1. Introduction • The more rigid the customer, the harder to design a cost efficient route Vehicle routing • Pricing strategies - Going-rate pricing m ethodologies to support - Preceived value pricing - Markup or cost-plus pricing costing and pricing decisions • Even when prices in the industry are not cost-based, information on incremental costs remains essential - Price floor Wout Dullaert, ITMMA, University of Antwerp - Determine profitability • Undesirable to have completely customer-specific prices Olli Bräysy, University of Jyväskylä - Transaction costs Frans Cruijssen, TNT Express - Fairness issues - Development of price structures Bruno De Borger, University of Antwerp • Area of application: heterogeneous vehicle routing problem with time windows (FSMVRPTW) 1 Fleet Size and Mix Vehicle Routing Problem (FSMVRP) 2. Short literature review • different vehicle types with different capacities and • Shared costs: when part of the costs cannot be traced acquisition costs back to a single customer or a single shipment. • Objective: find a fleet composition and a corresponding - Common costs: routing plan that minimizes the sum of routing and - Joint costs: vehicle costs. • Game-theory in cost-allocation • Practical applications of FSMVRP • DEA • … • Various models exist in the literature depending on • In vehicle routing incremental costs of a customer - how the variable costs and fleet size are issued depends on customer characteristics and on the other - whether there are limits on the number of vehicles of customers’ characteristics each type - Best known objective function (Liu & Shen 1999): Vehicle cost + ”En route time” (constant sum of service time is excluded in reporting) 2 3
• Campbell and Savelsbergh (2005, Trans. Sci.): home • Computational testing on randomly generated problem delivery problem instances - Whether or not to accept delivery request upon arrival - impact of o.a. varying the number of allowable time slots, revenue per request and time slot width - All accepted requests must be serviced - Stringent time windows have a significant impact on - Which time slot maximizes expected total profits routing costs and suggest pricing slots as interesting • Insertion based heuristics in GRASP framework research avenue - All accepted delivery requests are inserted where they maximize profits (select among k best locations) - Check whether the delivery requests can be serviced in one of the time slots of the customer (different criteria to assess the profitability) - To estimate expected profits: compare best value of VRPTW with or without customer involved 4 5 • Campbell and Savelsbergh (2006, Trans. Sci.) • Confessore et al. (2007,IJPR) - Solomon’s sequential insertion heurist I1 - Adjust Campbell and Savelsbergh (2005) - Artificial 100 customer problem instances - maximize total profit=total revenue − total costs − total - 16 scenarios with varying percentage of customers with tight incentives paid to affect the probabilities of the time time windows and the time window width of the remaining more slots. flexible customers as well - assumptions made on customer behavior (e.g. likelihood that customers select a time slot and effect of incentives = + TC a a RelativeTW Size 0 1 on customers’ behavior) - Simulation runs to assess the impact of future costs - 78% to 97% of the variability in total cost for the 16 scenarios based on simple insertions in GRASP setting considered 6 7
3. A MSDA heuristic (Bräysy et al. 2007) • Our research objectives • Multi-Start Deterministic Annealing (MSDA) - Estimating the incremental cost of a customer • 3 phases, embedded in restart loop - Identifying cost drivers • Phase 1: Initial solution • Phase 2: Route elimination • Need for: • Phase 3: Iterative improvement - Powerful heuristic to calculate the cheapest solution for - 4 local search operators all customers - Variable Neighborhood Descent until local optimum - Different heuristics to estimate the solution after - Threshold Accepting until iteration limit, or no removing a customer (trade off computation time and improvement limit solution quality) • First accept • Adaptive memory of good arcs, utilized upon restart 8 9 Phase 1: generation of Phase 2: route initial solutions elimination • Based on Savings (Clarke & Wright 1964) • Based on simple insertions, procedure ELIM • Savings based on total cost • All routes considered for depletion, in random order • Merging route R1 into R2, all insertion points in R2 are • Customers tried in decreasing order of criticality tried ( ) η • Probabilistic insertion in one of the 3 best improving d ( ) ς = + η i c points ( ) η − i 0 i b a • Each route initialized with smallest possible vehicle type i i • Greedy upgrade of vehicle type when needed • Best feasible insertion point w.r.t. total cost • Cutoff when insertion cost exceeds elimination savings • Upon restart, some arcs from the final solution kept • ELIM is run until quiescence 10 11
Phase 3: iterative improvement • 4 local search operators iterated, First Accept • The procedure is restarted a given number of times • ICROSS • Adaptive memory of arcs appearing in elite solutions - Cross-exchange with reversal of segments • New initial solution, start with current - Heterogeneous fleet - Remove arcs not present in x % of the best solutions - Limited segment length (e.g. 70 %) • IOPT: Or-opt extended with segment reversal • ELIM: As in Phase 2 (every second iteration) - Random removal of remaining arcs (e.g. 50 % probability) • SPLIT: All possible splits (every third iteration) • Route sequence shuffled before each iteration • Iterate until - local optimum, or no improvement over given # iterations • Threshold Accepting - all moves except SPLIT - until iteration limit 12 13 Summary of Results MSDA - Four settings • MSDA Quick (~ 6.5 CPU seconds) • MSDA Quick - Outperformed on several individual instances - 1.6 % better than Dell’Amico et al., 13 times faster - 200 iterations, 2 restarts, 1 run - 7.4 % better than Liu & Shen, 18 % slower • MSDA Medium-1 • MSDA Medium-1 (~ 70 CPU seconds) - 1.000 iterations, 2 restarts, 1 run - 3.6 % better than Dell’Amico et al., 140 best known, 134 new • MSDA Medium-3 • MSDA Medium-3 (~ 210 CPU seconds) - 1.000 iterations, 2 restarts, 3 runs - 157 best known, 151 new • MSDA Best - on average 3.9 % better than Dell’Amico et al., 2.5 times slower - 10.000 iterations, 2 restarts, 3 runs - on average 9.6 % better than Liu & Shen, 13 times slower • MSDA Best (~ 660 CPU seconds) • Remaining parameter values are identical - 167 best known, 165 new - 1.1 % performance improvement over MSDA Medium-3, 10 times slower • Type 2 instances are improved the most 14 15
8 approaches to estimating the incremental cost 1. Full re-optimization: 5. Single route optimization • Construct new solution from scratch • Consider using smaller vehicle after removing customer • Use MSDA for 1000 iterations from route 2. DA500: MSDA for 500 iterations • Try IOPT to reduce distance in the route • Maintain current structure after removing customer 6. Close re-optimization I • MSDA for 500 iterations • Similar to Local optimization (4) 3. DA100: MSDA for 100 iterations • Look for improvements within route from which • Current solution structure + MSDA for 100 iterations customer was removed + from the neighboring routes 4. Local optimization • Distance limited adjusted during search, only consider • Current solution structure customers within current distance limit • MSDA for maximum 1000 iterations • Use of 2 of 4 local search operators (ICROSS and IOPT) • No threshold for stage 3 local search: search ends when local optimum is found 16 17 Computational testing 7. Close re-optimization II • Liu and Shen (1999) benchmark: • Similar to close re-optimization II - 56 problem instances • Set of routes is extended by the routes that are closest - R, C, RC subsets & 1 and 2 subsets to the route in which an improvement was obtained in - 100 customers per problem instance the previous step 8. Simple removal • Restrict hardest problem instances: C103, C204, R104, • Relink route without changing the sequence of customers R209, RC202, RC101 • Java JDK (5.0), AMD Athlon 2600+ (512 MB RAM) computer • Short run and long run cost estimates: the higher, the more powerful the cost estimator 18 19
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