## 1 Introduction

^{1}(resp. Sars-CoV-2). In times of social distancing, well matched supply and demand is important to avoid unnecessary gathering of people. Online appointment systems can decrease the number of customers lining up in a long queue not knowing whether they will be treated after a reasonable waiting time, especially after a reopening (e.g., after a pandemic lockdown). The pandemic situation has even pushed virtual health care forward. “This crisis has forced us to change how we deliver health care more in 20 days than we had in 20 years” (Dr. Robert McLean in Span 2020) and Dr. Meeta Shah conjectures “kind of a turning point for virtual health care” (Dr. Meeta Shah in Abelson 2020). In any case (virtual or physical service), offering the booking possibility online comes with the decision which appointments to offer in detail. Some providers ask potential customers about their time preferences before offering appointment times. Irrespective of possible time preferences, the following question is raised: Should the provider solely offer one appointment time, e.g., the one that is closest to the preference (if known), several times to choose from, or even all available appointments?

## 2 Related Literature

## 3 Survey

### 3.1 Design

^{2}As an incentive, subjects received bonus points for the exam if they completed the survey.

Within-subject | ||||||
---|---|---|---|---|---|---|

Treatment | # of obs | First choice | Second choice | Third choice | First independent choice matched data | |

T1 | a: | 46 | 2 vs. 8 | 2 vs. 32 | 8 vs. 32 | 2 vs. 8 |

b: | 47 | 8 vs. 2 | 32 vs. 2 | 32 vs. 8 | ||

T2 | a: | 42 | 2 vs. 32 | 2 vs. 8 | 8 vs. 32 | 2 vs. 32 |

b: | 42 | 32 vs. 2 | 8 vs. 2 | 32 vs. 8 | ||

T3 | a: | 41 | 8 vs. 32 | 8 vs. 2 | 2 vs. 32 | 8 vs. 32 |

b: | 30 | 32 vs. 8 | 2 vs. 8 | 2 vs. 32 |

^{3}

### 3.2 Survey Results

2 vs. 8 slots (T1) | 2 vs. 32 slots (T2) | 8 vs. 32 slots (T3) | |
---|---|---|---|

Choice frequency | 47% vs. 53% | 57% vs. 43% | 75% vs. 25% |

# observations | 93 | 84 | 71 |

### 3.3 Choice Motives

^{4}

Frequency in % (Cohen’s kappa) | Quality | Choice overload | Flexibility | Scarcity | Less wait time | Other |

42.95 (0.9713) | 4.88 (0.5497) | 38.03 (0.9402) | 10.35 (0.6290) | 9.13 (0.9154) | 23.78 (0.6920) |

### 3.4 Strength of the Quality Inference Effect

^{5}We set the alternative specific constant to zero, \(asc=0\), as no differences between the alternatives are given in our survey, apart from the number of slots that are offered. We thus cannot find any alternative specific anchor that requires the integration of an \(asc\neq 0\). The upper bound is \(b=32\) and the lower bound \(a=2\) (the lowest offer set we consider in our survey, thus \(b\geq o\geq a\) in our setting). The \(\hat{\beta }\)-coefficient measures the strength of the effect relative to asc and the error term ϵ.

## 4 Model Formulation

_{n}, with \(\kappa =\sum _{n\in N}\kappa _{n}\). Each slot belongs to one unique slot type n and can be booked for at most one customer. The booking status of the workday is denoted by \(\vec{s}=\left(s_{1,},\ldots ,s_{N}\right)\), whereby \(s_{n}\in \left\{0{,}1,\ldots ,\kappa _{n}\right\}\) states how many slots of type n are available. The offered appointments are denoted by \(\vec{o}=\left(o_{1},\ldots ,o_{N}\right)\), whereby \(o_{n}\in \left\{0{,}1,\ldots ,s_{n}\right\}\) states how many slots of type n are offered. It follows that \(o_{n}\leq s_{n}\), i.e., only available slots can be offered. It the following we will refer to those slots being available but not offered as blocked slots.

_{n}, with the utility \(U_{n}=w_{n}-\beta \cdot \left(\frac{ o-a}{b-a}\right)^{+}+\epsilon _{n}\), follow a Gumbel distribution (Train 2009).

_{n}, captures its mean utility, also known as the alternative specific constant. We assume the weights to be exogenous. A survey similar to Liu et al. (2019) could help gather further information on time preferences. Similar to our survey with one slot type, we assume that the size of the offer set \(o={\sum }_{i=1}^{N}o_{i}\) has a negative linear impact on the choice probability of all offered slots. Thus, the observable attribute of each slot of type n consists of the weight w

_{n}and the size of the offer set (number of offered slots) in a relative formulation, multiplied by the parameter β, which indicates the strength of the effect.

_{n}, we multiply each exponentiated utility of slot type n by the number of offered slots o

_{n}.

## 5 Generalizable Insights

### 5.1 Policy

### 5.2 Necessary Condition for Blocking Being an Optimal Action

## 6 Numerical Study

### 6.1 Setup

_{h}), whereas slot type \(n=2\) contains the less preferred slots (in the following w

_{l}). We consider the levels \(w_{h}\in \{1{,}2,5\}\), while keeping \(w_{l}=1\). This results in instances where slots are homogenous to the customer (\(w_{h}=w_{l}\)), in other words, where we have one single slot type, and instances with heterogeneous slot preferences (\(w_{h}> w_{l}\)). Note that w

_{0}(no-choice) is normalized to 0. Besides varying the overall capacity and the weights, we build different ratios of the capacities of the two slot types. The assignment of the slots to the two slot types follows a typical time-of-day preference distribution, in which appointments in the morning, at noon and after work are preferred to appointments in the forenoon and in the afternoon. Accordingly, we get \(\kappa _{1}=3\) and \(\kappa _{2}=2\) for a system with \(\kappa =5\) appointments. The capacities per slot type of all considered system sizes are given in Table 4.

_{h}) vs. low-weighted slots (w

_{l}), in absolute values

\(\kappa\) | \(3\) | \(4\) | \(5\) | \(6\) |
---|---|---|---|---|

Ratio \(w_{h}\colon w_{l}\) | 2 : 1 | 2 : 2 | 3 : 2 | 4 : 2 |

### 6.2 Results: Optimal Dynamic Policy

_{h}) and arrival rate (λ)

Parameter | \(\beta =0\) (%) | \(\beta =0.5\) (%) | \(\beta =1\) (%) | \(\beta =1.5\) (%) | \(\beta =3\) (%) | Overall (%) |
---|---|---|---|---|---|---|

\(w_{h}=1\) | 0 | 0 | 0 | 3 | 30 | 7 |

\(w_{h}=2\) | 0 | 0 | 1 | 2 | 19 | 4 |

\(w_{h}=5\) | 0 | 0 | 0 | 0 | 2 | 0 |

\(\lambda =0.2\) | 0 | 1 | 3 | 8 | 35 | 10 |

\(\lambda =0.5\) | 0 | 0 | 1 | 4 | 25 | 6 |

\(\lambda =0.8\) | 0 | 0 | 0 | 2 | 15 | 4 |

### 6.3 Decision Rules

#### 6.3.1 Definition

_{low}blocks only less preferred slots with a lower slot weight (w

_{h}). All preferred slots (with a higher slot weight), are always offered to the potential customer. If at least one preferred slot is yet unbooked in period t, all less preferred slots are blocked. As soon as all preferred slots are booked, the less preferred slots are unblocked successively, one at a time. (2) R

_{one}offers one slot at a time. With different slot types, the slot weights are decisive for the order. First, the preferred slots are successively offered. When all preferred slots are booked, the less preferred slots are offered, again one at a time. Note that we consider offering one slot at a time because \(a=\max \left\{0.25;1\right\}=1\) in all instances. A higher lower bound would increase the number of offered slots proportionally. (3) R

_{myopic}is the more complex decision rule. It checks in each period myopically all possible blocking options for each booking status and chooses the one with the lowest probability for the outside option. For homogenous slot types, this is the optimal policy (see Theorem 1).

#### 6.3.2 Results: Decision Rules

_{low}and R

_{one}may be worse than not blocking at all for low β-values. Only the myopic rule performs well over all instances. In-line with our analysis of the optimal rule in Sect. 6.2, we observe that blocking has the greatest influence when slot weight differences are small, i.e., the highest deviations captured in Fig. 4 shift to the right with increasing slot weight differences. With a high slot weight difference (\(w_{h}=5\)), all three decision rules show only minor deviations from the optimum.

_{one}:

_{one}does not perform well for low and moderate β-values.

_{low}:

_{low}can be considered as an alternative to the myopic rule. Only if the quality inference effect is very strong, R

_{one}should be considered as an alternative.

_{one}regarding the order of slot types that are to be offered) is dominated in our numerical study by a strategy that first blocks less preferred slots. Intuitively, the main purpose of blocking slots is to trigger demand. Yet, if preferred slots are blocked, the no-choice probability increases more than if a less preferred slot is blocked.

#### 6.3.3 Sensitivity Analysis

_{a}(a: actual size of effect) and β

_{e}(e: estimated size of effect). Our previous analysis is nested with \(\beta =\beta _{e}=\beta _{a}\). Underestimation is formalized by \(\beta _{e}\leq \beta _{a}\) and vice versa. We assume that the service provider makes the blocking decision in line with the optimal decision in point t given state s if the quality effect were β

_{e}and denote this as \(\vec{o}_{\left(t,s\middle| \beta _{e}\right)}\). We denote the corresponding expected profits for an actual quality inference effect of β

_{a}by \(V_{\tau ,{\beta _{e}}}(s| \beta _{a})\). We calculate the performance losses of differences of estimated and actual quality inference effect by \(\Updelta \left[{\%}\right]=\left(V_{\tau ,{\beta _{e}}}\left(s|\beta _{a}\right)-V_{\tau ,{\beta _{a}}}\left(s|\beta _{a}\right)\right)/V_{\tau ,{\beta _{a}}}\left(s|\beta _{a}\right)\). We again test values \(\beta _{e},\beta _{a}\in \left\{0{,}0.5{,}1,\ldots ,6\right\}.\)

_{a}level as estimated from our survey data, the potential gains and losses appear balanced, while for strong quality inference effects, there is no downside risk.

## 7 Discussion and Limitations

^{6}and similar laws might be in place in other legislative areas. However, this law does not oblige service providers to announce all conceivable appointments at each point in time. Still, the instrument of blocking time slots should be used carefully to not evoke negative long-term consequences when pretending to be more booked than being demanded. Offering a subset of the available appointments to a customer should only be a preselection of time slots. Asking for time preferences could be a helpful instrument to offer an interesting subset instead of deceiving customers by pretending to be well booked.