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Designing distribution patterns for long-term inventory routing with constant demand rates
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Int. J. Production Economics 112 (2008) 255–263 www.elsevier.com/locate/ijpe
Designing distribution patterns for long-term inventory routing with constant demand rates Birger Raa, El-Houssaine Aghezzaf Department of Industrial Management, Ghent University, Technologiepark 903, 9052 Zwijnaarde-Gent, Belgium Received 5 December 2005; accepted 28 August 2006 Available online 12 April 2007
Abstract This paper proposes a practical solution approach for the challenging optimization problem of minimizing overall costs in an integrated distribution and inventory control system. Constant customer demand rates are assumed and therefore a long-term, cyclic planning approach is adopted. The concept of distribution patterns, consisting of vehicles performing multiple tours with possibly different frequencies, is used to extend the traditional concept of a single tour per vehicle. A heuristic is proposed that is capable of solving a cyclical distribution problem involving real-life features, such as customer capacity restrictions, loading and unloading extra times and prespecified minimum times between consecutive deliveries. r 2007 Elsevier B.V. All rights reserved. Keywords: Inventory routing; Vendor-managed inventory; Cyclic planning; Delivery scheduling; Column generation
1. Introduction One of the main objectives in ‘supply chain management’ (SCM) is the coordination of the decisions in different stages of the supply chain. It has been proven that this coordination leads to a better overall performance (Vidal and Goetschalckx, 1997). Our paper focuses on integrating decisions of a distributor and its customers. If customers make their inventory data available, the distributor no longer has to wait for customer orders, but instead can decide independently on when to deliver which quantities to which customers, thus having the possibility of combining Corresponding author. Tel.: +32 9 264 54 97; fax: +32 9 264 58 47. E-mail address: [email protected] (B. Raa).
customers into more cost efficient vehicle routes. This concept of coordination between the distributor and customers is known as ‘vendor-managed inventory’ (VMI). We discuss a VMI problem, which includes some real-life features such as customer capacity restrictions, loading and unloading extra times and minimum time between consecutive deliveries. The total cost of the distribution system consists of the following four components: (1) a fixed cost per vehicle, (2) a variable cost per kilometer traveled, (3) a fixed cost per delivery and (4) variable stockholding costs. Demand rates at the customers are assumed to be constant. Therefore, a cyclic-planning approach is adopted, in which a vehicle repeatedly travels along the same routes to replenish customers. In the cyclic planning, customers impose that the time between consecutive deliveries is
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.08.023
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constant. This means that the timing of the different replenishment tours that a vehicle makes to its customers has to be determined carefully. In Section 2, the problem is further described and the concept of ‘distribution patterns’ is introduced. Section 3 gives a small example to illustrate this concept and the overall complexity of the problem. In Section 4, a heuristic solution approach in a column generation framework is presented. This solution approach is tested and evaluated in Section 5. Section 6, finally, holds some conclusions and ideas for future research. 2. Distribution patterns If a vehicle with finite capacity is assigned the task of replenishing a set of customers, an obvious solution is to find the shortest tour that visits these customers and the depot. In cyclic planning, the vehicle must repeatedly make this tour with a cycle time chosen such that the cost rate of this tour is minimized under the time and capacity constraints at hand; e.g. Anily and Federgruen (1990) and Viswanathan and Mathur (1997) limit vehicle use to a single tour per vehicle in their cyclic solution approaches for integrated distribution and inventory management. Another way of organizing the replenishment of a set of customers is to divide them into subsets and let the vehicle make a separate tour to each of the customer subsets. This is the concept of a multitour. Compared to the single tour, the multi-tour solution uses the vehicle capacity more than once to replenish the same set of customers. Bigger delivery quantities are thus possible at the cost of traveling additional distance to reload the vehicle at the depot (Aghezzaf et al., 2006). It is possible to assign frequencies to the different tours, such that a short tour to customers with high demand rates is performed more often than a longer tour to customers with lower demand rates. Such a routing scheme for a vehicle, consisting of multiple tours with different frequencies, is referred to as a ‘distribution pattern’. Thus, a distribution pattern with cycle time T covering the set of customers S consists of a set of tours visiting disjoint customer subsets SiCS, i ¼ 1,y,n, where each tour i has a frequency ki, meaning that tour i is made ki times during one cycle of length T. There are a number of elements that complicate the search for the cost-minimizing solution. On the
distributor side, the question is whether the vehicle fleet size is fixed or variable. Another issue is the fleet homogeneity. In this paper, we assume that all available vehicles are identical and that determining the vehicle fleet size is part of the problem, which is why fixed vehicle costs are included in our model. Other constraining elements are on the customer side. Namely, a customer j can impose a maximal visiting frequency fj (e.g. no more than two deliveries per week) to the distributor. Further, a customer has limited stock-keeping facilities, which limits the delivery quantity for that customer and thus imposes a minimal visiting frequency. A distribution pattern has a maximal cycle time Tmax derived from the limited vehicle capacity and the limited customer stock-keeping capacity. Every customer has a demand rate dj, expressed in units per period. During one cycle of length T, customer j in subset Si receives ki times a quantity of Tdj/ki. The customer stock-keeping capacity kj (expressed in units) has to be large enough to contain this quantity, or stated differently, Tpkikj/dj. Together, the P customers in tour i have a demand rate of jASidj and the vehicle can deliver at most ki times a full load k (expressed in units). P Therefore, the cycle time T can be at most kik/ jASidj and the maximal cycle time is given by: 0 0 1 1 ki kj C B B ki k C T max ¼ min@mini¼1;...;n @ P A; mini¼1;...;n;j2Si A dj dj j2Si
(1) There is also a minimal cycle time Tmin for every distribution pattern imposed by the time a vehicle needs to make all tours. If customers impose a maximum visiting frequency, this may restrict the minimal cycle time. If TTSP(Si) denotes the travel time to complete the shortest tour through the subset of customers Si and the depot; tj the delivery time at customer j; t0 the time to reload the vehicle at the depot and fj the maximum allowed visiting frequency of customer j, then the minimal cycle time of the distribution pattern is: ! X X T min ¼ max ki T TSPðSi Þ þ t0 þ tj ; i¼1;...;n
maxi¼1;...;n;j2Si
ki fj
!
j2S i
ð2Þ
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2.1. Scheduling the tours of a distribution pattern Once customers that are assigned to a vehicle are partitioned into subsets of customers that are replenished in separate tours, and tour frequencies are determined, a delivery schedule has to be made. In this delivery schedule, every tour i has to be made ki times at points in time that are perfectly spread. Obviously, the sum of travel, loading and unloading times gives a lower bound for the length of the schedule, denoted Tsched: ! X X T sched X ki T TSPðSi Þ þ t0 þ tj (3) i¼1;...;n
j2S i
When all tour frequencies are equal to 1, the above inequality becomes an equality. In that case, every tour is made once per cycle and the time between consecutive deliveries is automatically constant since it is exactly the cycle time. In general, however, with tour frequencies higher than 1, tours are made more than once per cycle. The constraint that consecutive deliveries have to be equidistant in time then introduces gaps into the schedule, such that the above relationship becomes a strict inequality. The gaps in the schedule indicate that the vehicle has to wait in the depot before starting its next tour. Due to the gaps in the schedule, Tsched may be higher than the minimal cycle time Tmin. The interval of feasible cycle times thus becomes [max(Tmin, Tsched), Tmax]. To have more chances of getting close to the optimal cycle time TEOQ (see below), we want this interval to be as large as possible, so the objective of the scheduling is to minimize the ‘idle time’ of the vehicle, which is the same as minimizing Tsched.
The fourth and last cost component is the stockholding cost per cycle at the customers. The quantity delivered to customer j in Si during a cycle is qj ¼ Tdj, so the average stock level during a cycle is qj/2ki, because this quantity comes in ki separate deliveries that are perfectly spread in time. The stock-holding cost per cycle is then Zj(qj/2ki) T ( ¼ holding cost per unit per period average number of units in stock cycle length) ¼ (Zjdj/ 2ki) T2 euro per cycle. With a cycle time of T hours, the cost rate C of a distribution pattern is ! P P P k i j0 þ jj ki C TSPðSi Þ C ¼cþ
i¼1;...;n
i¼1;...;n
þ T X X Zj d j . þT 2ki i¼1;...;n j2S
j2S i
T ð4Þ
i
Obviously, the cost rate C of the distribution pattern depends on the cycle time T. Each distribution pattern thus has a theoretically optimal cycle time, for which the cost rate is minimal. For this theoretically optimal cycle time, holding costs (which increase with cycle time) balance with the sum of delivery-handling costs and transportation costs (which decrease as cycle time increases). This can be viewed as an extension of the EOQ-model and therefore this theoretically optimal cycle time is referred to as the ‘EOQ cycle time’: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u P P P u ki C TSPðSi Þ þ k i j0 þ jj u ui¼1;...;n i¼1;...;n j2Si u P P . T EOQ ¼ t Z d =2k j j
i
i¼1;...;n j2S i
(5)
2.2. Cost rate of a distribution pattern The cost rate C of a distribution pattern has four components. The first component is the fixed operating cost of the vehicle, which is c euro per period. This fixed cost is accounted for regardless of whether the vehicle is traveling or not. The second cost P component is the transportation cost, given by i ¼ 1,y,nkiCTSP(Si) euro per cycle, where CTSP(Si) is the transportation cost of the shortest (cheapest) tour through the subset of customers Si and the depot. The third cost component is the delivery-handling cost of jj euro per delivery for customer j in Si, and j0 for each time the vehicle is reloaded at the depot.
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This EOQ cycle time is infeasible if it is greater than the maximal cycle time Tmax (or smaller than Tsched or Tmin). In that case, the actual optimal cycle time of the distribution pattern is its maximal cycle time Tmax (or schedule time Tsched or minimal cycle time Tmin).
3. Illustrative example In the four-customer example below, the complexity of the problem is illustrated. Distances and demand rates are shown in Fig. 1. A vehicle with a capacity of 200 units is available for product
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replenishment from the depot. Delivery times and costs are ignored for ease of exposition. The best single-tour solution starts at the depot and goes to customer 2, on to customers 1, 3 and 4 next and then back to the depot. The travel time Tmin of this tour is 18 h, while Tmax is only 200/ 18 ¼ 11.11 h. This solution is therefore infeasible. Next, consider the multi-tour solution, called MT, in which customer 2 is served in a first tour, while customers 1, 3 and 4 are covered by another tour. The travel time Tmin of this solution is 19 h (4 h for the first tour+15 h for the second), while Tmax is the minimum of 200/10 and 200/8, i.e. 20 h. This multi-tour solution is thus feasible. Finally, consider the distribution pattern in which all customers are in a separate tour. If all these tours have frequency 1, then Tmin is 22 h and Tmax is 20 h, meaning that the solution is infeasible. Even if this solution would be feasible, it would not be a good solution, because the vehicle load would only be 20 items (or 10%) for the tour to customer 3. Therefore, the frequencies of the tours are adapted to use vehicle capacity more efficiently and thus obtain a better solution. To optimally use vehicle capacity, tour frequencies have to be chosen proportional to the demand rates, meaning 5 for customer 1, 10 for customer 2, 1 for customer 3 and 2 for customer 4. This gives us the distribution
Fig. 1. Illustrative four-customer example.
pattern solution DP1, with a Tmax of 200 h. The schedule for this distribution pattern with the least idle time is shown in Fig. 2. It takes 140 h, and thus has 54 h of idle time. Instead of using any possible tour frequency, we could restrict this to powers-of-two. The appropriate tour frequencies are then 4, 8, 2 and 1, respectively. This gives the second distribution pattern solution, DP2, with Tmax equal to 160 h. The resulting schedule takes 96 h and has only 24 h of idle time (see Fig. 3). The main advantage of powers-of-two frequencies is that for any two frequencies, one is always an integer multiple of the other. This results in relatively easy schedules. Powers-of-two frequencies are often used in scheduling multiple products on a single machine (see e.g. Hahm and Yano, 1995). Now, let us compare the different solutions. In MT, the vehicle is driving at least 95% of its time ( ¼ 19/20), in DP1 43% ( ¼ 86/200) and in DP2 45% ( ¼ 72/160). This means that in the multi-tour solution MT, transportation costs are more than doubled compared to the multi-frequency solutions DP1 and DP2. However, in DP1 and DP2, much more stock is being held: 400 and 360 units on average over all customers vs. only 180 in MT. Thus, in DP1 and DP2, stock costs are doubled compared to MT. The relative weight of transportation costs is usually much higher than that of holding costs, which means that MT is the preferred solution only if holding costs are very high and/or transportation is very cheap. The difference between DP1 and DP2 is only very small, and choosing among them is more of a strategic question. On these one hand, the powers-of-two approach gives relatively easier delivery schedules, while on the other hand, allowing any frequency sometimes gives more opportunities for cost reduction.
Fig. 2. Schedule for DP1.
Fig. 3. Schedule for DP2.
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4. Solution approach To find an overall solution for the integrated distribution and inventory management problem, the following nested subproblems need to be solved.
Customers have to be partitioned over vehicles. For each vehicle, the set of customers that is assigned to it has to be partitioned over different tours. For each partition of customers over tours considered, appropriate tour frequencies have to be determined. For each partition of customers over tours and each combination of tour frequencies, a delivery schedule needs to be made to check feasibility.
In the following paragraphs, the solution approach for each of these subproblems is explained. 4.1. Partitioning customers over vehicles To partition customers over vehicles, a column generation approach is used. 1. The set of columns is initialized with a separate distribution pattern for each of the customers. Thus, in the initial solution, every customer is visited in a separate tour, by a separate vehicle. 2. The LP-relaxed version of the partitioning problem is solved with the available set of columns. Dual prices for all customers are deducted from this LP-solution. 3. New distribution patterns with negative reduced cost rates (i.e. cost rate minus dual prices of all visited customers) are generated with a savings heuristic (see below) and added to the set of columns for the master-partitioning problem. 4. If one or more new columns with negative reduced cost rates have been added in Step 3, Steps 2 and 3 are repeated. Else, the column generation is finished, and the integer version of the partitioning master problem is solved, giving the final solution. The savings heuristic, used to find new columns (Step 3), is adapted from the heuristic proposed by Clarke and Wright (1964). 3a. A list is initialized with a separate distribution pattern for each customer.
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3b. For each possible pair of distribution patterns, a new distribution pattern is constructed that combines the two. If this results in a saving (i.e. the reduced cost rate of the new distribution pattern is smaller than the sum of the reduced cost rates of the two constituent distribution patterns), this new distribution pattern is kept. 3c. The distribution pattern combination from Step 3b with the largest saving is selected. The two constituent distribution patterns are removed from the list and replaced by the new distribution pattern combining them. If this distribution pattern has a negative reduced cost rate, it is added as a new column in the master-partitioning problem. 3d. Steps 3b and 3c are repeated as long as savings can be obtained by combining distribution patterns. Note that multiple columns can be added to the master-partitioning problem in a single iteration of the savings heuristic. 4.2. Partitioning customers of a vehicle over different tours Partitioning customers over different tours is done when combining distribution patterns (Step 3b above), according to the following greedy heuristic. First, the distribution pattern is generated that contains all tours of both constituent distribution patterns. Then, within this distribution pattern, tour combinations are evaluated. Two tours are merged (using a cheapest insertion heuristic) and tour frequencies recalculated. If the cost rate of the resulting distribution pattern is cheaper, the tour combination is retained. This is done as long as cost rate reducing tour combinations can be found. Thus, customer partitioning over different tours within a distribution pattern is performed by a savings heuristic (using tour combination) within a savings heuristic (using distribution pattern combination). 4.3. Determining tour frequencies To determine the tour frequencies, the following simple procedure is used. All tours start with a frequency of one. Iteratively, the frequency of the tour in the distribution pattern with the smallest maximal cycle time is increased, as long as this results in a cost decrease.
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4.4. Scheduling the deliveries Once customers are assigned to a vehicle, partitioned into tours and tour frequencies are chosen, the optimal cycle time and the resulting cost rate have to be determined. For this, a delivery schedule of minimal length needs to be constructed (see Section 2.1). In this delivery schedule, the tour to Si has to be made ki times in a minimal amount of time, such that the time between consecutive iterations of the same tour is constant. Below is a greedy algorithm for scheduling the tours in a distribution pattern within a minimal time span, the so-called ‘schedule time’. This algorithm needs to be quick and efficient, since it is applied for every combination of tour frequencies, in every partition of customers over tours. (1) Initially, Tsched is set to Tmin and the schedule is empty. (2) If all the tours have been added to the schedule, go to Step 4. Else, insert the next remaining tour into the schedule, i.e. the tour with highest frequency (and with the longest travel time in case of a tie). (a) Determine the largest gap in the existing schedule. (b) Schedule the new tour at the beginning of this gap, and calculate its other starting times from there.
(3) Check the schedule for overlaps. If there are overlaps due to the newly added tour, increase Tsched to avoid these. If Tsched is increased above Tmax, go to Step 4. Else, return to Step 2. (4) The schedule is complete. Check if the final Tsched is smaller than Tmax. 5. Computational results To evaluate the proposed solution approach explained above, a large number of tests were run on problem instances with varying characteristics. Customer locations are generated randomly within a square of 100 100 km, with the depot located in the center of this square. Customer demand rates are generated randomly between 1 and 30 units per day. Available vehicles have an average speed of 50 km per hour and a transportation cost of 1 euro per km. Holding costs are the same for all customers, 1 euro per unit per day. The parameters that vary over the problem instances are the following:
Number of customers Three subsets of problem instances are generated, with 25, 50 and 100 customers, respectively. Ten instances are generated per subset. Vehicle capacity Two vehicle types are tested. The first vehicle has a capacity of 50 units and a fixed cost of 20 euro
Fig. 4. Effect of the customer capacity restriction on the total cost rate per customer.
Table 1 Results for the 120 problem instances No. of Customer capacity restriction customers No
Yes
Vehicle capacity
Vehicle capacity
50 Fix Trans Order Hold
Time Total
50 Fix Trans Order Hold
Time Total
100 Fix Trans Order Hold Time Total
Fix Trans Order Hold Time
210.06 171.18 182.30 183.13 181.22 186.26 176.50 191.32 187.65 149.25
60 40 40 40 40 40 40 40 40 40
91.32 77.07 84.98 83.53 81.81 92.39 74.81 92.49 88.02 59.15
20.84 20.25 20.19 20.70 17.65 20.25 15.53 18.16 17.98 15.10
37.90 33.85 37.14 38.90 41.76 33.63 46.17 40.66 41.66 35.00
13 10 17 10 12 4 15 16 15 10
176.29 163.56 168.25 173.14 167.65 168.84 136.44 174.71 171.61 120.14
60 60 60 60 60 60 30 60 60 30
57.11 47.13 51.10 51.58 50.05 53.83 49.25 53.90 48.97 40.65
20.96 16.30 20.49 18.44 18.60 19.19 19.58 16.04 15.45 15.30
38.23 40.13 36.66 43.12 39.00 35.83 37.62 44.76 47.18 34.19
9 5 10 7 6 3 4 9 5 3
370.73 354.84 375.51 338.38 320.48 359.71 349.34 371.92 331.52 280.23
100 100 100 80 80 100 100 100 100 80
192.35 185.32 199.46 174.15 165.38 188.40 172.43 196.84 158.97 134.57
66.42 57.75 64.73 73.53 63.83 59.50 65.28 63.63 59.63 55.88
11.96 11.77 11.33 10.71 11.28 11.82 11.63 11.45 12.92 9.78
1 2 2 0 1 1 1 1 2 1
420.73 404.84 424.43 376.97 358.84 409.68 376.02 422.42 381.52 327.48
150 150 150 120 120 150 120 150 150 120
192.35 181.74 198.65 172.74 163.67 187.86 172.99 195.37 158.97 139.22
66.42 61.86 64.38 73.53 63.50 60.13 72.04 66.00 59.63 59.38
11.96 11.24 11.41 10.71 11.67 11.69 10.99 11.04 12.92 8.89
1 1 1 1 1 2 2 1 1 1
50
352.76 392.97 356.76 379.18 401.09 334.57 323.62 348.97 351.94 399.74
80 100 80 100 100 80 80 80 80 100
162.18 177.54 164.53 164.08 178.17 145.06 139.38 158.26 156.60 182.72
39.95 42.66 43.10 39.46 41.70 37.69 39.03 41.95 36.19 39.63
70.63 72.77 69.13 75.64 81.21 71.81 65.21 68.76 79.15 77.39
53 74 96 103 47 61 100 63 94 69
293.16 312.92 298.99 304.29 319.63 285.63 249.96 294.86 293.24 315.71
90 90 90 90 90 90 60 90 90 90
95.11 107.72 98.56 102.27 109.41 87.56 87.41 96.17 93.54 109.24
42.02 43.35 43.63 42.97 42.78 39.93 40.31 41.95 36.35 40.48
66.04 71.85 66.80 69.05 77.44 68.15 62.24 66.74 73.35 75.99
73 29 52 41 42 39 82 27 35 32
578.52 582.41 573.02 566.21 585.38 515.20 521.50 573.79 571.93 622.77
140 140 140 140 140 120 140 140 160 160
289.20 289.39 280.08 280.60 283.77 252.55 250.73 277.70 275.11 309.22
127.48 128.53 130.53 121.38 137.70 120.40 107.39 134.63 112.98 130.38
21.84 24.49 22.42 24.24 23.91 22.25 23.38 21.47 23.85 23.18
11 14 15 24 9 14 12 14 7 7
635.01 638.90 627.95 622.60 612.86 545.20 557.60 634.42 599.21 665.07
210 210 210 210 180 180 180 210 210 210
281.69 277.14 260.90 267.55 268.96 227.83 239.35 277.43 254.20 297.03
120.50 127.30 134.50 120.75 140.75 113.06 116.16 124.38 110.50 135.88
22.82 24.46 22.54 24.30 23.14 24.31 22.09 22.61 24.50 22.17
12 12 8 9 11 11 11 13 13 6
100
690.07 739.29 742.52 703.51 696.64 730.06 713.68 681.69 711.10 700.21
160 160 180 160 160 160 160 160 160 160
304.56 345.64 338.07 318.13 321.51 344.05 321.52 305.94 318.52 309.88
79.10 88.26 84.29 84.68 88.55 86.74 86.01 85.23 88.44 90.30
146.41 1069 548.93 150 182.03 86.69 130.21 540 145.39 449 583.94 150 202.03 88.38 143.53 240 140.16 494 569.81 150 195.13 85.79 138.90 401 140.71 496 559.75 150 187.04 83.68 139.03 423 126.59 1058 552.60 150 184.18 79.78 138.65 275 139.28 597 574.62 150 202.48 86.38 135.77 453 146.15 748 572.78 150 191.72 85.48 145.58 249 130.52 760 547.29 150 181.05 83.20 133.05 475 144.14 529 567.42 150 184.75 87.24 145.43 501 140.04 814 566.09 150 181.62 86.86 147.62 401
967.24 1090.80 1028.44 1010.21 1005.98 1091.90 1029.74 981.25 1016.47 999.84
240 260 240 240 240 260 240 220 240 220
446.34 531.67 498.86 480.86 474.08 533.25 480.12 462.49 469.05 466.11
233.02 250.41 240.31 241.05 246.90 251.71 262.46 254.20 259.06 266.24
47.88 48.72 49.26 48.30 45.00 46.94 47.16 44.56 48.36 47.50
98 206 226 206 170 92 184 163 163 148
967.24 1090.80 1028.44 1010.21 1005.98 1091.90 1029.74 981.25 1016.47 999.84
270 300 300 300 300 330 270 300 300 270
362.81 444.60 399.50 412.26 409.08 460.65 390.65 374.55 391.52 366.72
246.06 259.88 248.28 241.88 240.69 244.44 274.50 250.00 258.50 271.50
45.39 46.94 48.08 47.87 47.07 47.03 46.30 47.20 48.23 46.39
82 113 98 166 136 95 114 117 130 100
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per hour, the second is a 100-unit vehicle that has a fixed cost of 30 euro per hour. Customer capacity restriction All problem instances are solved twice. The first time, no customer capacity restrictions are imposed, while the second time, there is a capacity restriction of 10 units for each customer.
The heuristic solution approach described in Section 4 was programmed with MS Visual C++.NET 2003, using ILOG Concert Technology 2.0 and ILOG CPLEX 9.0 for the column generation. Computational testing was done on a 2.0 GHz Intel Centrino processor with 1 GB of RAM.
Table 2 Average cost rates per customer CCR
Cap
Nr
Total
Fix
Trans
Order
Hold
Time
No No No No No No Yes Yes Yes Yes Yes Yes
50 50 50 100 100 100 50 50 50 100 100 100
25 50 100 25 50 100 25 50 100 25 50 100
7.28 7.28 7.11 6.48 5.94 5.64 13.81 11.38 10.22 15.61 12.28 9.96
1.68 1.76 1.62 2.16 1.74 1.50 3.76 2.84 2.40 5.52 4.02 2.94
3.30 3.26 3.23 2.01 1.97 1.89 7.07 5.58 4.84 7.05 5.30 4.01
0.75 0.80 0.86 0.72 0.83 0.85 2.52 2.50 2.51 2.59 2.49 2.54
1.55 1.46 1.40 1.59 1.40 1.40 0.46 0.46 0.47 0.45 0.47 0.47
0.49 1.52 7.01 0.24 0.90 3.96 0.05 0.25 1.66 0.05 0.21 1.15
Table 1 shows the results for all 120 problem instances. In Table 2, averages per scenario are shown. These average cost rates have been divided by the number of customers for easier comparison. From Table 2, some obvious conclusions can be drawn, such as the decrease of both fixed vehicle and transportation cost rate per customer with increasing number of customers. In Fig. 4, the effect of imposing customer capacity restrictions on the total cost rate per customer is shown. When there are no such restrictions (left side of Fig. 4), there is not much variation with the number of customers: with a vehicle capacity of 50 units, the cost rate per customer is a bit more than 7 euro per hour, while with a vehicle capacity of 100 units, it is around 6 euro per hour per customer. When customer capacity restrictions are active, there is a big difference among the cost rates of the different instances. For the 25 customer instances, the solution with 100-unit vehicles is more expensive than the solution with 50-unit vehicles. For the 50 customer instances, the 100-unit vehicle solution is still more expensive, but the difference is smaller. For the 100 customer instances, the solution with the 100-unit vehicles becomes cheaper, as it is in all instances without customer capacity restriction. These differences can be explained by the difference in vehicle filling opportunities. When customers impose capacity restrictions, more customers have to be visited in a single tour, to efficiently use vehicle capacity. With 25 customers, there are
Fig. 5. Effect of the customer capacity restriction on calculation times.
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not that many possibilities for efficiently partitioning customers over tours. With growing number of customers, the number of possibilities increases and better partitions are actually found, such that eventually the vehicles with a capacity of 100 units again become the cheaper alternative. Calculation times are obviously related to the number of customers, but apart from that, calculation times also depend heavily on the customer capacity restriction (see Fig. 5). When customer capacity restrictions are imposed, vehicle capacity can only be used efficiently by visiting more customers per tour. This results in a smaller number of tours in the solutions, meaning that less tour combinations have to be examined, less tour frequencies have to be determined and schedules with less tours have to be made. This explains why computation times are much smaller when the customer capacity restrictions are active. 6. Conclusion Cyclic planning is a promising approach for longterm integrated distribution and inventory management, especially in the case of constant demand rates. This paper presents a solution approach for this challenging inventory-routing problem. Apart from actually proposing cyclic delivery schedules, the approach can also be used to select the appropriate type of vehicle and to quantify the effect of customer imposed restrictions. Further research will consist of (i) adapting the existing solution approach to richer problems including driving-time restrictions for the vehicles and their drivers, delivery time windows at the customers, heterogeneous vehicle fleets, multiple depots, etc. and (ii) investigating the possibility of including replenishment and inventory decisions at the depot(s) in the cyclic strategy.
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In the cyclic delivery schedules, vehicles can have quite some idle time. This idle time gives the opportunity to serve irregular customers with unpredictable demands, customers that cannot be considered in the cyclic framework. This indicates that the cyclic solution approach remains valid even if there are a number of customers that do not have stable demand rates. Future research can start from this conclusion in trying to assess the validity of cyclic distribution planning under more uncertain conditions. Implementing cyclic planning brings stability to the supply chain: the distributor can reuse the same distribution schedule and the customers always know when and with what quantity they will be replenished. Therefore, an interesting research theme is to quantify the effects of introducing cyclic planning on supply chain stability by examining the impact on e.g. safety stock levels and the bullwhip effect. References Aghezzaf, E.H., Raa, B., Van Landeghem, H., 2006. Modeling inventory routing problems in supply chains of high consumption products. European Journal of Operational Research 169 (3), 1048–1063. Anily, S., Federgruen, A., 1990. One Warehouse multiple customer systems with vehicle routing costs. Management Science 36 (1), 92–114. Clarke, G., Wright, J.W., 1964. Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12 (4), 568–581. Hahm, J., Yano, C.A., 1995. The economic lot and delivery scheduling problem: Powers of two policies. Management Science 29 (3), 222–241. Vidal, C.J., Goetschalckx, M., 1997. Strategic productiondistribution models: A critical review with emphasis on global supply chain models. European Journal of Operational Research 98 (1), 1–18. Viswanathan, S., Mathur, K., 1997. Integrating routing and inventory decisions in one-warehouse multi-customer multiproduct distribution systems. Management Science 43 (3), 294–312.
Int. J. Production Economics 112 (2008) 255–263 www.elsevier.com/locate/ijpe
Designing distribution patterns for long-term inventory routing with constant demand rates Birger Raa, El-Houssaine Aghezzaf Department of Industrial Management, Ghent University, Technologiepark 903, 9052 Zwijnaarde-Gent, Belgium Received 5 December 2005; accepted 28 August 2006 Available online 12 April 2007
Abstract This paper proposes a practical solution approach for the challenging optimization problem of minimizing overall costs in an integrated distribution and inventory control system. Constant customer demand rates are assumed and therefore a long-term, cyclic planning approach is adopted. The concept of distribution patterns, consisting of vehicles performing multiple tours with possibly different frequencies, is used to extend the traditional concept of a single tour per vehicle. A heuristic is proposed that is capable of solving a cyclical distribution problem involving real-life features, such as customer capacity restrictions, loading and unloading extra times and prespecified minimum times between consecutive deliveries. r 2007 Elsevier B.V. All rights reserved. Keywords: Inventory routing; Vendor-managed inventory; Cyclic planning; Delivery scheduling; Column generation
1. Introduction One of the main objectives in ‘supply chain management’ (SCM) is the coordination of the decisions in different stages of the supply chain. It has been proven that this coordination leads to a better overall performance (Vidal and Goetschalckx, 1997). Our paper focuses on integrating decisions of a distributor and its customers. If customers make their inventory data available, the distributor no longer has to wait for customer orders, but instead can decide independently on when to deliver which quantities to which customers, thus having the possibility of combining Corresponding author. Tel.: +32 9 264 54 97; fax: +32 9 264 58 47. E-mail address: [email protected] (B. Raa).
customers into more cost efficient vehicle routes. This concept of coordination between the distributor and customers is known as ‘vendor-managed inventory’ (VMI). We discuss a VMI problem, which includes some real-life features such as customer capacity restrictions, loading and unloading extra times and minimum time between consecutive deliveries. The total cost of the distribution system consists of the following four components: (1) a fixed cost per vehicle, (2) a variable cost per kilometer traveled, (3) a fixed cost per delivery and (4) variable stockholding costs. Demand rates at the customers are assumed to be constant. Therefore, a cyclic-planning approach is adopted, in which a vehicle repeatedly travels along the same routes to replenish customers. In the cyclic planning, customers impose that the time between consecutive deliveries is
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2006.08.023
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constant. This means that the timing of the different replenishment tours that a vehicle makes to its customers has to be determined carefully. In Section 2, the problem is further described and the concept of ‘distribution patterns’ is introduced. Section 3 gives a small example to illustrate this concept and the overall complexity of the problem. In Section 4, a heuristic solution approach in a column generation framework is presented. This solution approach is tested and evaluated in Section 5. Section 6, finally, holds some conclusions and ideas for future research. 2. Distribution patterns If a vehicle with finite capacity is assigned the task of replenishing a set of customers, an obvious solution is to find the shortest tour that visits these customers and the depot. In cyclic planning, the vehicle must repeatedly make this tour with a cycle time chosen such that the cost rate of this tour is minimized under the time and capacity constraints at hand; e.g. Anily and Federgruen (1990) and Viswanathan and Mathur (1997) limit vehicle use to a single tour per vehicle in their cyclic solution approaches for integrated distribution and inventory management. Another way of organizing the replenishment of a set of customers is to divide them into subsets and let the vehicle make a separate tour to each of the customer subsets. This is the concept of a multitour. Compared to the single tour, the multi-tour solution uses the vehicle capacity more than once to replenish the same set of customers. Bigger delivery quantities are thus possible at the cost of traveling additional distance to reload the vehicle at the depot (Aghezzaf et al., 2006). It is possible to assign frequencies to the different tours, such that a short tour to customers with high demand rates is performed more often than a longer tour to customers with lower demand rates. Such a routing scheme for a vehicle, consisting of multiple tours with different frequencies, is referred to as a ‘distribution pattern’. Thus, a distribution pattern with cycle time T covering the set of customers S consists of a set of tours visiting disjoint customer subsets SiCS, i ¼ 1,y,n, where each tour i has a frequency ki, meaning that tour i is made ki times during one cycle of length T. There are a number of elements that complicate the search for the cost-minimizing solution. On the
distributor side, the question is whether the vehicle fleet size is fixed or variable. Another issue is the fleet homogeneity. In this paper, we assume that all available vehicles are identical and that determining the vehicle fleet size is part of the problem, which is why fixed vehicle costs are included in our model. Other constraining elements are on the customer side. Namely, a customer j can impose a maximal visiting frequency fj (e.g. no more than two deliveries per week) to the distributor. Further, a customer has limited stock-keeping facilities, which limits the delivery quantity for that customer and thus imposes a minimal visiting frequency. A distribution pattern has a maximal cycle time Tmax derived from the limited vehicle capacity and the limited customer stock-keeping capacity. Every customer has a demand rate dj, expressed in units per period. During one cycle of length T, customer j in subset Si receives ki times a quantity of Tdj/ki. The customer stock-keeping capacity kj (expressed in units) has to be large enough to contain this quantity, or stated differently, Tpkikj/dj. Together, the P customers in tour i have a demand rate of jASidj and the vehicle can deliver at most ki times a full load k (expressed in units). P Therefore, the cycle time T can be at most kik/ jASidj and the maximal cycle time is given by: 0 0 1 1 ki kj C B B ki k C T max ¼ min@mini¼1;...;n @ P A; mini¼1;...;n;j2Si A dj dj j2Si
(1) There is also a minimal cycle time Tmin for every distribution pattern imposed by the time a vehicle needs to make all tours. If customers impose a maximum visiting frequency, this may restrict the minimal cycle time. If TTSP(Si) denotes the travel time to complete the shortest tour through the subset of customers Si and the depot; tj the delivery time at customer j; t0 the time to reload the vehicle at the depot and fj the maximum allowed visiting frequency of customer j, then the minimal cycle time of the distribution pattern is: ! X X T min ¼ max ki T TSPðSi Þ þ t0 þ tj ; i¼1;...;n
maxi¼1;...;n;j2Si
ki fj
!
j2S i
ð2Þ
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2.1. Scheduling the tours of a distribution pattern Once customers that are assigned to a vehicle are partitioned into subsets of customers that are replenished in separate tours, and tour frequencies are determined, a delivery schedule has to be made. In this delivery schedule, every tour i has to be made ki times at points in time that are perfectly spread. Obviously, the sum of travel, loading and unloading times gives a lower bound for the length of the schedule, denoted Tsched: ! X X T sched X ki T TSPðSi Þ þ t0 þ tj (3) i¼1;...;n
j2S i
When all tour frequencies are equal to 1, the above inequality becomes an equality. In that case, every tour is made once per cycle and the time between consecutive deliveries is automatically constant since it is exactly the cycle time. In general, however, with tour frequencies higher than 1, tours are made more than once per cycle. The constraint that consecutive deliveries have to be equidistant in time then introduces gaps into the schedule, such that the above relationship becomes a strict inequality. The gaps in the schedule indicate that the vehicle has to wait in the depot before starting its next tour. Due to the gaps in the schedule, Tsched may be higher than the minimal cycle time Tmin. The interval of feasible cycle times thus becomes [max(Tmin, Tsched), Tmax]. To have more chances of getting close to the optimal cycle time TEOQ (see below), we want this interval to be as large as possible, so the objective of the scheduling is to minimize the ‘idle time’ of the vehicle, which is the same as minimizing Tsched.
The fourth and last cost component is the stockholding cost per cycle at the customers. The quantity delivered to customer j in Si during a cycle is qj ¼ Tdj, so the average stock level during a cycle is qj/2ki, because this quantity comes in ki separate deliveries that are perfectly spread in time. The stock-holding cost per cycle is then Zj(qj/2ki) T ( ¼ holding cost per unit per period average number of units in stock cycle length) ¼ (Zjdj/ 2ki) T2 euro per cycle. With a cycle time of T hours, the cost rate C of a distribution pattern is ! P P P k i j0 þ jj ki C TSPðSi Þ C ¼cþ
i¼1;...;n
i¼1;...;n
þ T X X Zj d j . þT 2ki i¼1;...;n j2S
j2S i
T ð4Þ
i
Obviously, the cost rate C of the distribution pattern depends on the cycle time T. Each distribution pattern thus has a theoretically optimal cycle time, for which the cost rate is minimal. For this theoretically optimal cycle time, holding costs (which increase with cycle time) balance with the sum of delivery-handling costs and transportation costs (which decrease as cycle time increases). This can be viewed as an extension of the EOQ-model and therefore this theoretically optimal cycle time is referred to as the ‘EOQ cycle time’: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u P P P u ki C TSPðSi Þ þ k i j0 þ jj u ui¼1;...;n i¼1;...;n j2Si u P P . T EOQ ¼ t Z d =2k j j
i
i¼1;...;n j2S i
(5)
2.2. Cost rate of a distribution pattern The cost rate C of a distribution pattern has four components. The first component is the fixed operating cost of the vehicle, which is c euro per period. This fixed cost is accounted for regardless of whether the vehicle is traveling or not. The second cost P component is the transportation cost, given by i ¼ 1,y,nkiCTSP(Si) euro per cycle, where CTSP(Si) is the transportation cost of the shortest (cheapest) tour through the subset of customers Si and the depot. The third cost component is the delivery-handling cost of jj euro per delivery for customer j in Si, and j0 for each time the vehicle is reloaded at the depot.
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This EOQ cycle time is infeasible if it is greater than the maximal cycle time Tmax (or smaller than Tsched or Tmin). In that case, the actual optimal cycle time of the distribution pattern is its maximal cycle time Tmax (or schedule time Tsched or minimal cycle time Tmin).
3. Illustrative example In the four-customer example below, the complexity of the problem is illustrated. Distances and demand rates are shown in Fig. 1. A vehicle with a capacity of 200 units is available for product
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replenishment from the depot. Delivery times and costs are ignored for ease of exposition. The best single-tour solution starts at the depot and goes to customer 2, on to customers 1, 3 and 4 next and then back to the depot. The travel time Tmin of this tour is 18 h, while Tmax is only 200/ 18 ¼ 11.11 h. This solution is therefore infeasible. Next, consider the multi-tour solution, called MT, in which customer 2 is served in a first tour, while customers 1, 3 and 4 are covered by another tour. The travel time Tmin of this solution is 19 h (4 h for the first tour+15 h for the second), while Tmax is the minimum of 200/10 and 200/8, i.e. 20 h. This multi-tour solution is thus feasible. Finally, consider the distribution pattern in which all customers are in a separate tour. If all these tours have frequency 1, then Tmin is 22 h and Tmax is 20 h, meaning that the solution is infeasible. Even if this solution would be feasible, it would not be a good solution, because the vehicle load would only be 20 items (or 10%) for the tour to customer 3. Therefore, the frequencies of the tours are adapted to use vehicle capacity more efficiently and thus obtain a better solution. To optimally use vehicle capacity, tour frequencies have to be chosen proportional to the demand rates, meaning 5 for customer 1, 10 for customer 2, 1 for customer 3 and 2 for customer 4. This gives us the distribution
Fig. 1. Illustrative four-customer example.
pattern solution DP1, with a Tmax of 200 h. The schedule for this distribution pattern with the least idle time is shown in Fig. 2. It takes 140 h, and thus has 54 h of idle time. Instead of using any possible tour frequency, we could restrict this to powers-of-two. The appropriate tour frequencies are then 4, 8, 2 and 1, respectively. This gives the second distribution pattern solution, DP2, with Tmax equal to 160 h. The resulting schedule takes 96 h and has only 24 h of idle time (see Fig. 3). The main advantage of powers-of-two frequencies is that for any two frequencies, one is always an integer multiple of the other. This results in relatively easy schedules. Powers-of-two frequencies are often used in scheduling multiple products on a single machine (see e.g. Hahm and Yano, 1995). Now, let us compare the different solutions. In MT, the vehicle is driving at least 95% of its time ( ¼ 19/20), in DP1 43% ( ¼ 86/200) and in DP2 45% ( ¼ 72/160). This means that in the multi-tour solution MT, transportation costs are more than doubled compared to the multi-frequency solutions DP1 and DP2. However, in DP1 and DP2, much more stock is being held: 400 and 360 units on average over all customers vs. only 180 in MT. Thus, in DP1 and DP2, stock costs are doubled compared to MT. The relative weight of transportation costs is usually much higher than that of holding costs, which means that MT is the preferred solution only if holding costs are very high and/or transportation is very cheap. The difference between DP1 and DP2 is only very small, and choosing among them is more of a strategic question. On these one hand, the powers-of-two approach gives relatively easier delivery schedules, while on the other hand, allowing any frequency sometimes gives more opportunities for cost reduction.
Fig. 2. Schedule for DP1.
Fig. 3. Schedule for DP2.
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4. Solution approach To find an overall solution for the integrated distribution and inventory management problem, the following nested subproblems need to be solved.
Customers have to be partitioned over vehicles. For each vehicle, the set of customers that is assigned to it has to be partitioned over different tours. For each partition of customers over tours considered, appropriate tour frequencies have to be determined. For each partition of customers over tours and each combination of tour frequencies, a delivery schedule needs to be made to check feasibility.
In the following paragraphs, the solution approach for each of these subproblems is explained. 4.1. Partitioning customers over vehicles To partition customers over vehicles, a column generation approach is used. 1. The set of columns is initialized with a separate distribution pattern for each of the customers. Thus, in the initial solution, every customer is visited in a separate tour, by a separate vehicle. 2. The LP-relaxed version of the partitioning problem is solved with the available set of columns. Dual prices for all customers are deducted from this LP-solution. 3. New distribution patterns with negative reduced cost rates (i.e. cost rate minus dual prices of all visited customers) are generated with a savings heuristic (see below) and added to the set of columns for the master-partitioning problem. 4. If one or more new columns with negative reduced cost rates have been added in Step 3, Steps 2 and 3 are repeated. Else, the column generation is finished, and the integer version of the partitioning master problem is solved, giving the final solution. The savings heuristic, used to find new columns (Step 3), is adapted from the heuristic proposed by Clarke and Wright (1964). 3a. A list is initialized with a separate distribution pattern for each customer.
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3b. For each possible pair of distribution patterns, a new distribution pattern is constructed that combines the two. If this results in a saving (i.e. the reduced cost rate of the new distribution pattern is smaller than the sum of the reduced cost rates of the two constituent distribution patterns), this new distribution pattern is kept. 3c. The distribution pattern combination from Step 3b with the largest saving is selected. The two constituent distribution patterns are removed from the list and replaced by the new distribution pattern combining them. If this distribution pattern has a negative reduced cost rate, it is added as a new column in the master-partitioning problem. 3d. Steps 3b and 3c are repeated as long as savings can be obtained by combining distribution patterns. Note that multiple columns can be added to the master-partitioning problem in a single iteration of the savings heuristic. 4.2. Partitioning customers of a vehicle over different tours Partitioning customers over different tours is done when combining distribution patterns (Step 3b above), according to the following greedy heuristic. First, the distribution pattern is generated that contains all tours of both constituent distribution patterns. Then, within this distribution pattern, tour combinations are evaluated. Two tours are merged (using a cheapest insertion heuristic) and tour frequencies recalculated. If the cost rate of the resulting distribution pattern is cheaper, the tour combination is retained. This is done as long as cost rate reducing tour combinations can be found. Thus, customer partitioning over different tours within a distribution pattern is performed by a savings heuristic (using tour combination) within a savings heuristic (using distribution pattern combination). 4.3. Determining tour frequencies To determine the tour frequencies, the following simple procedure is used. All tours start with a frequency of one. Iteratively, the frequency of the tour in the distribution pattern with the smallest maximal cycle time is increased, as long as this results in a cost decrease.
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4.4. Scheduling the deliveries Once customers are assigned to a vehicle, partitioned into tours and tour frequencies are chosen, the optimal cycle time and the resulting cost rate have to be determined. For this, a delivery schedule of minimal length needs to be constructed (see Section 2.1). In this delivery schedule, the tour to Si has to be made ki times in a minimal amount of time, such that the time between consecutive iterations of the same tour is constant. Below is a greedy algorithm for scheduling the tours in a distribution pattern within a minimal time span, the so-called ‘schedule time’. This algorithm needs to be quick and efficient, since it is applied for every combination of tour frequencies, in every partition of customers over tours. (1) Initially, Tsched is set to Tmin and the schedule is empty. (2) If all the tours have been added to the schedule, go to Step 4. Else, insert the next remaining tour into the schedule, i.e. the tour with highest frequency (and with the longest travel time in case of a tie). (a) Determine the largest gap in the existing schedule. (b) Schedule the new tour at the beginning of this gap, and calculate its other starting times from there.
(3) Check the schedule for overlaps. If there are overlaps due to the newly added tour, increase Tsched to avoid these. If Tsched is increased above Tmax, go to Step 4. Else, return to Step 2. (4) The schedule is complete. Check if the final Tsched is smaller than Tmax. 5. Computational results To evaluate the proposed solution approach explained above, a large number of tests were run on problem instances with varying characteristics. Customer locations are generated randomly within a square of 100 100 km, with the depot located in the center of this square. Customer demand rates are generated randomly between 1 and 30 units per day. Available vehicles have an average speed of 50 km per hour and a transportation cost of 1 euro per km. Holding costs are the same for all customers, 1 euro per unit per day. The parameters that vary over the problem instances are the following:
Number of customers Three subsets of problem instances are generated, with 25, 50 and 100 customers, respectively. Ten instances are generated per subset. Vehicle capacity Two vehicle types are tested. The first vehicle has a capacity of 50 units and a fixed cost of 20 euro
Fig. 4. Effect of the customer capacity restriction on the total cost rate per customer.
Table 1 Results for the 120 problem instances No. of Customer capacity restriction customers No
Yes
Vehicle capacity
Vehicle capacity
50 Fix Trans Order Hold
Time Total
50 Fix Trans Order Hold
Time Total
100 Fix Trans Order Hold Time Total
Fix Trans Order Hold Time
210.06 171.18 182.30 183.13 181.22 186.26 176.50 191.32 187.65 149.25
60 40 40 40 40 40 40 40 40 40
91.32 77.07 84.98 83.53 81.81 92.39 74.81 92.49 88.02 59.15
20.84 20.25 20.19 20.70 17.65 20.25 15.53 18.16 17.98 15.10
37.90 33.85 37.14 38.90 41.76 33.63 46.17 40.66 41.66 35.00
13 10 17 10 12 4 15 16 15 10
176.29 163.56 168.25 173.14 167.65 168.84 136.44 174.71 171.61 120.14
60 60 60 60 60 60 30 60 60 30
57.11 47.13 51.10 51.58 50.05 53.83 49.25 53.90 48.97 40.65
20.96 16.30 20.49 18.44 18.60 19.19 19.58 16.04 15.45 15.30
38.23 40.13 36.66 43.12 39.00 35.83 37.62 44.76 47.18 34.19
9 5 10 7 6 3 4 9 5 3
370.73 354.84 375.51 338.38 320.48 359.71 349.34 371.92 331.52 280.23
100 100 100 80 80 100 100 100 100 80
192.35 185.32 199.46 174.15 165.38 188.40 172.43 196.84 158.97 134.57
66.42 57.75 64.73 73.53 63.83 59.50 65.28 63.63 59.63 55.88
11.96 11.77 11.33 10.71 11.28 11.82 11.63 11.45 12.92 9.78
1 2 2 0 1 1 1 1 2 1
420.73 404.84 424.43 376.97 358.84 409.68 376.02 422.42 381.52 327.48
150 150 150 120 120 150 120 150 150 120
192.35 181.74 198.65 172.74 163.67 187.86 172.99 195.37 158.97 139.22
66.42 61.86 64.38 73.53 63.50 60.13 72.04 66.00 59.63 59.38
11.96 11.24 11.41 10.71 11.67 11.69 10.99 11.04 12.92 8.89
1 1 1 1 1 2 2 1 1 1
50
352.76 392.97 356.76 379.18 401.09 334.57 323.62 348.97 351.94 399.74
80 100 80 100 100 80 80 80 80 100
162.18 177.54 164.53 164.08 178.17 145.06 139.38 158.26 156.60 182.72
39.95 42.66 43.10 39.46 41.70 37.69 39.03 41.95 36.19 39.63
70.63 72.77 69.13 75.64 81.21 71.81 65.21 68.76 79.15 77.39
53 74 96 103 47 61 100 63 94 69
293.16 312.92 298.99 304.29 319.63 285.63 249.96 294.86 293.24 315.71
90 90 90 90 90 90 60 90 90 90
95.11 107.72 98.56 102.27 109.41 87.56 87.41 96.17 93.54 109.24
42.02 43.35 43.63 42.97 42.78 39.93 40.31 41.95 36.35 40.48
66.04 71.85 66.80 69.05 77.44 68.15 62.24 66.74 73.35 75.99
73 29 52 41 42 39 82 27 35 32
578.52 582.41 573.02 566.21 585.38 515.20 521.50 573.79 571.93 622.77
140 140 140 140 140 120 140 140 160 160
289.20 289.39 280.08 280.60 283.77 252.55 250.73 277.70 275.11 309.22
127.48 128.53 130.53 121.38 137.70 120.40 107.39 134.63 112.98 130.38
21.84 24.49 22.42 24.24 23.91 22.25 23.38 21.47 23.85 23.18
11 14 15 24 9 14 12 14 7 7
635.01 638.90 627.95 622.60 612.86 545.20 557.60 634.42 599.21 665.07
210 210 210 210 180 180 180 210 210 210
281.69 277.14 260.90 267.55 268.96 227.83 239.35 277.43 254.20 297.03
120.50 127.30 134.50 120.75 140.75 113.06 116.16 124.38 110.50 135.88
22.82 24.46 22.54 24.30 23.14 24.31 22.09 22.61 24.50 22.17
12 12 8 9 11 11 11 13 13 6
100
690.07 739.29 742.52 703.51 696.64 730.06 713.68 681.69 711.10 700.21
160 160 180 160 160 160 160 160 160 160
304.56 345.64 338.07 318.13 321.51 344.05 321.52 305.94 318.52 309.88
79.10 88.26 84.29 84.68 88.55 86.74 86.01 85.23 88.44 90.30
146.41 1069 548.93 150 182.03 86.69 130.21 540 145.39 449 583.94 150 202.03 88.38 143.53 240 140.16 494 569.81 150 195.13 85.79 138.90 401 140.71 496 559.75 150 187.04 83.68 139.03 423 126.59 1058 552.60 150 184.18 79.78 138.65 275 139.28 597 574.62 150 202.48 86.38 135.77 453 146.15 748 572.78 150 191.72 85.48 145.58 249 130.52 760 547.29 150 181.05 83.20 133.05 475 144.14 529 567.42 150 184.75 87.24 145.43 501 140.04 814 566.09 150 181.62 86.86 147.62 401
967.24 1090.80 1028.44 1010.21 1005.98 1091.90 1029.74 981.25 1016.47 999.84
240 260 240 240 240 260 240 220 240 220
446.34 531.67 498.86 480.86 474.08 533.25 480.12 462.49 469.05 466.11
233.02 250.41 240.31 241.05 246.90 251.71 262.46 254.20 259.06 266.24
47.88 48.72 49.26 48.30 45.00 46.94 47.16 44.56 48.36 47.50
98 206 226 206 170 92 184 163 163 148
967.24 1090.80 1028.44 1010.21 1005.98 1091.90 1029.74 981.25 1016.47 999.84
270 300 300 300 300 330 270 300 300 270
362.81 444.60 399.50 412.26 409.08 460.65 390.65 374.55 391.52 366.72
246.06 259.88 248.28 241.88 240.69 244.44 274.50 250.00 258.50 271.50
45.39 46.94 48.08 47.87 47.07 47.03 46.30 47.20 48.23 46.39
82 113 98 166 136 95 114 117 130 100
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per hour, the second is a 100-unit vehicle that has a fixed cost of 30 euro per hour. Customer capacity restriction All problem instances are solved twice. The first time, no customer capacity restrictions are imposed, while the second time, there is a capacity restriction of 10 units for each customer.
The heuristic solution approach described in Section 4 was programmed with MS Visual C++.NET 2003, using ILOG Concert Technology 2.0 and ILOG CPLEX 9.0 for the column generation. Computational testing was done on a 2.0 GHz Intel Centrino processor with 1 GB of RAM.
Table 2 Average cost rates per customer CCR
Cap
Nr
Total
Fix
Trans
Order
Hold
Time
No No No No No No Yes Yes Yes Yes Yes Yes
50 50 50 100 100 100 50 50 50 100 100 100
25 50 100 25 50 100 25 50 100 25 50 100
7.28 7.28 7.11 6.48 5.94 5.64 13.81 11.38 10.22 15.61 12.28 9.96
1.68 1.76 1.62 2.16 1.74 1.50 3.76 2.84 2.40 5.52 4.02 2.94
3.30 3.26 3.23 2.01 1.97 1.89 7.07 5.58 4.84 7.05 5.30 4.01
0.75 0.80 0.86 0.72 0.83 0.85 2.52 2.50 2.51 2.59 2.49 2.54
1.55 1.46 1.40 1.59 1.40 1.40 0.46 0.46 0.47 0.45 0.47 0.47
0.49 1.52 7.01 0.24 0.90 3.96 0.05 0.25 1.66 0.05 0.21 1.15
Table 1 shows the results for all 120 problem instances. In Table 2, averages per scenario are shown. These average cost rates have been divided by the number of customers for easier comparison. From Table 2, some obvious conclusions can be drawn, such as the decrease of both fixed vehicle and transportation cost rate per customer with increasing number of customers. In Fig. 4, the effect of imposing customer capacity restrictions on the total cost rate per customer is shown. When there are no such restrictions (left side of Fig. 4), there is not much variation with the number of customers: with a vehicle capacity of 50 units, the cost rate per customer is a bit more than 7 euro per hour, while with a vehicle capacity of 100 units, it is around 6 euro per hour per customer. When customer capacity restrictions are active, there is a big difference among the cost rates of the different instances. For the 25 customer instances, the solution with 100-unit vehicles is more expensive than the solution with 50-unit vehicles. For the 50 customer instances, the 100-unit vehicle solution is still more expensive, but the difference is smaller. For the 100 customer instances, the solution with the 100-unit vehicles becomes cheaper, as it is in all instances without customer capacity restriction. These differences can be explained by the difference in vehicle filling opportunities. When customers impose capacity restrictions, more customers have to be visited in a single tour, to efficiently use vehicle capacity. With 25 customers, there are
Fig. 5. Effect of the customer capacity restriction on calculation times.
ARTICLE IN PRESS B. Raa, E.-H. Aghezzaf / Int. J. Production Economics 112 (2008) 255–263
not that many possibilities for efficiently partitioning customers over tours. With growing number of customers, the number of possibilities increases and better partitions are actually found, such that eventually the vehicles with a capacity of 100 units again become the cheaper alternative. Calculation times are obviously related to the number of customers, but apart from that, calculation times also depend heavily on the customer capacity restriction (see Fig. 5). When customer capacity restrictions are imposed, vehicle capacity can only be used efficiently by visiting more customers per tour. This results in a smaller number of tours in the solutions, meaning that less tour combinations have to be examined, less tour frequencies have to be determined and schedules with less tours have to be made. This explains why computation times are much smaller when the customer capacity restrictions are active. 6. Conclusion Cyclic planning is a promising approach for longterm integrated distribution and inventory management, especially in the case of constant demand rates. This paper presents a solution approach for this challenging inventory-routing problem. Apart from actually proposing cyclic delivery schedules, the approach can also be used to select the appropriate type of vehicle and to quantify the effect of customer imposed restrictions. Further research will consist of (i) adapting the existing solution approach to richer problems including driving-time restrictions for the vehicles and their drivers, delivery time windows at the customers, heterogeneous vehicle fleets, multiple depots, etc. and (ii) investigating the possibility of including replenishment and inventory decisions at the depot(s) in the cyclic strategy.
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In the cyclic delivery schedules, vehicles can have quite some idle time. This idle time gives the opportunity to serve irregular customers with unpredictable demands, customers that cannot be considered in the cyclic framework. This indicates that the cyclic solution approach remains valid even if there are a number of customers that do not have stable demand rates. Future research can start from this conclusion in trying to assess the validity of cyclic distribution planning under more uncertain conditions. Implementing cyclic planning brings stability to the supply chain: the distributor can reuse the same distribution schedule and the customers always know when and with what quantity they will be replenished. Therefore, an interesting research theme is to quantify the effects of introducing cyclic planning on supply chain stability by examining the impact on e.g. safety stock levels and the bullwhip effect. References Aghezzaf, E.H., Raa, B., Van Landeghem, H., 2006. Modeling inventory routing problems in supply chains of high consumption products. European Journal of Operational Research 169 (3), 1048–1063. Anily, S., Federgruen, A., 1990. One Warehouse multiple customer systems with vehicle routing costs. Management Science 36 (1), 92–114. Clarke, G., Wright, J.W., 1964. Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12 (4), 568–581. Hahm, J., Yano, C.A., 1995. The economic lot and delivery scheduling problem: Powers of two policies. Management Science 29 (3), 222–241. Vidal, C.J., Goetschalckx, M., 1997. Strategic productiondistribution models: A critical review with emphasis on global supply chain models. European Journal of Operational Research 98 (1), 1–18. Viswanathan, S., Mathur, K., 1997. Integrating routing and inventory decisions in one-warehouse multi-customer multiproduct distribution systems. Management Science 43 (3), 294–312.