A robust optimization approach for an artillery fire-scheduling problem under uncertain threat

Accepted Manuscript A robust optimization approach for an artillery fire-scheduling problem under uncertain threat Yong Baek Choi, Suk Ho Jin, Kyung Sup Kim, Byung Do Chung PII: DOI: Reference:

S0360-8352(18)30387-5 https://doi.org/10.1016/j.cie.2018.08.015 CAIE 5364

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Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

10 August 2017 9 August 2018 11 August 2018

Please cite this article as: Choi, Y.B., Jin, S.H., Kim, K.S., Chung, B.D., A robust optimization approach for an artillery fire-scheduling problem under uncertain threat, Computers & Industrial Engineering (2018), doi: https:// doi.org/10.1016/j.cie.2018.08.015

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A robust optimization approach for an artillery fire-scheduling problem under uncertain threat A robust optimization approach for an artillery fire-scheduling problem under uncertain threat Yong Baek Choi, Suk Ho Jin, Kyung Sup Kim, and Byung Do Chung * Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Korea

* Corresponding author. Tel.: +82 2 2123 3875; fax: +82 2 364 7807.

E-mail addresses: [email protected] (Y.B. Choi), [email protected] (S.H. Jin), [email protected] (K.S. Kim), [email protected] (B.D. Chung).

Abstract In this study, the determination of an artillery firing sequence is considered in order to minimize the enemy threat to friendly forces prior to the conclusion of the firing operation. A deterministic model and robust counterpart are developed to deal with the deterministic and uncertain enemy threat levels, respectively. In both cases, the optimal strategy is demonstrated to be firing based on the threat removal rate per unit time. Moreover, a two-phase approach is developed based on the concept of cardinality-constrained uncertainty. In this approach, the firing sequence is determined in the first phase and then, in the second phase, the robustness of the uncertain factors is adjusted by using the modified problem to evaluate the total threat exposure of friendly units to the enemy. A set of problems is generated and tested using the proposed model and strategy. The results of numerical

experiments demonstrate that the proposed strategy outperforms conventional fire-scheduling approaches. Additionally, the price of robustness is considered by adjusting the value of uncertain factors. Keywords: Artillery firing sequence; Robust optimization; Threat; Uncertainty;

1. Introduction War is an act conducted when a nation or a group attempts to subject some other party to its will through various coercive means such as military action. As a result of war, a great amount of damage and losses are experienced in both human terms and property value. Among the most critical types of military force used in fighting a war, artillery possesses tremendous destructive power central to achieving the objectives of a military conflict. The mission of an artillery unit is to conduct precision firing on a remote target, destroying enemy units to support the advance of the friendly force’s infantry units. Accordingly, artillery units are expected to conduct the fast and accurate firing on a large number of enemy targets. There are many types of enemy targets for artillery units, such as infantry, artillery, communications, and intelligence units, as well as command and communications facilities. Once a war begins, these various enemy units conduct their mission for the benefit of the enemy, attempting to inflict significant damage on friendly forces. As a result of the threat presented by these enemy units, artillery is often among the first resources called upon in a conflict. However, one artillery unit cannot fire multiple shots at multiple enemy targets simultaneously, and therefore, it must fire upon enemy targets in a sequence. The artillery unit must choose a sequence of firing to compliment the operation of other friendly units. The basic strategy of any war is to accomplish the objective of an operation with minimum losses to friendly forces. Therefore, it is critical to select a firing sequence that minimizes the losses of friendly military resources during its conduct. War is not fought in a completely known situation: in most cases, complete information on the characteristics of the enemy units, as well as other factors, can be uncertain. This uncertainty is the result of the desire of the belligerents to minimize the exposure of their power to the enemy, and even deceive the enemy, making it difficult for the enemy to act in certainty. Accordingly, it is critical to develop the capability to determine a sequence of firing in an uncertain situation with limited information on enemy deployment. To date, most artillery studies have focused on the allocation of multiple targets to multiple artillery units (Cetin & Esen, 2006; Cha & Kim, 2010; Kwon, Lee, & Park, 1997). There have been no studies on the sequence of firing for single or multiple artillery units, especially with respect to

minimizing the losses to friendly units by considering the degree of threat presented by enemy targets in an uncertain environment. In the real world, there are many uncertain factors. However, most studies of robust optimization problems have considered only the most important uncertain factors. This focus upon the more uncertain factors is directly related to the solution robustness issue. For example, Bertsimas and Thiele (2006) and Kim and Chung (2017) focused on demand uncertainty, an external uncertainty describing opposing judgments made by decision makers, from among various possible uncertainties including demand, environmental, and system uncertainties. In the modeling of artillery fire scheduling, the enemy threat level and threat removal rate are among the parameters that can be considered uncertain. The threat level of a target is indicated by a quantitative value determined by the type and size of the enemy target as well as a qualitative value determined by the degree of training and morale of the enemy target. On the other hand, the threat removal rates are determined solely by quantitative values based on numerous experiments and historical data. In this application, the threat level is the most difficult parameter for friendly forces to comprehensively quantify because it is predicated upon the status of the enemy forces, which is typically unknown. Therefore, in the model constructed in this study, the uncertainty of the enemy threat level has been considered. In this paper, the degree of threat presented by an enemy target is defined as the capability of the target to inflict damage on the military power of friendly forces during a given unit of time. Optimization models for determining the ideal firing sequence are developed in order to minimize the threat of the enemy to friendly units prior to the conclusion of firing operations. Additionally, this paper proposes a method for determining the firing sequence of an artillery unit in such a way that the threat to friendly units is minimized even when information on the enemy target is insufficient. The main contributions of this paper are as follows: - A deterministic optimization model is developed for determining an artillery firing sequence in order to minimize the total enemy threat to friendly units by considering the target’s threat level to be a dynamic value that can be determined from the target’s survival time, the magnitude of which decreases as the attack time against the target increases. - A robust optimization (RO) model is developed for determining an artillery firing sequence in order to minimize the total threat of the enemy under a scenario in which the enemy’s threat level is uncertain. - A two-phase method is proposed to solve the robust counterpart and evaluate the total threat exposure of friendly units when the enemy threat is uncertain. In consideration of the problem, a firing sequence is determined in a first phase and then in a second phase in which the robustness of

the uncertain factors is adjusted using the modified problem to evaluate the exposure of friendly units to the total enemy threat. This paper is organized as follows: In Section 2, we review the previous research related to the determination of firing sequences and robust optimization. In Section 3, the deterministic problem is described and an optimization model is developed. Next, we describe uncertain threat levels and develop a robust counterpart in which we propose a two-phase method to solve the problem. In Section 4, we provide results from numerical experiments, and concluding remarks are given in Section 5.

2 Literature review 2.1 Targeting and scheduling problem for field artillery The assignment and scheduling problem as related to military operations can be classified into artillery firing, aerial transport of troops and materials (Kim & Kim, 2011), radar tasks (Taner, Karasan, & Yavuzturk, 2012), and military UAV mission (Semiz, 2015; Alotaibi, Rosenberger, Mattingly, Punugu & Visoldilokpun, 2018) types. This study is related to the artillery firing assignment and scheduling problem type. The problems related to artillery firing can be classified into two types: the first type is the weapon target assignment problem (WTAP), which is concerned with the assignment of multiple enemy targets to several available weapons systems, and the second type is the fire scheduling problem (FSP), which is concerned with the firing schedule or sequence of firing on the assigned targets. The WTAP has been studied by many researchers. Manne (1958) studied the problem of weapons system and target assignment for the first time, suggesting that although the target assignment problem is probabilistic and non-linear, it can be approximated with a linear model. Lloyd and Witsenhausen (1986) showed that the weapons system target allocation problem is NP-complete and that the computation time for the optimal algorithm increases exponentially as the problem size increases. Lee et al. (2002), and Erdem and Ozdemirel (2003) solved the WTAP with a modified evolution algorithm. Lee and Lee (2003) solved the WTAP with a hybrid algorithm using ant colony optimization (ACO) and a genetic algorithm to enhance the searching function. Ahuja et al. (2007) modified the WTAP into integer programming and network flow problems. They developed a lower bounding scheme and suggested an exact algorithm and a heuristic algorithm, which together help to determine the optimal solution. Unlike the WTAP, the FSP has seldom been previously studied. The FSP was first introduced by Kwon et al. (1997), who modified the WTAP with a non-linear constraint into a knapsack problem

using Lagrangian relaxation and proposed a greedy heuristic to minimize the makespan of the FSP. Cha and Kim (2010) determined that the probability of target destruction by an artillery unit would decrease as time passes and developed a heuristic with the purpose of minimizing the total threat of the surviving targets after the end of the artillery firing. The degree of threat of the target was considered as a constant value in this model. However, as the friendly unit continues to attempt to destroy a target, the target will eventually be destroyed, subsequently decreasing the threat level of the target. Therefore, it is reasonable to consider the threat level of the target as a dynamic value rather than a constant. In this study, the threat level of the target is treated as an uncertain value, the magnitude of which decreases as attack time increases. In this study, the model for determining the firing sequence that minimizes the total threat of enemy units can be considered as a job scheduling problem for a single machine with common due dates related to earliness and tardiness penalties. In this model, the artillery unit is the machine, the enemy target that the artillery unit must destroy is the job or task, and the time required for the artillery unit to destroy an enemy target can be considered the process time. If the common due date for the destruction of the enemy target is set as the war start time, all tasks are immediately overdue, and thus the problem becomes a special situation in which tardiness exists for all tasks. At the beginning of a war, friendly forces are immediately exposed to threats from enemy targets, and these threats can be considered to be the tardiness penalty. If every tardiness penalty is assigned the same value, the single machine scheduling problem that minimizes the total tardiness penalty is equivalent to a flowtime problem. In this case, the shortest processing time (SPT) schedule is determined to minimize the total tardiness penalty(French, 1982). However, every target threat is not of the same value. In this study, deterministic and robust optimization scheduling models are proposed that minimize the total threat of the enemy units. A job scheduling problem for a single machine can be classified into categories with and without uncertainty. In the absence of uncertainty, extant job scheduling problems for a single machine with common due dates, accounting for earliness and tardiness penalties, are known as NP-hard problems (Hall & Posner, 1991). Ying (2008), Feldmann and Biskup (2003), Hino et al. (2005), and Lin et al. (2007) have all proposed heuristic approaches to solve this problem considering processing time, earliness, and tardiness penalties as certain parameters. In the presence of uncertainty, stochastic, fuzzy, and robust models have been studied for single machine scheduling problems. Soroush (1999) studied a stochastic single machine problem that proposed two efficient heuristics to determine duedates and candidates for an optimal sequence when the processing time is a normally distributed random variable. Baker (2014) developed a branch and bound algorithm to find optimal solutions to a single-machine stochastic scheduling problem with earliness and tardiness costs, assuming that the processing times follow normal distributions. Wang et al. (2002) studied a ready time scheduling

problem with fuzzy job processing times. They assumed the fuzzy processing time to be a continuous triangular fuzzy number and then, by defining the job completion likelihood profile, converted from the fuzzy constraint equations into a crisp mathematical representation. Toksan and Ank (2017) considered single machine scheduling problems using fuzzy model parameters. In order to model the uncertainty of fuzzy parameters such as processing time and learning effect, they used a likelihood profile that depended on the possibility and necessity measures of fuzzy parameters. Umang et al. (2014) studied heuristics based on a variable neighborhood search and iterated a local search to obtain a robust solution for a single machine scheduling problem with uncertainty in release times and processing times. Drwal (2018) developed a new mix integer linear program and proposed a polynomial time algorithm to minimize the weighted number of late jobs under interval due-date uncertainty. In many studies related to stochastic, fuzzy, and robust models for the single machine scheduling problem, processing time, release time, and due-date have been assumed as uncertain parameters. In this study, we propose a robust optimization approach that considers the target threat level, equivalent to a tardiness penalty, as an uncertain parameter.

2.2 Robust Optimization Robust optimization (RO) was first studied by Soyster (1973), who suggested an inexact linear programming model in which the solution feasibility is guaranteed whatever value the input data has within the pre-determined uncertainty set. The solution produced by this approach, however, is too conservative. In the last fifteen years, the inexact linear programming model has been further developed and applied by many researchers in an attempt to find a less conservative solution (Ben-Tal, El Ghaoui, & Nemirovski, 2009; Bertsimas, Brown, & Caramanis, 2011). For example, Ben-Tal and Nemirovski (1999) demonstrated a robust counterpart of the linear programming problem using an ellipsoidal uncertainty set that can be reformulated as a computationally tractable second-order conic programming model. Later, Bertsimas and Sim (2003) developed a methodology that can be directly applied to discrete optimization and can control the robustness of the solution. Bertsimas and Sim (2004) showed that the robust counterpart of the linear programming model with an uncertain data set can be reformulated into a linear programming model using the concept of cardinality-constrained uncertainty. The greatest advantage of RO is that it does not require the probability distribution or an exact scenario to consider the uncertain data. Accordingly, RO has gained credibility as an important optimization approach when data collection is limited and the prediction of the scenario is difficult. Robust optimization has been studied and developed in many research areas including vehicle routing (Wu, Hifi, & Bederina, 2017; Gounaris, Wiesemann, & Floudas, 2013), supply chain management

(Ivanov, Dolgui, & Sokolov, 2016; Pishvaee, Rabbani, & Torabi, 2011), location and allocation (Baohua & Shiwei, 2009; De Rosa, Hartmann, Gebhard, & Wollenweber, 2014), and portfolio optimization (Quaranta & Zaffaroni, 2008; Kim, Kim, & Fabozzi, 2014). However, RO has not been considered in the domain of military operations, in which all situations and conditions are uncertain. To the best of our knowledge, RO has not been studied and applied to the problem of fire scheduling. In this study, we develop an RO model to determine an artillery firing sequence in consideration of the uncertainty of the enemy threat. Moreover, based on the properties of the fire scheduling problem, a traditional robust counterpart, as developed by Bertsimas and Sim (2004), is converted into a new approach based on a two-phase method that determines a robust optimal solution in the first phase and a robust objective value that considers the cardinality-constrained uncertainty in the second phase.

3. Models and methods The mission of an artillery unit is to suppress and destroy enemy capabilities by indirectly firing on remote targets. An artillery unit is supposed to fire on an enemy target as requested by forward field units or as instructed by the commanding authorities. In wartime, the number of requested targets can number in the dozens. If an artillery unit has sufficient assets to fire upon targets as they are requested, the unit need not determine a firing sequence. However, if the number of targets is greater than the number of units can fire upon them, the problem of determining a firing sequence becomes critical: one artillery unit cannot fire at multiple enemy targets simultaneously, but instead must fire on each enemy target sequentially. In a sequential firing sequence, the undestroyed enemy targets are capable of conducting operations to inflict losses or damage upon friendly military power in many ways depending on the degree of the enemy threat. Because there are typically various types and sizes of enemy targets that require different amounts of time to destroy, the total threat posed by the enemy targets can change depending on the selected firing sequence. Accordingly, the firing sequence should be determined in such a way that the enemy’s total threat to friendly units is minimized. In this paper, we have made the following assumptions: (1) The subject artillery unit can only fire at one enemy target at a time, and only moves to the next target once the current target is destroyed. (2) The initial threat level of a target and the threat removal rate per unit time resulting from attacking the target is determined by the type and size of the target. (3) An enemy target inflicts damage on friendly units at the same level until it is attacked. (4) The time required for the artillery unit to destroy the enemy target is determined by the target’s

threat level and the removal rate of the threat. (5) The threat level of enemy targets under artillery attack decreases linearly. (6) A target is destroyed when the threat level of the target becomes zero. The notations used in this paper are defined as follows: Notation

Definition

Decision variables Firing start time of target j ′

If target j has precedence over target j’ 1, otherwise 0

Parameters Target Nominal threat level of target j Uncertain threat level of target j Maximum threat level of target j The difference between

and

,

The threat removal rate of target j per unit time Amount of time required to destroy target j, Firing start time of nth target A large positive value

3.1 Deterministic model Suppose that there are n targets to be fired in the current operation. The enemy targets are varied in terms of type and size, and the artillery unit is well aware of the characteristics of all enemy targets and their respective threat levels. At the beginning of the current operation, all enemy targets are capable of inflicting damage on the military power of friendly forces. The deterministic model for determining the firing sequence in order to minimize the total threat of enemy units is:

(DET)

(1)

subject to (2)

(3) (4) (5)

The objective function in Eq. (1) minimizes the total threat to friendly units until all enemy targets are destroyed. In this problem, the threat at any given time is calculated as a function of the initial threat level and the elapsed time since the operation began. The cumulative threat of target j before the firing begins is given by

. The elapsed time from the start of firing to the destruction of target j is

obtained by dividing the threat level ( ) of target j by the threat removal rate per unit time

. The

initial threat level of target j linearly decreases to 0 after the duration required for its destruction. Accordingly, the cumulative threat to friendly units of the enemy target j between the commencement of firing and the destruction of the target is

. Equations (2), (3), and (4) determine the sequence of

firing at the targets and the firing start time. Equation (4) illustrates that the decision variable binary. When target

is fired upon prior to target ′,

′

′

is

is 1, and Eq. (2) sets the firing start time

for target ′ after the destruction of target , then Eq. (2) is enforced; otherwise,

′

is 0 and Eq. (2)

is discounted and Eq. (3) is enforced. However, a large positive value for M makes it true for any target

and target ′. In Eq. (5),

must be a nonnegative variable because it denotes the start time

of firing upon target . Theorem 1 states that the firing sequence required to minimize the total enemy threat is determined by the threat removal rate

.

[Theorem 1] When an artillery unit determines the firing sequence for an enemy target, the total enemy threat to friendly units is minimized when the sequence is conducted in the order determined by the threat

removal rate per unit time. [Proof] 『Let T be the total enemy threat to friendly units in time t. Suppose that there are two targets j and j’ which should be fired upon in the required firing sequence within the time t. Here, the threat removal rate per unit time is The total enemy threat to friendly units when target j is fired upon prior to target j’ is as follows in Fig. 1 and Eq. (6):

Fig. 1. The total enemy threat to friendly units when target j is fired upon prior to target j’

(6)

The total enemy threat to friendly units when the target j’ is fired upon prior to target j is as follows in Fig. 2 and Eq. (7):

Fig. 2. The total enemy threat to friendly units when target j’ is fired upon prior to target j

(7)

As the threat removal rate is defined as

or

, Eq. (6) – Eq. (7) =

. Accordingly, the total enemy threat can be observed to decrease more quickly when target j is fired upon prior to target j’, demonstrating that the total enemy threat to friendly units is minimized when firing is first directed at the target with the larger threat removal rate per unit time 』

3.2 Robust optimization model In a war, information about enemy forces contains many uncertain factors. Chiefly, each participant

in the war attempts to deceive their opponents with respect to their own military power, making deterministic evaluation of threat level difficult. Additionally, topographical, climactic, and information gathering errors can also result in uncertainty when assessing the threat level of a target. When information on a target is uncertain, any prediction of military losses occurring to friendly forces by the end of a firing operation will be uncertain. The prediction of the total loss of military power of friendly forces is very critical because the planning of subsequent operations is largely determined by the remaining level of military power among friendly forces. To address this issue, in this paper we assume that the uncertain threat level uncertainty set

of an enemy target belongs to a crude

, resulting in the following optimization model considering the uncertainty set:

(RO)

(8)

subject to (9)

(10)

(11) (12) (13)

The RO model is a semi-infinite problem and has infinitely many constraints. It cannot be directly solved, and thus requires a tractable robust counterpart formulation with a finite number of constraints. 3.2.1 Robust counterpart with box uncertainty First, the uncertain threat of the enemy target can be assumed to be a box set as follows.

(14)

In Eq. (14),

and

are the nominal value and maximum value, respectively, of the threat of

enemy target j. If the uncertain threat of the target is defined as above, the RO model can be reformulated as a deterministic problem for the maximum threat level. In the left side of Eq. (9), the

uncertain threat of target j, the starting time of firing on target j, and the threat removal rate for target j per unit time are all non-negative variables. Additionally, when the uncertain threat of target j increases, the left side of the equation can be seen to act as an increasing quadratic function, that is, . Then,

,

if and only if

(Bertsimas, Brown, & Caramains, 2011). Similarly, the left side of Eqs. (10) and (11) can be reformulated, resulting in a tractable robust counterpart of RO, RC1.

(RC1)

(15)

subject to (16)

′

′ ′ ′

′ ′

′

′

′

(17)

′

′

(18)

′

(19) (20)

The equations above provide a solution that can be realized for whatever values are present in the pre-determined uncertainty box set. This is a worst-case analysis, which may be very unlikely, and thus, this solution may be excessively conservative. 3.2.2 Robust counterpart with cardinality-constrained uncertainty The objective value of the robust counterpart with a given box uncertainty set occurs when all conditions are concurrently their most critical, which is unlikely in reality. To remedy this issue, a robust counterpart based on cardinality-constrained uncertainty is applied to the problem. This approach, proposed by Bertsimas and Sim (2003) and Bertsimas and Sim (2004), has been successfully applied to many areas of mathematical modelling, overcoming the conservatism of the robust optimal solution by adjusting the robustness with a parameter

in each constraint, allowing

up to   of the uncertainty coefficient to be altered. We arrive at the following RC2 model by employing the cardinality-constrained uncertainty set:

(RC 2)

(21)

subject to (22)

where

＼

′

′

′

′

′

′

(23)

′

′

(24)

′

′

(25) (26)

The above approach attempts to overcome the conservatism inherent in the worst case approach by introducing the parameter

to constraints with uncertain elements. However, Eqs. (10) and (11) in

the RO model have only one uncertain element in each constraint, and so it is not possible to adjust the robustness by defining a value of

that is greater than one. Therefore, Eqs. (23) and (24) in RC2

still select the worst-case parameter, resulting, once again, in a conservative model. With this restriction, a decision maker cannot take advantage of

, requiring a different reformulation approach

(Bertsimas, Nasrabadi, & Stiller, 2013; Solyalı, Cordeau, & Laporte, 2015). Based on Theorem 2, a new approach can be developed and can provide the same results as the methodology of Bertsimas and Sim (2003) and Bertsimas and Sim (2004). [Theorem 2] Even when the degree of threat from the enemy target is uncertain, the firing sequence is determined in the order of the threat removal rate per unit time. [Proof] 『Let T be the total threat of military power to friendly forces up to a time t. There are two different targets, j and j’, for whom the firing sequence is to be determined. The threat level of target j is uncertain, with an interval of uncertainty of

. Because the threat level of the target is uncertain, the

time required to destroy target j is also uncertain with the value of of threat removal per unit time is

′

. We will assume that the rate

.

If target j is attacked prior to target j’, the total enemy threat to friendly forces is:

′

′

′

′

′

′

′

′ ′

′

(27) ′

If target j’ is attacked prior to target j, the total enemy threat to friendly forces is:

′

′

′

′

′

′ ′

′

′

′

′

′

′

Note that

(28)

′

′

′

since

′

and

.

′

Therefore, even though the information on target j is uncertain, the optimal firing sequence determined using the deterministic model is the same. This method can also be proven to successfully produce a firing sequence even if the information on target j’ is the more uncertain of the two targets j and j’ where (

′

, or if the information on both targets j and j’ is uncertain.』

Next, we propose a two-phase method for determining a robust optimal solution, and calculate a robust objective value using a cardinality-constrained uncertainty set. In the first phase, based on Theorem 2, we can easily determine the robust optimal firing sequence for the RC2 model under an uncertain threat level in order to minimize the total enemy threat to friendly forces. In the second phase, we solve a new optimization model to calculate the robust objective function value by adjusting the robustness using the parameter

. The robust objective function value obtained in the

second phase is the best worst-case value that friendly forces can face for the given cardinalityconstrained uncertainty set.

The optimization model used in the second phase is developed as follows. In the RC2 model, Eqs. (23) and (24) are constraints for finding the optimal firing sequence. However, we have already obtained an optimal sequence from Phase 1 based on Theorem 2. Therefore, the value of

′

is given and the

constraint Eqs. (23) and (24) can be reformulated. With the firing sequence determined, we can find the relationship between the firing start time and the inequality Eqs. (23) and (24), which can be reformulated as the equality Eq. (29).

(29)

Equation (30) shows the generalized form of Eq. (29). ′ ′

′

′ ′ ′

The decision variable

′

′

′

(30)

′

in the objective function of the RC2 model is replaced by Eq. (30), and Eqs.

(32) and (33) are added to control the robustness. The new model for the second phase is then developed as follows:

(RC3)

(31)

subject to (32) (33)

In Eq. (31), the robustness can be adjusted by the parameter determined depending on the value of

and the objective function value is

being a real number between 0 and n. Note that when

RC3 provides the objective value of the deterministic problem, and

is 0,

provides the same result

as the objective value of RC1 with the box uncertainty set.

4. Numerical experiments In this section, we solve the models developed in this paper and compare their performance. First, the deterministic solution is compared with other solutions such as the “largest threat first” and “largest firing time first” rules. Next, we consider the threat level uncertainty and compare the solutions from RC1 and RC3 for various values of Γ. In order to select parameter values, random numbers were generated as specified in Kwon et al. (1996) and Cha and Kim (2010). For example, the threat level and removal rate

were randomly selected from uniform distributions with ranges of [3,10] and

[1,5], respectively. The numerical experiments were conducted using the GAMS and CPLEX software packages on an Intel(R) Core i5-6600 3.3 GHz CPU computer with 8 GB of RAM.

4.1 Results of the deterministic model When the information on the enemy is certain, the total enemy threat to friendly forces is evaluated depending on the strategy for determining the sequence of firing on the enemy targets. A comparison will be made between strategies for determining the firing sequence. Largest of firing first on the target having the biggest threat, Largest the target requiring more time to destroy, Smallest requiring less time to destroy, and Largest

(LF) is the strategy

(LP) is the strategy of firing first on

(SP) is the strategy of firing first on the target

(LR) is the strategy derived from Theorem 1 in Section

3.1. The number of targets was set to 5, 10, 15, and 20. The numerical experiments were repeated 100 times for each case. To analyze the performance of each strategy in determining the sequence of firing on the enemy targets, the percentage differences between the objective values are determined using Eq. (34). (34)

Table 1. Objective values of and percentage difference between firing sequence decision strategies Objective value n

LR (DET)

LF

LP

Percentage difference SP

5

174.0

216.8

254.0

180.9

24.6

45.9

3.9

10

672.9

889.5

1069.8

704.2

32.2

59.0

4.6

15

1426.5

1866.4

2227.3

1497.4

30.8

56.1

5.0

20

2574.4

3388.3

4095.0

2701.0

31.6

59.1

4.9

Fig. 3. Comparison of objective values of firing sequence decision strategies

Fig. 4. Percentage difference in the objective values of firing sequence decision strategies

Table 1 compares the performance of each decision method. The LR strategy minimizes the total enemy threat to friendly units regardless of the number of enemy targets. The objective values for determining the firing sequence increase from LR to SP to LF to LP. The objective value difference between LR and SP is small, while that between LF and LP is relatively larger. In the proposed model, if every target threat level is of the same value, the artillery fire scheduling problem is equivalent to a flow time problem. In this case, because the SPT schedule minimizes the objective value, SP performs better than LF and LP. When the number of targets increases, the total enemy threat to friendly forces increases nonlinearly, regardless of the method used to determine the firing sequence: as the number of targets increases, the increase in threat gets bigger faster. This relationship suggests that care should be taken not to assign too many targets to a single artillery unit when allocating enemy targets. There was no significant difference in computation time according to the strategy used to determine the sequence of firing on the enemy targets; in every case the computation time was less than a few seconds.

4.2 Results of the robust counterpart model With uncertain information on the enemy, the firing sequence required to minimize the total enemy threat to friendly forces is found and the change in the objective function value is observed depending on Γ from RC3. In this experiment, the uncertain parameter , where

, and

belongs to the uncertainty set

is the uncertainty level selected from the set

{0.2, 0.4, 0.6, 0.8, 1.0}. Additionally, we solved RC1 with the box uncertainty set. In this test, n is 15 and

is between 0.2 and 1.0.

Table 2. Objective values for the robust counterpart firing sequence decision strategy for different values of θ and Γ

Γ

Θ

0

1

2

3

4

5

6

7

8

9

10

0.2

1426.5

1493.2

1554.4

1610.9

1663.5

1713.1

1759.5

1802.4

1842.5

1880.6

1916.1

0.4

1426.5

1561.8

1688.7

1807.6

1920.1

2027.4

2128.6

2223.2

2312.5

2397.8

2478.2

0.6

1426.5

1632.5

1829.4

2016.9

2196.5

2369.4

2533.9

2688.9

2836.4

2978.4

3112.9

0.8

1426.5

1705.1

1976.7

2238.6

2492.6

2739.1

2975.4

3199.7

3414.4

3622.3

3820.2

1.0

1426.5

1779.8

2130.5

2473.0

2808.4

3136.6

3453.2

3755.6

4046.5

4329.6

4600.0

11

12

14

15

RC 1(n=15)

0.2

1949.3

1979.9

2008.0

2033.2

2054.2

2054.2

0.4

2553.8

2624.0

2688.7

2747.0

2796.0

2796.0

0.6

3240.1

3358.8

3468.6

3568.0

3652.0

3652.0

0.8

4008.2

4184.2

4347.7

4496.3

4622.0

4622.0

1.0

4858.1

5100.4

5326.2

5531.8

5706.2

5706.2

Γ

Θ

Fig. 5. Comparison of objective values for the robust counterpart firing sequence decision strategy for each given theta value

Fig. 6. Comparison in the change of objective values for robust counterpart firing sequence decision strategy for each given theta value

Table 2 shows that when the uncertainty degree

is 0, the objective value is 1426.5 regardless of

the uncertainty level θ, which is the same as the objective value provided by the DET model (the LR strategy) when n = 15 in Table 1. This means that even though value in the uncertain set

, which is the threat of target j, has a

, the objective value has the value provided by

the deterministic model when

is 0. Also, we can observe that when

is n, the objective value is

the same as that provided by RC1 with the box uncertainty set. By adjusting the value of

from n to

0, we can predict the best worst case with the robust optimal solution and find the price of robustness (Bertsimas & Sim, 2004). The computation time was less than a few seconds. Figure 5 shows that when the threat presented by the target is uncertain, the objective values increase as the uncertainty degree uncertainty degree degree

increases, but at a progressively slower rate. This means that when the

is small, the change in the objective value is large, while when the uncertainly

is large, the change in the objective value is small, as shown in Fig. 6. As for the effect of

the uncertainty level θ on the objective value, as the uncertainty level

increases, the difference

between the objective values with respect to θ increases as shown in Fig. 5. For example, when θ = 0.2 and

, the objective value is 1.44 times that for θ = 0.2 and

value in the case of θ = 1.0 and

is 4.00 times that for θ = 0.2 and

, while the objective , indicating that as

the information on the threat of the enemy target becomes more accurate, the prediction of the total enemy threat to friendly forces becomes more accurate. Therefore, it is best to expend the effort to minimize the uncertainty level of the information on the enemy.

4.3 Results of simulation experiments In this section, we compare the LR strategy from Theorem 2 with the LF, LP, and SP strategies under conditions of threat level uncertainty. The simulation experiments were conducted for 5, 10, 15, and 20 targets. The degree of robustness,

, was set to 3, 5, 7, and 10 for each case. Similar to Section

4.2, we employed the uncertainty set

, where

selected from the set of {0.2, 0.4, 0.6, 0.8, 1.0}. For each value of n,

and

is

, and θ, 100 scenarios were

randomly generated, and 30 simulations were conducted for each scenario to test the firing sequence with the selected uncertainty, resulting in 3000 tests in total. Table 3. Objective value and percentage difference between firing sequence decision strategy simulations Objective value θ

n

5

10

15

20

3

5

7

10

Mean

Percentage

SD

difference

Max

LR

LF

LP

SP

LR

LF

LP

SP

LR

LF

LP

SP

0.2

201.7

251.1

294.1

209.6

7.0

8.6

10.1

7.2

215.8

268.3

314.5

223.8

24.5

45.8

3.9

0.4

231.6

288.1

337.3

240.6

15.1

18.3

21.6

15.5

262.2

325.3

381.4

271.5

24.4

45.6

3.9

0.6

263.8

327.8

383.6

273.9

24.1

29.3

34.5

24.8

313.3

387.7

454.7

323.8

24.3

45.4

3.8

0.8

298.1

370.1

432.8

309.4

34.2

41.5

48.8

35.2

369.0

455.6

534.3

380.8

24.1

45.2

3.8

1.0

334.8

415.1

485.1

347.3

45.3

54.9

64.6

46.6

429.4

528.9

620.3

442.6

24.0

44.9

3.7

0.2

767.6

1013.7

1219.5

801.1

16.0

21.8

26.2

16.5

799.9

1058.7

1273.3

834.8

32.1

58.9

4.4

0.4

868.8

1146.1

1379.2

906.8

34.0

46.4

55.7

35.0

938.3

1242.3

1494.0

979.2

31.9

58.7

4.4

0.6

976.6

1286.8

1548.6

1019.5

54.2

73.6

88.5

55.8

1088.0

1440.3

1732.0

1135.5

31.8

58.6

4.4

0.8

1090.9

1435.8

1727.9

1138.9

76.4

103.6

124.4

78.7

1249.1

1652.8

1987.3

1303.7

31.6

58.4

4.4

1.0

1211.9

1593.1

1917.1

1265.3

100.7

136.2

163.7

103.7

1421.6

1879.7

2259.9

1483.7

31.5

58.2

4.4

0.2

1617.4

2116.9

2525.9

1697.1

26.1

34.8

41.1

26.8

1671.6

2189.0

2613.9

1752.2

30.9

56.2

4.9

0.4

1820.7

2383.4

2843.3

1909.7

55.4

73.7

87.1

57.0

1936.7

2536.8

3030.9

2027.6

30.9

56.2

4.9

0.6

2036.5

2665.8

3179.6

2135.3

87.9

116.7

138.0

90.5

2221.7

2909.9

3478.1

2323.6

30.9

56.1

4.9

0.8

2264.7

2964.2

3534.6

2373.7

123.7

164.0

193.9

127.3

2526.7

3308.3

3955.6

2640.2

30.9

56.1

4.8

1.0

2505.3

3278.5

3908.3

2625.2

162.8

215.3

254.6

167.5

2851.6

3732.0

4463.4

2977.4

30.9

56.0

4.8

0.2

2955.8

3889.7

4702.1

3101.2

43.5

60.9

73.0

44.6

3042.7

4016.5

4853.7

3192.1

31.6

59.1

4.9

0.4

3364.3

4426.1

5351.3

3529.7

92.8

129.8

155.5

95.2

3551.0

4697.4

5676.0

3725.1

31.6

59.1

4.9

0.6

3799.8

4997.4

6042.5

3986.5

148.1

206.6

247.6

151.9

4099.2

5431.0

6561.9

4299.8

31.5

59.0

4.9

0.8

4262.3

5603.5

6775.7

4471.6

209.2

291.4

349.3

214.6

4687.4

6217.4

7511.5

4916.4

31.5

59.0

4.9

1.0

4751.9

6244.6

7551.0

4985.1

276.2

384.1

460.5

283.3

5315.6

7056.5

8524.6

5574.8

31.4

58.9

4.9

As shown in Table 3, under the uncertain threat of the enemy target, the objective values for determining the firing sequence increase from LR to SP to LF to LP in the same manner as when the

threat of the enemy target is deterministic, indicating that when the threat of the enemy target is uncertain, the best method for determining the firing sequence to minimize the total enemy threat to friendly forces is the LR method, which prioritizes the target providing the largest removal rate of enemy threat per unit time. Among the alternative strategies, the performance of SP was the best. The percentage difference between the proposed method and the alternative strategies LP and LF ranges from 24.0% to 59.1%, but the difference between the proposed method and the alternative SP ranges from 3.7% to 4.9%. Of particular note, it can be observed that the mean objective values from the LF and LP methods were bigger than the maximum objective value from the LR method. The robust objective function value obtained from the RC3 method is the guaranteed worst-case value. For example, when n = 15, Γ = 7, and θ = 0.2, the robust objective function value in Table 2 is 1802.4 and the realized maximum objective function value from the simulation experiments in Table 3 is 1671.6, indicating that the objective value provided by RO modeling addresses the worst possible situation at all times. More importantly, in some cases, the average performance of the LF and LP methods in Table 3 are even worse than the guaranteed worst-case performance of the RC method. For example, when n = 15, Γ = 7, and θ = 0.2, the average performances of the LF and LR methods are 2116.9 and 2525.9, respectively. The LR method can also be observed to have the smallest standard deviation in terms of realized objective value of all the methods evaluated, demonstrating that when predicting the total enemy threat to friendly forces prior to the completion of the firing sequence with uncertain information on the enemy, the LR method provides the most consistent value. There was no significant difference in computation time according to the strategy used to determine the sequence of firing on the enemy targets; in every case the computation time was less than a few seconds.

5. Conclusions The situation along the front of a war is usually uncertain, particularly in the case of information on the enemy, which is subject to misleading efforts and is thus often hard to firmly evaluate and predict. Accordingly, under the uncertain situation of war, there is a need for a method to robustly evaluate the potential loss of friendly military power during operations. The robust optimization model for determining an artillery firing sequence proposed in this paper provides a best worst-case solution that works well even under the worst possible scenario or when information on the enemy is uncertain, and robustly predicts the total enemy threat to friendly forces during firing operations, thus preserving the military power of friendly forces as much as possible. This paper proposes a method for determining an artillery firing sequence that can minimize the total enemy threat to friendly forces over the time span of the artillery firing operation. In evaluating potential methods, we considered both deterministic and uncertain cases. In a case with an uncertain

threat level, the constraints in the robust counterpart have one uncertain parameter, meaning that the cardinality-constrained uncertainty set proposed by Bertsimas and Sim (2003) and Bertsimas and Sim (2004) cannot be effectively applied. The robust counterpart then becomes a robust counterpart with box uncertainty, and as a result may be too conservative. Therefore, we developed a two-phase approach reliant on the properties of the fire scheduling problem and the concept of cardinalityconstrained uncertainty. It was demonstrated that the method of Largest

(LR), placing higher priority on the target with the

higher removal rate, is the optimal solution from Theorem 1 for the deterministic case, and from Theorem 2 for the uncertain case. Using numerical experiments, we first demonstrated that the LR method outperformed other strategies such as “largest threat level first” and “largest firing time first” in both deterministic and uncertain cases. Additionally, we demonstrated that the two-phase approach can be used to reveal the change in the robust objective function value. From the experiments, we determined the price of robustness by adjusting the value of

, an approach that can be used even if

each constraint has only a single uncertain parameter. The robust objective function value is critical when assessing the military power of friendly forces once the artillery operation has concluded because a single artillery operation is a single event at a specific time among many events occurring over the span of a war, and the result of a current artillery operation on both friendly and enemy forces has a significant impact on the magnitude and character of subsequent operations. The method proposed in this paper is a practical and effective methodology that can be used by military decision makers to plan the various operations and effectively take actions in any unforeseen emergency. In future work, a method for determining the firing scheduling of artillery, taking into account the setup time required when switching fire to the next target, should be considered. This type of problem could be solved using meta-heuristic methods, such as genetic algorithms, ant colony algorithms, or the tabu search with the integration of the robust optimization approach (Bertsimas, Nohadani, & Teo, 2007; Chung, Yao, Friesz, & Liu, 2012). Additionally, a method for determining the firing scheduling of artillery that simultaneously takes into account the enemy threat and the threat removal rate as uncertain parameters should be considered, making the model more realistic.

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Fig. 1. The total enemy threat to friendly units when target j is fired upon prior to target j’

Fig. 2. The total enemy threat to friendly units when target j’ is fired upon prior to target j

Objective Values

5000 4000 3000

Largest LR

2000

Largest LF Largest LP

1000

Smallest SP

0 5

10

15

20

Target size (n)

Fig. 3. Comparison of objective values of firing sequence decision strategies

Percentage difference

70 60 50 40

𝛿_LF

30

𝛿_LP

20

𝛿_SP

10 0 5

10

15

20

Target size (n)

Fig. 4. Percentage difference in the objective values of firing sequence decision strategies

6000.0 Objective values

5000.0 4000.0

𝜃=0.2

3000.0

𝜃=0.4

2000.0

𝜃=0.6

1000.0

𝜃=0.8 𝜃=1.0

0.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fig. 5. Comparison of objective values for the robust counterpart firing sequence decision strategy for each given theta value

Changes in Objective values

400 350 300 250

𝜃=0.2

200

𝜃=0.4

150

𝜃=0.6

100

𝜃=0.8

50

𝜃=1.0

0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

changes

Fig. 6. Comparison in the change of objective values for robust counterpart firing sequence decision strategy for each given theta value

Highlights. - The threat of an enemy target in a firing sequencing problem is considered.

- A deterministic optimization firing model to minimize enemy threat is derived. - A robust optimization firing model under an uncertain enemy threat is derived. - A strategy is proposed to minimize deterministic and uncertain objective values. - A method is proposed to obtain a less conservative solution in uncertain cases.