LNG Inventory Routing With Roaming Delivery
Locations And Location Dependent Demand
Paul van der Kolk
Student number: S2766205
Master’s Thesis Technology and Operations Management
Rijksuniversiteit Groningen
LNG Inventory Routing With Roaming Delivery
Locations And Location Dependent Demand
Abstract
This paper considers a new variant for the Inventory Routing Problem (IRP) of Liquefied Natural Gas (LNG). In this problem an actor is responsible for the distribution of LNG from a liquefaction plant to customers. The customers reside at different ports during different time windows and each customer has to be visited at one of these locations. The characteristic that each customer can be visited at varying ports has only recently been researched in the context of trunk delivery. In the context of refuelling ships with LNG this characteristic causes demand to increase for each location further in time. Branch-and-cut is used to solve this problem. The results show that uniquely defined valid inequal-ities regarding time windows of each customer are most effective for reducing the problem. The model is capable of solving instances with 50 customers or in-stances with at most approximately 100 customer locations within a 30 minute period. As a result, this model can be used as a tool to investigate the prof-itability of investing in storage facilities at ports.
Acknowledgements
This thesis is the result of research of the past 6 months. It is the last work to fulfill the graduation requirements of the Master’s Degree, Technology and Operations Management.
The research started off as a tough project but with the help of my supervisors, dr. X. Zhu and dr. E. Ursavas, I was able to model the problem on paper as well as in a software program. The last two months have been the most difficult period. This had to do with the more technical aspect of this thesis, the development of an effective heuristic. In hindsight, I might not have had enough experience with this aspect of my thesis to be able to deliver high quality work. Unfortunately, this thesis does not fully reflect the effort I have made to develop a heuristic but that seems to be part of any thesis.
1
Introduction
Liquefied natural gas (LNG) is a highly valuable resource and demand is rapidly growing. The year 2017 shows the largest annual volume increase since 2010 of 10% (International Group of Liquefied Natural Gas Importers, 2018). LNG is commonly used as fuel by vessels. The transfer of LNG can be done by ship-to-ship transfer (STS) and involves the transfer of the fuel by ships positioned alongside each other. It also includes the routing and scheduling of the LNG vessels (i.e. the ships that transport the LNG). This involves deciding at what location the LNG vessel meets the demand. Since the demand consists of mul-tiple customers (i.e. ships) this problem becomes quite complex. Additionally, each ship can potentially be refuelled at varying locations which further com-plicates the problem.
The importance of solving this problem is twofold. First, the shipping indus-try is capital intensive and a moderate improvement can already cause a large increase in profit (Agra et al. 2018). Second, the problem is difficult to solve because of the high degree of freedom in its variables (Agra et al. 2018). The problem of routing and scheduling is related to the vehicle routing problem which has been extensively researched. This problem consist of determining the optimal set of routes for a fleet of vehicles in order to deliver to a given set of customers. Many variants have been proposed since the problem first appeared in literature. However, for the problem in this paper the current variants do not fully suffice. Since this problem deals with varying locations at which a ship can be refuelled, it has the most resemblance with the vehicle routing problem with roaming delivery locations proposed by Reyes et al. (2017). This model includes multiple delivery location per customer each within a different time window. However, since the model in the article of Reyes et al. (2017) focuses on demand that is static (i.e. the customer order quantity is independent to the location of delivery) it does not suffice for the routing of a LNG vessel. Moving ships will require more LNG when its demand is met in a later time window since the ship used more of its available LNG.
The aim of this paper is to present a solution method that can solve small scale instances. The paper proposes the LNG Inventory Routing with Roaming Deliv-ery Locations and Location Dependent Demand and uses pure branch-and-cut to solve it. This will be an extension to current literature and has a practical impact in the routing of LNG vessels.
2
Literature
The model presented in this paper is derived from literature regarding the LNG Inventory Routing Problem (IRP) and secondly, from literature regarding the Vehicle Routing Problem with Roaming Delivery Locations. Literature regard-ing both models will be explored after which the contributions of this paper are explained.
2.1
Maritime inventory routing
A basic Maritime Inventory Routing Problem (MIRP) involves the transporta-tion of a single product from loading ports to unloading ports, with each port having a given inventory storage capacity, a pre-specified production or con-sumption rate, a restriction on the number of visits to the port, and a limitation on the quantity of product to be loaded or unloaded (Christiansen et al. 2007). A very similar description is given by Al-Khayyal & Hwang (2007) and Goel et al. (2015). In general, path-based models can be exploited well since sailing times are relatively long compared to the total planning period, such that paths include relatively few legs (Grønhaug et al. 2010).
Most of the research regarding the MIRP deals with static and deterministic problems (Agra et al. 2018). However, a few papers have been published that include uncertainty regarding sailing times and port times by for example Agra et al. (2013), Halvorsen-Weare & Fagerholt (2011) and Halvorsen-Weare et al. (2013). Interesting are the variants with time windows (Rakke et al. (2011) and Christiansen & Nygreen (2005)), where port visits must be during known time periods, for example only on weekdays and not during weekends.
2.1.1 LNG inventory routing
As for the MIR, time windows have been taken into account into the LNG IRP as well. More specifically, Kobayashi & Kubo (2010) developed a model with time windows that restrict the arrival time, so that unloading starts during the day when ports are open. The same type of time window is used by Norstad et al. (2011) and Halvorsen-Weare & Fagerholt (2013). In contrast to Chris-tiansen & Fagerholt (2002) where they attempt to consider penalty costs for risky arrival times in order to create a robust schedule.
2.2
VRP with Roaming Delivery Locations
Dantzig & Ramser (1959) were the first to propose the Vehicle Routing Problem (VRP). This problem is concerned with finding a set of routes for a number of identical vehicles such that each customer is visited exactly once. The aim is to minimize the routing costs. Many variants have been proposed since then (Pillac et al. 2013). A relevant variant of the vehicle routing problem (VRP) is the VRP with roaming delivery locations (VRPRDL). Reyes et al. (2017) developed a model for last mile deliveries that included the possibility for trunk deliveries along with home delivery. In this particular problem, a customer can be visited at its car or home during different time windows and only one of these locations has to be visited. The VRPRDL is similar to Time Dependent Vehicle Routing Problem (TDVRP) where the cost (or travel time) to go from one location to another depends on the departure time, first formulated by Malandraki & Daskin (1992). However, an important difference is that a customer can be visited at different locations so that the travel cost is not only dependent on the departure time but also on the location travelled to. An important aspect of the VRPRDL is that the demand of each customer is independent of the location of delivery, the package that is delivered is the same at all locations of each customer.
2.2.1 Variable dependent demand
2.2.2 Theoretical contribution
3
Problem Description and
Mathematical Model
3.1
Problem description
This section provides a detailed description upon which the model is built. The problem is based on a real life setting. Assumptions will be explicitly stated. The producer and distributor of the LNG has a single liquefaction plant from which a LNG vessel loads and departs. This LNG vessel will always be filled up.
From the liquefaction plant, the LNG vessel travels to the customers. The customer remains at varying locations during different time windows and each customer should be visited at one of these locations. It is assumed that the vessel of the customer will always be filled up. Bunkering will be done by ship-to-ship (STS) transfer. This is a transfer of cargo between ships positioned alongside each other. It is assumed that bunkering takes place at a port while stationary, i.e. there is a set of discrete locations at which each customer can be bunkered.
Subsequently, the required amount of fuel of each customer is dependent on the location at which the customer is visited. A customer that is visited at a different location later in time will require more fuel. The amount required is assumed to be known for each location.
A route can consist of visiting multiple customers as long as the capacity of the LNG vessel allows it. A visit to the liquefaction plant will fill up the LNG vessel. Inventory constraints at the liquefaction plant is excluded, it is assumed that there is enough LNG for each loading of the LNG vessel. During travel, LNG in the tank boils off which is used to fuel the LNG vessel. This results in the capacity of LNG vessel to lower during travel.
The waterway network is dense and travel between each port is possible, the travel time satisfy the triangle inequality, i.e. direct travel from one place to another is always faster than travel via another port. The distance between each port is known.
The goal is to maximize the profit which consists of the costs associated with the routes and the sales associated with the satisfied demand.
3.2
Mathematical formulation
Let G = (N, A) with N = {0, 1, ..., n} be a complete directed graph in which nodes correspond with a location that can be visited. Each arc (i, j) ∈ A has an associated distance wij.
The set of customers that require a delivery during the planning period [0,T] is represented by C. Let Nc ⊆ N denote the set of locations that customer c
will visit. For each location i ∈ Nc for c ∈ C, there is a non-overlapping time
window [ej, lj] and a bunkering time si. The delivery to a customer c ∈ C is
characterized by a demand quantity qi for different nodes i ∈ Nc. Each arc
costs of the arc i, j, denoted as uij.
One vessel is available for the transfer of LNG. The vessel can return to the depot to fill up capacity. Let Nd ⊆ N denote the depot. By duplicating this
node multiple visits to this depot during a single route are possible. The time it takes to refuel the LNG vessel is fixed and is denoted by si, ∀ i ∈ Nd.
The goal is to find the most profitable set of delivery routes visiting each cus-tomer at one of the locations in the cuscus-tomer’s itinerary, during the time that the customer is at that location, and such that the demand delivered on a route is no more than the capacity of the vessel.
3.3
Assumptions
The model is built upon a few key assumption, which are the following: 1. Visiting a customer will always fully meet its demand, it is not allowed to partially meet the demand for capacity or time reasons.
2. A customer can only be visited at one of its locations and not at multiple locations.
3. It is assumed that refilling the LNG vessel will take a fixed amount of time regardless of the remaining capacity after a trip. Implementing this will not have a significant impact and therefore is excluded to reduce complexity. 4. Demand, travel time and travel costs are known beforehand for each node respectively each arc.
3.4
Decision Variables
xij = 1 if the LNG vessel travels from i to j, 0 otherwise, ∀ i, j ∈ N
dc = departure time after service at customer or depot ∀ c ∈ C and c ∈ Nd
fc = remaining capacity after service at customer or depot ∀ c ∈ C and c ∈ Nd
3.5
Parameters
Q = Capacity of the LNG vessel qj = Demand at node j ∈ Nc
tij = Travel time of arc i, j ∈ N
mij = Fuel usage for travel over arc j ∈ N
uij = Value of arc from i to j which is the sale value of LNG at j minus cost of
travel
si = Bunkering or refuelling time at node i ∈ N
[ej, lj] = Arrival and departure time for each node j ∈ Nc
3.6
Objective function
maxX i∈N X j∈N uijxij (1)3.7
Subject to
X j∈N xij = X j∈N xj i ∀ i ∈ N (2) X j∈N xij ≤ 1 ∀ i ∈ N (3) X i∈N X j∈Nc xij = 1 ∀ c ∈ C (4) dc+ X i∈Nc X j∈Nc 0 (tij+ sj)xij≤ dc0+ M (1 − X i∈Nc X j∈Nc 0 xij) ∀ c, c0∈ C, ∀ c, c0∈ Nd, c06= a (5) dc≥ (ej+ sj) X i∈N xij ∀ j ∈ Nc, ∀ c ∈ C (6) dc+ X i∈Nc X j∈Nc 0 tijxij ≤ X i∈Nc X j∈Nc 0 (lj− sj)xij ∀ c, c 0 ∈ C, ∀ c, c0∈ Nd (7) fc− Q(1 − X i∈Nc X j∈Nc0 xij) ≥ fc0− X i∈Nc X j∈Nc0 (mij+ qj)xij ∀ c, c0∈ C, ∀ c, c0∈ Nd (8) fc≤ Q ∀ c ∈ C, Nd (9) xij∈ {0, 1} ∀ i, j ∈ N (10)4
Branch-and-cut algorithm
The vehicle routing problem with roaming delivery locations, upon which the problem of this paper is based, has been very recently developed by Reyes et al. (2017) and Ozbaygin et al. (2017). Reyes et al. (2017) developed a construction and improvement heuristic while Ozbaygin et al. (2017) used branch and price to reduce solving time. Branch-and-cut algorithms for the vehicle routing problem with time windows were introduced by Kohl & Madsen (1997) and have been further developed in the last years by Bard et al. (2002), Lysgaard (2006) and Kallehauge et al. (2007). It includes valid inequalities, for example k-path and infeasible paths inequalities. The pure branch-and-cut algorithm requires a very large number of cuts to close the integrality gap, and these cuts cannot be easily identified (Desaulniers et al. 2008). These cuts are however the key component of the branch-and-cut. The next sections will present valid inequalities that have been used for this model. In the ’Results’ section, the most effective ones will be compared.
4.1
Duplicated depot nodes
The model uses duplicated depot nodes to allow multiple visits to the depot for refuelling. However, this also increases the amount of solutions tremendously since each depot node can be used at any time during the solution. By restrict-ing travel to these depot nodes except one, identical solutions will be eliminated. These solutions have effectively with the same route, using only different dupli-cated depot nodes. The solution times will be reduced because of this. Assume that jc depicts the duplicated depot node linked to customer c.
X i∈Nc X j∈Nd xij = 0 ∀ c ∈ C, j <> jc (11)
4.2
Infeasible paths
As a result of the duplicated depot nodes, no travel between theses nodes is allowed which leads to the following inequality.
xij= 0 ∀ i, j ∈ Nd (12)
In addition, since each depot node is linked to a customer, travel from a depot node back to the linked customer is impossible.
X
i∈Nd
X
j∈Nc
A customer has multiple nodes at which refuelling can take place. Constraint 4 ensures one travel to one of these nodes. However, in the LP relaxation fractional travels will occur. Therefore, in the LP relaxation travel between nodes of same customer is not automatically restricted. The valid inequality is as follows.
X
i∈Nc
X
j∈Nc
xij= 0 ∀ c ∈ C (14)
Secondly, if travel occurs from i to j then travel from j to i is not allowed. Constraint 5 ensures this but for the LP relaxation the following constraint can be added.
xij+ xj i≤ 1 ∀ i, j ∈ N (15)
Another set of infeasible paths are based on the time windows associated with those paths, also called reachability cuts. These were introduced by Lysgaard (2006). Basically, reachability cuts are derived by considering for each customer which arcs can possibly, taking time windows into account, be traversed. Con-straint 6 and 7 makes sure that the departure times respect the time windows. However, in the LP relaxation a new constraint can be added to exclude travel that is not feasible based on the time windows.
xij= 0 ∀ i, j ∈ Nc, ei+ sj+ tij> lj− sj (16)
4.3
K-path inequality
Basically, the k-path inequality calculates the minimum number of vehicles needed to visit all the customers. These inequalities were introduced by Kohl et al. (1999). It can be adjusted to fit the model by calculating the minimum number of visits to a depot, based on the capacity of the LNG vessel and the minimum fuel required by each customer. For example, if each customer re-quires 20 units of LNG and the capacity of the LNG vessel is 60, at most 3 customers can be visited before travel to a depot is required. When including fuel cost for travel into the equation, the minimum amount can be decreased further. If there are a total of 10 customer, at least 4 visits to a depot are required (including the depot at which the route starts). Assume that Dsis the
The maximization of the LP relaxation is solved by maximizing the arcs that are profitable and minimizing the arcs that are not. Based on the minimum amount of depot visits the minimal sum of xij can be calculated which is 1 travel for
each customer + 1 travel for the minimum required depot visits. Assume that n stands for the total number of customers
X
i∈N
X
j∈N
xij≥ n + dDs/Qe (18)
4.4
Departure times inequalities
Constraint 5, 6 and 7 are not very effective for the LP relaxation. Adding the following inequalities will restrict departure times to the time windows of each customer. These inequalities are unique to the problem of this paper since it can only be applied to models with varying customer locations.
dc≥ min j∈Nc ej ∀ c ∈ C (19) dc≤ max j∈Nc sj ∀ c ∈ C (20)
Since the duplicated depot nodes can only be travelled to from one customer, the preceding two constraints can also be implemented for these depot nodes. Assume that dcd denotes the departure time for the depot node linked to
cus-tomer c as the result of constraint 11. Let mc denote the the travel time to and
5
Network construction
This chapter describes the generation of the information needed to run the model. The information regards the nodes which represent ports and customers which includes their locations, demand and time windows.
5.1
Node generation
X- and y-coordinates of nodes are randomly generated between 0 and 10. Dis-tances are then calculated based on euclidean disDis-tances and are rounded to the nearest integer. The cost are only dependent on the distance and therefore the triangular inequality with regards to costs is satisfied. The maximum distance between two nodes is 14. Travel time for each arc is set to 1 for each unit of distance.
5.2
Customer location generation
For each instance the amount of customers, their amount of delivery locations and their availability are key components of each instance ((Reyes et al. 2017) and (Ozbaygin et al. 2017)). For this problem, an additionally important param-eter is the demand in relation to the capacity of the LNG vessel. The capacity of the LNG vessel is fixed to 200 while demand of each customer is randomly generated between 20 and 30 for the first location and for each succeeding loca-tion between 3 and 8 is added. The model will try to find a route prioritizing visits to the last customer location since potential profit will be the greatest. However, in most instances it will not be possible to visit all these locations due to conflicting time windows. The next section will include an analysis of the effect of the capacity on the solving speed.
5.3
Time windows
The start of a time window for each location of a customer is randomly generated while departure is randomly generated between 10 and 20 hours after arrival. Arrival at succeeding locations is between 5 and 15 hours later. Since this model requires to meet demand of all customers, the time windows are set so that every customer can be visited, otherwise no feasible solution exists.
5.4
Valid inequalities
6
Computational results
The aim of this study is (1) to present a model that produce routing scheduling for LNG vessels and (2) to present valid inequalities that reduce solution times to a reasonable point. The first point has been addressed in previous sections. This chapter will present computational results to show that solution times are reasonable and will provide with a comparison of the most important valid inequalities.
During computational analyses, it was noted that the capacity of the LNG vessel had impact on the solution times of the model. Therefore, an additional analysis has been done to investigate its effect.
6.1
Performance valid inequalities
Preliminary analysis of the effect of the valid inequalities showed that valid in-equality 11, 12 and 19, 20, 21 and 22 show the most improvement. The effect of these valid inequalities will be compared by running the same instance with different valid inequalities enabled. Multiple instances will be generated and used for comparison. Table 6.1 shows the instances that are used to compare the 3 most relevant valid inequalities.
Table 6.1: Characteristics of instances for VI comparison Instance #Cust #Loc avg RDL
1 10 22 2.20
2 10 26 2.60
3 12 27 2.25
4 12 32 2.67
Table 6.2 shows the solution times of the previously shown instances for five cases; one case with none of the most relevant valid inequalities active, three cases in which one of the sets of valid inequalities is active and one case with all valid inequalities active.
Table 6.2: Comparison valid inequalities
No VI VI 11 VI 12 VI 19, 20, 21 & 22 All VI
Inst time gap time gap time gap time gap time gap
1 1,800 2.24% 1,800 13.84% 2.9 0% 0.5 0% 0.4 0%
2 1,800 2.81% 1,800 9.45% 90.8 0% 2.0 0% 0.1 0%
3 1,800 * 1,800 74.31% 1,800 10.63% 400.0 0% 0.6 0%
4 1,800 79.80% 1,800 18.10% 1,800 * 1,800 3.97% 1.6 0% * No integer feasible solution has been found
shortest solution times and smallest gap. The reason for this is that this set of valid inequalities depict a general outline of the route. It eliminates routes between customers that are far apart, time wise, which would make the route infeasible when further branched. As a result, a large portion of routes are not included in the branch-and-bound. This general outline is not achieved via the constraints. This is because the LP relaxation is bypassing the 1 − xij part of
the constraints 5 and 7 with fractional values for xij. These valid inequalities
tighten the departure times and therefore the LP relaxation solutions.
6.2
Instances
The instances are generated according to chapter 5. Table 6.3 below presents the characteristics of the instances that are used to run the model.
Table 6.3: Characteristics of instances
Availability Instance #Cust #Loc avg RDL min avg max
6.3
Performance model
All instances are run with all the valid inequalities active at the top node. Table 6.4 below shows the information of each instance, the solution time and when the optimal solution is not found within 1,800 seconds, the integrality gap, shown as a percentage. A more detailed table is shown in the appendix A, it includes the amount of constraints, variable, the first lower bound and the time it was ob-tained, the upper bound of the relaxation and the gap with the first lower bound. Table 6.4: Solution times instances
Solution Instance #Cust #Loc avg RDL time gap
1 25 53 2.12 2.7 0% 2 25 65 2.60 6.5 0% 3 25 79 3.16 38.0 0% 4 25 84 3.36 136.5 0% 5 25 89 3.56 1,277.1 0% 6 25 97 3.88 1,800* 11.34% 7 30 68 2.27 15.8 0% 8 30 81 2.70 1,800* 1.40% 9 30 92 3.07 1,800* 0.70% 10 30 96 3.20 1,800* 2.10% 11 35 99 2.83 362.9 0% 12 35 111 3.17 580.5 0% 13 35 117 3.34 1,800* 9.46% 14 35 135 3.86 1,800* 18.56% 15 40 93 2.33 535.0 0% 16 40 98 2.45 1,800* 1.12% 17 40 115 2.88 1,800* 1.11% 18 40 126 3.15 1,800* 6.07% 19 45 98 2.18 113.2 0% 20 45 105 2.33 1,800* 1.33% 21 45 116 2.58 1,800* 5.56% 22 45 164 3.64 1,800* 10.00% 23 50 94 1.88 166.1 0% 24 50 104 2.08 142.4 0% 25 50 123 2.46 1,800* 8.87% 26 50 147 2.94 1,800* 4.68%
customers’ time windows. If this is the case, the most effective set of valid inequalities (departure time inequalities) is less effective when time windows of customers overlap more. This overlap will occur more often when the average roaming delivery locations is higher.
6.4
Capacity LNG vessel
The parameter of the capacity of the LNG vessel is set to 200 for the model. This results in the potential to visit between 6 and 10 customers without re-fuelling if fuel usage of the route is not taken into account. However, capacity of LNG vessels vary in practice and four instances have been tested by using a capacity of 150 (5-7 customers) and 250 (8-12 customers). The expectation is that the larger the capacity, the shorter the solution time will be. The reason is that with a larger capacity, less travel to the depot node for refuelling is re-quired therefore the best feasible solution will potentially be closer to the upper bound of the LP relaxation. Table 6.5 will show the results of the analysis. The instance number refers to the instance in table 6.3.
Table 6.5 Capacity analysis
Q = 150 Q = 200 Q = 250 Instance time gap time gap time gap
4 1,800 1,84% 136.5 0% 26.7 0% 11 1,800 2,72% 362.9 0% 221,0 0% 19 1,800 0,59% 113.2 0% 224,4 0%
23 127.0 0% 166.1 0% 51.7 0%
7
Conclusion
In this paper, the vehicle routing problem with roaming delivery locations and location dependent demand has been presented. The LNG-IRP and the VR-PRDL have been used as basis and adjustments have been made to the for-mulations, the most important one being the changes in demand. This, in combination with the time windows effected the model significantly and unique valid inequalities were necessary to reduce solution times of small-medium in-stances.
A computational experiment showed that 3 sets of valid inequalities significantly reduce solution times, especially when combined. Two of the three sets were based on existing literature, the k-path inequalities and the reachability cuts. However, the most effective set of valid inequalities was unique. It restricts the departure times of each customer to the corresponding time windows of all of its locations. It is the most effective since the LP relaxation does not work well with the constraints regarding the departure times. These valid inequalities are unique to the problem of this paper as it deals with a unique characteristic, namely the time windows of all locations of a single customer. In other models, a customer only has one location and one time window.
8
Discussion
Computation experiments showed that the model is able to solve instance of 50 customers with 100 customer locations with unique time windows. A solution technique that is more effective can increase this. This model is developed for small scale instances where ports lack a storage facility for LNG. However, it might be the case that a large scale instances deals with ports without storage facilities. A more effective solution technique is required to solve these larger instances.
The model in this paper is relatively simple and it has been simplified to reduce complexity. A few additions can be implemented at the cost of solving speed. For example, this model does not penalize idle time at ports, while minimization of idle time at ports is a high preferential issue of planners (Christiansen & Fagerholt 2002). This could be an addition to this model. Examples of this penalizing are using soft time windows or hard limits on inventory levels of customers, see Taillard et al. (1997) and Christiansen & Nygreen (1998). Another relevant addition is routing for multiple vehicles. This can be combined by changing the constraint that each customer has to be visited once, to allow for multiple visits to a single customer by for example multiple vehicles. It is interesting to see whether this provides more profitable routes. In this case, the assumption that each visit requires full satisfaction of demand should be replaced by a variable that indicates the amount of fuel sold at each location. As mentioned in the problem description, customers are refuelling while at a port and thus stationary. This is required for safety reasons. However, ship to ship transfer can also be applied while at sea. This model can be extended to include this possibility when dealing with other goods. This will require dynamic travel time for these locations since arriving at a customer later in time will also result in this customer being at a (relative slightly) different location.
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Appendix A Solution times with extra information
Solution
Inst #Cust #Loc RDL time gap cons vars LB time LB UB gap
