Flexible Service Capacity: Optimal Investment and the Impact of Demand Correlation

TL;DR: The effects of increasing demand correlation on the optimal solution are analyzed, which makes precise conjectures based on numerical experiments that have existed in the literature for some time.

Abstract: We consider a firm that provides multiple services using both specialized and flexible capacity. The problem is formulated as a two-stage, single-period stochastic program. The firm invests in capacity before the actual demand is known and optimally assigns capacity to customers when demand is realized. Sample applications include a car rental company's use of mid-sized cars to satisfy unexpectedly high demand for compact cars and an airline's use of business-class seats to satisfy economy-class demand. We obtain an analytical solution for a particular case, when services may be upgraded by one class. The simple form of the solution allows us to compare the optimal capacities explicitly with a solution that does not anticipate flexibility. Given that demand follows a multivariate normal distribution, we analytically characterize the effects of increasing demand correlation on the optimal solution. For the case with two customer classes, the effects of demand correlation are intuitive: Increasing correlation induces a shift from flexible to dedicated capacity. When there are three or more classes, there are also adjustments to the resources not directly affected by the correlation change. As correlation rises, these changes follow an alternating pattern (for example, if the optimal capacity of one resource rises, then the optimal capacity of the adjacent resource falls). These results make precise conjectures based on numerical experiments that have existed in the literature for some time.

Summary

1. Introduction

• Consider a telecommunications company that must construct daily service assignments for its hardware technicians and must also, in the long run, determine the number of technicians to hire and train.
• The authors assume that the demand follows a continuous distribution and that the firm has multiple flexible resources that accept customer upgrades.
• Pasternak and Drezner (1991) examine the value of substitution in a two-product inventory system.

2. Model formulation

• Consider a company that provides services indexed by i = 1,…,n, for which each service has an associated revenue per unit (e. g., billing rate) pi and a penalty cost per unit for not fulfilling the demand for this service Ci.
• The authors objective is to find the optimal capacity for each resource in order to maximize expected profit under the assumption that once demand {di} is observed, the authors allocate optimally the existing resources to fulfill demand.
• FD may incorporate both 'peak' and 'lull' demand days, with the probability of each demand scenario determined by that scenario's relative frequency within the long-term period.
• First, the single period model does not capture situations in which customers order a rental car, for example, for several days.
• Their model can be altered to incorporate a parameter designating the proportion of customers willing to upgrade, although in the following text, to keep the number of parameters manageable, the authors shall assume that all customers will accept upgrades.

3. Single Level Upgrades

• The authors now make the following additional assumptions about the structure of the problem.
• Proposition 2. Suppose that there are n service classes and n resource classes with one level upgrades and where conditions (2) apply.
• For their case the greedy algorithm results in an analytic expression for the objective, (3), which in turn allows us to derive (4)-(6) by differentiating (3).
• Moreover, the relationship between the optimal capacity levels and the news-vendor quantities is quite specific: Proposition 3. x1*≥ x1NV, xn*≤ xnNV.
• The proof for the lower bound is similar.

4. Correlation and Capacity

• A car rental company may experience a heavy load throughout the week due to business travelers who tend to rent mid-sized and luxury cars.
• When daily demands are aggregated into a general distribution, demand among certain cars classes will be negatively correlated.
• Initially, the authors will make no assumptions about the distribution of demand.
• In fact, in Section 4.1 the authors show that their objective function, as defined in (3), is concave for any demand distribution, including those with negative support.

4.1 Impact of Correlation for an Arbitrary Demand Distribution

• In Section 4.2 the authors will assume that the products follow the multivariate Normal demand distribution.
• The objective function Γ(x) as expressed in (3) is concave for any demand distribution.
• The authors might also expect that the direction of capacity change (i.e., decrease or increase) as correlation rises will be reversed for neighboring capacities.
• The authors will prove statements (i) and (iii), while the proofs of statements (ii) and (iv) are analogous.

4.2 Impact of Correlation for the Normal Demand Distribution

• To characterize this relationship, the authors will assume that demand is normally distributed.
• This assumption allows us to use a standard result from probability theory, Slepian's inequality (see Tong, 1980, pg. 10).
• The proposition states that the shift will lie somewhere in the halfplane, in regions I, II, or III.
• 21 A shift to region I indicates a decline in the relatively expensive capacity while inexpensive capacity rises.

4.3 Impact of Correlation for Specified Parameters

• Thus far the authors have not made any assumptions about parameters of the demand distribution.
• In each case the authors will require moderate restrictions on the correlation coefficient to obtain their results.
• Proposition 8. Suppose demand is Normally distributed.
• The authors will later verify through numerical experiments that it is not possible to further restrict the direction of the capacity shift.
• The authors will demonstrate that shifts to any of the regions are possible, given appropriate parameters.

5. Algorithms and Numerical Experiments

• Here the authors describe an algorithm for finding the optimal capacities .*ix.
• This method estimates the derivative of convergence using the three successive iterations.
• The authors now demonstrate that, as noted in section 3, the relationship between the optimal capacity levels and the news-vendor quantities is not obvious when there are more than two demand classes.
• The authors again assume that demand classes follow a multivariate Normal distribution with correlation 27 coefficients ρ23=0 and ρ12∈(-1,1).
• Finally, the authors examine the impact of demand variability on the optimal capacities.

6. Conclusions and Future Research

• By restricting service upgrades to at most one class, the authors have found a relatively simple method to determine the optimal capacity of flexible resources.
• While the authors have described the problem in terms of service delivery (automobile rental, flexible staffing), the method is also applicable to 28 inventory problems in which inventory may be used to satisfy multiple demand streams (see BAA).
• The authors found that this does not necessarily hold in the general case.
• The authors have also determined that a change in correlation between adjacent demand streams has an 'alternating' impact on the optimal capacity, for capacities of adjacent resources move in opposite directions.
• Their formulation allows a more general structure for substitution.

Figures

Figure 3: Change in capacity, given an increase in ρ.

Table 2. Parameters for the automobile rental example – three car classes.

Figure 6: Both optimal capacities Figure 7: Both optimal capacities may rise with correlation may fall with correlation

Figure 4: Resource Capacities Figure 5: Increase in profit as a function of correlation as a function of the correlation

Figure 1: Transportation problem for n types of service and n resources.

Table 1. Parameters for car rental example – two car classes.

Figure 3: Change in capacity, given an increase in ρ.

Figure 8: Optimal capacities for three resources as a function of correlation coefficient ρ12

Figure 2. Direction and magnitude of the change in optimal capacity.

Full text

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Flexible Service Capacity:
Optimal Investment and the Impact of Demand Correlation
Serguei Netessine
The Wharton School
of Business Administration
University of Pennsylvania
Philadelphia, PA 19104
netessin@wharton.upenn.edu
Gregory Dobson
William E. Simon Graduate School of
Business Administration
University of Rochester,
Rochester, NY 14627
dobson@simon.rochester.edu
Robert A. Shumsky
∗
William E. Simon Graduate School of
Business Administration
University of Rochester,
Rochester, NY 14627
shumsky@simon.rochester.edu
.
September, 2000
∗
Corresponding author. Phone (716) 275-7956, FAX (716) 273-1140
Forthcoming in Operations Research

1
Flexible Service Capacity:
Optimal Investment and the Impact of Demand Correlation
Abstract: We consider a firm that provides multiple services using both
specialized and flexible capacity. The problem is formulated as a two-stage
single-period stochastic program. The firm invests in capacity before the actual
demand is known and optimally assigns capacity to customers when demand is
realized. Sample applications include a car-rental company’s use of mid-sized
cars to satisfy unexpectedly high demand for compact cars and an airline’s use of
business-class seats to satisfy economy-class demand. We obtain an analytical
solution for a particular case, when services may be upgraded by one class. The
simple form of the solution allows us to compare the optimal capacities explicitly
with a solution that does not anticipate flexibility. Given that demand follows a
multivariate Normal distribution, we analytically characterize the effects of
increasing demand correlation on the optimal solution. For the case with two
customer classes, the effects of demand correlation are intuitive: increasing
correlation induces a shift from flexible to dedicated capacity. When there are
three or more classes, there are also adjustments to the resources not directly
affected by the correlation change. As correlation rises, these changes follow an
alternating pattern (for example, if the optimal capacity of one resource rises, then
the optimal capacity of the adjacent resource falls). These results make precise
conjectures based on numerical experiments that have existed in the literature for
some time.

1
1. Introduction
Consider a telecommunications company that must construct daily service assignments for its
hardware technicians and must also, in the long run, determine the number of technicians to hire
and train. There are three major functions that the technicians perform. They are, in order of
increasing complexity, equipment installation, testing, and repair. Technicians from the repair
department are able to handle the other two jobs, those in the testing department can also
accomplish installation, and members of the installation department are not yet cross-trained in
any other job. When constructing the daily schedule, if the company runs out of technicians in
one department, it may try to borrow a specialist with expertise at the next highest level. Due to
union rules and employee preferences, borrowing from a department with skills two levels higher
may not be feasible. In the long run, this limited flexibility may be taken into account when
specifying the number of technicians needed to staff each department.
This paper considers such environments, in which customers may be upgraded to a higher
level of service at no cost to the customer. We consider the short-term assignment problem,
when actual demand is known, as well as the long-term capacity decision in the face of uncertain
demand. This model applies to problems in which customers may not be aware of the upgrade
(as in the hardware technician example, above), as well as service environments in which the
customer experiences the upgrade directly. Examples include commercial aviation (economy to
business-class upgrades), time-shared executive jets (the use of larger, faster jets to substitute for
smaller, slower aircraft, as described by Keskinocak, 1999) and the car rental industry
(customers who request a compact car may be offered a mid-sized car).
We formulate the problem as a one-period model (in Section 2, after describing the
formulation, we will discuss the limitations of such a model for real-world service applications).
The firm must commit to resource capacities at the beginning of the period, when demand is

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Flexible service capacity: optimal investment and the impact of demand correlation" ?

The authors consider a firm that provides multiple services using both specialized and flexible capacity.