A Two-Stage Stochastic Mixed-Integer Programming Approach to the SmartHouse Scheduling Problem

SHUNSUKE OZOE,1 YOICHI TANAKA,2 and MASAO FUKUSHIMA11Kyoto University, Japan

2Toho Gas Co., Ltd., Japan

SUMMARY

A “smart house” is a highly energy-optimized houseequipped with photovoltaic (PV) systems, electric batterysystems, fuel cell (FC) cogeneration systems, electric vehi-cles (EVs), and so on. Smart houses are attracting muchattention recently because of their enhanced ability to saveenergy by making full use of renewable energy and byachieving power grid stability despite an increased powerdraw for installed PV systems. Yet running a smart house’spower system, with its multiple power sources and powerstorages, is no simple task. In this paper, we consider theproblem of power scheduling for a smart house with a PVsystem, an FC cogeneration system, and an EV. We formu-late the problem as a mixed-integer programming problem,and then extend it to a stochastic programming probleminvolving recourse costs to cope with uncertain electricitydemand, heat demand, and PV power generation. Using ourmethod, we seek to achieve the optimal power schedulerunning at the minimum expected operation cost. We pre-sent some results of numerical experiments with data onreal-life demands and PV power generation to show theeffectiveness of our method. © 2013 Wiley Periodicals, Inc.Electr Eng Jpn, 186(4): 48–58, 2014; Published online inWiley Online Library (wileyonlinelibrary.com). DOI10.1002/eej.22336

Key words: smart house; optimal scheduling; sto-chastic programming; recourse cost.

1. Introduction

A smart house is a house that is equipped with aphotovoltaic (PV) system, an electric battery, a fuel cell(FC) cogeneration system, and other electric appliances.Smart houses are currently attracting much attention since

they are expected to achieve reduction of residential CO2

emission and improve the quality of life by efficient energymanagement of the whole house. Various trials for smarthouses have been carried out in Japan [1].

The optimal control of energy devices such as a PVsystem, a battery, and an FC cogeneration system is one ofthe technical challenges of the smart house. The optimalcontrol of energy devices includes subsecond power controland long-term operation planning for over 24 hours. Thispaper deals in particular with the latter operation planning.To the best of the authors’ knowledge, such attempts haveyet to be reported in the literature.

Since the energy system of a smart house can be seenas a kind of distributed power system, it is natural to employand extend a scheduling method for the cogeneration sys-tem, which is also regarded as a kind of distributed powersystem.

Studies on scheduling problems of cogeneration sys-tems have been conducted for more than 20 years [2]. Manyof these studies employ a mathematical programming ap-proach to minimize the total operation cost during a givenscheduling period under some constraints regarding electricequipment and energy balances. We will formulate thescheduling problem of a smart house, which is equippedwith a PV system, a battery, an FC cogeneration system,and an electric vehicle (EV), as a mixed-integer program-ming model, and seek the operation schedule that mini-mizes the total running cost.

Most of the previous studies on cogeneration systemscheduling problems assume that demands for electricityand heat are deterministic even though these are uncertainin reality. To overcome these difficulties, Tanaka andFukushima [3] proposed a model based on stochastic pro-gramming [4] that treats demands for electricity and heatas random variables. Stochastic programming has also beenapplied to unit commitment problems under uncertainty[5–8].

© 2013 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 186, No. 4, 2014Translated from Denki Gakkai Ronbunshi, Vol. 131-B, No. 11, November 2011, pp. 885–895

48

In this paper, we formulate a two-stage stochasticprogramming [9, 10] model of a smart house, in which solarpower and demands for electricity and heat are treated asrandom variables. We also report some results of numericalexperiments with real-life data of solar power and demandsfor electricity and heat.

2. Mathematical Model of Smart House

In this section, we formulate a mathematical modelof a smart house, assuming that solar power and demandsfor electricity and heat are deterministically given.

To begin with, we describe the model of a smart houseconsidered in this paper. As shown in Fig. 1, the smart houseis composed of six devices: a solar panel, a fuel cell, a heatstorage tank, a gas boiler, a storage battery, and an electricvehicle (EV). In the diagram, EH is the electric powerdemand, Θ is the heat demand, z– is the purchased electric-ity, z+ is the sold electricity, s is the PV generated power, r+

is the charged electricity, r– is the discharged electricity, v+

is the charged electricity of the EV, v– is the dischargedelectricity, x is the gas consumption of the FC, pFC is thepower output of the FC, qFC

in is the heat output of the FC,qFC

out is the heat output of the heat storage tank, y– is the heatsupplied by the gas boiler, and y+ is the released heat.

We consider an optimization problem where the totaloperation cost of the smart house during a given period isminimized under the constraints on electrical devices, en-ergy balances, and electricity trading. We call this periodthe scheduling period, and this problem the smart housescheduling problem. We divide the scheduling period intoH equally spaced subperiods and call the h-th subperiod (h= 1, 2, . . . , H) period h. We denote the set of the periodindices as H = {1, 2, . . . , H}.

2.1 Objective function

2.1.1 Cost for fuel cell

Let x(h) denote the gas consumed by the fuel cellduring period h. Then the total gas cost for the fuel cell iswritten as

where C1 is the unit cost of gas.While the fuel cell is in stand-by condition, a stand-by

cost is required. We define 0-1 variables δFC(h) as

Then the total stand-by cost of the fuel cell can be writtenas

where C2 is the stand-by cost per period.When the fuel cell starts up, a start-up cost is neces-

sary. We define 0-1 variables αFC(h) as

Then the total start-up cost of the fuel cell is written as

where C3 is the cost for a one-time start-up of the fuel cell.Similarly, we define variables βFC(h) which take a

value of 1 if the FC stops at the beginning of the period, anda value of 0 in the other cases. Then the total stop cost ofthe fuel cell can be written as

where C4 is the cost for a one-time stoppage of the fuel cell.Since the objective function to be minimized contains

function (3), we can rewrite (2) as

For convenience, we define δFC(0) = 0. Note that we can regardαFC(h) as continuous variables rather than 0-1 variables.

(2)

(4)

(3)

Fig. 1. Mathematical model of smart house. [Colorfigure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

(5)

(6)

(1)

49

Similarly, βFC(h) can be written as

Similarly to αFC(h), we can consider βFC(h) to be continu-ous variables rather than 0-1 variables.

2.1.2 Cost and profit from electricity trading

Let z–(h) and z+(h) denote, respectively, purchasedelectricity and sold electricity during period h. Then thetotal cost of purchased electricity is written as

and the total profit of sold electricity is written as

where C5 and C6 are the unit cost of purchased electricityand the unit profit of sold electricity, respectively.

2.1.3 Cost for gas boiler

Let y–(h) denote heat produced by the gas boilerduring period h. Then the total cost of the gas boiler iswritten as

where C7 is the unit cost of heat.Summing up the costs and profit considered so far,

the objective function is represented as

Note that function (9) is included in the objective functionwith the negative sign, since it represents a profit.

2.2 Constraints

2.2.1 Constraints associated with storagebattery of EV

Let V(h) and V(H + 1) denote the stored electricity inthe storage battery of the EV at the beginning of period h

and at the end of the scheduling period, respectively. LetVmax and Vmin denote the upper and lower limits of thestorage battery, respectively, and V0 denote the initialamount of electricity stored in the storage battery, that is,

where V0 satisfies Vmin ≤ V0 ≤ Vmax. The stored electricityin the storage battery of the EV is also limited to

Let v+(h) and v–(h) denote the charged electricity inthe storage battery and discharged electricity into the powergrid, respectively. Let ηV

C and ηVD denote the charge and

discharge efficiencies, respectively. Moreover, let EV(h)denote the electricity consumption of the EV during periodh. Then the stored electricity in the storage battery of theEV satisfies the following recurrence relation:

We cannot charge and discharge at the same time.This condition can be written as

where δCH+ (h) and δCH

− (h) are 0-1 variables defined as

and δEV(h) are constants defined as

The electricity charge v+(h) must take a value of 0 ifthe EV is charged [δCH

− (h) = 1] or a value lying between vmin

and vmax otherwise [δCH− (h) = 0]. Such conditions can be

written as

(9)

(10)

(11)

(15)

(16)

(17)

(12)

(13)

(14)

(7)

(8)

50

where λEV+1 (h) and λEV+

2 (h) are auxiliary variables. Thecorresponding conditions on the electricity discharge v–(h)can be written similarly.

2.2.2 Constraints associated with storagebattery

The constraints on the house battery are similar tothose of the EV battery. However, unlike the EV case, thehouse battery does not consume electric power, and so theelectric balance can be written as

2.2.3 Constraints on input and output of fuelcell

The gas consumption x(h) takes a value of 0 when theFC is on stand-by [δFC(h) = 0] or a value lying between xmin

and xmax when the FC is active [δFC(h) = 1]. This conditioncan be written as

where λFC1 (h) and λFC

2 (h) are auxiliary variables.We assume that the gas consumption x(h) and the

power output pFC(h) of the fuel cell are related as follows[11]:

where a and pFC0 are positive constants. Furthermore, thegas consumption x(h) and the hot water output qFC

in (h) of thefuel cell are related as follows [11]:

where b and qFC0in are positive constants.

2.2.4 Constraints on heat storage tank

Let QFC(h) and QFC(H + 1) denote the stored heat inthe hot water tank during period h and at the end of thescheduling period, respectively. Let Qmax

FC and QminFC denote,

respectively, the upper and lower limits of the heat storagetank and Q0

FC denote the stored heat at the beginning of thescheduling period, that is,

where 0 ≤ QminFC ≤ Q0

FC ≤ QmaxFC . The heat storage is also lim-

ited to

We assume that the heat loss µFC(h) from the heatstorage tank is proportional to the stored heat QFC(h).Therefore, the following equation holds:

where ηHWFC is a constant satisfying 0 < ηHW

FC < 1.Let qFC

in (h) denote the hot water output from the fuelcell and qFC

out(h) denote the hot water output from the heatstorage tank. Then we have the following recurrence rela-tion:

2.2.5 Constraints on electricity balance

Let pFC(h) denote the power output of the fuel cell.Let z+(h) and z–(h) denote, respectively, the purchased andsold electricity. In addition, let s(h) and EH(h) denote givenconstants that represent the solar power and the heat de-mand during period h, respectively. Since heat balance mustbe satisfied, we have the following constraints:

2.2.6 Constraints on heat balance

Let y–(h) and y+(h) denote the hot water output of thegas boiler and the released heat during period h. Let Θ(h)denote a given constant that represents the heat demand.Since heat balance must be satisfied, we have the followingconstraints:

2.2.7 Constraints on electricity trading

Since we cannot sell electricity when we buy it, atleast one of z–(h) and z+(h) must be equal to 0, that is,

Since we cannot sell more electricity than solarpower, the following must be satisfied:

Let zmax+ denote the upper limit on the electricity sale

per period. Then we have

(22)

(18)

(23)

(24)

(25)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(19)

(20)

(21)

(26)

51

To convert condition (30) to the linear constraints, weintroduce 0-1 variables δESW (h) as

and let M > 0 be a sufficiently large constant. Then con-straint (31) can be rewritten as

2.2.8 Constraints on hot water output andreleased heat

Since we cannot generate and release heat at the sametime, at least one of the hot water output y–(h) and thereleased heat y+(h) during period h must be equal to 0, thatis,

In view of constraints (29) and (30), it can be deduced fromthe special feature of the objective function that (37) holdsat an optimal solution even if we do not explicitly include(37) in the constraints.

2.3 Formulation of deterministic model

Summing up the discussion so far, the deterministicmodel of the smart house is formulated as follows:

3. Stochastic Model

In the previous section, we have formulated themodel assuming that solar power s(h) and demands forelectricity EH(h) and heat Θ(h) are deterministically given.However, they are uncertain in reality. For example, solarpower depends on the weather. Demand for electricitydepends on when and how long we watch television. De-mand for heat depends on when and how we take a shower.

For this reason, we propose another mathematical modelwhere solar power and demands for electricity and heat aretreated as random variables.

3.1 Recourse constraints

3.1.1 Recourse constraints on electricitybalance

In constraints (27) on electricity demand, we assumethat solar power s(h) on the left-hand side and electricitydemand E(h) on the right-hand side are random variables.Since we must make our schedule before we know therealizations of s(h) and E(h), (27) do not always hold whenthe realizations become known. However, (27) must besatisfied during all periods. Therefore, we must keep (27)satisfied by taking corrective actions. This kind of action iscalled recourse and the cost for recourse is called recoursecost.

If the left-hand side of (27) is larger than the right-hand side when the realization values of s(h) and E(h) aredisclosed, electricity is in short supply. In this case, we musttake corrective actions such as purchasing electricity, reduc-ing electricity sale, or increasing the power output of thefuel cell. We choose one or some of the actions as recourse.Let e–(h) denote the recourse variables for supplying power.

On the other hand, if the left-hand side of (27) is lessthan the right-hand side when the realization values of s(h)and E(h) are disclosed, electricity is in excess supply. In thiscase, we must take corrective actions such as increasingelectricity sales, reducing electricity purchases, decreasingpower output of the fuel cell, or discarding electricity. Lete+(h) denote the recourse variables to reduce power supply.

Then we can write the recourse constraints on elec-tricity balance as

3.1.2 Recourse constraints on heat balance

In constraints (29) on heat demand, we assume thatthe heat demand Θ(h) on the right-hand side is a randomvariable. Since we must make our schedule before we knowits realization, the (29) do not always hold when the reali-zation becomes known. However, (29) must be satisfiedduring all periods. Therefore, we must keep (29) satisfiedby taking corrective actions.

If the left-hand side of (29) is smaller than the right-hand side when the realization value of Θ(h) is disclosed,hot water is in short supply. In this case, we must take

(34)

(37)

(35)

(36)

(38)

(39)

52

corrective actions such as increasing output of the gasboiler, increasing hot water output of the fuel cell, orreleasing heat. Let θ–(h) denote the recourse variables forheat supply.

On the other hand, if the left-hand side of (29) isgreater than the right-hand side when the realization of Θ(h)is disclosed, hot water is in excess supply. In this case, wemust take corrective actions such as reducing electricitypurchase, decreasing power output of the fuel cell, or dis-carding electricity. Let θ+(h) denote the recourse variablesto reduce heat supply.

Then we can write the recourse constraints on heatbalance as

3.1.3 Recourse constraints associated withsolar power and electricity sale

In constraints (32) on solar power and electricity sale,we assume that solar power s(h) on the right-hand side is arandom variable. Since we must make a schedule before weknow its realization, (32) do not always hold when therealization becomes known. However, (32) must be satis-fied during all periods. Therefore, we must keep themsatisfied by any means. If they are not satisfied, we musttake corrective actions. Let ζ(h) denote the recourse vari-ables to reduce electricity sale. Then we can write therecourse constraints on solar power and electricity sale as

3.2 Formulation of stochastic model

In two-stage stochastic programming, the expectedvalues of the recourse costs are added to the objectivefunction as a penalty. Let Ce

− and Ce+ denote the unit costs

of recourses e−(h) and e+(h), respectively, and Cθ− and Cθ

+

denote the unit costs of recourses θ−(h) and θ+(h), respec-tively. Moreover, let Cζ denote the unit cost of recourse ζ(h).Then the stochastic model of the smart house can be repre-sented as follows:

subject to (1), (5)–(8), (10)–(26), (28), (30), (32)–(36), (38)–(43).

3.3 Formulation of scenario-based stochasticmodel

We assume that solar power s(h) and demands forelectricity EH(h) and heat Θ(h) depend on a discrete prob-ability distribution with a finite sample space. Specifically,we assume that the vector of these random variables

takes a finite number of realization values

with probability pi, where i denotes a scenario number, n isthe number of scenarios, and I is the set of scenario indices,that is, I = {1, 2, . . . , n}.

Since EH(h) and s(h) are random variables, e−(h) ande+(h) become random variables through (38). In addition,since Θ(h) is also a random variable, θ−(h) and θ+(h) be-come random variables through (40). Similarly, ζ(h) be-comes a random variable through (42).

Let ei−(h), ei

+(h), θi−(h), θi

+(h), and ζi(h) denote therealizations of e−(h), e+(h), θ−(h), θ+(h) and ζ(h), respec-tively. Then (38) and (39) are written as

whereas (40) and (41) can be written as

In addition, (42) and (43) can be written as

Summing up the discussion so far, the scenario-basedstochastic model is formulated as follows:

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

53

subject to (1), (5)–(8), (10)–(26), (28), (30), (33)–(36), (44)–(49).

4. Numerical Experiments

In this section, we report some numerical results. Weset the scheduling period as 3 days and a subperiod as anhour, and hence H = {1, 2, . . . , 72}. All programs werecoded in MATLAB 7.4, and run on a machine with 3.0-GHzCore 2 Duo CPU and 3.2 GB of RAM. We used IBM ILOGCPLEX 11.0 to solve mixed-integer programming prob-lems.

4.1 Numerical results for stochastic model

4.1.1 Scenario generation

The real-life data taken from Monday to Wednesdayof 15 weeks from November 20, 2006 to February 28, 2007were used to generate scenarios of solar power and demandsfor electricity and heat. Specifically, we made 15 sets ofdata, each consisting of the demand data from Monday toWednesday of each week, and then classified those 15 setsinto three classes using the k-means method. As a result,three clusters with 6, 5, and 4 sets respectively were ob-tained. Then we chose the centroid of each cluster as ascenario, thereby generating three scenarios.

The generated scenarios are shown in Figs. 2 to 4.These three scenarios are supposed to occur with prob-abilities p1 = 6/15, p2 = 5/15, and p3 = 4/15, respectively.

4.1.2 Optimal solutions of stochastic model

We carry out numerical experiments for the scenario-based stochastic model. The values of the constants used in

Fig. 3. Scenarios of electricity demand Θ(h). [Colorfigure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Fig. 4. Scenarios of solar power s(h). [Color figure canbe viewed in the online issue, which is available at

wileyonlinelibrary.com.]

Fig. 2. Scenarios of heat demand EH(h). [Color figurecan be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

54

the experiment are shown in Table 1. The model has 2232constraints besides the upper and lower bound conditions,and 2232 continuous variables and 504 0-1 variables. Com-putation time was 0.2072 second. We let the discharge fromEV always be zero [v–(h) = 0, Wh ∈ H], since house batterynever reaches its maximum capacity in our real-life data.

The charged electricity and discharged electricity ofthe storage battery at the optimal solution are shown in Fig.5. It is seen that both charged electricity and dischargedelectricity are small, and the electricity charge amount ismuch smaller than the lower limit on the storage battery.This is mainly because the purchase cost and sale cost ofelectricity are assumed to be time-independent and con-stant. This suggests that, unless time-varying electricity

Table 1. Constants used in the experiment

Fig. 5. R(h),r−(h), and r+(h) at the optimal solutions.[Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Fig. 6. z−(h) and z+(h) at the optimal solutions. [Colorfigure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

55

price is introduced, a large-capacity battery will not help toreduce the energy cost significantly.

The purchased electricity and sold electricity at theoptimal solution are shown in Fig 6. It is seen that soldelectricity z+(h) takes a value between the maximum andthe minimum limits of PV power generation. Energy costcan be reduced by increasing the sold electricity, becausethe unit profit C6 of electricity sale is larger than the unitcost C5 of electricity purchase. However, if FC powergeneration is planned based on the PV output of the sce-nario with the maximum PV power generation, then therecourse cost for surplus electricity becomes larger for ascenario with smaller PV power generation.

Figure 7 shows that the recourse for the shortage ofelectricity occurs in every scenario.

4.2 Comparative experiments of deterministicmodel and stochastic model

We compare the operation cost, the recourse cost, andthe total cost of the scenario-based stochastic model withthose of the deterministic model. The real-life data used inthe scenario generation

are employed to compare these two models, where H = 72and J = {1, 2, . . . , 15}. We denote an optimal solution ofeach model as x*(h), y+*(h), y−∗(h), and so on. Then, theoperation cost is given by

We calculate the recourse costs for each model by thefollowing procedure: First, calculate ej

−(h) and ej+(h) from

the constraints on electricity balance

and then calculate θj−(h) and θj

+(h) from the constraints onheat balance

From constraints (42) and (43) on solar power and electric-ity sales, the following inequalities hold at the optimalsolution:

The total recourse cost is calculated by

The operation cost, the recourse cost, and the totalcost are shown in Figs. 8, 9, and 10, respectively. As can beseen from Fig. 8, the stochastic model constantly giveslower operation costs. For the current unit price C6 = 0.049yen of power sales, the optimal value of the stochasticmodel is 5.7845 × 103 yen and the optimal value of thedeterministic model is 8.6094 × 103 yen. Thus, the total costin the stochastic model is lower than that in the determinis-tic model by 32.8%.

As shown in Fig. 8, the operation cost of the stochas-tic model is lower than the deterministic model. On theother hand, the recourse cost of the stochastic model ishigher than the deterministic model as seen from Fig. 9. Inthe deterministic model, the optimal solution is obtainedfrom the average of 15 sets of electric power demands, heatdemands, and PV power generations, and it turns out thatthe energy is supplied excessively in many subperiods ofthe whole scheduling period. In the stochastic model, the

(50)

(51)

Fig. 7. ei−(h) at the optimal solutions. [Color figure can

be viewed in the online issue, which is available atwileyonlinelibrary.com.]

56

optimal solution is obtained from three scenarios, and theschedule with minimum total cost is found by taking intoaccount both the operation cost and the recourse cost.

As seen in Fig. 10, when C6 is larger than C5 = 0.019,the total cost increases as C6 decreases. However, when C6

becomes smaller than C5, the sold electricity z+(h) is re-duced and the effect of the value of C6 becomes small.When C6 is very small, z+(h) are close to zero in both thedeterministic model and the stochastic model. Hence, thedifference in the total costs between these models becomessmall.

5. Conclusion

In this paper, we have presented a mathematicalmodel of a smart house that is composed of a solar powersystem, a fuel cell, a gas boiler, a storage battery, and anelectric vehicle, and then formulated the smart house sched-uling problem as a mixed-integer programming problem.To deal with the uncertainty in solar power and demandsfor electricity and heat, we have reformulated the model asa two-stage stochastic mixed-integer programming prob-lem, assuming that these are random variables. Further-more, we have carried out numerical experiments andexamined the effectiveness of the proposed model.

REFERENCES

1. Ministry of Economy, Trade and Industry of Japan.Smart house trial project report 2009.

2. Salgado F, Pedrero P. Short-term operation planningon cogeneration systems: A survey. Electric PowerSyst Res 2008;78:835–848.

3. Tanaka Y, Fukushima M. Optimal operation of cogen-eration systems by stochastic programming. IEEJTrans Power Energy 2009;129:765–775.

4. Birge JR, Louveaux F. Introduction to stochasticprogramming. Springer-Verlag; 1997.

5. Nowak MP, Schultz R, Westphalen M. A stochasticinteger programming model for incorporating day-ahead trading of electricity into hydro-thermal unitcommitment. Optim Eng 2005;6:163–176.

Fig. 8. Comparison of the stochastic model and thedeterministic model: operation costs. [Color figure can

be viewed in the online issue, which is available atwileyonlinelibrary.com.]

Fig. 9. Comparison of the stochastic model and thedeterministic model: recourse costs. [Color figure can be

viewed in the online issue, which is available atwileyonlinelibrary.com.]

Fig. 10. Comparison of the stochastic model and thedeterministic model: total costs. [Color figure can be

viewed in the online issue, which is available atwileyonlinelibrary.com.]

57

6. Nurnberg R, Römisch W. A two-stage planningmodel for power scheduling in a hydrothermal sys-tem under uncertainty. Optim Eng 2002;3:355–378.

7. Römisch W, Vigerske S. Recent progress in two-stagemixed-integer stochastic programming with applica-tions to power production planning. In Handbook ofpower systems I, Energy systems, 2010, p 177–208.

8. Takriti S, Birge JR, Long E. A stochastic model forthe unit commitment problem. IEEE Trans PowerSyst 1996;3:1497–1508.

9. Dantzig GB. Linear programming under uncertainty.Manage Sci 1955;1:197–206.

10. Beale EML. On minimizing a convex function sub-ject to linear inequalities. J R Stat Soc Ser B1955;17:173–184.

11. Ito K, Yokoyama R. Optimal design of cogenerationsystem. Sangyo Tosho; 1990.

AUTHORS (from left to right)

Shunsuke Ozoe (nonmember) graduated from the Faculty of Engineering, Osaka University, in 2009. He completed theM.E. program at the Graduate School of Informatics, Kyoto University, in 2011.

Yoichi Tanaka (member) completed the M.E. program at Waseda University (Graduate School of Science and Engineering)in 1999 and joined Toho Gas Co., Ltd. (Technical Group, Product Development Department). He completed the doctoral programat the Graduate School of Informatics, Kyoto University, in 2009. His research interests are in energy systems.

Masao Fukushima (nonmember) completed the M.E. program at the Graduate School of Engineering, Kyoto University.After working for Mitsubishi Chemical Corp., and then at Kyoto University as an instructor, lecturer, and associate professor,he was appointed a professor at Nara Institute of Science and Technology in 1993. Since 1996, he has been a professor in theGraduate School of Informatics, Kyoto University. He also held visiting professorships at the University of Waterloo, Universityof Namur, and University of New South Wales. His research interests include mathematical programming and variationalinequalities. He holds a D.Eng. degree, and is a member of ORSJ, ISCIE, MOS, INFORMS, and SIAM.

58