Model

As explained, we are interested in the cost-effectiveness of advanced planning and routing systems in different humanitarian contexts. Since costs will be highly context-and organization-specific context-and are sometimes hard to quantify, we shall focus our analyses on context-specific effectiveness. Decision makers can then weigh the results against the specific cost structures they encounter.

The aim of our analyses is to provide general insights into how effectiveness relates to planning system characteristics and contextual factors and to indicate the order or magnitude of effectiveness gains that could be achieved when implementing one system instead of another. In line with this purpose, we present a simple model that enables us to obtain results in an exact manner.

2.3.1 Assumptions

Our analyses assume the general request flow depicted in Figure 2.1. Requests are classified into two classes, those of higher urgency (H) and those of lower urgency (L),

and may incur request delay, planning delay, queuing delay, and field delay. Delays are largely affected by the planning system. For a given system, let us consider a random request of urgency class i ∈ I = {H, L}. We denote the four types of delay incurred by random variables X_i^R, X_i^P, X_i^Q, and X_i^F and the total delay by random variable Xi. The latter is characterized by some probability density function fX_i(xi).

Furthermore, function u_i(x_i) quantifies the total expected disutility a class i request incurs when its total delay equals xi time units. Combining the probability function and the disutility function yields the total expected disutility incurred by a random request in the system, which we use as an indicator of effectiveness:

U = πH

Z ∞ 0

fX_H(xH)uH(xH)dxH+ πL

Z ∞ 0

fX_L(xL)uL(xL)dxL (2.1)

Here, πi denotes the probability that a request belongs to urgency class i. In what follows, we develop an explicit expression for U based on the following simplifying assumptions:

Assumption 2.1. Class i requests evolve according to a Poisson process with rate λi

Assumption 2.2. The sum of request delay and planning delay, X_i^RP = X_i^R+ X_i^P, is exponentially distributed with parameter λ^RP = λ^R+ λ^P

Assumption 2.3. Field delay is exponentially distributed with parameter λ^F = µ^T+ µ^S

Assumption 2.4. Variables X_i^RP, X_i^Q, and X_i^F are independent

Here, parameters λ^R, λ^P, µ^T, and µ^S denote the reciprocal of the mean request delay, planning delay, travel delay, and time on site, respectively.

2.3.2 Queuing Delay Distribution

We split the planning system into a part in which request and planning delays are incurred (part 1) and a part in which queuing delays and field delays are incurred

(part 2). The system can be seen as a simple queuing network in which servers rep-resent vehicles and clients reprep-resent requests. Due to Assumptions 2.1 and 2.2, part 1 becomes an M/M/∞ queuing system where requests simply incur an exponentially distributed service time.

Let us consider the case where requests are assigned to vehicles on a first-come-first-served (FCFS) basis. Because of Assumption 2.3 and because the arrival process for part 2 is Poisson (Mirasol, 1963), this part becomes an M/M/C queuing system, in which C denotes the number of vehicles. Queuing theory states that X_i^Q is ex-ponentially distributed with some parameter λ^Q given that an arriving request has to wait (because all servers are busy). The latter occurs with some probability Π.

Expressions for Π and λ^Q are given in Appendix 2.B.

As an alternative, one could first assess the urgency of an arriving request and then assign it to a priority queue. Requests assigned to a high priority queue then have priority over those assigned to the low priority queue when a vehicle becomes available. Let qij denote the probability that a class i request ends up in queue j ∈ J = {H, L}. This parameter hence measures the quality of urgency assessments. The probability that a request is of urgency class i and ends up in queue j is denoted by πij

and equals πiqij. Part 2 then represents an M/M/C priority queuing system. Due to the analytical complexity of the exact queuing delay distribution (see Davis, 1966), we propose a simple but close approximation by making the following simplifying assumption:

Assumption 2.5. Variable X_i^Qequals 0 with probability (1−Π) and is exponentially distributed with parameter λ^Qj with probability q_ijΠ.

Though this assumption does not misrepresent mean delay and delay probability Π, it does slightly misrepresent the delay distributions. Expressions for Π and λ^Qj are provided in Appendix 2.B, as is an analysis of the quality of the approximation.

2.3.3 Mean Field Delay

Recall that field delay is the sum of travel delay and time on site. We assume that the latter is not affected by the planning system, and hence that µ^S represents some

context-specific constant. Expected travel delay for any request depends on system characteristics and contextual factors like the road network. To keep our results general, we estimate expected delays for networks with certain properties rather than delays for one specific road network. Specifically, we analyze expected delays per request in random graphs in the square [0, τ ]². Here, τ denotes a scaling factor such that expected travel times equal Euclidian distances. Let N denote the number of destinations to be visited, K the number of destinations per trip, η ∈ [0, 1] some measure of network sparsity (see Appendix 2.C), and δ ∈ [0, 1] the level of travel time stochasticity. Regarding the latter, we assume that the real travel time ˜t_ij for arc (i, j) deviates up to a fraction δ from the expected travel time, i.e. Euclidian distance, t_ij according to a uniform distribution. Finally, ϕ represents the specific routing procedure used. Appendix 2.C shows that Tϕ(N, K, η, δ) – the expected travel delay per request (i.e., _µ¹_T), given the indicated parameters – is well estimated by the following formula:

Tϕ(N, K, η, δ) ≈ Tϕ(N, K, 1, 0) 1 + βϕ1(1 − η)^βϕ2

1 − βϕ3δ^βϕ4

(2.2) Here, βϕ1, βϕ2, ... are some constants and Tϕ(N, K, 1, 0) the expected travel delay in a full graph. We shall fit this function for three routing procedures. The first represents the case where routing decisions are only constrained by the number of destinations per route. In the second, routing must also adhere to precedence constraints. Specifi-cally, as many vehicles as possible must visit at least one high priority destination and each vehicle must first visit all high priority destinations assigned to it. (Recall that a destination corresponds to a high or low priority request.) The third represents the case where routing decisions must also adhere to clustering constraints. Specifically, we first divide the region into four quadrants and then optimize routing within each cluster while adhering to the aforementioned precedence and capacity constraints.

We refer to the three vehicle routing problem (VRP) variants corresponding to these procedures as the CVRP, the PC-CVRP, and the CL-PC-CVRP. Acronym CVRP stands for the “capacitated vehicle routing problem” and acronyms PC and CL stand for “precedence constrained” and “clustered”, respectively.

2.3.4 Disutility Functions

Holguín-Veras et al. (2013) argue that disutility (which they refer to as “deprivation costs”) is expected to be a monotonic, non-linear, and convex function of delay.

Moreover, they show that the willingness to pay for water when the time without water equals t is well described by a function of the form:

u(t) = γ(e^κt− 1) (2.3)

Here, γ and κ denote some strictly positive constants. The authors also propose several proxy functions including fixed, variable, and infinite penalty functions. Al-ternative disutility functions may include an upper bound on disutility. One example is the logistic (s-shaped) function, which is commonly used to measure the evolution of the disease burden resulting from an epidemic (see, e.g., De Vries et al., 2016).

2.3.5 Expected Disutility

Figure 2.3 summarizes our assumptions about the delay distributions. In Appendix 2.B, we prove a simple, closed-form expression for the expected disutility incurred by a random request under the additional assumption that disutility is quantified by func-tion (2.3). The expression takes the following as input: (1) a parametrizafunc-tion of the disutility function, (2) the mean field delay, (3) the mean planning and request delay, (4) the probability that a class i request ends up in queue j, (5) the probability that queuing delay is incurred, and (6) the conditional average queuing delay. Here, for a given routing procedure, mean field delay is a simple function of network sparsity, travel time stochasticity, the number of destinations, the number of destinations per trip, and a travel time scaling factor τ . Furthermore, (5) and (6) are simple functions of the rate at which class i requests evolve, the number of vehicles, and the mean field delay.

Request delay

Planning delay

Queuing delay

Travel delay

Time on site

𝑋_𝑖^𝑅𝑃~exp(𝜆^𝑅𝑃) 𝑋_𝑖^𝑄~cond. exp(𝜆^𝑄𝑗, 𝑞_𝑖𝑗) 𝑋_𝑖^𝐹~exp(𝜆^𝐹)

Figure 2.3: Model assumptions about the distribution of delays incurred between generation and fulfillment of a request.