O.R. Applications
Optimal pricing for service facilities with self-optimizing customers

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Abstract

Many service industries (e.g., walk-in clinics, vehicle inspection facilities, and data-processing centers) have customers who choose among congested facilities, and select the facility with the lowest combination of travel cost plus congestion cost at the facility. In general, customers over-utilize attractive facilities, causing higher costs than if customers were assigned to facilities to minimize total costs. Optimal facility prices induce customers to select facilities that minimize total cost. We find optimal facility prices and show they equal charging customers for the impact (net costs and benefits) they cause for others. We explore a rich flexibility that allows a range of optimal prices, useful when negotiating the implementation of facility fees. Facility prices can be positive or negative (price discounts), and can be adjusted to be all positive, or to provide net subsidy or net revenue. We contribute to unifying and generalizing several disparate streams of research.

Introduction

For many service facilities, temporary imbalance of supply and demand causes customers to wait in queue and otherwise be inconvenienced by congestion before receiving service. Where customers dislike waiting, such congestion increases the costs to customers of using the facility. When customers have a choice among facilities, they typically act in their own self-interest by “self-optimizing” and seeking service in a manner that is cheapest for them.

For a variety of service systems it has been shown that the self-optimizing customers cause higher overall costs by over-utilizing certain facilities. Managers face the challenge of reducing costs and congestion (and complaints) at the most popular facilities. This could be done by investing in additional capacity at popular facilities. Instead, we show how to better utilize existing capacity by redirecting demand from overly congested facilities to less congested facilities.

One way to redirect is to compel customers to use only specified facility(s) to minimize total cost. This would be inconsistent with good customer service since customers generally prefer to make their own choices. A better way to redirect demand is to use differential facility prices to motivate customers to choose different facilities. We show how to use prices to induce self-optimizing customers to utilize facilities in the lowest-cost, “socially optimal” manner.

We consider customers (or aggregate customer demand points such as neighborhoods) and facilities that are spatially distributed, where customers face a travel cost to each facility. Customers select facility(s) to minimize the sum of travel cost to a facility and the expected cost to receive service at the facility, including the inconvenience of queueing delay caused by other customers choosing the same facility. (Customers achieve a Nash equilibrium, where none can better himself given the actions of all the others.) Examples of such situations include patients visiting walk-in clinics for health-maintenance organizations, drivers visiting motor vehicle safety inspection facilities or automobile emissions testing facilities, and batch jobs submitted to one of several mainframe computers (in this case travel costs are replaced by machine-specific job setup costs).

We use differential facility prices (or “tolls”) to induce customers to utilize facilities in a way that minimizes total cost. Facility tolls (where each service request at a facility pays the toll specific to that facility) could usefully be applied in many practical situations where facilities share a common owner or have fees set by a governing authority. Fee-charging vehicle test stations for emission control or safety inspection could impose a high toll at overly congested stations and low toll or negative toll (a discount) at under-utilized stations. Walk-in clinics run by health-maintenance organizations, which commonly charge a usage fee (sometimes called a “co-payment”), could shift demand by charging facility-specific fees. Batch-based data-processing centers (which remain common despite the PC revolution) have customers whose job costs (for job preparation and/or processing) differ by machine. These centers typically already charge different fees for the use of different machines, so implementing a toll would merely entail altering a fee schedule. Negative tolls can be implemented as facility-specific fee reductions to avoid any difficulties were facilities to pay customers for receiving service.

We discuss the properties of the self-optimizing and socially optimal facility utilizations, show that self-optimizing customers impose extra costs, compute facility tolls that induce self-optimizing customers to achieve the optimal facility utilization, evaluate optimal toll flexibility, devise a method to flexibly determine optimal tolls, and explore revenue flexibility.

Section snippets

Literature survey

We address service systems where self-optimizing customers receive service from facilities with different service capabilities, seeking the best combination of ease of access to facilities with congestion at facilities. Prior research has generally been done in restrictive settings, often with the equivalent of a single customer or a single facility.

Examples of research on self-optimizing queue models of service systems include studies of outpatient clinics, hospital beds, public vs. private

Contribution

This paper makes contributions in several areas.

We extend the stream of research on input control models by considering multiple facilities and by incorporating customers with distinct facility preferences (e.g. travel costs) which they consider when self-optimizing. We reinterpret the customer preference model used in the input control work of Bell and Stidham (1983) as a special case of the customer choice facility utilization approach. We generalize the results of Bell and Stidham (1983)

Facility utilization model

We consider spatially distributed customers who receive service from spatially distributed facilities. Service times are homogeneous with any distribution. Any service price (excluding tolls) charged to customers for service is the same at all facilities and is henceforth excluded from consideration. Customers have distinct preferences for facilities (e.g. travel cost to and from the facility) called a “local cost”. The expected cost of receiving service at a facility (the “queueing cost”) is

Optimal tolls

In general, the self-optimizing equilibrium entails greater total cost than the socially optimal assignment. (Lee and Cohen (1985a) described a “highly restrictive” case where the two regimes have equal cost and the reader can generalize these to analogous restrictive cases here.) Our goal is to use tolls to reduce total cost of service by inducing customers to make different choices. We impose a facility toll, Fj, on service requests at each facility j, causing each self-optimizing customer i

Characterizing toll flexibility

Implementation of facility tolls is likely to entail a negotiation. Managers need to understand and be able to exploit the available flexibility during the negotiation process.

We create a method flexibly to specify optimal tolls (Section 7). As discussed in the literature survey, prior research has not explored flexibility in setting the optimal toll. Toll flexibility requires careful characterization because of the interaction of multiple relative tolls. Optimal relative tolls among some

Exploiting toll flexibility

We present an interactive method for setting optimal tolls that exploits relative flexibility. The final toll values are likely to be the outcome of a negotiation process. We use a decision-maker (DM) as a proxy for this negotiation. To start, the DM selects a pair of facilities for which he will specify a relative toll. Algorithm 1 is used to compute the actual bounds, and the DM specifies a relative toll that satisfies these bounds. The DM then selects another facility pair, and the process

Optimal facility tolls

We extract optimal facility tolls (Fj's) from the kernel of optimal relative tolls (Gpr's). Writing (5), Gpr=FpFr, for the J−1 relative tolls in the kernel yields J−1 independent linear equations in J unknown Fj's. Setting one Fj to an arbitrary value, we can easily solve for the remaining Fj's.

We now complete our 3-facility example. The first set of flexible relative tolls (G12=0.5,G23=1) with F2=0 yields F1=0.5,F3=−1. The alternative set of flexible relative tolls (G12=−3,G23=2.5) with F2=0

Revenue flexibility and toll magnitude

Since Algorithm 2 bases facility tolls on an initial arbitrary value, the tolls can be adjusted by adding a constant to every facility toll, leaving the relative tolls (and hence the impact of the tolls) unchanged. For example, in some applications a negative facility toll may be undesirable or nonsensical, or in other cases it may be desirable to have a toll of zero at a particular facility. Since positive facility tolls represent payments by customers to the agent and negative tolls represent

Numerical example

Table 2, Table 3 show data for a numerical example of toll computation. Total demand is Λ=1050. Consistent with previous research, we let Qj(Λj) be the expected queue delay associated with the M/G/1 queue: Qi(Λj)=Λj(aj)/2(1−Λjsj) when Λjsj<1, and ∞ otherwise, where aj is the second moment of service time at j and sj is the expected service time at j (Gross and Harris, 1997). The capacity of each facility j is 1/sj. (We emphasize that our results hold for any increasing strictly convex Qj(Λj),

Conclusions

This paper considers spatially distributed self-optimizing customers who choose which of many congested service facilities to frequent, minimizing their costs due to overall facility usage (e.g. congestion) and local costs (e.g. travel to the facility). Examples include walk-in health clinics, motor vehicle testing and inspection stations, batch-based data-processing centers (with local costs corresponding to machine-specific setups), and even bribery (Economist, 1993) where bribes serve as

Acknowledgements

We gratefully acknowledge support provided by the United States National Science Foundation Grant DDM-8858355, and by the Canadian Natural Sciences and Engineering Research Council Grant OGP0172794. Comments by two anonymous referees improved the paper. Any errors of omission or commission are the responsibility of the authors.

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