transportation model is fairly easy, even when the cost structure is considerably more complex. • We illustrate one such extension in example 5.2.
Assignment models • Assignment models are used to assign, on a one-to-one basis, members of one set to members of another set in a least-cost (or least-time) manner. • The prototype assignment model is the assignment of machines to jobs. • For example, suppose there are four jobs and five machines. Every pairing of a machine and a job has a given job completion time. The problem is to assign the machines to the jobs so that the total time to complete all jobs is minimized.
Assignment models continued • To see how this is a network problem, recall the transportation problem of sending goods from suppliers to customers. • Now think of the machines as the suppliers, the jobs as the customers, and the job completion times as the unit shipping costs. • The capacity of any machine represents the most jobs it can handle. The “demand” of any job is the number of times it must be done, usually 1. Finally, there is an arc from every machine to every job it can handle, and the allowable flows on these arcs are all 0 or 1 - a particular machine is either paired with a particular job (a flow of 1) or it isn’t (a flow of 0). • Therefore, this assignment problem can be modeled exactly like the transportation problem.
Assignment models continued • An example of this model appears below.
Assignment models continued • The optimal solution above indicates, by the 1s and 0s in the changing cells, which machines are assigned to which jobs. • Specifically, machine 2 is assigned to job 4, machine 3 is assigned to job 3, machine 4 is assigned to jobs 1 and 2, and machines 1 and 5 are not assigned to any jobs. With this optimal assignment, it takes 14 time units to complete all jobs. • The following example is a somewhat different and less obvious type of assignment problem.
Other logistics models • The objective of many real-world network models is to ship goods from one set of locations to another set of locations at minimum cost, subject to various constraints. There are many variations of these models. • The general logistics problem is similar to the transportation problem except for two possible differences. • First, arc capacities are often imposed on some or all of the arcs. These become simple upper bound constraints in the model. • Second and more significantly, inflows and outflows can be associated with any node. Nodes are generally categorized as origins, destinations, and transshipment points.
Other logistics models continued • An origin is a location that starts with a certain supply (or possibly a capacity for supplying). A destination is the opposite; it requires a certain amount to end up there. A transshipment point is a location where goods simply pass through. • The best way to think of these categories is in terms of net inflow and net outflow. • The net inflow for any node is defined as total inflow minus total outflow for that node. • The net outflow is the negative of this, total outflow minus total inflow.
Other logistics models continued • Using the above convention, an origin is a node with positive net outflow, a destination is a node with positive net inflow, and a transshipment point is a node with net outflow (and net inflow) equal to 0. • It is important to realize that inflows are sometimes allowed to origins, but their net outflows must be positive. • Similarly, outflows from destinations are sometimes allowed, but their net inflows must be positive.
Other logistics models continued • There are typically two types of constraints in logistics models (other than nonnegativity of flows). • The first type represents the arc capacity constraints, which are simple upper bounds on the arc flows. • The second type represents the flow balance constraints, one for each node. • For an origin, this constraint is typically of the form Net Outflow = Original Supply or possibly Net Outflow < Capacity. • For a destination, it is typically of the form Net Inflow >Demand or possibly Net Inflow = Demand. • For a transshipment point, it is of the form Net Inflow = 0 (which is equivalent to Net Outflow = 0).
Other logistics models continued • It is easy to visualize these constraints in a graphical representation of the network by examining the flows of the arrows leading into and out of the various nodes. • We illustrate a typical logistics model in the example 5.4.
Variations of the model • There are many variations of the RedBrand shipping problem that can be handled by a network formulation. We consider two possible variations. • First, suppose RedBrand ships two products along the given network. We assume that the unit shipping costs are the same for either product, but the arc capacity (300) represents the maximum flow of both products that can flow on any arc. • In this sense the two products are competing for arc capacity. Each plant has a separate production capacity for each product and each customer has a separate demand for each product.
Shortest path models • In many applications, the objective is to find the shortest path between two points in a network. • Sometimes this problem occurs in a geographical context where, for example, the objective is to find the shortest path on interstate freeways from Seattle to Miami. • There are also problems that do not look like shortest path problems but can be modeled in the same way. We look at one possibility where the objective is to find an optimal schedule for replacing equipment.
Shortest path models continued • The typical shortest path problem is a special case of the network flow problem from the previous section. • To see why this is the case, suppose that you want to find the shortest path between node 1 and node N in a network. • To find this shortest path, you create a network flow model where the supply for node 1 is 1, and the demand for node N is 1. All other nodes are transshipment nodes.
Shortest path models continued • If an arc joins two nodes in the network, the “shipping cost” is equal to the length of the arc. • The “flow” through each arc in the network (in the optimal solution) is either 1 or 0, depending on whether the shortest path includes the arc. • No arc capacities are required in the model. • The value of the objective is then equal to the sum of the distances of the arcs involved in the path.
Geographical shortest path models • Example 5.5 illustrates the shortest path model in the context of a geographic network.
Equipment replacement models • Although shortest path problems often involve traveling through a network, this is not always the case. • For example, when should you trade your car in for a new car? As a car gets older, the maintenance cost per year increases, and it might become worthwhile to buy a new car. • If your goal is to minimize the average annual cost of owning a car (ignoring the time value of money), then it is possible to set up a shortest path representation of this problem.
Network models in the airline industry • We conclude this chapter with two network models that apply to the airline industry. (The airline industry is famous for using management science in a variety of ways to help manage operations and save on costs.) • Neither of these problems looks like a network at first glance, but some creative thinking reveals underlying network structures. • The first problem turns out to be an assignment model; the second is similar to the RedBrandlogistics model. Note that these two examples are considerably more difficult than any covered so far in this chapter.
