Assigning delivery routes to drivers under variable customer demands

Transportation Research Part E 43 (2007) 157–172 www.elsevier.com/locate/tre

Assigning delivery routes to drivers under variable customer demands Michael A. Haughton

*

School of Business and Economics, Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5 Received 8 February 2005; received in revised form 5 July 2005; accepted 3 September 2005

Abstract This study covers situations in which random day-to-day customer demands complicate decisions made by managers of vehicle routing/dispatch operations. The paper proposes and analyzes a rule to guide the decision of interest: daily assignment of delivery routes to drivers. The rule targets customer service goals by trying to maximize the likelihood that each customer will continue to be served by the driver who is most familiar with that customer. Statistical analysis of the rule yields several managerial implications about pursuing maximum customer-driver familiarity. Among these is the potentially problematic wide disparity in a customer’s level of familiarity with different drivers.  2005 Elsevier Ltd. All rights reserved. Keywords: Vehicle routing; Simulation experiments; Workforce assignment; Regression modeling

1. Introduction This paper’s primary purpose is to analyze how a decision rule for assigning drivers to delivery routes performs under a range of vehicle routing scenarios. The context for the study involves customers’ demands fluctuating randomly from day-to-day. Such fluctuations complicate the already challenging logistics problem of designing delivery routes for vehicle routing and dispatch operations since one must also address the issue of which driver to assign to each route. The driver-to-route assignment rule analyzed here is one which ensures that daily, every customer has an above average chance of being served by the driver who is historically most familiar with that customer. The supporting logic for this approach is that increasing a driver’s familiarity with the specifics of a given customer, not only increases the efficiency with which the driver can serve the customer but is also a mark of praiseworthy customer service. The key criteria in the analysis of the proposed assignment rule focus on the interrelated issues of customer service and the potential level of driver performance. More specifically, this study defines a context as exhibiting tight driver–customer relationships when customers typically get a large average number of visits per

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1366-5545/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2005.09.009

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driver from a few drivers rather than a few visits per driver from a large number of drivers; i.e., relationships that, through high levels of customer–driver familiarity, are undiluted, and thus likely to have depth. While it is intuitive that the aforementioned approach would facilitate such outcomes, the magnitude of the familiarity that is achievable in situations of randomly fluctuating customer demands remains unknown. Therefore, this paper’s analysis will focus on quantifying it. The practical relevance of the driver–customer familiarity issues that motivate this study’s analysis to inform the selection of assignment rules is persuasively underscored in the following response by Evan MacKinnon (CEO of MacKinnon Transport) to questions of what he thought contributed to his company’s selection as one of Canada’s 50 Best Managed Companies in 2004: ‘‘One of the biggest advantages our driver retention provides our shipper clients is consistency of service—familiar faces (my emphasis) showing up at their facilities. Our drivers get to know the shippers and their receivers and the company policies and safety requirements very well.’’ Excerpted from Canadian Transportation and Logistics, April 2004; p. 49 This paper’s focus relates to two broad segments of the scientific literature: vehicle routing/dispatch under probabilistic customer demands and workforce scheduling. The former category has largely focused on developing solution methodologies for the mathematical programming representation of the routing problem. Thus, many relevant matters concerning the assignment and scheduling of drivers remain unexplored. For example, the potential problem of instability in the day-to-day duties of drivers, while long acknowledged in the literature—e.g., Bertsimas (1992) and Waters (1989)—has only recently been subjected to formal measurement (Haughton, 2002). The literature segment on workforce scheduling is also dominated by efforts to develop heuristic/algorithmic solutions. See, for example, Xu and Chiu (2001), Ahire et al. (2000), Gans and Zhou (2002) and Brusco (1998). In that literature, issues concerning the link between the workforce assignment/schedule and the worker’s learning (via, e.g., the familiarity that comes with frequent performance of a task and short intervals between stints of working on the task) appear to have been broached only peripherally and/or as a matter for future research; e.g., Quintana and Ortiz (2002). Thus, neither of these two literature categories has dealt with the core question of this paper: characterizing the impact of a driver assignment policy on two interrelated issues in contexts of random variations in customer demands: customer service and driver performance, both of which are likely to be enhanced when the driver becomes more knowledgeable of the customer through frequent delivery visits to that customer. In addressing these issues, the remainder of the paper is organized into four sections. The first of these further articulates both the problem context and the aforementioned driver-to-route assignment policy. That section clarifies how the study will extend the extant literature. Section 3 then describes the key characteristics of the research methodology. Section 4 presents the results on the proposed assignment rule, as well as results on an alternative rule that seeks to overcome some of its potential drawbacks. That section also explains the implications of the findings for driver-to-route assignment decisions in probabilistic customer demand settings. The paper’s concluding section is a summary of the study’s main contributions and an outline of its more promising extensions. 2. Problem context Consider a situation in which N customers, each with delivery requirements (demands) that fluctuate each day according to some known distribution. The customers are to be served by a fleet of vehicles, each with capacity Q units. For simplicity, assume that the number of available vehicles is enough to handle the highest realistically attainable value of total demand across all customers on any day; i.e., the fleet size must be at least P d[ di]/Qe, where di is the ith customer’s demand, and d e means rounded up to the next integer. At the depot where vehicles begin and end their delivery trips to customers, the number of available delivery vehicle drivers, denoted J, is assumed to be similarly adequate. Each day, the duties of dispatch/routing personnel can be conceptualized as two-phased. The first phase is to design routes that minimize transportation costs (comprising both driver costs such as wages and vehicle costs such as fuel- and travel-related wear and tear). Ideally, the optimal route design—comprising number of routes (customer groupings), the specific customers on each route, and the intra-route sequences of visits to customers—is likely to be determined by using a

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computer-based vehicle routing algorithm. With the route design phase completed, the other phase is to decide which driver should serve each route. That is, denoting each driver (and vehicle) as j, j = 1 to J and each route as r, r = 1 to R (R 6 J), the issue is to select from among the J!/(J  R)! possible assignments. An implicit assumption is that neither the vehicle-driver nor the vehicle-route assignment task is of immediate concern; e.g., with a homogeneous fleet of vehicles, each driver is competent to operate any of those vehicles. Additionally, outside of costs that might be related to the level of customer–driver familiarity, costs are not affected by the different assignments of drivers to routes. Of interest in this study is how the proposed assignment rule affects the following statistics that would be relevant from a customer’s perspective: (a) On average across customers, the mean proportion of visits a customer gets from each driver that visits the customer. This is the response variable of utmost interest in this paper, and, for subsequent discussion its notation is Y , where in a given scenario, Yi = 1/Ji is the per driver proportion of visit customer i receives from each driver P that visits customer i (the inverse of the Ji drivers that the ith customer encounters), and Y ¼ E½Y i  ¼ Y i =N is the average across all N customers. Thus, a value of 0.50 for Y means that, on average, half of every customer’s visits are accounted for by each driver the customer encounters. It is worth noting that this does not necessarily translate to each customer facing an average of 2 (=1/0.5) drivers because while the average customer’s expected number of encountered drivers (E[Ji]) is equal to E[1/Yi], it need not be the same as 1=Y . However, since E[1/Yi], as expected, proved to have a statistically significant (negative) correlation with Y , it suffices to focus the paper on the latter metric. The metric is a gauge of how much familiarity exists, on average, between customers and drivers; i.e., larger proportions suggest greater depth of familiarity. (b) On average, the proportion of a customer’s visits that are handled by a customer’s ‘‘favourite’’ driver (i.e., the driver making the greatest number of visits to the customer), as well as by the customer’s second and third most frequently encountered driver. This can be more formally specified as follows:For cusðT Þ tomer i, if cij denotes the cumulative number of visits from driver j up to day T, then one can state ðT Þ ðT Þ ðT Þ ðT Þ the order statistics cið1Þ ; cið2Þ ; cið3Þ ; . . . ; where ciðkÞ is the kth largest non-zero value (i.e., the kth largest number of visits by the drivers that customer i encounters). Also, define N(k) as the number of customers that encounter at least k drivers, where N(k) = N for k = 1 (all customers must encounter at least one driver) and N(k) 6 N for k > 1 (not all customers are guaranteed to encounter more than one driver). So if Sk denotes the set of customers that encounter at least k drivers then, across all customers, the average proportion of visits from customers’ kth most frequently encountered driver is given by: P ðT Þ i2S k ciðkÞ ð1Þ TN ðkÞ (c) Across all customers, the largest and smallest values for the proportion of visits per encountered driver. Given the observations for Yi, i = 1, 2, . . . , N, one can obtain the two required proportions by specifying the ordered statistics Y(1), Y(2), . . . , Y(N), where Y(k) is the kth largest observation of Yi across all N customers. Therefore, Y(1) and Y(N) state, respectively, the highest (Y[max]) and lowest (Y[min]) levels of driver–customer familiarity experienced. 2.1. The customer–driver assignment rule This priority rule can be formally described by first defining the following terms: ðtÞ

X rj ¼ 1 if route r is served by driver j on day t, 0 if otherwise; t = 1, 2, . . . , (T + 1), where T + 1 is the current day. ðtÞ xij ¼ 1 if customer i is served by driver j on day t, (i.e., customer i is on route r), 0 if otherwise. PT ðtÞ ðT Þ cij ¼ t¼1 xij ¼ cumulative number of visits to customer i by driver j up to day T (earlier defined in presenting the expression in (1)). P ðT Þ ðT Þ C rj ¼ i2nr cij ¼ cumulative number of visits to all customers on route r by driver j up to day T; nr defines the set of customers on route r; i.e., i 2 nr .

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Since the problem on the current day, T + 1, is to maximize the familiarity between customer and driver by assigning each route to the driver who is most familiar with (has made the most previous visits to) the set of customers on that route, it can be formulated as: X X ðT Þ ðT þ1Þ C rj X rj ð2Þ maximize X

r

j

ðT þ1Þ

6 1 8j

ð3aÞ

ðT þ1Þ

¼1

ð3bÞ

X rj

r

X

X rj

8r

j

In this integer programming formulation, the constraints in (3a) and (3b) ensure that, respectively, each driver serves no more than one route, and that each route is served by exactly one driver. While consistent with the standard formulation of worker-to-task assignment problems in the literature, the above formulation has a subtle but important difference. Specifically, in typical formulations in the literature, the coefficient for the criterion to be optimized—e.g., the worker’s experience-influenced competence in the set of tasks to be comðT Þ pleted—is treated as exogenous. However, in this study, the coefficient C rj is endogenous. That is, being a ð1Þ ð2Þ ðT Þ multi-period problem, the assignment solutions in all previous periods ðX rj ; X rj ; . . . ; X rj Þ are explicitly used to quantify the driver’s current experience in each task; i.e., the task of serving a particular customer. Since there would be no experience to reference in the first period, the solution for that period is to randomly match the routes with the drivers. It is worth noting that while experience (knowledge acquisition) and competence are not identical concepts, the former is generally accepted as a determinant of the latter; i.e., the acquired knowledge increases the likelihood of competent performance. The following illustration of the proposed assignment rule is based on information in Fig. 1 and Table 1. In this illustration, 12 customers alphabetically labeled A through K (i = 1 for customer A, 2 for customer B, . . . , 12 for customer K) are to be served and three drivers are available: j = 1 for Andy, 2 for Bobby, 3 for Charlie. Part a of Fig. 1 maps an illustrative optimal solution from the vehicle routing phase of the depot personnel’s duties on Day 6 (T + 1 = 6); e.g., in this solution, one of the three routes (k = 1, 2, 3) has the trip sequence Depot ! A ! B ! C ! D ! Depot. With that routing solution in place for Day 6, the next phase of their duties involves determining which driver to assign to which route (customer grouping). Part a of Table 1 shows each driver’s cumulative number of visits to each customer up to Day 5, and these are summed across customers on the same route to show each driver’s cumulative number of visits to each customer grouping; i.e., ð5Þ C rj for r = 1, 2, 3. Inserting these coefficients into the stated integer programming formulation would yield the following driver-to-route assignment solution for Day 6: Andy-Route 1, Bobby-Route 3, and Charlie-Route 2. Part b of Fig. 1 shows a map of the solution that might be optimal for the actual customer demands on Day 7. Notice, for example, that the optimal customer groupings on Day 7 are different from those on Day 6. Because of day-to-day demand fluctuations, such day-to-day differences in optimal routing solutions are to be expected. Moving from the routing phase to the driver-to-route assignment phase for Day 7, the procedure

a

b

N

N F

F

B

B

E

E

A

A G

G

C H

W

E

Depot I

L

J

K

S

C H

D

D

W

E

Depot L

I

J

K

S

Fig. 1. Illustration of proposed assignment rule: required route design information. (a) Optimal routes for Day 6 and (b) optimal routes for Day 7.

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Table 1 Proposed assignment rule: familiarity-optimizing procedure Driver

No. of visits up to day 5 on Route 1 ð5Þ

Totals ¼ C 1j

To each customer A Part a: Relevant data for Day 6 Andy 5 Bobby 0 Charlie 0

B

C

D

1 2 2

1 2 2

1 3 1

8 7 5

No. of visits up to day 5 on Route 2 ð5Þ

Totals ¼ C 2j

To each customer

Andy Bobby Charlie

E

F

G

H

0 4 1

3 0 2

4 1 0

1 0 4

8 5 7

No. of visits up to day 5 on Route 3 ð5Þ

Totals ¼ C 3j

To each customer

Andy Bobby Charlie

I

J

K

L

3 1 1

2 1 2

1 1 3

0 5 0

6 8 6

Part b: Relevant data for Day 7 No. of visits up to day 6 on Route 1 ð6Þ

Totals ¼ C 1j

To each customer

Andy Bobby Charlie

L

C

B

D

0 6 0

2 2 2

2 2 2

2 3 1

6 13 5

No. of visits up to day 6 on Route 2 ð6Þ

Totals ¼ C 2j

To each customer

Andy Bobby Charlie

A

E

F

G

6 0 0

0 4 2

3 0 3

4 1 1

13 5 6

No. of visits up to day 6 on Route 3 ð6Þ

Totals ¼ C 3j

To each customer

Andy Bobby Charlie

H

I

J

K

1 0 5

3 2 1

2 2 2

1 2 3

7 6 11

Note: At the end of day 7, the statistics for the assignment rule would be: fY ; Y ð1Þ ; Y ðNÞ g ¼ f0:472; 1:000; 0:333g, with the mean visit proportions from a customer’s three most frequently encountered drivers being 0.631, 0.271, 0.214.

ð6Þ

would be to update the coefficients for C rj on the basis of assignments made on Day 6 then rerun the integer program on Day 7. This procedure, which would be repeated each subsequent day, yields the optimal Day 7 assignment of Andy-Route 2, Bobby-Route 1, and Charlie-Route 3. This illustration reflects the potentially consequential impacts of daily demand fluctuations on the optimum routing solution (changes in the optimum

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customer groupings from one day to the next) and, ultimately, on the customer–driver assignments. Case in point is that in order to maximize total driver–customer familiarity, the group of customers to which Andy was assigned on Day 6 (A, B, C, and D) has only one customer in common with the group to which he is assigned on Day 7 (A, E, F, G). The difference in assignment from Day 6 to Day 7 is similarly dramatic for both Bobby (from I, J, K, L to L, C, B, D) and Charlie (from visiting E, F, G, H to H, I, J, K). Table 1 shows what the resulting statistics (Y , etc.) would be at the end of Day 7. 3. Research methodology The base methodology involved simulation experiments, and to reflect the realities of making deliveries in an actual road network (one way streets, speed limits, etc.) the context for the experiments was a 100 · 100 km2 multi-city service region of southwestern Ontario in Canada. The customer addresses of interest in the region (1000 of them) were approximately uniformly distributed throughout the service region. Applying simulation as the chosen research methodology comprised the standard tasks of selecting the main effect factors, designing the experiments, obtaining the data, and analyzing/modeling the data. The three chosen factors were the variance of customer demand (using the gamma distribution, so variance = ab2), the capacity of each delivery vehicle (Q units), and the number of customers to be served (N). The need to examine a wide range of values for demand variance around a fixed mean demand without encountering negative demand values was the basis for using the gamma distribution for the experiments. Therefore, the selected combinations of a and b were such that each customer’s mean and variance of demand were always, respectively, ab = 100 and ab2 = 100b. The basis for these three factors is the logic (which was confirmed by preliminary experiments) that they affect the metrics used to analyze the proposed driver-to-route assignment approach. For example, greater demand variance leads to more dramatic changes in the optimal routes from one day to the next, and is expected to make it more difficult for drivers to consistently serve (and thus become familiar with) any specified set of customers. Similarly, smaller capacity vehicles necessitate more delivery routes (and drivers), making it harder for customers to encounter few drivers and thus have a large proportion of their visits from each driver. Larger numbers of customers, by increasing the required number of drivers, are expected to produce a similar outcome. The details of these effects are discussed in the findings. Table 2 shows the experimental levels of these three factors. A full factorial experimental design of 140 combinations was used: seven (7) levels for demand variability by five (5) levels for vehicle capacity by four (4) levels for the number of customers. With five (5) replications of each combination (see Table 2) this resulted in 700 routing scenarios. Each replicate focused on a separate randomly selected set of N delivery addresses from the available 1000 addresses. Each scenario was evaluated over a period of 250 simulated days, each day

Table 2 Experimental conditions Factor

Levels/values

Variability of customer demand (b) seven variability levels around a fixed mean of 100 units)

Each customer’s daily demand is independent and follows an identical gamma distribution with ab = 100 (to ensure a fixed mean of 100) so variance (ab2) = 100b. Seven (7) levels of b examined: 1, 4, 16, 25, 36, 64, 100

Capacity of delivery vehicles in number of units of the product: Q

Five (5) levels: 500, 1000, 2000, 4000, 5000

Number of customers or delivery addresses served by the Depot: N

Four (4) levels: 100, 200, 400, 500. The N delivery addresses were selected from 1000 actual addresses that are spatially distributed in an approximately uniform pattern within a 100 · 100 km2 multi-city service region in southwestern Ontario, Canada. The experiment was replicated five times by repeating the procedure of randomly selecting N addresses from the total of 1000

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representing a separate set of demand instances (according to the gamma distribution’s specified parameters) for the set of N customers. The simulation run length of 250 days was determined on the basis of the procedures outlined in Law and Kelton (1991). Specifically, the run length was incrementally raised until the desired precision—estimating the mean for Y to within a ±0.02 error margin with at least a 95% confidence level—was reached. With this level of precision (the ±0.02 error margin averaged 3.5% of the estimated mean), the combination of run length (250 days) and number of replicates (5) was considered adequate. For each of the 700 scenarios, the routing phase of the experiments, i.e., daily reoptimization of the routes on the basis of each day’s customer demands was done with Roadshow, a commercial grade vehicle routing software program. Then, based on the routing solution, the daily route-to-driver assignment phase involving the integer programming problem was handled with Java programming. Dev C++ was used to code the routines for calculating the response variable statistics for the driver-to-route assignment rule. Analysis of variance (ANOVA) techniques comprised the core statistical tool to which the data were subjected in order to derive relevant managerial insights on the proposed assignment rule and to signal further analysis. One of these insights concerned the potential for an alternative assignment rule with some potential to overcome possible drawbacks of the proposed rule while, at least partially, retaining its more desirable elements. The further complementary analysis included modeling the simulation data in order to provide a means for exploring predictions to situations beyond those covered in the experiments. 4. Findings and discussion The structure of the discussion in this section will begin with the overall impacts of the three main effects factors on the performance of the proposed assignment rule with respect to the response variable of primary focus in this paper: on average across customers, the average proportion of visits received per encountered driver; i.e., Y . The section closes with presentation of predictive models for Y and other response variables. Concerning Y , Table 3 shows the ANOVA results as well as the descriptive statistics. The results show that all three main effects factors (demand variability, vehicle capacity, and number of customers) and all twoway and three-way interactions have statistically significant effects on Y . These results are graphically captured in the main effects plots in Fig. 2 and the interaction plots in Fig. 3. The vertical axes values in Fig. 2 show that under the proposed rule, a customers’ average level of familiarity with the drivers encountered lies between 0.471 and 1.000. Starting with b, the measure of demand variability, the discussions and explanations of how the factors affect the differences follows.

Table 3 ANOVA results and descriptive statistics on the average proportion of visits per driver ðY Þ for the proposed driver-route assignment rule Source

DF

Seq SS

Part a: ANOVA for Y , using adjusted SS for tests b 6 1.304 Q 4 3.895 N 3 1.113 b*Q 24 0.423 b*N 18 0.112 Q*N 12 0.855 b*Q*N 72 0.672 Error 560 0.605 Total

699

Adj SS

Adj MS

F

P

1.304 3.895 1.113 0.423 0.112 0.855 0.672 0.605

0.217 0.974 0.371 0.018 0.006 0.071 0.009 0.001

201.05 900.83 343.18 16.30 5.75 65.93 8.64

0.000 0.000 0.000 0.000 0.000 0.000 0.000

8.980

Part b: Descriptive statistics on Y Obs. Mean Median

TrMean

StDev

Min

Max

Quartiles First

Third

700

0.627

0.117

0.471

1.000

0.550

0.720

0.638

0.598

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M.A. Haughton / Transportation Research Part E 43 (2007) 157–172 β

N

Q

0.74

0.69

Y

0.64

0.59

0.54 1

4

0

9 16 36 64 00 1

50

00

10

00

20

00

00

40

50

0

0

20

10

0

40

0

50

Fig. 2. Main effects plots of customers’ average proportion of visits per driver ðY Þ.

00

0

50

10

00

20

00

40

00

50

0

10

0

20

0

40

0.9

Beta 100 64 36 16 9 4 1

0

50

0.7

β

0.5 0.9

Capacity 5000 4000 2000

0.7

Q

1000 500

0.5

N Customers

Fig. 3. Interaction plots of customers’ average proportion of visits per driver ðY Þ.

4.1. The effect of demand variability (b) Fig. 2 shows that the increase in b from 1 to 100 decreases Y by an average of approximately 15% points. The logic behind the effect of increases in b on Y is that greater demand variability requires a larger pool of available drivers, which makes it more probable that a customer’s visits will be spread/averaged over a larger number of different drivers. Second, and more significantly, this reduces the percentage of customers that stick to one driver. Thus, a noticeable percentage of customers that have Yi = 1 at a given value of b would now experience the largest possible reduction in Yi: from Yi = 1 to Yi 6 0.5; i.e., go from being served by 1 driver to

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Table 4 Illustrative impact of demand variability on the proposed assignment rule (for N = 500, Q = 4000) Number of drivers encountered

1 2 3 4 5 6 7 8 9 10

Averagea proportion of customers encountering that number of drivers For b = 1

For b = 100

0.841 0.092 0.043 0.012 0.010 0.002 * * * *

0.361 0.120 0.104 0.103 0.096 0.080 0.060 0.060 0.010 0.006

a Averaged across the five replicates to yield Y ¼ 0:907 at b = 1 and 0.532 at b = 100. The required number of drivers for each case were 13 for b = 1 and 15 for b = 100.

at least two drivers. Using the study data at {N, Q} = {500, 4000} for an increase in b from 1 to 100, Table 4 provides an illustration of how b is inversely related to Y . In the illustration, the increase of b from 1 to 100 at {N, Q} = {500, 4000} reduced Y from 0.907 to 0.532, a decrease of 0.375. 4.2. The effects of vehicle capacity (Q) and number of customers (N) Vehicle capacity (Q) and number of customers (N) share the commonality that they require directly opposing changes in order to move the response variable statistics in a given direction. For example, increasing (decreasing) the number of customers for a given vehicle capacity is similar to decreasing (increasing) to vehicle capacity for a given number of customers. That is because both result in a larger (smaller) number of routes and drivers and, consequently, a reduction (increase) in the depth of customer–driver familiarity. Although, as Fig. 3 indicates, the results from opposing changes in N and Q differ in their quantitative specifics, they are similar enough for a discussion dominated by a focus on Q to suffice. Concerning the impact of Q on Y , larger vehicles generally improve that metric. That is, by reducing the required number of routes (and drivers), larger vehicles lower the number of drivers that have a positive probability of visiting any given customer. However, as the middle graph in Fig. 3 shows, this pattern of improvement is not completely consistent for the proposed rule. The oddity, conspicuous in the comparison of Q = 4000 and Q = 5000, is that increased vehicle capacity can reduce Y . The explanation is that a larger capacity can sometimes produce more variability in the day-to-day customer groupings. The following example illustrates. Suppose that the distribution of total customer demand varies between a maximum of 9000 and a minimum of 11,000 units per day. For Q = 5000 units, the required number of vehicles per day is likely to vary between 2 and 3 (i.e., between d9000–5000e = 2 and d11,000–5000e = 3) but for Q = 4000 units, the required number of vehicles is likely to remain fixed at 3. So, for a 100-customer problem, the customer groupings for Q = 5000, unlike for Q = 4000, are prone to large fluctuations (between 33 and 50 customers), this makes it more difficult for many customers to be consistently served by the same driver. Since b influences the range of values within which demand fluctuates, this peculiarity signals an interactive effect of Q and b on the performance of the proposed assignment rule. The left-most interaction plot in Fig. 3 provides additional confirmation of this effect. That is, except for large demand variances (b P 64 in these experiments), there are threshold vehicle capacities (Q = 4000 here) beyond which further capacity increases can, instead of yielding the intuitive result of increasing average customer–driver familiarity, actually decrease it. The plot in the lower right-hand panel illustrates this another way by showing that the curve for Q = 5000 does not consistently stay above the curve for Q = 4000; i.e., Q = 5000 does not always yield higher familiarity levels than Q = 4000. The fact that the aforementioned idiosyncrasy does not hold at high levels of demand variance is borne out in the following counter-example to the

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one presented earlier. Suppose that demand fluctuations produce a wider range of values for total demand, say, 7500–12,500 units. This would tend to produce more stable day-to-day customer groupings at Q = 5000 (2–3 routes per day) than at Q = 4000 (2–4 routes per day). 4.3. Assessing the efficacy of the assignment rule As indicated earlier, the other assessment metrics apart from overall mean familiarity results (Y ) were (i) the mean proportion of visits that a customer gets from the three drivers with whom the customer is most familiar (see expression (1)) and (ii) across all customers, the maximum and minimum levels of familiarity experienced (Y[max] and Y[min]). Tables 5 and 6 present these findings at the eight combinations of extreme values for the three main effects factors. The tables add some perspective to the findings by way of comparison with results on a simple benchmark assignment rule that involves equitable distribution of the routes across drivers; i.e., instead of accounting for historical familiarity, each day, for any given route, r, each driver has the same probability of being assigned to it. Table 5 clearly shows that under the proposed rule a customer’s most familiar driver will handle a very large average proportion of the customer’s visits (no less than 80%). Further these values are significantly larger than the corresponding values for the benchmark (this held true for all 700 combinations of the main effects factors). Table 6 provides another angle from which to view the deep familiarity between driver and customer under the proposed assignment rule. The data in that table show that in each of the aforementioned eight extreme value scenarios for the main effects factors, the lowest observed mean visit proportion experienced by an individual customer (Y[min]) is always greater than or equal to the largest value for the benchmark. In other words, no customer will have a lower mean level of familiarity with the encountered drivers than the customer would have under the benchmark rule. Also, unlike the case with the benchmark, the proposed rule makes it possible for a customer to be served by just one driver (Y[max]1). In fact, across all scenarios in the experiment, between 20% and 64% of the customers get served by the same driver every day. Interestingly, some of the very statistics in Tables 5 and 6 that are meant to support the rule’s desirability raise two concerns. First, with each customer’s visits so likely to be dominated by one driver, there may be some loss of flexibility in being able to continue serving a customer whose most frequently encountered

Table 5 Mean visit proportions for the proposed assignment rule at extreme values of demand variability (b), vehicle capacity (Q) and number of customers (N) b

1 1 1 1 100 100 100 100

Q

500 500 5000 5000 500 500 5000 5000

N

100 500 100 500 100 500 100 500

J

21 102 3 11 29 117 3 12

Overall means Y

0.614 0.549 0.802 0.695 0.513 0.510 0.734 0.583

(0.048) (0.010) (0.333) (0.091) (0.039) (0.010) (0.333) (0.083)

Means by frequency of encounter with the customer Most frequent

2nd most frequent

3rd most frequent

0.87 0.81 0.94 0.92 0.85 0.80 0.93 0.86

0.24 0.17 0.10 0.14 0.26 0.15 0.22 0.18

0.07 0.08 0.08 0.05 0.04 0.07 0.06 0.07

(0.11) (0.04) (0.42) (0.15) (0.09) (0.04) (0.42) (0.12)

(0.09) (0.03) (0.37) (0.12) (0.07) (0.03) (0.36) (0.11)

(0.07) (0.02) (0.31) (0.10) (0.06) (0.02) (0.32) (0.10)

Notes: 1. The fifth column in the table shows the overall averages and the 6th through 8th columns disaggregate these into averages for the typical customer’s three most frequently encountered drivers. As a frame of reference for these averages, each cell shows, in parentheses, what the averages would be if the assignments were made in such a way that daily, for each customer grouping (route), every driver has the same probability of being assigned to it. 2. Each value of J, the actual required number of drivers was not preset but was the results of the experiments: the observed maximum required number of drivers across all 1250 simulated days (five replicates by 250 days per replicate). This overall maximum was consistently equal to the replicate-specific maxima for all 140 scenarios examined. 3. Averages for, say, the 2nd favourite driver are calculated only for the subset of customers who encountered two drivers (see expression (1) on p. 5 for the computational formula); therefore, the row totals of the means by frequency of encounter need not always be 61.00 (100%).

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Table 6 Single customer best and worst visit proportions at extreme values of demand variability (b), vehicle capacity (Q) and number of customers (N) b

Q

N

J

Smallest and largest visit proportions of any single customer Y[max]

1 1 1 1 100 100 100 100

500 500 5000 5000 500 500 5000 5000

100 500 100 500 100 500 100 500

21 102 3 11 29 117 3 12

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Y[min] (0.050) (0.013) (0.333) (0.100) (0.037) (0.011) (0.333) (0.083)

0.125 0.100 0.333 0.125 0.091 0.083 0.333 0.100

(0.048) (0.010) (0.333) (0.091) (0.034) (0.009) (0.333) (0.083)

Notes: 1. The values are identified as the maximum (Y[max]) and minimum (Y[min]) across all five replicates of the experiment. 2. As a frame of reference for these proportions, the parenthetical value in each cell shows what the averages would be if assignments were made in such a way that daily, for each customer grouping (route), every driver has the same probability of being assigned to it.

(‘‘favourite’’) driver becomes unavailable for a day or longer. Even in cases where a customer encounters multiple drivers, the driver that makes the second largest proportion of the customer’s visits (‘‘the second favourite driver’’) makes such a small proportion of these visits that he might not be an able substitute for the customer’s favourite driver. A second concern, which is derivable from Table 6, is the conspicuous disparity across customers in terms of depth of familiarity with drivers. As an example, for the scenario {b, Q, N} = {100, 500, 100}, while some customers encounter the same driver each day (Yi = 1), some have Yi as low as 0.091; i.e., they have more diluted driver–customer familiarity since they encounter eleven (1/ 0.091) different drivers over the 250 days. These two caveats on the desirability of the proposed rule beg the question of whether one can configure an alternative assignment rule that simultaneously provides adequate balance in the allocation of visits among the drivers a customer encounters and a high/undiluted level of familiarity between the customer and those drivers. This assignment rule might involve fixed assignment of specific drivers to specific geographic segments of (territories within) the service region. This can be approached by partitioning the J available drivers into groups of D drivers and allocating each group to a pre-specified geographic segment of the service region. In this study, J was determined from the simulation as the largest number of drivers used across all 250 simulated days and across all five replicates for each of the 140 combinations of the main effects variables (as it turned out, this overall (cross-replicate) maximum for each combination was consistently equal to the maximum for each replicate). With J determined, there would be dJ/De1 geographic segments, averaging approximately N/dit J/De customers per segment. On any given day, each customer within any specified segment would be visited only by a driver from the set of D drivers that is uniquely assigned to that segment; i.e., not by any of the other J  D drivers (who are altogether responsible for the other dit J/De  1 segments). So with each of these drivers operating within an essentially fixed D-driver rotation for each customer (i.e., averaging D days between consecutive visits to the same customer) each customer’s average proportion of visits per driver would be ffi1/D. Thus, this rule can be viewed as being based on the principle of worker flexibility via task rotation (e.g., Brusco and Johns, 1998), where, increases in D extend the scope of each driver’s task rotation across a larger geographic segment. Adhering to this system would entail tolerating some routing/dispatch inefficiency, which can be viewed as coming from two sources. One source relates directly to the inability of the system to reduce resource usage in response to demand contractions and the other to its limited capacity to avert recourse action in instances of

1 Some segments were, of necessity, fractional segments/territories; e.g., for J = 21 and D = 2, one of the 11 resulting segments would be a ‘‘half’’ segment and therefore be the responsibility of one driver (half of D) and thus contain about half the typical number of customers in each of the 10 full 2-driver segments.

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capacity shortfall (caused by demand surges). The first source can be explained by way of illustration for the case of D = 1. At D = 1 each of the J drivers is solely responsible for a geographic segment (territory) comprising N/J customers so this territory-based assignment rule would involve dispatching J vehicles every day. However, on some days demand contractions might be such that fewer than J vehicles would suffice if some customers in one geographic segment can be grouped with those from an adjoining segment. While this policy of maintaining the territory-based assignments prohibits such resource-saving moves (because each driver can serve only the customers in his assigned territory), complete route reoptimizing does not: it deploys an unrestricted search for feasible customer groupings. Therefore, the resulting routing efficiency that is sacrificed under this rule can be measured as the mean daily difference between J and the optimum number of vehicles (6J) specified in the solution based on complete route reoptimization. Depending on demand volatility, increases in D (i.e., larger geographic segments with more customers shared by more drivers) will increase the likelihood of the system being able to reduce the efficiency sacrifice since there will be greater flexibility in grouping customers for deliveries. The second source concerns how this rule would have to be structured to take recourse action to resolve a segment failure2 (an instance of segment demand exceeding segment capacity of DQ units). The resolution, which can be viewed as expedient in light of the rule being used, is to dispatch one of the segment’s D preassigned drivers on an extra route (say, an afternoon trip) to tend to the set of customers whose demands were unfulfilled. No such failures occur with route reoptimization since the routes are reconstituted each day to account for the demands on the day under consideration. Thus, the second source of efficiency sacrifice for this alternative assignment rule can be measured as the per day number of vehicles that are dispatched to resolve those failures. Table 7 condenses the findings from the simulation experiments to show the magnitude of each source of sacrifices in routing efficiency for values of D from 1 through 5 (at the eight extreme value combinations of the main effects factors). In each cell of the table, the first source appears before the second source, which is followed by the total for both sources. As an illustration of the efficiency sacrifice reflected in these totals, consider the total for the factor combination (b, Q, N, D) = (100, 500, 500, 1). Table 7 shows that for that combination, the sacrifice in terms of excess vehicle requirements of 58.81 is just over 50% of the worst-case vehicle requirement for daily route reoptimization (i.e., the worst-case requirement of 117 vehicles, which transpired on days when total demand peaked). In terms of the question of what is the number of drivers that should be exclusively assigned to make deliveries to each customer, the crucial information in Table 7 is that the larger that number is, the more efficient routing operations can be. This results from having larger geographic segments to work with in attempting to advantageously reorganize the delivery routes in response to daily demand changes. That is, larger values of D increase the segments towards the size of the entire service region, which is what global optimization requires. The corollary of this is that at D = J, the policy becomes identical to route reoptimization in terms of maximum routing efficiency. However, that accomplishment thwarts the goal of undiluted customer–driver familiarity. By showing reliable estimates of the sacrificed routing efficiency, Table 7 provides a guide for making tradeoffs between that goal and vehicle routing efficiency. Two additional managerially relevant matters concerning the results from this aspect of the simulation experiments are in order. First, is that for any given value of D, the sacrificed efficiency increase with increases in b (demand variability). This is consistent with the notion that a stable/unresponsive system will be more inefficient when demand fluctuations are wilder. The second matter concerns possible alternatives to the simulated method of redressing segment failures: dispatching one of a segment’s D pre-assigned drivers on a trip later in the day to look after the affected customers. The truth is that, as a practical matter, the extra vehicles dispatched to redress segment failures (the second figure in each cell of Table 7) is eliminable, or at least, reducible. For example, instead of using same-day (afternoon) dispatch, dispatch/routing personnel may simply add the shortfall quantities to the affected customers’ next-day demands. This tactic would lower the 2 Although the term ‘‘route failure’’ is commonly used in vehicle routing research, its use is in reference to the problem of failure on a single route (route demand > Q units). However, since the unit of analysis here is the segment comprising multiple (D) routes, the problem of failure on a single route has no meaning except for the special case of D = 1. Here, the concept of failure is only relevant if segment demand exceeds segment capacity of DQ units. To reflect this fact, this paper uses the term segment failure to substitute for the more traditional term of ‘‘route failure’’.

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Table 7 Extra vehicles used in fixed driver-to-territory assignments at extreme values of demand variability (b), vehicle capacity (Q) and number of customers (N) Q

b

N

J

Number of drivers (D) assigned to each geographic segment 1

2

3

4

5

1

500

100

21

0.57 8.00 8.57

0.57 3.00 3.57

0.57 1.02 1.59

0.57 0.54 1.11

0.57 0.08 0.65

1

500

500

102

1.51 46.00 47.51

1.51 20.50 22.01

1.51 12.04 13.55

1.51 8.26 9.77

1.51 5.85 7.36

1

5000

100

3

0.50 0.00 0.50

0.50 0.00 0.50

1

5000

500

11

0.51 0.00 0.51

0.51 0.00 0.51

0.51 0.00 0.51

0.51 0.00 0.51

0.51 0.00 0.51

100

500

100

29

8.50 6.00 14.50

5.18 1.90 7.08

3.69 0.82 4.51

2.96 0.56 3.52

2.29 0.21 2.50

100

500

500

117

16.58 42.23 58.81

9.90 18.40 28.30

7.57 10.62 18.19

5.41 7.39 12.80

5.41 5.35 10.76

100

5000

100

3

0.53 0.00 0.53

0.53 0.00 0.53

Zero since D = J

N/A since D P J

N/A since D P J

100

5000

500

12

1.51 1.19 2.70

1.51 0.21 1.72

Zero since D = J

1.45 0.05 1.50

N/A since D P J

1.20 0.02 1.22

N/A since D P J

0.94 0.00 0.94

Notes: Each cell shows the daily extra number of vehicles dispatched under territory-based driver assignment (vis-a`-vis daily (global) route reoptimization). The top number in each cell is the extra vehicles due to the assignment method’s limited capacity to reduce vehicle usage in response to demand contractions and the second number is the number of vehicles dispatched to redress segment failure.

related transportation cost and would be feasible in situations where customers see it as no more than minor inconvenience in having to wait a day (or not much longer) for only a portion of a given day’s order. Another option for lowering the related transportation cost is to make occasional and minor deviations from the driver–customer assignment policy by having a driver with extra capacity in an adjacent geographic sub-segment make up the shortfall. As indicated below, these variants of the operating procedures presented here seem worthy of future research. 4.4. Predictive models for the simulation data To model the performance of the proposed rule on customer–driver familiarity, regression analysis was used to develop Eqs. (4) and (5) below. These modeled the behavior of two cross-customer averages: (i) the proportion of visits received per encountered driver (Y ) and (ii) the proportion of total visits received from the customer’s favourite (most frequently encountered) driver. These equations were found to provide the best fit based on use of the ANOVA results and plots to identify and test a wide range of variable transformations and interactions. Quality of fit and its standard companion regression modeling goals of homoscedasticity, error term independence and normality guided the identification of promising interactions of the independent

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Table 8 Statistical analysis of regression Eqs. (4) and (5) Eq. (4): Y ¼ 0:541 þ 0:140eg1 þ 0:295eg2 þ e1 Predictor Coefficient

Std. error

T

P

Constant EXPG1 (eg1) EXPG2 (eg2)

0.0026 0.0113 0.0119

210.04 12.41 24.82

0.000 0.000 0.000

0.541 0.140 0.295

S = 0.04876, R-Sq = 81.5%, R-Sq(adj) = 81.5% Analysis of variance Source

DF

SS

MS

F

P

Regression Residual error Lack of fit Pure error

2 697 74 623

7.323 1.657 0.186 1.471

3.6616 0.0024 0.0025 0.0024

1540.29

0.000

1.06

0.344

Total

8.980 P

Eq. (5): Predictor

ðT Þ c i2S 1 ið1Þ

TN

¼ 0:821 þ 0:048eg1 þ 0:134eg2 þ e2 Coefficient Std. error

Constant EXPG1 (eg1) EXPG2 (eg2)

0.821 0.048 0.134

T

P

0.0010 0.0044 0.0047

814.37 10.88 28.90

0.000 0.000 0.000

S = 0.01908, R-Sq = 83.6%, R-Sq(adj) = 83.6% Analysis of variance Source

DF

SS

MS

F

P

Regression Residual error Lack of fit Pure error

2 697 74 623

1.296 0.254 0.032 0.222

0.6480 0.00036 0.00043 0.00036

1779.70

0.000

1.20

0.137

Total

1.550

factors and transformations of both the response and predictor variables. Table 8 presents the key statistical output concerning the equations and shows that they provide an adequate fit for the data; e.g., at any reasonable testing level, the p-values for lack of fit support the hypothesis that there is no lack of fit. In fact, using the criterion that the ‘‘R2’’ value that should be relevant when pure error (inter-replicate) variation exists is the percentage of explainable (rather than total) variation (Draper and Smith, p. 42), the models account for 97.5% (Eq. (4)) and 97.6% (Eq. (5)) of the explainable variation (variation other than pure error): Y ¼ 0:541 þ 0:140eg1 þ 0:295eg2 þ e1 P ðT Þ i2S 1 cið1Þ ¼ 0:821 þ 0:048eg1 þ 0:134eg2 þ e2 TN

ð4Þ ð5Þ

where e = the natural logarithm base of 2.718. e1, e2 are the regression estimation errors. g1 = dVhigh/Qe  dVlow/Qe; g2 = (Vhigh  Vlow)/Q. Vhigh P and Vlow are such that: P Prob[ di > Vhigh] = 0.0001 = Prob[ di < Vlow]; di being the ith customer’s demand. In the above regression equations, the predictors (g1 and g2), which represent transformations of an interaction of the factors, are, respectively, un-rounded and integer-rounded estimates of the range in the day-to-

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day number of required routes. The intuition behind these variables is that, as discussed in analyzing the impact of the various factors, wide ranges translate to greater instability in the day-to-day customer groupings, which, in turn, reduces customer–driver familiarity. The range of required routes was estimated by approximation of the upper and lower limits on total demand across all N customers.3 Though seemingly the more natural choice, the integer rounded version of this range (g2) did not quite capture the idea that even with a fixed number of routes (g2 = 0), demand fluctuations can still necessitate day-to-day changes in customer regrouping. At the same time the un-rounded version (g1) by itself led to downwardly biased predictions by over-stating the effects of those fluctuations. Thus, simultaneous inclusion of both variables resulted in better regression models than use of either one by itself. Regarding the modeling of the extra vehicle requirement of fixed driver-to-territory assignment, study of the simulation data suggested a better approach than regression analysis. The modeling approach and the quality of fit of the model are presented in Appendix A. 5. Conclusion The primary contributions of this paper to the literature are its introduction and assessment of a rule for assigning drivers to delivery routes in probabilistic customer demand settings. The rule’s design was motivated by considerations that are keyed to the customer service objectives of vehicle routing and dispatch operations; e.g., ensuring tight relationships between a driver and the customers that the driver makes deliveries to. The study quantifies how factors such as demand variability and vehicle capacity affect the extent to which the rule actually engenders those types of relationships. For example, the study found that even with very volatile fluctuations in day-to-day customer demands, each customer can have a high level of familiarity with one driver. Specifically, an average of at least 80% of each customer’s delivery visits is handled by one driver. Further, at least 20% of customers are consistently visited by the same driver. Reflection on some possible negative side effects of the rule led to the examination of an alternative customer–driver assignment. Of particular concern is that the wide disparity in familiarity levels between a customer’s most frequently encountered driver and the next most frequently encountered driver could create problems on days when the former driver is unavailable. This alternative is designed to build and broaden driver–customer familiarity by partitioning the service region into geographic segments and establishing rigid assignments of a pre-specified set of drivers to each segment. This study estimated the additional mean daily vehicle requirements of adhering to this tactic instead of completely reoptimizing the routes each day. The additional requirement was found to be capable of reaching as high as 50% of the worst-case vehicle requirement for daily route reoptimization. Following are four of this study’s extensions that seem promising. The first is to quantify how related operating procedures such as next-day (instead of same-day) deliveries to resolve segment failures and limited detours by drivers from their pre-assigned territories might impact this cost. The second is to examine alternative assignment tactics that address concerns about potentially undesirable features of excessive driver–customer familiarity; e.g., kickbacks. Third, this study does not cover all the complexities of manpower scheduling so the impact of those complexities on the baseline insights from this study remain an open question for future research. Finally, it would also be interesting to explore the development of closed-form expressions for bounds on this study’s simulation-based estimates. Acknowledgment Grant No. 261513-03 from the Natural Sciences and Engineering Research Council (NSERC) of Canada supported this research.

3 For the gamma distributed total demand across all N customers (parameters Na and b), the true range of values for that demand for P P computing g1 and g2 should be based on the more absolute probability limit of 0; i.e., Prob[ di > Vhigh] = 0 = Prob[ di < Vlow]. However, the limit of 0.0001 yielded regression equations that fit the data better than the more absolute limit of 0.

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Appendix A. Estimation model for the extra vehicles used by fixed driver-to-territory assignments This estimation procedure, which can be summarized into three steps, yielded accurate estimates: they deviated from the observed simulation values by an average of 0.094 vehicles/day (the maximum was 1.32 vehicles/ day). Step 1: For each (b, Q, N) combination, estimate J (the required number of drivers (routes) to satisfy maximum total demand) as b J ¼ dV high =Qe P where Vhigh is the value of total demand that has a 0.0001 probability of being exceeded and total demand ( di, i = 1, . . . , N) is gamma distributed with parameters Na and b (as with the regression models discussed earlier, a probability of 0.0001 yielded a better fitting model than the absolute probability of zero). Based on b J , estimate route reoptimization’s mean daily vehicle dispatches as ( ) J^ N X X P ðk  1ÞQ < d i 6 kQ k ðA:1Þ k¼1

i¼1

Step 2: Approximate the customer groupings for single-driver territories (D = 1) as u territories with eN = b J u customers each and v territories with €eN = b J^ u customers each, where u and v are determined by solving simultaneous equations: u þ v ¼ b J and eN = b J uu þ €eN = b J ^uv. Note that for D > 1 territories (segments) are formed by joining adjacent territories created for D = 1. Step 3: Estimate each segment’s average daily vehicle requirements as ! D X P fðk  1ÞQ < segment demand 6 kQgk þ P fsegment demand > DQgD k¼1

þ P fsegment demand > DQg

ðA:2Þ

In this expression, the third term (the probability of a segment failure) equates to the probability of a vehicle having to be dispatched to redress the failure. The implicit (and reasonable) assumption is that the capacity of a single vehicle (Q units) is sufficient to redress the shortfall of (segment demand—DQ) units. The result of the expression in (A.2) is then summed to over all territories to estimate the average daily vehicle requirement of the fixed territory assignment, so the extra vehicle requirement is the excess of that sum over the value of the expression in (A.1). References Ahire, S., Greenwood, G., Gupta, A., Terwilliger, M., 2000. Workforce-constrained preventive maintenance scheduling using evolution strategies. Decision Sciences 31 (4), 833–859. Bertsimas, D., 1992. A vehicle routing problem with stochastic demand. Operations Research 40 (3), 574–585. Brusco, M.J., 1998. Solving personnel tour scheduling problems using the dual all-integer cutting plane. IIE Transactions 30 (9), 835–844. Brusco, M.J., Johns, T.R., 1998. Staffing a multiskilled workforce with varying levels of productivity: an analysis of cross-training policies. Decision Sciences 29 (2), 499–515. Draper, N.R., Smith, H., 1981. Applied Regression Analysis, second ed. John Wiley and Sons. Gans, N., Zhou, Y.-P., 2002. Managing learning and turnover in employee staffing. Operations Research 50 (6), 991–1006. Haughton, M., 2002. Measuring and managing the learning requirements of route reoptimization on delivery vehicle drivers. Journal of Business Logistics 23 (2), 45–66. Law, A., Kelton, W.D., 1991. Simulation Modeling and Analysis, second ed. McGraw-Hill. Quintana, R., Ortiz, J.G., 2002. Increasing the effectiveness and cost-efficiency of corrective maintenance using relay-type assignment. Journal of Quality in Maintenance Engineering 8 (1), 40–61. Waters, C.D.J., 1989. Vehicle scheduling problems with uncertainty and omitted customers. Journal of the Operational Research Society 40 (5), 1099–1108. Xu, J., Chiu, S.Y., 2001. Effective heuristic procedures for a field technician scheduling problem. Journal of Heuristics 7 (5), 495–509.