An efficient system of incentivizing truck drivers to accept night loads

ABSTRACT Night deliveries are increasingly used by motor carriers to cut costs, but many carriers struggle to secure night truck drivers because many drivers dislike night driving. Monetary incentives are offered to drivers to accept night loads, but the incentive system currently used in practice is inefficient, as it not only pays more incentives than necessary but also does not incentivize the right drivers, both of which increase costs. This paper develops a new incentive system that allows carriers to secure night drivers at lower costs. Based on the interviews conducted with seven motor carriers, as well as the analysis of large night delivery data, we propose a bi-level programming approach that (1) generates a unique incentive for each driver based on his/her night driving performance, and (2) implicitly aligns the incentive paid to each driver with his/her reservation price. Simulation experiments conducted with real-world data showed that by using our approach carriers can not only avoid over-incentivizing drivers, but also (1) incentive only the right set of drivers who can bring cost savings and (2) encourage other drivers to improve night driving. Results also showed that our approach may give cost savings of up to 2.5% over the current system.


Introduction
Night deliveries are increasingly used by motor carriers (truckload and less-than-truckload carriers), especially for business-to-business (B2B) logistics, as it is known that operating trucks at night reduces the cost of carriers.Advantages of night deliveries include: (1) reduced fuel burn (higher road speed due to reduced congestion), (2) reduced parking fine (ease of finding legal parking spaces at night), and (3) reduced fleet sizes (decreased need for trucks because each truck can be used for both day and night operations) (e.g.Holguín-Veras et al. 2011;McPhee et al. 2015).
Night deliveries, however, have disadvantages too.According to our interviews conducted with seven motor carriers, one disadvantage is that carriers struggle to find truck drivers for night loads (where a load is a batch of shipments delivered by a single truck), as many drivers dislike night driving for safety and other reasons (Moonaghi et al. 2015).Some carrier managers indicated that 'To fill night loads we must either offer pretty good monetary incentives to internal drivers or hire external drivers through brokers, whose wages are at least 20% higher than our drivers, both of which offset the merits of night deliveries'.This condition suggests that to benefit from night deliveries, carriers must have an efficient incentive raise gets the first load), which ignores the night driving performance.
The above implies that the incentive system currently used by motor carriers is an inefficient one that not only pays more incentives than necessary but also does not incentivize the right drivers, both of which increase costs.Such system inefficiencies limit the amount of benefit carriers can realise from night deliveries.This motivated us to develop a new method of incentivizing truck drivers that allows carriers to populate night loads with internal drivers at lower costs.The goal of this paper is to develop one such system that may be used by motor carriers as a decision tool.Our approach was developed by incorporating expert opinions obtained from a series of interviews with seven motor carriers, and was tested using the large night delivery data obtained from the industry.
The problem we address is the one that matches drivers to (day or night) loads using monetary incentives.We formulate this problem as a bi-level program and develop an interactive solution method that uses both the sensitivity report of the upper-level problem and the econometric estimation results of drivers' night driving records.An interesting aspect of our method is that it: (1) offers a unique incentive to each driver based on his/her night driving performance, and (2) implicitly aligns the incentive paid to each driver with the driver's reservation price by finding the amount that is reasonable from both the carrier's standpoint (sufficiently small such that it results in cost savings) and the driver's viewpoint (at or higher than the reservation price).We show by conducting a series of simulation experiments that our method outperforms the existing system.This paper makes three important contributions to the driver management literature.First, this study paves a new road for future research on driver management by considering for the first time the driver assignment problem that involves both the monetary incentives for night driving and the drivers' response behaviours to such incentive offers.Second, this study fills the gap between the (rather inefficient) current driver incentive system that merely focuses on populating night loads and the more efficient system that seeks not only to populate night loads but also to do so at lower costs.Third, this study provides several managerial implications that can help carrier managers understand how a driver incentive system ought to be designed to achieve cost efficiencies.Specifically, it gives valuable insights into what type of drivers should (should not) be incentivized to accept night loads, by how much each such driver should be incentivized to be profitable, and in what order (sequence) each such driver should be incentivized to minimise the assignment costs.

Literature review
This section reviews the driver-to-load assignment studies.The studies reviewed here belong mainly to the drayage (short truck transportation performed in the container shipping industry) literature, because most driver-to-load assignment studies are found in this area.We also reviewed other literatures that are either conceptually or methodologically relevant to the driver-toload assignment problems, which include auction studies, assignment problems with incentives, bid-price control studies, and bi-level programming studies.Reviews of these studies, as well as the descriptions on why and how they are related to this study, can be found in online supplement.
Studies in the drayage literature considered a variety of topics including truck routing (often with time windows and hours of service constraints), management of empty hauls, assignment of pick-up/drop-off terminals, and scheduling of trucks and truck drivers.The studies that considered the driver-to-load assignment problem include the following (see Pérez Rivera and Mes 2019 for detailed reviews).Taylor et al. (2002) considered the choice of an initial ramp for truck-rail freight transport.Verma and Verter (2010) and Verma, Verter, and Zufferey (2012) developed a drayage model of hazardous materials for truck-rail networks.Wang and Regan (2002) and Escudero et al. (2013) analysed how delays at terminals trigger the re-scheduling of freight transports.Smilowitz (2006), Francis, Zhang, and Smilowitz (2007), and Caris and Janssens (2009) studied scheduling flexibilities at the terminals with empty containers.Nossack and Pesch (2013) and Braekers, Caris, and Janssens (2013) considered the problems with multiple terminals having homogenous truck fleets.Imai, Nishimura, and Current (2007) studied a similar problem with no time windows.Markovi, Drobnjak, and Schonfeld (2014) studied random delays at intermodal terminals, while Pérez Rivera and Mes (2017) considered a problem with terminal assignment costs.
One limitation of these studies is that they treated truck drivers as 'homogeneous' (with respect to driving performance).For this reason, many of them don't even distinguish trucks and drivers -i.e. a 'truck' is treated the same as a 'driver'.As such, these studies are not actually matching a driver with a load.Note that this 'homogeneous truck driver' assumption is unlikely to hold in practice, especially when solving the problems involving night loads, because as discussed earlier drivers are highly heterogeneous with respect to their night driving performances.
We found only three studies that considered matching a driver directly with a load.Trappey and Ho (2002) developed a decision support system (DSS) and algorithms for assigning jobs to workers in a distribution centre.Assigning a truck driver to a shipment (load) is a part of this large DSS.Their algorithm assigns drivers to loads based on a set of simple rules (e.g.drivers' assigned region).Pazour and Neubert (2013) considered a special type of drayage problem where intermodal containers are transferred from one rail ramp to another.This problem seeks to assign drivers to loads such that the maximum loads can be covered with minimum empty moves.They developed a heuristic that favours the routes balancing bi-directional moves between origin-destination pairs.
The above two studies, however, considered only daytime loads.Perhaps the only study that considered the problem of matching a driver with a load that involves night driving is Chen et al. (2018).They considered the mixed-shift vehicle routing problem, where a driver can be assigned to either a short shift (12-hour single driver task, which can be day or night shift) or a long shift (24-hour team driving task).The objective is to minimise the driver cost and empty miles.This study, however, assumed implicitly that drivers are always willing to accept night loads.In practice, this assumption does not hold because many drivers reject night loads for a variety of reasons.Additionally, they did not consider monetary incentives for night driving, or the drivers' responses to such incentives, which are important factors that must be considered by carriers with night loads.
In short, our literature review suggests that limited studies exist that considered the problem of matching drivers to loads, especially night loads.Studies that considered such problems either treated truck drivers as commodities (homogeneous) or ignored the drivers' night-load-acceptance behaviours.Ideally, decisions involving night-load assignments should be made by considering each driver's unique night driving performance, while also considering his/her load-acceptance behaviour.We consider a problem that incorporates both of these aspects and develop a solution technique.

Nature of the problem
The problem we consider is an assignment problem that matches drivers to loads, in which monetary incentives are used to encourage drivers to accept night loads.It seeks the minimal-cost solution that fills every load (day and night) with an available driver (internal or external).This problem, called the load assignment problem with incentives (LAPI), is a daily operational problem.The nature of the problem and the solution procedure used in practice are described below.Note that although the following reflects the procedure used by many motor carriers we interviewed (i.e. it represents a 'popular' system), it may not reflect a system that is used by all motor carriers.
The problem is defined by a set of loads (day and night) to be filled with drivers, and a set of available internal drivers.Given the problem, a carrier announces the initial night incentive l for internal drivers, expressed as a % of daytime wage, and the drivers respond with their decisions on whether they accept night loads.The carrier then solves an assignment problem that populates the loads with drivers at minimal-cost, taking into account the drivers' night-driving choices.If the solution fills every load with a driver, the solution is final, but otherwise the carrier raises l by the amount l + in an attempt to fill vacant loads.This raise applies to all drivers, including those who accepted night loads before the raise.As drivers respond to the raise, the carrier adjusts the assignment solution using the first-come, first-served rule (e.g. if there are 10 loads to be filled with drivers, the first 10 responders get these 10 loads).If the above procedure fails to fill all loads, the vacant loads are filled with external (more expensive) drivers that are hired through a broker.
In theory, the above incentive raise and re-assignment process can be repeated iteratively until all or nearly all loads are filled with internal drivers.In reality, however, the incentive raise is performed only once because carriers have limited time to wait for driver responses.Note that this interactive process between a carrier and its drivers can be viewed as a Stackelberg game.Thus, following the works that studied this game (e.g.Federgruen, Lall, and Simsek 2019;Mu, Jiang, and Leng 2017;Yang et al. 2020), we formulate the LAPI as a bi-level program in which the upper-level problem reflects the decision of a carrier (leader) and the lower-level problem reflects that of each driver (follower).

Assumptions
We adopt the following assumptions, which are deemed reasonable by practitioners.First, the planning horizon is one day, i.e. the set of loads included in one problem consist of the loads to be dispatched on a given day (t).Second, the problem for day t is solved on day t−2 (which means that a carrier has up to 48 h to adjust the solution).Third, there are three types of drivers, namely, day drivers who accept day loads only, night drivers who accept night loads only, and hybrid drivers who accept day and night loads.Fourth, a driver's decision to accept night loads can vary from one day to the next.For example, a hybrid driver who is normally willing to accept night loads may not be willing on day t if he/she has a family event in the evening of day t.Fifth, when a carrier can populate only a subset of the loads with its internal drivers, it can find sufficient external drivers to fill the vacant loads (according to practitioners finding external drivers is usually not difficult except when the economy is very strong).Sixth, carriers can implement a performance-based pay system where two drivers' payments for the same load can be different based on their performance records (such a system is already used by many truckload carriers; e.g.Lilja 2018).

No-Extra incentive problem
The notations used in our problem formulation below are summarised in Table 1.We first consider the problem in which no incentive beyond l is allowed.In this case, the LAPI reduces to a simple linear assignment problem.Let L = 1, 2, . . .j, . . .be the set of day and night loads on day t, and D = 1, 2, . . .i, . . .be the set of internal and external drivers available on day t.L consists of two subsets L d (day loads) and L n (night loads), while D consists of three subsets D d (day drivers), D n (night drivers), and D h (hybrid drivers).We assume that |L| ≤ |D|, and that the elements of D are prescreened such that those drivers who cannot legally drive on day t due to hours-of-service regulation are already excluded.Also let c ij be the expected cost of load j ∈ L if driver i ∈ D is assigned to j (c ij includes all the time-dependent costs of operating a vehicle because the longer it takes for i to haul j, the higher the c ij ).Note that c ij of an internal driver can be estimated from the internal driving records (details are discussed later in §6.3), while that of an external driver is set by a broker (beyond a carrier's control).We assume that l is included in c ij ∀ i ∈ D n ∪ D h , j ∈ L n .Finally, let x ij be a binary decision variable such that it is 1 if i is assigned to j (0 otherwise) and z ij be a binary constant indicating a driver's load preference (1 if i is willing to accept j, 0 otherwise).
The problem with no incentives (denoted P1) can be expressed as a linear program as: Subject to: where z ij values of constraints (3) (determined exogenously) must satisfy the following conditions.
Minimal benefit a carrier wishes to attain by Transit time of q LAVG q Average transit time of the lane (q's lane) NIGHT q Load dummy variable (1 if q is a night load, 0 otherwise) DRIVER i q Driver dummy variable (1 if the driver of q is i, 0 otherwise) DAY t q Time dummy variable (1 if q was hauled on day t, 0 otherwise) γ , φ, ρ, λ Regression parameters K 1 q Weight of load q K 2 q Amount of time q moved during the morning rush hour K 3 q Amount of time q moved during the afternoon rush hour n r Number of drivers a carrier can incentivize and get responses from within a time limit of 48 hrs.
Note that all elements of D h are internal drivers as external drivers are hired specifically for vacant loads (day driver is hired for a vacant day load and night driver is hired for a vacant night load).
The objective function (1) minimises the assignment cost (summed vehicle-operating cost of all loads).Constraints (2)-( 3) define the domains of decision variables and also ensure that a driver would not be assigned to non-preferred loads.Constraints (4) require that exactly one driver be assigned to every load.Constraints (5) prevent assigning more than one load to any driver (since each j ∈ L requires 8 to 12 work hours, assigning multiple loads to a driver violates the hours-of-service rule).Note that P1 is easy to solve as it is an assignment problem with no 'side constraints'.

Incorporating extra incentives
If a carrier uses incentives beyond l, the hybrid drivers' load preferences can no longer be treated as exogenous, as the intent of using extra incentives (during the second round of driver input solicitation) is to convince hybrid drivers to change their night-load preferences (indirectly control their choices).Thus, the full version of LAPI is a bi-level program that jointly considers the upper-level problem (driver-to-load assignment problem solved by a carrier) and the lower-level problem (loadacceptance problem solved by hybrid drivers), wherein both the incentives and the driver choices are treated as decision variables under the carrier's control (direct control of incentives and indirect control of driver choices).This means that to construct the LAPI, we must understand how drivers make load-acceptance decisions and how carriers can control such decisions with incentives.
Since the lower-level problem is a discrete-choice problem, it can be specified by using the utilitymaximization framework.Let p ij ≤ l + be the nonnegative monetary incentive offered beyond l to i ∈ D h for accepting j ∈ L n during the second driver-input solicitation round, and δ ij be a binary variable indicating the revised z ij given p ij (1 if i is willing to accept j with an extra incentive p ij , 0 otherwise).Note that δ ij applies only to the drivers for whom z ij = 0 (rejected j with l during the initial solicitation round).Although p ij = l + ∀ i, j under the current system, we allow p ij to vary across drivers and loads, as long as it does not exceed l + .We denote the utility of driver i ∈ D h for accepting j ∈ L n as u ij , and that of i for rejecting j (i.e.utility of doing something else) as u * ij .
Assuming linear utility functions and a positive impact of p ij on u ij , but not on u * ij (since p ij is an alternative-specific attribute -e.g.Koppelman and Chandra 2006), we express u ij and u * ij as: ) where σ ij , σ * ij , and β are unknown parameters, and ij and * ij are the non-systematic parts of utility functions (random errors).Note that σ ij and σ * ij reflect drivers' heterogeneity parameters (i's unique preference or aversion of load j), while β reflects drivers' price (incentive) elasticity parameter.Like many discrete-choice studies (e.g.Costa, Montemurro, and Giuliani 2019;Eissa and Hoynes 2004), we assume that the β does not vary across drivers (β represents the average elasticity of all drivers).We also assume that drivers make their load-acceptance choices by using the utility-maximization rule as: Ben-Akiva and Lerman (1985) have shown that, when decision makers are using the above rule, the choice probability (probability that i accepts j with incentive p ij ) can be expressed as: where . This is a standard logit function, which bounds the choice probability between 0 and 1.Note that, with perfect knowledge of the systematic parts of u ij and u * ij (no unobserved attributes or measurement errors exist in u ij and u * ij ), ij = * ij = 0 (Ben-Akiva and Lerman 1985) so that In such cases we have: (10) Eqn.(10) establishes the functional relationship between p ij and the driver choice when θ ij 's and β are perfectly known.Thus, it gives important insights into how a carrier can control (indirectly) the δ ij 's of hybrid drivers for night loads using p ij , when it has the full knowledge of θ ij 's and β values.

The bi-level program
If we assume (for now) that a carrier has the perfect knowledge of θ ij 's and β, the bi-level program to be solved by the carrier during the second round of driverinput solicitation is given by: Note that δ ij is a decision variable under a carrier's control that is used to adjust z ij 's of selected drivers in the favourable direction.Also note that although p ij and δ ij variables apply only to i ∈ D h and j ∈ L n , we treat them as generic variables and constrain their values for i ∈ D h and j ∈ L n .
Objective function ( 11) minimises the sum of assignment and incentive costs.Constraints ( 12)-( 14) define the domains of decision variables.Constraints ( 15) and ( 16) have the same interpretations as ( 4) and ( 5) respectively.Constraints ( 17) prevents assigning a driver to non-preferred loads, where i's preference on j is now defined by the sum of z ij (determined exogenously by i) and δ ij (decision variable under the carrier's control).Constraints ( 18)-( 19) allows the carrier to adjust only the z ij 's of hybrid drivers for night loads.Constraints ( 20)-( 21) define the relationship among δ ij , z ij , and p ij variables ∀ i ∈ D h , j ∈ L n , which require that in order for δ ij to be 1: (1) z ij must be 0, and (2) p ij must be sufficiently large to make the logit function larger than or equal to 0.5.This means that the carrier can adjust z ij 's of selected drivers (unwilling to accept j) in the favourable direction (accept j) by setting δ ij = 1, but by doing so the carrier must ensure that u ij ≥ u * ij .Since P2 requires the perfect knowledge of θ ij 's and β values, no uncertainties are involved in P2.

Properties of the LAPI (P2)
Since P2 is expressed as a single-level problem (as constraint (21) allows us to incorporate the lower-level optimality condition into the upper-level problem), it can, in theory, be solved to optimality by using commercial solvers.In reality, however, solving P2 is difficult not only because it is complex (mixed-integer nonlinear program), but also because θ ij 's and β are unknown to carrier managers (decision makers of the bi-level program).In particular, knowing θ ij values is difficult because they reflect drivers' time-varying individual heterogeneities (e.g. a driver's personal schedule on a given day).This means that the conventional assumption used implicitly by many bi-level programming studies (the upper-level decision maker knows the lower-level parameters -e.g.Moore and Bard 1990) is violated in our study, which makes it difficult to solve P2 directly.

Solution method: framework
Since P2 cannot be solved directly, we study properties of the optimal P2 solutions in this section to gain insights into how similar solutions can be generated when the following restrictions apply: (1) carriers have no knowledge of θ ij and β, (2) only the drivers can solve the lower-level problem, and (3) carriers can only solve simple linear programs, such as P1, to obtain solutions.Our goal is to answer the following question: 'If we are to generate a solution similar to the optimal P2 solution, which driver and/or load should be incentivized and how much incentive should be offered?'Findings from this section will be used to develop a solution algorithm in the next section.Proofs of the arguments made in this, and the next, sections can be found in online Appendix A.

Which driver/load to incentivize
Let (i, j) denote a combination of i and j, where i ∈ D h , j ∈ L n .The carrier can incentivize any (i, j) to reduce the P2 objective value.Since the expected cost saving attained by incentivizing (i, j) varies, often substantially, across i and j, it is intuitive that the optimal P2 solution incentivizes only a limited number of (i, j)'s which, if incentivized, are expected to bring positive cost savings.
The necessary, but not sufficient, conditions for a given (i, j) to be incentivized under the optimal P2 solution are that: (1) z ij = 0, and (2) C p←0 (z ij = 0) − C p←0 (z ij = 1) > 0, where C p←0 (z ij = 0) and C p←0 (z ij = 1) are the objective values of the optimal P2 solution obtained by setting z ij = 0 and z ij = 1 respectively, while fixing all p ij 's to zeros (p ij = 0 ∀ i ∈ D, j ∈ L).This means that if, for any (i, j) for which z ij = 0, the change of z ij from 0 to 1 (assuming this can be done with no extra incentive) results in no cost reduction at all, (i, j) will not be incentivized.Notice that C p←0 (z ij = 0) − C p←0 (z ij = 1) is the (negated) shadow price of z ij , i.e. it is the negated shadow price of constraint ( 17) for (i, j), after moving δ ij to the left-hand-side (we denote this shadow price as P2(z ij )).This condition suggests that the optimal P2 solution incentivizes only the (i, j)'s with negative P2(z ij )'s.
It can be shown that we can obtain P2(z ij )'s by solving P1 (i.e.we need not solve P2 to get P2(z ij )'s).Notice that when all p ij 's are zeros, all δ ij 's must also be zeros because in this case adjusting the right-hand-side of (21) by controlling p ij (making δ ij = 1) is not possible.This means that P2 reduces to P1 when p ij = 0 ∀ i ∈ D, j ∈ L, so that the shadow price of constraint (3) for (i, j) under the optimal P1 solution, which is denoted P1(z ij ), is the same as Theorem 4.1: The optimal P2 solution will incentivize only the subset of (i, j)'s that satisfy the following two conditions jointly: (1) z ij = 0, and ( 2) P1(z ij ) < 0, where i ∈ D h , j ∈ L n .

How much to incentivize
We now investigate by how much each (i, j) ∈ should be incentivized, where is the set of (i, j)'s that satisfy the conditions of Theorem 4.1.It is intuitive that under the optimal P2 solution, the extra incentive p ij offered to each (i, j) ∈ is the value that makes the right-hand-side of ( 21) equal to 1 (minimal p ij that will make u ij ≥ u * ij ; this p ij is denoted p ij min ).Thus, in the perfect world where θ ij and β are known, the optimal P2 solution would set p ij such that: (1) 2) p ij = 0 otherwise (does not incentivize (i, j)).But in reality where θ ij and β are unknown, p ij min is also unknown, so that we must use an alternative approach to determine p ij .
One alternative approach is to use p ij that maximises the probability that a carrier gets a positive benefit from incentivizing (i, j), which can be obtained by solving the following problem: where α ij > 0 is a parameter reflecting the minimal benefit a carrier wishes to attain by incentivizing (i, j) (α ij can vary, or be identical, across i and j -we assume α ij is identical ∀ i, j).Note that this problem maximises the probability that driver i accepts night load j with incentive p ij , while making sure that: (1) p ij must not be negative or exceed l + , and (2) the carrier's monetary gain (negated shadow price less incentive cost) is greater than or equal to α ij .Thus, the optimal p ij found by solving this problem maximises the probability that the carrier gets a positive benefit of at least α ij .
An interesting aspect of this (alternative) approach is that, when min , it generates the same p ij as that of the optimal P2 solution.This means that, while the approach does not always generate the same p ij as that of P2 optimal solution, it sometimes can, depending on the α ij value used.This, in turn, implies that, although carriers cannot intentionally set α ij to | P1(z ij )| − p ij min , they can test a variety of α ij values over time (trial and errors) to find a good α ij value that may allow them to obtain near-optimal p ij values in the long run.This condition suggests that this alternative approach may facilitate the production of solutions close to the optimal P2 solution, at least in the long run, without the knowledge of θ ij or β.Since it can be shown that the optimal solution of the problem ( 22)-( 24) is given by p ij = minl + , | P1(z ij )| − α ij , we have the following.
Theorem 4.2: The optimal incentive value p ij * , which maximises the probability that a carrier receives the minimum benefit of α ij by incentivizing (i, j) ∈ , is: attainable cost saving by incentivizing (i, j) is less-than α ij ), so that only (i, j)'s with positive p ij * 's should be incentivized.This implies that the set of (i, j)'s that are incentivized under this approach, denoted as , can have a smaller cardinality than .

Algorithm outline
Since the lower-level problem must be solved by drivers, we adopt an interactive procedure whereby the carrier solves the upper-level problem and the drivers solve the lower-level problem.Specifically, we use an iterative procedure in which an incentive is offered to one (i, j) at a time, rather than incentivizing all (i, j) ∈ simultaneously.This is because the shadow price P1(z ij ), from which p ij * is obtained, indicates the expected cost reduction of changing one particular z ij from 0 to 1 (see, e.g.Ho 2000), so that we do not know the effect (expected cost reduction) of jointly changing z ij 's from 0 to 1 for all (i, j)'s with positive p ij * 's (as such, simultaneously incentivizing all (i, j) ∈ can result in unexpected outcomes).The algorithm outline is presented below.

Algorithm 1
Do until there are no more (i, j) ∈ with positive p ij * 's 1. Incentivize (i, j) ∈ (ask i to accept j with incentive p ij * ) and wait for i's response 2. If i accepts j, change z ij and c ij such that z ij ← 1 and c ij ← c ij + p ij * (otherwise remove (i, j) from and skip steps 3 and 4) 3. Re-solve P1 4. Update p ij * 's of the remaining (i, j) ∈ with positive p ij * 's using the new P1(z ij ) obtained from P1 sensitivity report

Loop
In theory, (i, j) to incentivize at each iteration of Algorithm 1 can be chosen arbitrary since, as long as we repeat the process until there are no more positive p ij * 's, the final solution should be the same.This, however, is a naïve approach, because the number of attainable driver contacts n r (number of drivers a carrier can incentivize and get responses from) within the time limit of 48 h is limited.Note that if | | > n r , we may need to terminate the process before completing all the intended driver contacts, which can lower the solution quality.Thus, to obtain a good solution, we must contact (i, j)'s with the largest expected cost savings early in the process.As such, the (i, j) pair to contact at iteration r should have the highest value of . the expected benefit of incentivizing (i, j) at iteration r.It can be shown that the (i, j) pair with the largest | P1(z ij )| also has the largest value of both Theorem 5.1: To maximise a carrier's expected cost saving attainable within the time limit of 48 h, (i, j) ∈ with the largest | P1(z ij )| should be contacted (incentivized) at each iteration.
Since n r varies by case-by-case (as n r is a random variable given that each driver's response time to an incentive is stochastic), we later test the sensitivity of our method to the changes in n r .

Obtaining shadow price from P1
It is known that an assignment problem (P1) frequently produces degenerate solutions (Lin and Wen 2003), so that the resulting shadow prices are often misleading too (Ho 2000).Thus, for our method to work properly, we must find a way to obtain reliable shadow prices from P1.We use the following heuristic procedure to obtain what we call a pseudo shadow price (denoted PSP).

Loop
Once PSP is obtained it is used to calculate the incentive p * for i * (for accepting all j ∈ L * as a set) via the formula: p * = minl + , PSP − α ij .Note that this procedure 'empirically' determines the cost saving (shadow price) for each (i, j) pair, so that it is unaffected by the degeneracy of solutions.One potential issue with this procedure is that it does not give an individual shadow price when |L * | > 1; rather it gives a 'collective' shadow price (expected cost saving when z i * j ∀ j ∈ L * change to 1).This, however, may have minimal impact on the performance of our method because the method still works even when i * accepts only a subset of L * .In such cases a carrier can re-solve P1 by changing z ij 's of only the loads that are accepted by i * (from 0 to 1), so that i * will be assigned to a load only if the cost (P1 objective function value) becomes lower with this partial z ij changes.

The algorithm and features
The proposed algorithm that uses PSP to compute driver incentives is shown in Figure 1.Notice in Figure 1 that no part of P1 solution obtained in a given iteration r is 'locked' in future iterations (r + 1 and onward).Although this may seem confusing from the drivers' viewpoint, as a driver's load assignment can change from one iteration to another (i.e. each time a driver accepts an incentive offer), it actually is not since only the final assignment solution is shared with drivers.
Features of our algorithm are as follows.First, our method always provides feasible solutions as it simply seeks to improve the initial P1 solution (no incentives beyond l) by replacing external drivers with internal drivers using extra incentives.Thus, our solution is always feasible provided that the initial LAPI solution is feasible (which is always true given the fifth assumption discussed in §3.2).Second, our method can reduce the vehicle-operating cost of carriers by cutting the transit time of night loads, as it incentivizes only the drivers with high cost-saving potentials (high | P1(z ij )|'s for night loads).Third, our method can cut the carriers' labour cost by implicitly aligning the night incentive paid to each driver at or near the driver's reservation price.Recall that our method first determines a 'reasonable' incentive for each driver i to accept load j (not larger than cost saving expected from assigning i to j), and then let i decide whether to accept the offer.Since i accepts the offer only if it aligns with his/her reservation price, this process allows one to better align incentives to drivers' reservation prices than the current system, which raises incentives of all drivers simultaneously (often beyond expected savings), regardless of reservation prices.
Readers should note the following points.First, although our method adjusts incentives based on the drivers' load-acceptance behaviours, it should not trigger game playing by drivers (e.g.drivers always reject night loads during the initial input solicitation round).This is because when using our method, such game playing reduces the drivers' expected payoffs (see online Appendix B for details).Second, although our method rewards quick night deliveries, it should not trigger unsafe driving.This is because according to the interviewed carriers: (1) quick deliveries are the result of the drivers' choices of routes and stops (not the choices of speed), and (2) most carriers impose safety constraints on drivers (unsafe driving results in driver terminations).Third, although our method seeks to align the incentive paid to each driver with his/her reservation price, it should not create a feeling of unfairness among the drivers.This is because drivers can always reject the incentive offers.Note that if a driver accepts an offer, it is an indication that the incentive is 'fair' to the driver, so that the driver should not have a feeling of unfair treatment by the employer, even if the premium offered to the driver may differ from that offered to others.Premium differentiation based on reservation price is a practice widely used in many industries (see, e.g.Fiig, Guen, and Gauchet 2018).

Simulation experiments
We perform numerical experiments and contrast the performance of our method vis-a-vis that of the benchmark method, which mimics the current system.We also contrast our solution with the lower bound cost.Our testing is conducted by generating many LAPI instances and solving each instance three times − once using our method, once using the benchmark method, and once using the method that gives the lower bound cost.Each instance, which represents a real instance taken from practice, contains 200-461 loads and up to 700 drivers (internal and external drivers combined).We use CPLEX Callable Library 12.10 (coded in C++) to run the experiments.
We obtained both expert opinions on experimental design and the data used in our experiments from a medium-sized U.S. truckload carrier that performs night deliveries (denoted as Carrier X).Their data, which contained 231 days of shipping records (over 130,000 data of day and night loads), included load assignment decisions (each day), attributes of each load (e.g.origin, destination, stops made), and driving record of each internal driver (e.g.transit time of each load).Details of our simulation experiments are discussed below.Due to space limitation we discuss only selected details in this section.Readers can find further details, such as: (1) how each instance is generated, (2) how some key simulation parameters are determined, and (3) how the benchmark cost (BM) and the lower bound cost (LB) are computed for each instance, in online Appendix C.

Experimental design
We adjust the values of the following factors systematically: (1) drivers' incentive elasticity (β), (2) proportion of night loads that can be filled with internal drivers initially (ratio of the number of drivers willing to accept night loads to the number of night loads -denoted as fill rate), and (3) maximum incentive used in the second solicitation round (l + l + ).We employ a full factorial design of these three factors (3 × 3 × 4 design; see Table 2).For each factor combination (experimental cell), we generate and solve 30 LAPI instances (randomly draw a real instance from 231 daily instances included in the Carrier X data and adjust parameters to conform with Notes.Fill rate indicates the number of night loads that can be filled with internal drivers (night and hybrid drivers).Expert opinion refers to the inputs obtained from practitioners (mainly Carrier X managers).l + l + is capped at 20% (otherwise some internal drivers' wages can become higher than those of external drivers).θ ij 's are generated only for (i, j)'s with z ij = 0 using a truncated uniform distribution such that P(z ij = 1) ∈ (0, 0.5).z ij 's are controlled such that the number of night loads that can be filled with internal drivers equals the fill rate.ATRI (2018) = American Transportation Research Institute (Hooper and Murray 2018).ZipRecruiter: https://www.ziprecruiter.com/Salaries/How-Much-Does-a-Truck-Driver-Make-an-Hour.
the factorial design for the cell), and compute the average solution value for each method.This requires generating 1,080 instances and producing 3,240 solutions.We solve each instance only once because: (1) problem data, many of which are determined randomly, are generated only once per instance, and (2) no solution method (benchmark, proposed, or lower-bound cost) performs stochastic search.

Simulating drivers' response behaviours
We emulate each hybrid driver's response to an incentive raise as follows.First, when i, who is unwilling to accept night loads with the initial incentive l, is offered a higher incentive p * to accept j ∈ L n (or set of loads L * ) i's probability of accepting j with this new incentive (value of eqn.( 9)) is re-computed using the θ ij value generated for this (i, j) pair and p * .Second, z ij is updated such that z ij changes from 0 to 1 if the re-computed probability is at or above 0.5, but z ij remains at 0 otherwise.It is to be noted that in our experiments θ ij 's and β are used only to simulate the drivers' load-acceptance behaviours.That is, in our simulation θ ij 's and β are unknown to the decision maker (carrier), so that the carrier must make the load assignment decision for each instance without the knowledge of these parameters.Only when calculating the lower-bound cost (see Appendix C) we assume that the decision maker has the perfect knowledge of all parameters, including β and θ ij 's.

Estimating c ij values
We estimate c ij of internal drivers by calibrating the regression model below with the data obtained from Carrier X (regression details are discussed in online Appendix D).Each sample unit represents a load.Covariates were chosen based on the expert opinions obtained from Carrier X.
where LT q is the transit time of q, LAVG q is the average transit time of the lane (q's lane), NIGHT q is a load dummy variable (1 if q is a night load, 0 otherwise), DRIVER i q is a driver dummy variable (1 if the driver of q is i, 0 otherwise), DAY t q is a time dummy variable (1 if q was hauled on day t, 0 otherwise), T is the set of dates in the data, γ , φ, ρ, λ are regression parameters, and is an error term.The K r q variables capture the characteristics of load q, and include: (1) q's weight (K 1 q ), (2) amount of time q moved during the morning rush hour (K 2 q ), and (3) that during the afternoon rush hour (K 3 q ).Note that eqn.( 25) is a two-factor fixed effect model that controls driver and time effects.
Once the coefficients of eqn.( 25) are empirically estimated (by the ordinary least squares), we use the following formula to compute the value of c ij for each driver-load combination (i, j).
where π d and π n are the hourly cost of operating a truck (driver wage plus other variable costs) at daytime and that at nighttime respectively (initial incentive l is included in π n , as discussed earlier).
Regression results with 68,809 sample data are reported in Table 3 (dummy coefficients are omitted for space limitation).Results indicate that all of the coefficients attained expected signs (+ or -) with significant tvalues, and that the model fits the data well (R 2 = 0.9689) which suggests that it can do a nice job of predicting c ij values.The c ij 's of external drivers are determined randomly for each instance such that their average would be roughly 20% higher than that of internal drivers.

Choosing n r value
According to Carrier X, drivers usually respond to incentive offers quickly (within 15 min via electronic texting), but it sometimes takes longer than 6 h.Based on these inputs, we performed a small-scale simulation experiment to find out how long it would take to run the entire process (our algorithm) when it requires contacting (sequentially incentivizing and receiving responses from) 25, 50, 75, and 100 drivers.Details of this experiment are reported in online Appendix E. Results show that the average time required to contact 100 drivers is 33.3 h (with the 95% upper confidence limit of 38.8 h), which suggests that one should be able to contact 100 drivers in 48 h with high statistical confidence.Given this finding, which was consistent with carrier X's past experiences, we set n r = 75 (we discount 100 by 25% to be conservative).

General findings
Results are shown in Table 4, Table 5 and Table 6.The CPU time of running the algorithm was about 5.995 s on average, with a typical range of 0.835-14.628s (note: in our simulation, each time an incentive is offered the driver's response is determined instantaneously so that the entire process takes only a few seconds rather than hours).The CPU time of solving P1 ranged from 0.036 to 0.276 s, with an average of 0.158 s.The most important findings follow.
benchmark method in every instance generated and solved in our experiment.This implies that our method should always give positive benefits to carries, i.e. that the risk of losing money by adopting the proposed incentive system, in lieu of the current incentive system, is zero or trivial.Tables 4-6 show that the saving attained by our method over the benchmark method ranges (in dollar value) from $1,481 to $3,371 with an average of $2,337, and (in percentage) from 1.16% to 2.50% with an average of 1.69%.Second, our method captures a large proportion of the cost-saving opportunity (theoretically attainable cost saving which is given by the gap in cost between the benchmark method and the LB).Tables 4-6 show that the percentage of cost-saving opportunity captured by our method ranges from 23.40% to 82.41%, with an overall average of 72.28%.This suggests that our method captures over 50% of cost-saving opportunities in most instances, and that our solution may be reasonably close to the LB.This also suggests that although the cost saving achieved by our method over the benchmark method may look rather small (1.69% on average), it actually is not, as it captures more than 50% of the cost saving which is attainable only with the perfect information of logit parameters.
Third, our method may provide considerable benefits to motor carriers.Our results showed that by using our method in lieu of the current system, carriers can save 1.2% to 2.5% in operating cost.This level of cost saving, while seemingly small, represents substantial financial benefits.It is known that motor carriers operate on very thin operating profit margins (average of 3% -see, e.g.Biery 2018), so that a 1.2% to 2.5% saving in operating cost may increase the profit of carriers substantially.According to our rough estimates, these figures convert to $800,000 additional profit per year for Carrier X.For larger carriers, additional profits may be in tens of millions of dollars.
Fourth, in every instance our algorithm terminated before the number of contacted drivers reached n r (75).The results (not reported) showed that the number of contacted drivers in our experiment averaged 38, with the maximum of 53.This suggests that in all cases our algorithm converged rather quickly to the final solution,  and that in no cases it had to compromise the quality of solution, i.e. it was never forced to terminate before completing all the intended driver contacts.

Impact of individual factors (proposed vs. benchmark)
Figure 2, Figure 3 and Figure 4 show, separately for the three levels of fill rate (95%, 85%, and 75%), how the changes in maximum incentive (l + l + ) and the driver elasticity (β) affect the performance of our method (% cost saving attained vs. the benchmark method).Important findings are as follows.First, the curve depicting the relationship between our method's performance and the incentive level has a positive slope, regardless of the fill rate or driver elasticity (Figures 2-4).This means that our method works particularly well when the incentive level is high.This finding is theoretically sound.Recall that the cost of external drivers is 20% higher than that of an average internal driver, which means that when the incentive is high the cost of certain internal drivers (poor night performers, whose c ij values for night loads are higher than average) can become comparable to or even higher than that of external drivers, after adding the cost of incentives.Thus, when the incentive is high carriers must carefully choose, during the second driver-input solicitation round, which internal drivers to use to replace external drivers, since using some internal drivers can be expensive.Nonetheless, the benchmark method chooses the internal drivers to replace external drivers via the logic that ignores the assignment cost (first-come, first-served).In contrast, our method uses internal drivers with the highest costsaving potentials to replace external drivers.
Second, in all the three figures the 'high' elasticity curve lies above the 'medium' and 'low' elasticity curves (except when the fill rate is 85% and the incentive level is 8%).This suggests that our method generally works better when the driver elasticity is high than when it is low.This makes sense because when the elasticity is high a larger number of hybrid drivers are willing to accept night loads, so that our method benefits from a larger feasible region (it can replace external drivers with the best night performers picked from the larger pool of internal drivers).In contrast, the benchmark method may not benefit from higher elasticities, as it chooses the internal drivers to replace external drivers via the firstcome, first-served rule (i.e.night performance of the early responders needs not improve with the increase in the size of internal driver candidates).
Third, no obvious pattern (relationship) was found between the performance of our method and the fill rate.
We initially expected the performance of our method to worsen with a drop in the fill rate, because a lower fill rate means that our method must contact a larger number of internal drivers to reach the final solution, so that it faces a higher risk of being terminated by the stopping rule n r before completing all the intended driver contacts.Nonetheless, the drop in the fill rate did not negatively affect the performance of our method perhaps because, as discussed earlier, the driver contacts never reached n r before converging to the final solution (in the entire simulation).We later examine how the effectiveness of our method is affected by the use of smaller n r values.

Impact of individual factors (proposed vs. lower bound)
We also examined how experimental factors affect the performance of our method relative to the LB (optimality gap).Results (online Appendix G) showed that, while the performance of our method is unaffected by the elasticity or fill rate, it can be negatively affected by the incentive level (optimality gap increases monotonically as l + l + increases from 8% to 20%).This makes intuitive sense.Note that the extra incentive used by our method is given by p * = minl + , PSP − α ij , so that the amount by which the incentive of our method exceeds the minimum needed amount p ij min (this incentive gap is denoted η) can be expressed as η = l + minl + , PSP − α ij − p ij min .This means that as l + increases, η either increases or does not change (non-decreasing).Thus, when l + is high our method may over-incentivize drivers, which makes the cost of our method high relative to the LB.

Sensitivity analyses
We performed sensitivity analyses by reducing n r from 75 to 50 and 25.Results are reported in online Appendix H. Important findings follow.First, the decline of our method's performance caused by the reduction of n r to 50 is trivial.This is understandable because, given that the number of driver contacts did not exceed 53 previously (when n r = 75), reducing n r from 75 to 50 should only minimally affect the solution quality.Second, a larger performance decline was detected when n r was reduced to 25.However, the amount of decline was still small, with an average decline of 0.03% (maximum decline of 0.11%).This is perhaps because our method contacts the driver with the largest PSP at each iteration, so that the largest cost reductions are attained by the first few driver contacts while trivial cost reductions are attained by the final few contacts.This can be seen from Figure 5, which shows a typical relationship between the operating cost and the number of contacted drivers.The figure shows that about 90% of cost savings are attained by the first 50% of driver contacts, so that cutting the last 50% of driver contacts would only reduce the cost saving by roughly 10%.This suggests that our method's performance is quite robust to the reductions of n r .

Conclusions
This study has shown that the current driver incentive system is inefficient and needs not minimise cost.Our findings provide valuable insights into how the efficiency can be improved.

Theoretical and managerial implications
Our theoretical results (theorems) provide three important implications.First, each driver's incentive should be determined by the cost saving expected from utilising the driver, i.e. basically the incentive should not exceed the expected cost reduction.Second, drives who accepted night loads during the first input solicitation round should not be offered extra incentives during the second round.This will align incentives given to most drivers at or near their reservation prices.Third, drivers should be incentivized in a specific order (sequence) such that the drivers with higher expected cost savings should be incentivized before the drivers with lower expected cost savings.
Our experimental results provide two practical implications to carrier managers.First, avoid incentivizing or assigning drivers to night loads using the first-come, first-served rule.Instead, understand the night driving performance of drivers by analysing their historical driving records, and avoid incentivizing drivers whose night driving performances are poor.This will reduce the transit time of night loads and save operating costs.Second, avoid offering equal incentive raises to all drivers simultaneously during the second input solicitation round.This will not only pay more incentives than necessary to those drivers who would accept night loads without raises, but also encourage poor night drivers (who bring limited cost savings) to accept night loads.Instead, use a sequential approach wherein the driver-load combination with the largest expected cost saving gets the first raise.This will cut costs and also encourage drivers to improve their night driving records, as it sends a message that poor night drivers have limited chances of getting additional incentives.

Future research
This study has its limitations, which may need to be addressed by future research.First, we used a standard econometric method (two-factor fixed effect model) to predict c ij 's.Although this approach was useful in this study as we only needed to perform interpolation (predicting c ij 's of the drivers and the dates included in the calibration dataset, for which the fixed-effect model works well), the field application of our method requires extrapolation (predicting c ij 's beyond the drivers and the dates included in the calibration dataset), for which the fixed-effect model may not work well.Thus, future studies may consider using other methods to predict c ij 's (e.g.machine-learning).
Second, our method generates minimal-cost solutions, but such solutions may not always be preferred by carriers.In some cases, our method can use a moderate number of external drivers if extra incentives given to internal drivers (l + ) are set high (because in such cases some internal drivers can be more costly than external drivers).Carriers may often dislike such solutions as there are reasons, other than cost, that favour the use of internal drivers, such as the superior safety records of internal drivers' (many carriers hire only the drivers with good safety records as internal drivers).Future studies may incorporate such factors (driver-type preferences) into the proposed approach.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Algorithm 2
Do for each (i, j) ∈ with positive p ij * 's 1.From latest P1 sensitivity report, get all P1(z ij )'s for which z ij = 0, i∈D h , j∈L n 2. Find largest | P1(z ij )|.Set its value as SP * and its driver as i * (breaking ties arbitrarily) 3. Create the set of loads L * = j: | P1(z i * j )| = SP * 4. Adjust z ij 's of P1 such that z i * j = 1 ∀ j ∈ L * (other z ij 's do not change) 5. Re-solve P1 and record the decline in objective function value as PSP

Yoshinori
Suzuki is Land O'Lakes Endowed Professor of Supply Chain Management at the Ivy College of Business, Iowa State University.Dr. Suzuki's research interest centres on mathematical modelling of transportation and logistics problems.He has participated in many publicly and privately funded research projects, and has published over 50 research papers in academic journals.He has served as the Co-Editor-in-Chief of Transportation Journal for six years until 2021, and is currently serving as Senior Editor of Journal of Business Logistics.He is also the co-author of the book Transportation: A Global Supply Chain Perspective (10th edition).Meltem Denizel is Associate Professor of Supply Chain Management at the Ivy College of Business, Iowa State University.Dr. Denizel works in the area of Operations Management.Her current research interests focus on sustainable operations and product recovery systems through which she aims to contribute to the highly critical issue of sustainable use of scarce resources.She also continues her research in mathematical modelling and production lot sizing.Her previous works include flexible manufacturing and interdisciplinary research in marketing and operations, organisation studies and technology management.

Table 1 .
List of notations.

Table 2 .
Selected parameters used in experiments.