Large Neighborhood Search for LNG Inventory Routing

Vikas Goel∗

ExxonMobil Upstream Research Company

Kevin C. FurmanExxonMobil Upstream Research Company

Jin-Hwa Songformerly at ExxonMobil Corporate Strategic Research

Amr S. El-BakryExxonMobil Production Company

Abstract

Liquefied Natural Gas (LNG) is steadily becoming a common mode for commer-cializing natural gas. Due to the capital intensive nature of LNG projects, the optimaldesign of LNG supply chains is extremely important from a profitability perspective.Motivated by the need for a model that can assist in the design analysis of LNG supplychains, we address an LNG inventory routing problem where optimized ship scheduleshave to be developed for an LNG project. In this paper, we present an arc-flow for-mulation based on the MIP model of Song and Furman [17]. We also present a setof construction and improvement heuristics to solve this model efficiently. The heuris-tics are evaluated based on a set of realistic test instances that are very large relativeto the problem instances seen in recent literature related to this problem. Extensivecomputational results indicate that the proposed methods are computationally effi-cient in finding optimal or near optimal solutions and are substantially faster thanstate-of-the-art commercial optimization software.

1 Introduction

Natural gas is expected to be the world’s fastest growing fossil fuel with consumption increas-ing at an average rate of 1.6% per year from 2008 to 2035 [1]. In many cases, large reservesof natural gas are located in areas of little or no local demand. As a result, natural gas mustbe transported over long distances, either via pipelines or shipped as liquefied natural gas(LNG) in specially designed ships. To be transported as LNG, natural gas produced froma reservoir is first processed to remove impurities. The purified natural gas is liquefied by

∗Corresponding author; vikas.goel@exxonmobil.com

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cooling it to −163 ◦C at a liquefaction terminal (also referred to as a production terminal).The LNG is stored at dedicated storage facilities at the production terminal before beingloaded on LNG ships that transport the LNG to regasification terminals (also referred to asregas terminals). At the regas terminal, LNG is stored before being regasified (i.e., vapor-ized) and injected into the local natural gas pipeline grid. For a more thorough introductionto the LNG business, the reader is referred to Tusiana and Shearer [19].

LNG accounts for a growing share of world natural gas trade with world natural gasliquefaction capacity expected to double between 2009 and 2035 [1]. However, new LNGproduction projects are complex and extremely capital intensive. As a result, the optimaldesign of the supply chain of an LNG project is extremely important from a profitabilityperspective. From an operations perspective, managing an LNG project involves negotiatinga delivery schedule for each customer on an annual basis. This Annual Delivery Program(ADP) is negotiated to best accommodate the expected requirements of each party in a givenyear. Due to the negotiated nature of the schedule, the ADP can impose inefficiencies in theutilization of the shipping fleet and the overall LNG supply chain.

In this paper we present an optimization model that optimizes the LNG shipping scheduleand inventory management for an LNG project. The main motivation behind this model isto provide an objective approach for the analysis of key LNG supply chain design decisionssuch as LNG fleet size and composition, and terminal infrastructure during the conceptand detailed design phases for new LNG projects. The proposed model can also be usedto develop ADPs in operational LNG projects and to quantify the inefficiency in a schedulecaused by the negotiation process. The model can also be used to conduct “what-if” analysisto test how robust a schedule is to disruption events.

LNG inventory routing can be considered as a special case of maritime inventory rout-ing problems (MIRP). MIRP combines inventory management and ship routing, which aretypically treated separately in industrial practice. The reader is referred to the papers byChristiansen et al. [8, 7] and Andersson et al. [3] for a thorough review of maritime in-ventory routing literature. Christiansen and Fagerholt [5] define a basic maritime inventoryrouting problem as the transportation of a single product that is produced at loading portsand consumed at unloading ports where each port has a given inventory storage capacityand a production or consumption rate. The number of visits to a port and the quantity ofproduct to be loaded or unloaded are not predetermined. As discussed by Andersson et al.[3], nearly every paper concerning combined inventory management and routing addressesa new version of the problem. This is in contrast to the work on classical routing problemssuch as the vehicle routing problem (VRP) that have widely accepted definitions and as-sumptions. The LNG inventory routing problem (LNG IRP) studied in this article is basedon a real-world application and shares the fundamental properties of a single product MIRP.However, the LNG IRP addressed here includes several variations including variable pro-duction and consumption rates, LNG specific contractual obligations, and berth constraints.Most importantly, the LNG IRP seeks to generate schedules where each ship makes severalvoyages over a time horizon with both the number of voyages and the time horizon beingconsiderably larger than those considered by a typical MIRP. Andersson et al. [4] providean excellent overview of the business cases and common characteristics for LNG inventoryrouting.

Grønhaug and Christiansen [12] develop discrete time arc-flow and path-flow models for

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operational LNG inventory routing and test these models on a number of operational plan-ning cases with time horizons up to 60 days. Grønhaug et al. [13] develop a branch-and-pricemethod for the discrete time path-flow formulation of the LNG IRP and apply it to an ex-tended set of test cases with time horizons up to 75 days with improved computationalperformance over the previous effort [12]. While the above work focuses on optimizing in-ventory management at production terminals only together with the ship schedules, Fodstadet al. [9] develop a discrete-time arc-flow model for LNG IRP that also considers optimizingdecisions related to sales downstream of the regas terminals. Although atypical of the LNGindustry, they also include the capability for split loads in their model. The model presentedby Fodstat et al. is implemented in the LNGScheduler software [10] and shows improved runtimes compared to those reported by Grønhaug et al. [13] over the same test set. Fodstadet al. [10] apply this model to a set of instances with a time horizon of 181 days.

There has recently been some development of software applications for combined in-ventory management and ship scheduling for the LNG supply chain. van de Broecke andAdams [20] and Stremersch et al. [18] describe the design of such decision support tools.Fodstad et al. [9, 10] discuss the model, implementation and some computational testing forLNGScheduler.

The paper by Rakke et al. [15] appears to be a first attempt to address problems of de-veloping ADPs for large LNG projects. In this paper an arc-flow model and rolling horizonheuristic method are applied to problems with up to 46 ships and a one year planning hori-zon. Halvorsen-Weare and Fagerholt [14] develop a decomposition-based heuristic methodto solve an arc-flow model of the the LNG IRP, and address instances with a moderatenumber of ships and terminals with a time horizon ranging from 30 to 360 days. While theseauthors have addressed larger problems than previously reported in literature, their worklimits optimization of ship schedules together with inventory management at the productionterminals only. In this paper, we address large scale LNG IRPs where ship schedules areoptimized together with inventory management on both the production and regas termi-nals. This approach provides the ability to analyze supply chain designs and develop shipschedules from a broader general-interest perspective.

Christiansen et al. [6] develop a greedy construction heuristic and a solution improvinggenetic algorithm for an MIRP from the cement industry. They note that efficient neigh-borhood operators for improvement heuristics remain a challenge. The methods describedin this paper are inspired by the fundamental modeling and algorithmic concepts developedby Song and Furman [17] for the MIRP problem defined by Furman et al. [11]. The “twoship” neighborhood operator proposed by Song and Furman [17] is inspired by the classical2-opt neighborhood search where the shipping schedules of a pair of ships are optimized ateach iteration of the search. In this article, we study the application of the two-ship neigh-borhood operator and a new neighborhood operator to extremely large-scale instances of theLNG inventory routing problem. We also propose approaches to improve the efficiency ofthe two-ship search operator. Although the solution techniques presented here have beendeveloped and tested in the context of this specific LNG IRP, these can be generally appliedto a wide variety of combined inventory management and routing problems. For a reviewof very large neighborhood search (VLNS) methods, the reader is referred to the excellentsurvey by Ahuja et al. [2].

The rest of the paper is organized as follows. The formal problem description is introduced

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in Section 2. Section 3 presents a mixed-integer programming model for the problem. Section4 discusses various solution techniques used in the large scale neighborhood search procedure.In Section 5, computational results are presented for various algorithms described in thispaper. Section 6 illustrates how various practical analyses are enabled by such a model.Finally, in Section 7 the conclusions and future research directions are discussed.

2 Problem Description

We consider an LNG IRP from the perspective of an vertically integrated oil and gas com-pany. The goal of the problem is to find an optimal schedule for a heterogeneous pool ofships that deliver LNG from a set of production terminals to a set of regasification terminalssuch that constraints related to inventory storage, port operations and contractual obliga-tions are satisfied. While the problem aims at finding an optimal schedule for a given supplychain design that specifies fleet composition and size, terminal storage and berth facilities,the main motivation for solving this problem is based on a need to develop a model that canassist in analyzing alternate supply chain designs.

Formally, the problem considers a set of production and regas terminals with fixed storagecapacities. Each production terminal has a given production profile. On the other hand,operations at the regas terminals are more flexible such that regasification rates can beadjusted within specified bounds on a daily basis. Profiles for the minimum and maximumregas rates over the planning horizon are specified for each regas terminal. In addition,total LNG demand over the planning horizon at each regas terminal is specified. Finally,each terminal has a limited number of berths to load and unload cargoes. Each loading orunloading activity spans over one time period and occupies a berth for that duration.

The supply chain includes a set of heterogeneous ships. We restrict our attention tofull load, full discharge problems. In other words, we consider the case where each shipwill fully load at a single production terminal and fully discharge at a single regas terminal(i.e., there is no partial unloading a.k.a. split delivery). Several previous papers haveconsidered split delivery of LNG cargoes [9, 10, 12, 13, 4]. From a planning and designanalysis perspective, this assumption is reasonable since this is typically considered thepreferred mode of operation within the LNG industry.

Due to its low boiling point, the LNG loaded on a ship gradually boils-off (vaporizes)while the ship travels or while it waits at a terminal. The boil-off gas is commonly burntas fuel and causes the volume delivered to be less than the volume loaded on the ship. Werefer to the volumes loaded and discharged by a ship as the load and discharge capacities,respectively, for the ship. The difference between these capacities represents the averageboil-off loss based on a typical voyage between two terminals. While actual boil-off amountsexperienced during operations may differ from the above approximation, we consider ourapproximation reasonable since boil-off amounts represent a small fraction of the total cargosize and minor deviations from this approximation are not expected to significantly impactlong term supply chain design analysis.

We consider the terminal storage and contractual demand constraints to be soft con-straints. In other words, production in excess of storage capacity at a production terminalis considered as lost production. Similarly, regasification in excess of existing inventory lev-

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els at a regas terminal is considered as a stockout where the excess has to be purchasedfrom the spot market. The objective is to minimize the total penalties associated with lostproduction, stockouts, and unmet contractual demands at regas terminals.

This problem has several complicating characteristics including: heterogeneous fleets,seasonal travel times, limited berth availability at terminals, seasonality in production rates,and variable regas rates. As mentioned above, a solution to the above problem can supportkey capital decisions regarding the sizing of terminal infrastructure and fleets. A solutionto the above problem can also be used to evaluate the effects of pooling shipping fleets,to quantify key operations metrics including fleet utilization, and evaluate the ability of asupply chain design to meet contractual obligations.

In defining the mathematical model in the next section, all of the characteristics of theproblem description are fully developed.

3 Model

The proposed mathematical model is formulated as a mixed integer programming (MIP)problem.

3.1 Sets and Parameters

We first define a number of sets and parameters that are used in the mathematical model.

L : Production terminals. L = {1, 2, . . . , |L|}.R : Regas terminals. R = {1, 2, . . . , |R|}.J : All terminals. J = L ∪R.

V : Vessels (Ships). v = {1, 2, . . . , |V |}.Jv : Set of terminals that ship v can visit.

T : Planning horizon. T = {1, 2, . . . , |T |}cj : Storage capacity at terminal j.

cv,j : Volume loaded (discharged) by ship v at production (regas) terminal j.

Dr : Demand for LNG over planning horizon at regas terminal r.

bj : Number of berths at terminal j.

pl,t : Production rate during time period t at production terminal l.

I0j : Initial inventory at terminal j.

j0v : Initial destination terminal for ship v.

t0v : Arrival date at initial destination terminal for v.

dLr,t : Minimum regas rate of regas terminal r.

dUr,t : Maximum regas rate of regas terminal r.

τj,j′,t : travel time (including berth time) from terminal j to terminal j′ for voyage

starting at time t.

wj,t : Weight for penalizing lost production (stockout) at production (regas) terminal j.

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wDr : Weight for penalizing unmet demand at regas terminal r.

3.2 Network

This section describes the time-space network formulation for the LNG IRP described above.This network model is closely related to the model of Song and Furman [17]. As presentedby Savelsbergh and Song [16], a time-space network formulation can be viewed as an in-teger multi-commodity network flow formulation where a ship is a commodity and a noderepresents a possible visit to a terminal at a particular time.

Figure 1: Example of Time-Space Network Structure

The time-space network in Figure 1 includes source (SRC) and sink (SNK) nodes torepresent initial and final locations for the ships. The network also includes nodes (referredto as “regular” nodes) that represent each terminal at a given time period. For each shipv, there are five types of arcs; an arc from the SRC to SNK node represents that a ship isnot utilized; an arc from SRC to a regular node represents the arrival of a ship to its initialdestination; an arc from a regular node to SNK represents the final departure of the ship;waiting arcs allow a ship to wait at a terminal without occupying a berth. These arcs connectregular nodes corresponding to the terminal at successive time periods. Finally, a travel arcfrom a regular node n1 to a regular node n2 represents the loading (or unloading) activityat the terminal corresponding to node n1 immediately followed by travel to the terminalcorresponding to node n2.

If the schedule for a ship has enough slack, the ship can either wait at a productionterminal after loading a cargo, or travel to the destination regas terminal and wait before itunloads a cargo, or split the waiting time between the two terminals. From a ship schedulingperspective, these are symmetric solutions. Using a travel arc to represent a loading orunloading activity immediately followed by travel eliminates this symmetry. The formalnode and arc definitions are presented below.

N : Set of nodes. N = {(j, t)|j ∈ Jv} ∪ {SRC} ∪ {SNK}

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N : Set of regular nodes. N = N\{SRC, SNK}.Aφv : Arc from SRC node to SNK node for ship v.

= {v, (SRC), (SNK)}AIv : Set of arcs from source to initial destination for ship v.

= {v, (SRC), (j0v , t

0v)}

ADv : Set of waiting arcs for ship v.

= {v, (j, t), (j, t+ 1)|j ∈ Jv}ATv : Set of travel arcs for ship v.

= {v, (l, t), (r, t+ τl,r,t)|l ∈ Jv ∩ L, r ∈ Jv ∩R} ∪{v, (r, t), (l, t+ τr,l,t)|l ∈ Jv ∩ L, r ∈ Jv ∩R}

AFv : Set of arcs from regular nodes to SNK node for ship v.

= {v, (j, t), (SNK)|j ∈ Jv}Av : Set of arcs for ship v.

= Aφv ∪ AIv ∪ ADv ∪ ATv ∪ AFvA : Set of arcs. A = ∪vAvδ+n : Set of outgoing arcs from node n.

δ−n : Set of incoming arcs to node n.

Note that to ensure consistency, we assume that the travel times satisfy the following con-dition

τj,j′,t ≤ τj,j′,t+1 + 1 ∀t ∈ T,∀j, j′ ∈ J, j 6= j′

This condition ensures that the network does not allow a ship to arrive at its destinationterminal earlier by delaying its departure.

3.2.1 Decision Variables

A feasible schedule for this LNG IRP specifies whether a ship travels on any particular arcwithin the network model. The solution also specifies the regas rates at each regas terminalduring each time period. All of the other variables can be considered state variables astheir values will be a direct result of the binary variables on the arcs and the regas rates.The following variables have been defined in the mixed integer programming formulationpresented below.

Ij,t : Inventory level at terminal j at the end of time period t.

dr,t : Regas rate during time period t at regas terminal r.

ol,t : Lost production at production terminal l during time period t.

sr,t : Stockout at regas terminal r during time period t.

δDr : Unmet demand at regas terminal r during planning horizon;

xa : Binary variable for arc a.

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3.3 MIP Formulation

The mixed integer programming formulation which we will refer to as model P follows.

min∑

(l,t)∈N |l∈Lwl,tol,t +

∑(r,t)∈N |r∈R

wr,tsr,t +∑r∈R

wDr δDr (1)

s.t.∑

a∈Av∩δ+n

xa −∑

a∈Av∩δ−n

xa = 0, ∀v ∈ V, n ∈ N , (2)

∑a∈Av∩δ+SRC

xa = 1, ∀v ∈ V, (3)

∑a∈Av∩δ−SNK

xa = 1, ∀v ∈ V, (4)

Il,t = Il,t−1 + pl,t −∑v

∑a∈(AT

v ∪AFv )∩δ+

(l,t)

cv,lxa − ol,t ∀l ∈ L, t ∈ T (5)

Ir,t = Ir,t−1 − dr,t +∑v

∑a∈(AT

v ∪AFv )∩δ+

(r,t)

cv,rxa + sr,t ∀r ∈ R, t ∈ T, (6)

∑v

∑a∈(AT

v ∪AFv )∩δ+n

xa ≤ bj, ∀j ∈ J, t ∈ T, n = (j, t) ∈ N (7)

δDr ≥ Dr −∑t

∑v

∑a∈(AT

v ∪AFv )∩δ+

(r,t)

cv,rxa, ∀r ∈ R (8)

dLr ≤ dr,t ≤ dUr , ∀r ∈ R, ∀t ∈ {1, 2, . . . , |T |}, (9)

0 ≤ Ij,t ≤ cj, ∀j ∈ J, t ∈ T (10)

ol,t ≥ 0 ∀l ∈ L (11)

sr,t ≥ 0 ∀r ∈ R (12)

δDr ≥ 0 ∀r ∈ R (13)

xa ∈ {0, 1} ∀a ∈ A (14)

The objective is to minimize the sum of weighted lost production, stockout, and unmetdemands. Constraints (2)-(4) are network-flow conservation constraints for each ship. Con-straints (5) and (6) model the inventory balance at the production and regas terminals,respectively. Lost production and stockout variables provide slack for these constraints.Constraint (7) ensures that the number of ships loading or unloading a cargo at a terminalwithin a time period does not exceed the number of berths. Note that a ship can wait ata terminal without occupying a berth. Constraint (8) enforces a lower bound on the unmetdemand variable based on total deliveries scheduled for a regas terminal. Since the objectivefunction seeks to minimize the total unmet demand, this constraint will be tight in everyoptimal solution that has a positive unmet demand. The final set of constraints (9)-(14)ensures that all the variables satisfy their specific bounds.

It should be noted that for the above model to calculate the lost production and stock-out amounts correctly, the penalty parameters for lost production and stockouts should bemonotonically decreasing in time, i.e., wj,t > wj,t+1. This ensures that a solution will notinvolve lost production (stockout) until the inventory reaches capacity (falls to zero). Inother words, without this condition the model could report solutions where lost production

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or stockouts are reported in advance of the actual event. For example, consider a solutionwhere production terminal l experiences a loss of ε units in time period t. If wl,t−1 = wl,t,an equivalent solution can be constructed where the loss is distributed over periods t andt− 1. This would be unrealistic if the terminal inventory at t− 1 is lower than the storagecapacity. A monotonically decreasing penalty weight ensures that the lost production andstockout calculations are correct.

In its present form, the model is generic with respect to the length of the planning hori-zon and the time discretization. Based on planning conventions in the LNG industry andtypical loading and discharging durations, a year-long planning horizon with one day timediscretization is expected to be a reasonable scale for the problem. An important charac-teristic of this formulation is that it is easy to accommodate various practical constraints ofthe problem. Many of these constraints can be handled by adding, removing, and/or fixingnodes and arcs. For example, planned maintenance at a terminal that disallows any loadingor unloading activities can be accounted for by removing all travel arcs from that terminalfor the duration of the activity.

Due to the complex nature of the above problem, commercially available Mixed IntegerProgramming solvers cannot solve our real world problems within a reasonable amount oftime. In the following section, we discuss some of the solution methods developed in orderto overcome this obstacle.

4 Solution Methods

We develop local neighborhood search heuristics with the goal of finding good solutionsto the proposed model in a short amount of time. This approach can also be used toprovide a warm-start for an exact solution method. Algorithm 1 proposes a three-stepheuristic method for the above model. The first step of the heuristic uses a constructionheuristic to build an initial feasible solution to the model. The construction heuristic builds afeasible solution by iteratively scheduling ship departures for the entire fleet on a daily basis.Ship departures on any given day are scheduled using an urgency-based greedy heuristic.Two improvement heuristics that employ different neighborhood structures are then usedin sequence to improve this solution. The Time-Window Improvement Heuristic seeks toimprove the solution obtained from the construction heuristic by allowing for departuredates for each voyage in the current solution to change within a small time-window definedaround the current departure dates. The Two-Ship Improvement Heuristic seeks to improvethe existing solution by allowing a subset of the ships to change schedules over the entirehorizon. The subproblems generated by both improvement heuristics can be solved as Mixed-integer programs (MIPs) and solved using a commercial solver.

Algorithm 1 Solution Heuristic

Build initial solution using Construction Heuristic (CH)Improve solution using Time window improvement heuristic (TWIH)Improve solution using Two-ship improvement heuristic (2SIH)

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4.1 Construction Heuristic (CH)

The construction heuristic is based on a greedy approach that seeks to deliver as much LNGas possible and as soon as possible from the production terminals to the regas terminals.In addition, the construction heuristic seeks to minimize lost production and stockouts byprioritizing voyages to terminals that have the most urgent demand for shipping capacityduring the next few time periods. The heuristic uses this approach to schedule ship depar-tures at all terminals on a daily basis. A feasible solution is generated by fixing daily shipdeparture decisions iteratively over the entire planning horizon.

The construction heuristic (shown in Algorithm 2) begins by initializing the terminalinventory levels at the beginning of the planning horizon, and setting the initial arrival datesand locations for each ship. The heuristic builds a solution where the daily regas rate at anyregas terminal is equal to the mean regas rate for the terminal on the respective day.

The construction heuristic then iterates over the time periods to fix ship departuresfrom each terminal in any given time period. The heuristic uses a similar approach forscheduling departures from production and regas terminals. We present the explanation forscheduling departures from production terminals. Variations for scheduling departures fromregas terminals are presented parenthetically.

In order to schedule departures at terminal k during time period t, the heuristic firstcalculates the closing inventory level at terminal k for time period t using equation (5)(equation (6) for regas terminals) if no ship departures were to take place from terminal kduring time period t. The closing inventory is used to identify the set of ships Vk that can bedispatched from terminal k during time period t. The set Vk includes ships that are locatedat production (regas) terminal k at period t and that can be loaded (discharged) based ontheir load (discharge) capacities and the closing inventory at terminal k in period t.

The heuristic schedules departures of ships in set Vk from production (regas) terminalk as long as a berth is available at terminal k, and the terminal inventory permits loading(discharging) some ship in Vk. In scheduling the next departure, the heuristic first identifiesthe destination terminal with the most urgent need for shipping capacity. The need forshipping capacity at a production (regas) terminal is quantified as the difference betweenthe total planned production (regasification) at the terminal during a future time horizon,and the total shipping capacity already scheduled to reach that terminal during that timehorizon. The terminal k with the most urgent need for shipping capacity is selected as thedestination for the next departure from terminal k. Note that not all ships may be allowedto travel to all terminals. Further, our model restricts ship to travel from a production(regas) terminal to a regas (production) terminal only. Therefore, the selection of the nextdestination is restricted to terminals such that at least one ship in Vk can travel from terminalk to terminal k.

In order to maximize the delivery of LNG, the heuristic selects the ship v with thelargest loading (discharging) capacity among all ships in Vk that can travel from production(regas) terminal k to terminal k. The heuristic then schedules a departure for ship v fromterminal k to terminal k in period t and the closing inventory level at terminal k is updatedaccordingly. This approach is repeated until no more departures can be scheduled fromterminal k in period t. A complete schedule for the entire time horizon is built by applyingthis approach for all terminals over all time periods.

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Finally, lost production (stockout) volumes for all production (regas) terminals can becalculated based on the production (regasification) profiles together with the shipping sched-ule.

Algorithm 2 Construction Heuristic

Initialize ship locations and terminal inventory levels

Fix daily regas rates todL

r,t+dUr,t

2for all regas terminals r

for t = 1, 2, . . . , T dofor all terminals k do

Evaluate closing inventory level for period t assuming no ship departures at terminalk during period tVk := Set of ships that are located at terminal k in period t and that can be dispatchedbased on ship capacity and terminal inventorywhile another berth is available and Vk is not empty doK ′ := Set of terminals k′ such that at least one ship in Vk can be dispatched fromterminal k to terminal k′

Select destination terminal k ∈ K ′ that has the most urgent demand for shippingcapacitySelect ship v that has the largest capacity among ships that can be dispatched fromk to kSchedule departure of ship v from terminal k to k in period tUpdate closing inventory for period t based on scheduled departure

end whileEvaluate stockout or lost production at k at t

end forend for

4.2 Time-Window Improvement Heuristic (TWIH)

The time-window improvement heuristic is a large neighborhood search method. This heuris-tic searches in the neighborhood consisting of all solutions in which any voyage departure isdelayed or advanced by at most m days, compared to the departure date for that voyage inthe input solution. An optimal solution in this neighborhood can be obtained by solving amodified version of Model P that excludes travel arcs for all voyages that depart outside thetime windows defined by the above neighborhood. This modified version of model P can besolved using a commercial MIP solver.

4.3 Two-Ship Improvement Heuristic (2SIH)

The two-ship improvement heuristic (described in Algorithm 3) is based on the variableneighborhood search method. At each iteration, the heuristic selects a pair of ships anddefines the search neighborhood to consist of the set of feasible schedules for these ships.

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Schedules for all other ships are considered fixed. A subproblem is solved to optimize sched-ules within this neighborhood. The algorithm iteratively selects pairs of ships and solves thecorresponding subproblems for the defined neighborhoods. The heuristic is terminated whena better solution is not found for

(|V |2

)successive iterations. Since the objective function for

this problem is lower bounded by zero, the heuristic can also be terminated as soon as asolution with a zero objective value is obtained. From an implementation perspective, theoptimal solution for a subproblem induced by a pair of ships can be obtained by solving areduced version of Model P where variables for all other ships are fixed based on the currentsolution. We refer to the reduced model as a two-ship subproblem.

Algorithm 3 Two Ship Improvement Heuristic

Initialize solution; UB := objective value of solutioncount := 0while count ≤

(|V |2

)and UB > 0 do

Select ships v and v′

Fix schedules for all ships other than v and v′

Solve two-ship subproblemif better solution is found then

Update solution; UB := objective value of solutioncount := 0

elsecount := count+ 1

end ifend while

4.3.1 Ship-pair Selection

The efficiency of the two-ship improvement heuristic depends on the sequence of two-shipsubproblems solved during the algorithm. By selecting ship-pairs that are most likely to leadto a better solution, the number of subproblems solved and the overall CPU time can beminimized. We present three schemes for selecting two-ship subproblems during the two-shipimprovement heuristic.

Lexicographic selection For a problem where the fleet of ships is represented by the or-dered set

{v1, v2, v3, . . . , v|V |

}, we consider the selection scheme to be lexicographic if it selects

two-ship subproblems in the following order: (s1, s2), (s1, s3), . . . , (s1, s|V |), (s2, s3), . . . , (s|V |−1, s|V |).The lexicographic ordering scheme may be effective in a practical setting where the relativeimportance of ships is inherent in the order in which they are included in the fleet.

Metrics based selection In this method, we use the current solution together with prob-lem specific insights to identify the two-ship subproblem that is likely to achieve the largestimprovement in objective function. Specifically, we define two metrics based on the current

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solution for each ship-pair for which a subproblem can be solved next. These metrics repre-sent estimates for the maximum potential reduction and for the likelihood of achieving thatreduction in the objective function if the corresponding two-ship subproblem were optimized.The ship-pair that has the highest likelihood of achieving a reduction among the ship-pairsthat have the largest potential for reduction is selected.

Algorithm 4 provides a detailed specification for the selection method. Let π1v represent

the improvement in the objective function if the schedule for ship v is re-optimized. Weestimate π1

v as the total production and stockouts that would be eliminated if ship v couldbe rescheduled such that the sequence of terminals visited by v remains unchanged but theship arrives at all terminals as soon as possible; i.e., the re-optimized schedule does notinvolve any demurrage. The estimate for reduction in the objective function by solving thetwo-ship subproblem for ship-pair (v1, v2) (represented by Π1

(v1,v2)) is estimated as the greater

of π1v1

and π1v2

.We evaluate the likelihood of achieving a reduction in the objective function by solving

the two-ship subproblem for ship-pair (v1, v2) assuming that the sequence of terminals visitedby v1 and v2 in the re-optimized solution will remain unchanged compared to the currentsolution. However, departure dates for the two ships from a terminal might be swapped orre-ordered in order to reduce overall lost production and stockouts. In order to quantifythis likelihood, we evaluate π2

(v1,v2) as the number of times in the current solution where thedeparture for ship v1 is delayed due to the departure for v2 from the same terminal. Inevaluating π2

(v1,v2), we count cases where ship v1 is forced to demur at a terminal while shipv2 departs from that terminal. Cases where ship v2 is the last ship to depart from a terminalbefore v1 arrives at that terminal and is forced to demur are also included in the calculationof π2

(v1,v2). Finally, the likelihood of achieving a reduction in the objective function by solving

the two-ship subproblem for ship-pair (v1, v2) (represented by Π2(v1,v2)) is approximated as the

number of times the ship that is estimated to lead to bigger reduction in objective function isdelayed due to the other ship in the pair. Note that since Π1

(v1,v2) and Π2(v1,v2) are symmetric

with respect to (v1, v2), an arbitrary ordering of ships is assumed such that v1 < v2 can bedefined.

The set P consists of ship-pairs (v1, v2) for which Π1(v1,v2) is within the top α percent of

the range of Π2(·,·). The ship-pair in P that has the largest estimated likelihood of achieving

a reduction in the objective function is selected.The ship associated with the largest value for this metric is selected as v2.

Model based selection The restriction that at most two ships can change their schedulescan be represented by constraints (15)-(17). Binary variable yv represents whether or not shipv can change schedule. Constraint 15 restricts at most two ships to change their schedule.Constraint (16) determines the value of yv based on whether the schedule for ship v insolution represented by variable x is different from that in a given solution x.∑

v

yv ≤ 2 (15)

yv = 1⇔ ∃a ∈ δv|xa 6= xa∀v (16)

yv ∈ {0, 1}∀v (17)

13

Algorithm 4 Metrics based two-ship subproblem selection

for all ships v doπ1v := total lost production and stockouts that would be eliminated if all demurrage in

current schedule for ship v were eliminated with the sequence of terminals visited byship v remained unchanged

end forfor all ship pairs (v1, v2) such that v1 < v2 do

Π1(v1,v2) := max

{π1v1, π1

v2

}end forfor all ship pairs (v1, v2) doπ2

(v1,v2) := number of times in current solution where departure for v1 is delayed due todeparture for v2 from the same terminal

end forfor all ship pairs (v1, v2) such that v1 < v2 do

Π2(v1,v2) :=

{π2

(v1,v2) : if π1v1≤ π1

v2

π2(v2,v1) : if π1

v1> π1

v2

end forΠ1,max := max

(v1,v2)Π1

(v1,v2),Π1,min := min

(v1,v2)Π1

(v1,v2)

P :={

(v1, v2)|Π1(v1,v2) ∈

[Π1,max − α

100(Π1,max − Π1,min) ,Π1,max

]}(v∗1, v

∗2) := arg max

(v1,v2)∈PΠ2

(v1,v2)

return (v∗1, v∗2)

14

Dimensions Cont. 0-1 Best knownProblem (|L|, |R|, |V |) Variables Variables Constraints solution

P1 (1,2,6) 2557 12812 6504 1172P2 (1,3,8) 4019 16988 8635 294P3 (2,1,10) 2557 21029 9303 5650P4 (3,1,13) 2922 27547 12196 0P5 (1,4,15) 5115 31449 14269 1701P6 (4,1,4) 3652 8377 6406 151P7 (1,6,6) 5117 18335 10812 1565P8 (1,1,14) 1827 28830 11205 116P9 (1,2,17) 2558 35142 14061 126

P10 (1,4,27) 5115 95788 32744 29388P11 (1,5,14) 6228 34438 15625 0P12 (1,4,18) 3655 37752 16491 631P13 (1,8,40) 6579 132995 47364 1453P14 (1,10,69) 11691 232856 79425 53240

Table 1: Problem Instances

We define a restricted version of model P (referred to as rP) by incorporating constraints(15)-(17). The solution to rP will identify the two-ship subproblem that will lead to thebiggest improvement in the objective function relative to the existing solution x. The modelbased selection approach involves selecting the two ships that are associated with the largestvalues for y(·) in the solution of the linear relaxation of model rP.

For large problems, solving the linear relaxation for model rP can be expensive. It iseasy to see that the constraint set for Model rP includes a network-substructure that isseparable over the set of ships. However, the constraints representing terminal inventoryand berth limits, together with constraints that limit the number of ships that can changeschedule cause the model to be coupled over the set of ships. Decomposition methods,such as Dantzig-Wolfe or Lagrangean decomposition can be employed for solving the linearrelaxation of rP efficiently when solution in the full space is inefficient.

5 Computational Results

In this section, we present computational results to demonstrate the performance of theproposed heuristic. The results are based on an implementation of the model in AIMMS.All optimization models have been solved with CPLEX 11.1 on a Dell 7500 Windows PCwith a 2.93 GHz dual-quad core processor and 24 GB RAM.

15

5.1 Problem Set

Table 1 shows the sizes of the 14 problems on which the proposed heuristics were evaluated.The table reports the dimensions (|L|, |R|, |V |) for each problem where |L|, |R| and |V |represent the number of production terminals, regas terminals and ships, respectively. Allthe problems are defined over a one year planning horizon. These problems are realistic testcases allowing for a limited amount of pooling of the shipping fleet among multiple shippinglanes restricted via the allowed arcs in the network. The number of continuous and binaryvariables, the number of constraints and the best known solution for each problem are alsoreported. We classify the problems in to three categories based on the ease of finding goodsolutions using CPLEX.

• easy problems: includes P1, P2, P3, P4, P5. CPLEX generates the best knownsolutions for these problems in less than 1 CPU hour

• medium problems: includes P6, P7, P8, P9, P10. CPLEX generates a “good” (within15% of best known) solution for these problems within 1-5 CPU hours

• hard problems: includes P11, P12, P13, P14. CPLEX cannot generate a good solutionfor these problems within 10 CPU hours

5.2 CPLEX vs. Proposed Heuristic

We first compare the performances of CPLEX and the proposed heuristic. For this com-parison, the results presented for the heuristic are based on randomly selecting the nexttwo-ship subproblem to be solved at each iteration of the two-ship improvement heuristic(2SIH). The proposed heuristic is therefore referred to as H-Randm. To account for the effectof randomization, results for H-Randm are based on 100 runs for problems P1-P12 and 50runs for problems P13-P14.

Our implementation for the time-window improvement heuristic (TWIH) is based ondefining a time-window that allows each ship departure to be delayed or advanced by atmost four time-periods. Each two-ship subproblem is solved with a node-limit of 1000 nodes.The time-window improvement heuristic is solved with a node-limit of 5 nodes. Results forCPLEX are based on runs with CPU time limits of 5 hours, 10 hours and 20 hours for easy,medium and hard problems, respectively.

Table 2 compares the solutions generated by CPLEX with those generated by H-Randm.For ease of exposition, we report the solution quality in terms of the percent gap of asolution with respect to the best known solution. Specifically, for a solution X and best

known solution X, the gap is computed as X−XX× 100. Since the best known solutions for

problems P4 and P11 have a zero objective value, the solution quality for these cases isreported in terms of the actual objective values.

Table 2 reports the percent gaps relative to the best known solution for the first feasible(FF) and the best feasible solutions (BF) found by CPLEX, together with the percentgap between BF and the best bound when CPLEX terminates. The table also reportspercent gaps relative to the best known solution for the solutions found by H-Randm atthe end of the three stages in the H-Randm; i.e., construction heuristic (CH), time-window

16

Problem CPLEX H-Randm

FF BF CPLEX Gap CH CH + TWIH CH + TWIH + 2SIHMin Avg Max

(%) (%) (%) (%) (%) (%) (%) (%)P1 9.87 0.00 5.26 4.81 2.28 1.24 1.25 1.26P2 290.40 0.00 16.34 133.15 61.16 8.53 8.53 8.53P3 0.59 0.00 0.69 6.51 4.82 0.81 0.81 0.81

P4* [1831.07] [0.00] 0.00 [399.17] [288.24] [0.00] [0.00] [0.00]P5 16.97 0.00 1.61 84.34 69.37 7.19 15.70 25.62P6 1450.64 0.00 63.88 0.00 0.00 0.00 0.00 0.00P7 117.95 11.84 51.11 168.03 168.02 0.04 10.04 26.27P8 160.28 0.00 0.00 0.00 0.00 0.00 0.00 0.00P9 789.72 0.00 0.00 0.00 0.00 0.00 0.00 0.00

P10 12.75 6.89 7.34 5.52 3.21 0.00 0.96 1.94P11* [9580.88] [4012.99] 100.00 [504.34] [504.34] [0.00] [2.62] [130.87]P12 2471.06 466.62 99.71 3.95 3.95 0.01 0.09 0.47P13 449.88 449.88 82.79 352.74 347.40 2.46 13.11 29.90P14 522.23 522.23 85.12 26.64 24.70 0.00 1.42 2.91

Table 2: Solution quality comparison: CPLEX vs H-Randm

improvement heuristic (TWIH) and two ship improvement heuristic (2SIH). Since 2SIHinvolves randomization, the minimum, maximum and average percent gaps relative to thebest known solution are reported at the end of 2SIH.

Table 3 compares the CPU times required by CPLEX with those required by H-Randm.The CPU times reported in columns that are marked as scld are scaled by the averageCPU time taken by H-Randm. Columns that report actual CPU times are marked as act.The table reports CPU times taken by CPLEX to generate the first feasible (TFF) and thebest feasible (TBF) solutions, time required to generate a feasible solution better than theaverage solution for H-Randm (TRH), and the total run time for CPLEX. The table alsoreports the cumulative CPU time taken to complete the three stages of H-Randm, includingthe minimum, maximum and average CPU times for the overall heuristic.

Table 2 shows that CPLEX can prove optimality for only 3 problems and generate thebest known solution for 8 problems within the CPU time limits. We compare the performanceof CPLEX with H-Randm for the three problem classes individually.

For the easy problems, CPLEX is able to generate the best known solutions for allproblems. For problems P1, P3 and P4, H-Randm generates solutions within 1.26% of thebest known solution. However, CPLEX is much slower (2.55x to 70x) in generating its bestsolution than the average CPU time taken by H-Randm. Further, CPLEX is 1.78x-19.02xslower in generating a solution that is at least as good as that generated by H-Randm.

For problems P2 and P5, H-Randm generates solutions with objective value 8.5% to 25.6%worse than the best known solution and CPLEX. However, CPLEX is much slower in gener-

17

Problem CPLEX H-Randm

TFF TBF TRH Total CH CH + TWIH CH + TWIH + 2SIHMin Avg Max

(scld) (scld) (scld) (act) (scld) (scld) (scld) (act) (scld)P1 0.45 70.04 2.60 18000 0.01 0.02 0.64 254 1.61P2 1.45 25.28 17.14 18000 0.04 0.25 0.72 69 1.26P3 1.78 2.55 1.78 18000 0.04 0.24 0.91 76 1.20P4 1.75 18.68 19.02 2985 0.03 0.07 0.19 157 3.13P5 0.83 14.39 0.87 18000 0.01 0.08 0.54 340 1.95P6 0.18 151.58 131.94 36000 0.01 0.01 1.00 235 1.00P7 0.21 17.47 - 36000 0.00 0.01 0.48 1515 2.42P8 2.29 68.59 68.59 7589 0.03 0.04 1.00 111 1.00P9 3.12 98.72 98.72 18145 0.03 0.03 1.00 184 1.00

P10 3.31 3.31 - 36001 0.01 0.04 0.43 2666 2.99P11 5.15 143.70 - 72000 0.02 0.03 0.13 264 3.36P12 1.43 55.63 - 72001 0.01 0.15 0.61 829 1.73P13 1.03 1.03 - 72001 0.00 0.01 0.44 18804 2.51P14 2.76 2.76 - 72003 0.00 0.01 0.63 26115 1.93

Table 3: CPU time comparison: CPLEX vs H-Randm

ating its best solution (14.4x-25.3x) for these problems than H-Randm. Further, in the caseof P2, CPLEX is 17.1x slower than H-Randm in generating a solution that is at least as goodas that generated by H-Randm. Overall, it can be concluded that for the easy problems,CPLEX generates better solutions within the time limit provided but H-Randm generatesgood solutions much faster. Note that although the speed-up factors above are based on theaverage performance of H-Randm, similar conclusions can be drawn based on the worst caseperformance of H-Randm.

For 3 out of 5 medium problems, both CPLEX and H-Randm are able to generate the bestknown solutions. However, CPLEX is 68.6x to 131.9x slower than H-Randm. In the case ofP10, H-Randm always generates a better solution than CPLEX although CPLEX requires3.3x more CPU time than the average run-time of H-Randm. Finally, the solutions reportedby H-Randm for P7 demonstrate significant variance in quality (0.04%-26.27% gap) although67% runs of H-Randm generates a better solution than CPLEX.

For the hard problems, the best feasible solutions generated by CPLEX are clearly poorcompared to the best known solutions (466.62% to 522.23% gap for P12, P13 and P14) andalso much worse than the average (and worst) solutions generated by H-Randm. Further,CPLEX is slower (1.03x to 143.7x) in generating its best feasible solution than the averageCPU time required the H-Randm. In fact, for 3 out of 4 problems, CPLEX is slower ingenerating its best feasible solution than the worst performance of H-Randm.

Note that comparing TBF or TRH with the total CPU time for H-Randm gives a conser-vative estimate of the speedup delivered by H-Randm since the total CPU time for H-Randm

18

also includes the time required to prove convergence in addition to the time required to findthe best solution.

Table 2 also shows that CH and TWIH together (referred to as CH+TWIH) form a veryeffective heuristic for generating initial solutions. The CH+TWIH method provides bettersolutions than the first feasible solution generated by CPLEX for 11 out of 14 problems. For3 problems, CH+TWIH generates the best known solution while for all hard problems thatmethod generates better solutions than the best solution generated by CPLEX. Further,CH+TWIH is much faster than CPLEX in generating feasible solutions.

Finally, it is worth noting the large variations in CPU time for H-Randm for cases wherethe heuristic returns the same solution across all runs. Specifically, the ratio of maximumand minimum CPU time taken by H-Randm ranges from 1.00 (problem P6) to 16.6 (problemP4). This variation is amplified for problems where the heuristic finds a solution with azero objective value since the heuristic can stop immediately without having to completea full cycle of solves of two ship subproblems to prove that no further improvements canbe achieved. These large variations indicate the potential for reducing overall CPU timethrough an intelligent ordering of two ship subproblems during 2SIH.

5.3 Sub-problem Selection in 2SIH

Next we evaluate the effectiveness of the three schemes presented in Section 4.3.1 for selectingthe next two-ship subproblem during 2SIH. We refer to the heuristic when combined withthe lexicographic, metrics-based and model-based selection schemes as H-Lexic, H-Mtrcs,and H-Model, respectively. The performance of these two-ship selection schemes is com-pared against H-Randm. The same parameters and termination criteria for the constructionheuristic, the TWIH, and the two-ship subproblems as those presented above are used for allversions of the heuristic. In addition, we partition the set of iterations (two-ship subproblems

solved) during 2SIH in to a set of cycles, where each cycle refers to a set of(|V |2

)successive

iterations. To achieve search diversification during 2SIH, we do not allow for a subproblemfor the same ship-pair to be solved more than once during any cycle. This restriction isapplied for all versions of the heuristic. For H-Model, this constraint is included explicitlyin model rP.

In order to reduce the overall CPU time used for selecting two-ship subproblems inH-Model, we solve the relaxation of model rP only when a new solution is obtained. Similarly,in the case of H-Mtrcs, metrics for quantifying the improvement that a two-ship subproblemwill yield are re-used until a better solution is obtained. We use α = 20% for identifying thecandidate ship-pairs for selection during H-Mtrcs.

Note that results for P6, P8 and P9 are not included in this analysis since the best knownsolutions for these problems are generated by (CH+TWIH) itself. Also, results reported forthe three largest problems (P10, P14 and P13) with H-Model are based on the use of Dantzig-Wolfe decomposition to solve the linear relaxation of rP for selecting two-ship subproblemsduring 2SIH. Results reported for H-Model for all other problems are obtained by solving rPusing CLPEX.

Table 4 compares the quality of solutions obtained by the heuristic method with the fourordering schemes. Solution quality is reported in terms of the % gap of the objective valuerelative to the best known solution (except in the case of P4 and P11 since the best known

19

Problem H-Randm H-Lexic H-Mtrcs H-Model

Min Avg Max(%) (%) (%) (%) (%) (%)

P1 1.24 1.25 1.26 1.26 1.24 1.24P2 8.53 8.53 8.53 8.53 8.53 8.53P3 0.81 0.81 0.81 0.81 0.81 0.81

P4* [0.00] [0.00] [0.00] [0.00] [0.00] [0.00]P10 0.00 0.96 1.94 0.58 0.96 1.11

P11* [0.00] [2.62] [130.87] [0.00] [0.00] [0.00]P12 0.01 0.09 0.47 0.00 0.01 0.01P14 0.00 1.42 2.91 0.52 1.24 2.48P5 7.19 15.70 25.62 15.29 17.91 15.86P7 0.04 10.04 26.27 0.00 6.91 5.46

P13 2.46 13.11 29.90 18.51 20.24 0.00

Table 4: Solution quality comparison: Heuristic with different ordering schemes for 2SIH

solutions have zero objective values. The solution qualities for these cases are reported interms of the actual objective values).

Based on the variance in the quality of solutions generated by the heuristic with the fourordering schemes, we partition the cases in to two sets.

• Set 1 includes P1, P2, P3, P4, P10, P11, P12, P14. As shown in Table 4, the solutionsreturned by the heuristic with all four ordering methods are fairly similar (the %gaprelative to the best known solution ranges between 0% to 2.91%). Note that theoptimal solution for P11 has a zero objective value and 99 out of 100 runs of H-Randmreturned this solution. Since only one run returns a non-optimal solution, we considerthe solutions generated by H-Randm for P11 to be almost as good as the deterministicordering schemes. P11 is thus include in Set 1.

• Set 2 includes P5, P7, P13: solutions returned by the heuristic with different orderingschemes differ significantly.

5.3.1 Set 1: Deterministic schemes vs. average performance of H-Randm

We focus on comparing the effectiveness of the selection schemes in reducing CPU time forproblems in Set 1 since all selection schemes generate similar solutions for these problems.Data for cases in Set 2 is reported for completeness.

Table 5 compares the number of two-ship subproblems solved by the heuristic methodwith the four selection schemes. Columns that are marked as scld report the number of twoship subproblems solved scaled by the minimum number of two ship subproblems that needto be solved in the ideal case (also reported in Table 5). For a problem that has a solution

with a non-zero objective value, at least(|V |2

)sub-problems have to be solved, where s is the

20

Problem H-Randm H-Lexic H-Mtrcs H-Model MinMin Avg Max solves

(scld) (scld) (scld) (scld) (scld) (scld) (act)P1 1.00 1.34 1.87 1.00 1.00 1.00 15P2 1.00 1.52 1.96 1.89 1.39 1.14 28P3 1.00 1.07 1.36 1.09 1.11 1.00 45P4 1.00 15.80 70.00 8.00 6.00 2.00 1

P10 1.00 1.07 1.91 1.10 1.01 1.00 351P11 6.00 19.32 107.00 21.00 35.00 6.00 1P12 1.24 2.06 3.61 1.62 2.55 2.82 153P14 1.05 1.63 2.84 1.83 1.11 1.00 2346P5 1.06 2.07 4.29 2.16 1.43 1.28 105P7 1.20 2.51 5.00 3.20 2.33 1.67 15

P13 1.37 2.34 4.60 1.65 1.71 2.33 780

Table 5: Comparison of number of two ship subproblems solved: Heuristic with differentordering schemes for 2SIH

number of ships in the problem. For a case that has a solution with a zero objective value,at least one sub-problem has to be solved (assuming 2SIH does not start with the optimalsolution).

Table 6 compares the total CPU times taken by the heuristic method with the fourordering schemes. As before, columns that are marked as scld in Table 6 report CPU timesscaled by total CPU time taken on average by H-Randm.

Table 5 shows that H-Lexic, H-Mtrcs and H-Model solve fewer two-ship subproblems thanthe average number of subproblems solved by H-Randm for 3, 5 and 7 problems, respectively,out of 8 problems in Set 1. H-Model solves the minimal number of subproblems for 4 problemsand seems to be most robust in terms of solving the fewest subproblems.

However, as can be seen in Table 6, H-Model is faster than the average performance ofH-Randm in only 5 out of 8 problems in Set 1. The overhead of solving the LP relaxationof rP mitigates the benefits of better subproblem selection on 2 (P3 and P10) of the 7 theproblems where H-Model solves fewer subproblems than H-Randm. Table 7 shows the %of overall CPU time that is used by H-Model in solving the LP models for selecting thenext two-ship subproblem during 2SIH. Since H-Lexic and H-Mtrcs do not have significantoverhead for selecting the subproblems, their performance in reducing CPU time is similarto that in reducing number of two ship subproblems solved.

Table 8 reports the geometric means for speedup in CPU time and for reduction innumber of two-ship subproblems solved for H-Lexic, H-Mtrcs and H-Model relative to theaverage performance of H-Randm over problems in Set 1. The table also reports the speedupin CPU time (reduction in number of subproblems solved) that would be achieved if wecould pick for each problem individually a selection scheme that leads to the least CPU time(fewest subproblems solved) among all runs of H-Randm. This metric provides an estimate

21

Problem H-Randm H-Lexic H-Mtrcs H-Model

Min Avg Max(scld) (act) (scld) (scld) (scld) (scld)

P1 0.64 254 1.61 0.78 0.64 0.67P2 0.72 69 1.26 1.07 0.89 0.92P3 0.91 76 1.20 0.97 1.02 1.14P4 0.19 157 3.13 1.25 1.20 0.41

P10 0.43 2666 2.99 0.99 0.47 1.11P11 0.13 264 3.36 1.69 0.63 0.57P12 0.61 829 1.73 0.89 1.36 1.57P14 0.63 26115 1.93 1.06 0.66 0.71P5 0.54 340 1.95 0.98 0.80 0.91P7 0.48 1515 2.42 1.37 0.91 0.70

P13 0.44 18804 2.51 0.72 0.95 1.04

Table 6: CPU time comparison: Heuristic with different ordering schemes for 2SIH

Problem Total CPU time Subproblem selection(s) (%)

P1 171 4P2 63 15P3 87 18P4 64 14

P10 2960 35P11 150 20P12 1297 8P14 18501 8P5 311 29P7 1061 4

P13 19648 14

Table 7: % CPU time used in different stages of 2SIH for H-Model

22

Speed-up in CPU time Reduction in number of subproblemsH-Lexic 0.94 1.10H-Mtrcs 1.23 1.13H-Model 1.22 1.68

H-Randm (Min) 2.22 2.05

Table 8: Geometric mean for speedup in CPU time and reduction in number of two-shipsubproblems solved compared to average performance of H-Randm on problems in Set 1

of the upper-bound on the speedup (reduction in subproblems solved) that can be obtainedby selecting the optimal ordering scheme.

5.3.2 Set 1: Deterministic schemes vs performance range of H-Randm

As reflected in Table 6, the performance of H-Randm varies significantly across the randomizedruns. Figure 2 shows a cumulative frequency distribution for the total CPU time for runsof H-Randm on P11. The total CPU time taken by H-Model is also shown for reference. Thefigure clearly emphasizes that a large fraction (77%) of runs of H-Randm take more CPUtime than H-Model. We compare the performance of the deterministic ordering schemesagainst the performance range of H-Randm. We consider an ordering scheme to be effectivein reducing CPU time (number of two ship subproblems solved) if at least 50% runs of theH-Randm require more CPU time (two ship subproblems solved) than the ordering scheme.We restrict the analysis to cases in Set 1 since all ordering schemes return similar solutionsfor these problems.

Table 9 displays the % of runs of H-Randm where the total CPU time and the number oftwo-ship subproblems solved is greater than that for H-Lexic, H-Mtrcs and H-Model. Thistable confirms the conclusions from Tables 5 and 6 that H-Lexic, H-Mtrcs and H-Model areeffective in reducing the number of two-ship subproblems solved for 3, 5 and 7 problems,respectively, and in reducing overall CPU time for 3, 5 and 5 problems respectively.

The three problems where H-Model is unable to outperform more than 50% runs ofH-Randm in terms of total CPU time give interesting insights in to the trade-offs of designingthe heuristic.

• P12: the run with H-Model sees an extremely small improvement towards the end of2SIH which delays termination until another full cycle of two ship subproblem solveshas been completed without any further improvement. If looser termination criteriawere used such that the minor improvement were ignored in determining convergence,38% (instead of 3%) runs of H-Randm would have required more subproblem solvesthan H-Model.

• P10: H-Model outperforms 91% runs of H-Randm in terms of number of two shipsubproblems solved but can only outperform 30% of runs in terms of total CPU time(Table 9). This is because the two ship subproblems identified by H-Model are much

23

Figure 2: Cumulative frequency distribution for total CPU time for H-Randm on P11. CPUtime for H-Model is indicated by vertical line

Problem H-Lexic H-Mtrcs H-Model

time iter time iter time iterP1 79 90 93 90 86 90P2 34 11 74 64 67 87P3 68 29 28 22 2 78P4 29 60 31 71 88 95

P10 39 22 90 73 30 91P11 11 27 74 9 77 97P12 67 84 3 14 1 3P14 32 30 94 96 88 100P5 46 39 73 80 53 87P7 16 17 50 52 73 86

P13 76 86 52 84 40 42

Table 9: % runs of H-Randm where deterministic ordering schemes require less CPU timeand the number of two ship subproblems than H-Randm

24

Problem H-Lexic H-Mtrcs H-Model

P5 43 24 36P7 100 76 86

P13 22 18 100

Table 10: % runs of H-Randm where deterministic ordering schemes generate solution at leastas good as H-Randm

more difficult. The average CPU time for solving the two ship subproblems duringH-Model is 6.6 s compared to 1.98s with H-Mtrcs.

• P3: H-Model outperforms 78% runs of H-Randm in terms of the number of two shipsubproblems solved but can only outperform 2% of runs in terms of total CPU time.Interestingly, H-Model solves the minimum number of two ship subproblems for thisproblem. However, the overhead of determining the optimal ordering worsens its per-formance in terms of total CPU time relative to H-Randm.

5.3.3 Set 2: Solution quality for deterministic selection schemes vs. H-Randm

As shown in Table 4, H-Randm has a large variance in terms of quality of solutions producedfor problems in Set 2. Since Set 2 includes only 3 problems, we cannot make strong con-clusions regarding which scheme for two-ship subproblem selection leads to better solution.Further, as shown in Table 4, none of the deterministic ordering schemes is able to generatebetter solutions than H-Randm for all problems in Set 2. However, Table 4 does show thatsolutions generated by H-Lexic and H-Model lie in the lower to middle range of objectivevalues generated by H-Randm.

Also, Table 10 shows that H-Model generates better solutions than more than 50% runsof H-Randm in 2 out of 3 problems. H-Lexic and H-Mtrcs, on the other hand, are able toachieve that for 1 problem only.

5.4 Overall Comparison

In summary, Table 11 reports the ratio of CPU time taken by CPLEX to generate a solutionat least as good as that generated by one of the proposed heuristics, relative to the CPUtime taken by that heuristic. Problems for which CPLEX could not find an equivalent orbetter solution than the heuristics are marked as “N/A”. The table illustrates that for nearlyall instances, all of the heuristic methods case are either substantially faster than CPLEX,or find solutions better than CPLEX. The table reports the geometric mean of the speed-upattained by each heuristic over CPLEX for problems where CPLEX finds a solution at leastas good as the one found by the heuristic method. Based on the geometric means, H-Modelhas a slight edge over the other methods.

We have shown that the methods developed in this paper can address large-scale in-stances. Since the problem set presented here has not been used in published literature, we

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Problem Heur-Lexic Heur-Random (Avg) Heur-Metrics Heur-LPP1 3.32 2.60 4.04 3.88P2 16.06 17.14 19.23 18.62P3 1.83 1.78 1.74 1.56P4 15.19 19.02 15.90 46.85P5 0.89 0.87 1.03 0.95P6 131.94 131.94 131.34 130.79P7 N/A N/A N/A N/AP8 68.59 68.59 67.30 62.85P9 98.72 98.72 97.57 90.74P10 N/A N/A N/A N/AP11 N/A N/A N/A N/AP12 N/A N/A N/A N/AP13 N/A N/A N/A N/AP14 N/A N/A N/A N/A

Geometric Mean 13.61 13.60 14.47 15.72

Table 11: Speed-up for each deterministic version of heuristic relative to CPLEX

cannot provide a direct comparison of the solution quality reported by our methods withthat reported by other authors. However, based on the CPU times reported above, we canconclude that the performance of the proposed methods appear to be very computationallyefficient such that much larger cases are tractable compared to those addressed by Grønhaugand Christiansen [12] and Grønhaug et al. [13] and Fodstad et al. [9, 10]. The 2SIH alsoseems to be very competitive in terms of CPU time with the rolling horizon heuristic ofRakke et al. [15], except in instances of size comparable to instances P13 and P14. TheCPU times for the decomposition technique of Halvorsen-Weare and Fagerholt [14] are com-parable to those for the methods presented here in most of their test instances of a similarscale. However, it must be noted that unlike the proposed method, their decompositionmethod does not guarantee a feasible schedule.

6 Practical Considerations

The model and algorithms described in previous sections are the core elements of an ad-vanced decision support tool. However in the course of application of this technology, someadditional considerations have been adopted for use in practice.

6.1 Inventory Smoothening

The inventory profiles at terminals are good indicators of the robustness of an optimizedship schedule. Specifically, low inventory levels at production terminals and high inventorylevels at regas terminals provide flexibility to allow the system to return to the plan without

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disrupting production and regas operation when ships get delayed. A schedule with this kindof inventory profile is considered robust.

We present an optimization model to develop a robust schedule from a solution of modelP. Given a solution to model P with objective function value z∗, constraint (18) limits thesearch to solutions where the weighted sum of lost production, stockout, and unmet demandsis within a range of g (referred to as “give-up”) units of z∗.∑

(l,t)|l∈Lwl,tol,t +

∑(r,t)|r∈R

wr,tsr,t +∑r∈R

wDr δDr ≤ z∗ + g (18)

The inventory smoothening model consists of minimizing the objective function shown inequation (19) subject to constraints (2)-(14) and (18). wIl and wIr are weight factors forinventory levels at production and regas terminals.

min∑

(l,t)|l∈LwIl Il,t −

∑(r,t)|r∈R

wIrIr,t +∑

(l,t)|l∈Lwl,tol,t +

∑(r,t)|r∈R

wr,tsr,t +∑r∈R

wDr δDr (19)

In practice, it is sufficient to find a good solution to the inventory smoothening model. Thetwo-ship improvement heuristic algorithm can be used to solve this model.

6.2 Boundary Effects

It is obvious that the inventory profiles in a solution to the LNG IRP are highly sensitive tothe initial inventory levels and the initial locations of ships. From an operational perspectiveit is perfectly reasonable to assume that initial inventory levels at terminals and startinglocations for ships are input parameters to an optimization model. However, from a sup-ply chain design analysis perspective, this operational data is not explicitly known. In thissituation, the initial inventory levels and the initial locations of ships can be included as op-timization variables in the model. The resulting solution will provide an optimistic estimateof the ability of a supply chain design to avoid losses at production and regas terminals.

7 Conclusions

A new variation of the LNG IRP is defined for the purpose of design and analysis of LNGsupply chains. Unlike previously published literature, the proposed model optimizes shipschedule decisions together with inventory management at both production and regas termi-nals. Practical features such as seasonality in production, consumption and travel time areincorporated into the model, and a new feature related to variable regas rates is introduced.

An arc-flow formulation based on the MIP model of Song and Furman [17] is proposedfor this LNG IRP. In addition to the adaptation of the 2-ship improvement heuristic schemedeveloped by Song and Furman [17], new construction and improvement heuristics are pro-posed to address this LNG IRP. Further several sophisticated 2-ship selection methods aredeveloped.

The set of realistic test instances are very large relative to those problem instances seenin recent LNG IRP literature. Extensive computational results indicate that these methods

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are computationally efficient in finding the optimal or near optimal solutions substantiallyfaster than commercial optimization software.

Future research directions could include the development of more detailed models for usein operational planning and improved methods and analytical techniques for decision-makingunder uncertainty. Continuing research in advanced modeling and algorithm is necessary toaddress the largest test instances within practical time limits.

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