DESIGNING FOR MAINTENANCE: A GAME THEORETIC APPROACH

Transcription

1 Eng. Opt., 2002, Vol. 34, pp DESIGNING FOR MAINTENANCE: A GAME THEORETIC APPROACH GABRIEL HERNANDEZ a, CAROLYN CONNER SEEPERSAD b and FARROKH MISTREE b, * a Siemens Westinghouse Power Corporation, 4400 Alafaya Trail, MC DV 322, Orlando, FL , USA; b George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA , USA (Received 8 November 2001; In final form 8 May 2002) Maintenance management is the effective and economical use of resources to keep equipment in, or restore it to, a serviceable condition. In this paper, maintenance considerations are introduced during product design using a game theoretic approach. Specifically, a product designer and a maintenance manager are modeled as two players in a Leader Follower game, and strategies for designing product components are derived accordingly. To implement this approach, the compromise Decision Support Problem, with a deviation function adapted from linear physical programming, is used to model decisions mathematically. This approach is intended for distributed collaborative design in which modeling, computational or organizational factors hinder complete integration of all aspects of a design problem. Using this approach, the knowledge and expertise of each designer are fully utilized while keeping modeling and computational challenges at a tractable level. The approach is illustrated with a case example, namely, the design of a series of absorption chillers for an industrial complex. Keywords: Maintenance; Game theory; Decision support; Distributed design; Collaborative design LIST OF SYMBOLS A i ðx Þ BRC c di þ ; di DC DSP FðtÞ HC I qmy IC ir L qm Value of Goal i Best reply correspondence Tube unit cost Deviation variables in a compromise DSP Product design (capital) cost Decision support problem Failure cumulative probability density function Holding cost Number of tube type q and length L qm in inventory in year Y Inventory cost Interest rate Length of tube type q for subsystem m * Corresponding author. farrokh.mistree@me.gatech.edu ISSN X print; ISSN online # 2002 Taylor & Francis Ltd DOI: =

2 562 G. HERNANDEZ et al. m MC N qm NF qmy n tp NDT q m T TC _W w i;k X Y Z Subsystem Maintenance cost Number of tube type q for subsystem m Number of failures of tube q of length L qm in year Y Number of tube passes Number of different tubes utilized Tube selection for system m Temperature Cost of a tube unit (Cost per length Length) Power consumption Weights given to the achievement of objective i at desirability level k in the deviation function of a compromise DSP Vector of variables Year Value of deviation function in a Compromise DSP 1 FRAME OF REFERENCE: DISTRIBUTED COLLABORATIVE DESIGN Reliability of a mechanical system depends on its design, the quality of its components, and its maintenance. However, maintenance is typically considered during product design in an ad hoc, ineffective manner, leading to unnecessary life-cycle costs. Systematic and effective methods are needed for incorporating maintenance considerations early in the product design process. In principle, the decisions associated with product design and maintenance management can be integrated and solved as a single design problem, but the resulting formulations often become intractable. This centralized approach also limits opportunities for stakeholders to apply their knowledge, expertise, and skills in the solution process. Hence, the problem is typically decomposed, and the resulting sub-problems are distributed to decision-makers according to their expertise, and then they attempt to collaborate. The latter scenario is referred to as distributed collaborative design. In this paper, an approach is presented for distributed collaborative design based on game theory. In Section 2, the theoretical foundation of the collaborative approach is described, including the application of non-cooperative games and Nash equilibrium in engineering design. Then, a product designer and a maintenance manager are modeled as two players in an extensive form game in Section 3, and accordingly, a method for achieving efficacious collaboration between them is developed. In Section 4, the method is illustrated in a case example, namely, the design of a series of absorption chillers for an industrial complex. 2 NON-COOPERATIVE GAMES AND NASH EQUILIBRIUM IN ENGINEERING DESIGN From a systems perspective, distributed collaborative design can be modeled as a game. Related work in multi-disciplinary design using game theory is found in Refs. ½6; 10; 11; 17 19Š. A game consists of multiple decision-makers; each decision-maker controls a subset of system variables and seeks to pursue individual objectives subject to constraints [16]. Game theory is divided into cooperative and non-cooperative game theory. In cooperative game theory, decision-makers form coalitions by agreeing to cooperate with one another. Most of the research in cooperative game theory involves investigating the

3 DESIGN FOR MAINTENANCE 563 stability of these coalitions [16]. On the other hand, in non-cooperative game theory the unit of analysis is the individual participant in the game who is concerned with doing as well for himself=herself as possible subject to clearly defined rules and possibilities. If individuals undertake cooperative behavior, it is because such behavior is in their best interest. The authors believe that non-cooperative models, with their emphases on the individual, better capture the dynamics of distributed collaborative design. There are two models employed in non-cooperative game theory. The first and simpler one is called a strategic form or normal form game that is suitable for modeling interactions between individuals when the timing of actions is not relevant. The second type of model is the extensive form game. In an extensive form game, the timing of actions is an important factor in the outcome of the game. Hence, in extensive games, possible future actions and reactions must be considered at each stage of the game, whereas in strategic form games, strategies are not influenced by future actions. Once a collaborative design situation is modeled as either a strategic or an extensive game, the next step is to analyze the model, predict how the decision-makers should act and formulate appropriate strategies based on one of two solution techniques: dominance arguments or Nash equilibrium analysis [9]. Dominance arguments have limited applications of practical interest in design, whereas Nash equilibrium is applicable to a broad class of practical situations. Nash equilibrium is a profile of strategies such that each decision-maker s strategy is his=her optimal response to the other decision-makers strategies. In Nash equilibrium, no decision-maker has any incentive for changing his=her choices if none of the other decisionmakers changes strategy. Nash equilibria are the only consistent predictions of how the game will be played because only a Nash equilibrium has the property that each decisionmaker can predict it, predict that his=her opponents predict it, and so on [4]. The formal definition provided in Ref. [9] follows: Assume a finite player game, where the players are indexed i ¼ 1;...; n, and their respective available strategy sets are denoted by X i. A strategy profile x ¼ ðx 1 ;...; x i 1 ; x i ; x iþ1 ;...; x n Þ is a Nash equilibrium if, for each player i and x i 2 X i, z i ðx Þz i ðx 1 ;...; x i 1 ; x i ; x iþ1 ;...; x j Þ for all x i 2 X i where Z i is the value of the payoff that player i receives as a result of the strategies chosen by the players in the game. Assume that designers involved in a collaborative process are abstracted as players in a game. Each designer i controls a set of variables X i and seeks to optimize an objective function Z i. The designer s strategy in this game is the value of the variables under his=her control, X i. Each designer develops a strategy that best achieves his=her own objectives, given the possible strategies of other designers involved. The result of this game must be a Nash equilibrium, as previously discussed. Contrary to the connotation, it is possible to achieve cooperative (Pareto) efficient solutions with non-cooperative models [8]. This is explained in more detail in Section 3, in which an approach is described for facilitating collaboration between a product designer and a maintenance manager. 3 A FIVE-STEP, GAME THEORETIC APPROACH FOR DESIGNING FOR MAINTENANCE Maintenance management is the effective, efficient and economical use of resources to keep equipment in, or restore it to, a serviceable condition. It includes both corrective and

4 564 G. HERNANDEZ et al. preventive actions that typically involve a large number of decisions regarding allocation of resources, time, and space. Ideally, these considerations should be introduced early in the product design process. Recently, Chen and co-authors [2] proposed an approach for design for maintenance based on integrating decisions into a single Decision Support Problem (DSP). This is a plausible approach when the design and maintenance problems are simple enough to be integrated and solved as a single one. However, this approach is difficult to implement when the models and solution methods are large or complex. In this paper, an approach is developed for solving complex design for maintenance problems using a particular extensive game in which one decision-maker, in this case a product designer, acts as a leader to a second decision-maker, a maintenance manager. A particular kind of Nash equilibrium, known as Stackelberg equilibrium, is applicable to this situation. This leader=follower model was applied in engineering design in Ref. [10] to model design situations in which the influence of one discipline on another is strongly unidirectional (in time, authority or information). This approach was also applied in Ref. [6] to include manufacturing considerations during product design. The leader=follower model is described next. Consider a leader=follower relationship between two designers or design teams abstracted as players in a game. The actions of the two players are to set the values of the variables X 1 for player 1 and X 2 for player 2. Each player seeks to optimize his=her individual objective function, Z 1 ðx 1 ; X 2 Þ for Player 1 or Z 2 ðx 1 ; X 2 Þ for Player 2. Player 1, the leader, chooses the value of X 1 first, and Player 2 observes X 1 before choosing X 2. Player 2 will consider the value of X 1 when choosing X 2. Player 1, on the other hand, moves first and cannot condition his=her choice on Player 2 s. The expected outcome in this situation is known as Stackelberg equilibrium. Player 2 s strategy is to choose, for each X 1, the level of X 2 that optimizes his=her objective function, Z 2 ðx 1 ; X 2 Þ, so that X 2 belongs to his=her best reply correspondence, BRC 2 ðx 1 Þ (also called a rational reaction set [10]). This best reply correspondence is a set of solutions that a player constructs based on his=her expected best response to the actions of other players. Given that Player 1 expects the strategy of player 2tobeBRC 2 ðx 1 Þ, the choice of X 1 should be the solution to minimize Z 1 ðx 1 ; BRC 2 ðx 1 ÞÞ. Thus, a formulation of a strategy for Player 1, the leader, follows: Given The best reply correspondence of the follower, BRC 2 ðx 1 Þ Find The design variables X 1 Satisfy X 2 2 BRC 2 ðx 1 Þ Minimize The objective function Z 1 ðx 1 ; X 2 Þ For Player 2, the follower, who chooses X 2 only after player 1 has chosen X 1, the strategy is: Given The design variables of the leader, X 1 Find The design variables X 2 Minimize The objective function Z 2 ðx 1 ; X 2 Þ As noted by Fudenberg and Tirole [4], these are the only credible strategies if both players act rationally. In game-based design [11], each designer is a player who formulates and solves strategies as compromise Decision Support Problems (DSP s). The compromise DSP is a multi-objective decision model that is used to determine the values of design variables that satisfy a set of constraints and achieve a set of conflicting goals as closely as possible [14]. The mathematical form of the compromise DSP is summarized in Figure 1.

5 DESIGN FOR MAINTENANCE 565 FIGURE 1 Mathematical form of the compromise DSP [14]. The proposed approach is rigorous and enhances collaboration without resorting to large, intractable integrated optimization problems. Unfortunately, this approach has two difficulties: (1) determining analytical models of the BRC s is difficult in complex design problems, and (2) Nash equilibrium solutions are not necessarily Pareto efficient. First, the difficulty of obtaining BRC s for complex design problems is addressed. Based on previous work in Refs. ½6; 10; 11Š, response surfaces [15] may be used to develop approximate models of the BRC s using a four-step method illustrated in Figure 2. (1) Each designer k identifies the system variables of other designers (where the set of other designers is identified as k in Fig. 2) that affect designer k s performance significantly. This can be done systematically using screening experiments. (2) An experiment is designed with the variables identified in Step 1. (3) Each designer k formulates and solves a compromise DSP repeatedly for each combination of other designers system variable values included in the experiment designed in Step 2. Each time a compromise DSP is solved, the resulting minimum value of designer k s deviation function, Z k, is recorded. (4) Each designer k fits a response surface to the values recorded in Step 3. The resulting model is the best reply correspondence of the designer.

6 566 G. HERNANDEZ et al. FIGURE 2 Developing an approximation for the BRCs. The BRCs employed here are slightly different from the BRCs defined in game theory and utilized in Refs. ½6; 10; 11Š. The BRCs in this paper are not models that map design variables as functions of other decision-makers design variables. Instead, the value of the objective function is approximated directly (an approach previously used in Ref. [6]). Next, the issue of Pareto optimality is addressed. In a non-cooperative game, if designers agree to cooperate and pursue a common objective, the resulting solution is Pareto efficient. How could such a common objective be determined and a corresponding objective function formulated? One possibility is to formulate an objective function as a weighted sum of the individual designers objectives, but determining appropriate weights is difficult. Here a common objective is developed from the individual objectives by utilizing the concepts of physical programming ½12; 13Š to formulate a deviation function for a compromise DSP. Instead of specifying ad hoc weights, the individuals agree on six desirability levels for each objective: (1) ideal (or highly desirable), (2) desirable, (3) tolerable, (4) undesirable, (5) highly undesirable and (6) unacceptable. Then, a series of goal functions are formulated as follows [5]: A i ðx Þ max½a i ðx Þ G i;kþ1 ; 0Š þ di;k G dþ i;k ¼ 1 for minimization ð1aþ i;k A i ðx Þ min½a i ðx Þ G i;kþ1 ; 0Š þ di;k G dþ i;k ¼ 1 for maximization ð1bþ i;k where A i is the value of the goal i to be minimized (1a) or maximized (1b), and G i;k is the target value for this same goal for the desirability level k ¼ 1;...; 5. The sixth level value, the unacceptable one, becomes an additional constraint, i.e. A i ðx ÞG i;6 ð2þ

7 DESIGN FOR MAINTENANCE 567 Then, a common deviation function to be minimized is formulated as follows: Z ¼ Xm X 5 i¼1 k¼1 ðw i;k d i;k þ wþ i;k dþ i;k Þ ð3þ where m is the number of goals and the value of the weights w i;k are obtained by applying an algorithm described in Ref. [5]. The flexible formulation of multiple goals using various degrees of desirability, specified by the designers themselves, makes this approach suitable for collaborative distributed design, in which capturing the knowledge and expertise of the various designers and achieving at least intermediate priority levels for all of them is important. It is more reasonable for a number of designers to agree on ideal (or highly desirable), desirable, tolerable and unacceptable values of the relevant criteria rather than ad hoc weights. Note that minimization of the function Z in Eq. (3) will be Pareto efficient as long as at least one of the deviation variables is larger than zero (i.e. as long as ideal levels of all objectives are not achievable simultaneously). Given a common deviation function, Z, that all the designers agree to optimize, how do they proceed to solve the design problem in a coordinated manner? Using a leader=follower game model and the concepts described in this section, a five-step method for approaching design for maintenance problems is constructed as follows, with a product designer as the leader and a maintenance manager as the follower. The process is illustrated in Figure Define a common objective function Z to be minimized: 1.1. Identify the relevant goals for both the product designer and the maintenance manager Define six levels of desirability for each of the goals identified in Step 1.1: (1) ideal, (2) desirable, (3) tolerable, (4) undesirable, (5) highly undesirable and (6) unacceptable Apply the algorithm described in Ref. [5] to formulate a common objective function Z (Eq. (3)). 2. The maintenance manager (follower) adopts the minimization of his=her portion of the objective function as his=her objective and formulates a compromise DSP accordingly. 3. Using the compromise DSP formulated in Step 2, the follower (maintenance manager) develops a BRC as a function of the product designer s variables using the method shown previously in Figure 2 (Sec. 3): BRC M : Z maint ¼ f ðx design Þ ð4þ 4. The product designer (leader) formulates a Compromise DSP with the common objective function developed in Step 1 as an objective, and a constraint of satisfying the follower s BRC. The product designer solves the DSP and finds the values of his=her design variables, X design. 5. Using X design, the maintenance manager solves the compromise DSP formulated in Step 2 and finds X maint. FIGURE 3 Solving the designing for maintenance problem as a leader=follower game.

8 568 G. HERNANDEZ et al. With this approach, the product design and maintenance problems are solved within a distributed collaborative framework rather than integrated into a single, large (usually intractable) optimization problem. Hence, coordination is achieved by incorporating the best reply correspondence function of maintenance as a goal in product design. Different solution algorithms can be employed for different decisions to increase the efficiency and effectiveness of the solution process. For example, linear programming may be appropriate for solving the maintenance problem but not the product design problem. Individual knowledge and expertise is fully utilized while keeping computational problems at a tractable level, and the solution is Pareto efficient. This approach for solving a coupled design for maintenance problem is illustrated with a case example in Section 4. 4 CASE EXAMPLE: DESIGN FOR MAINTENANCE OF A SERIES OF ABSORPTION CHILLERS Consider the design of four absorption chillers for air conditioning and cooling of various processes in an industrial complex. The designer of these pieces of equipment seeks to minimize the capital cost and energy consumption of these chillers while meeting refrigeration requirements. After the equipment is designed, a maintenance manager must specify an inventory of spare parts for preventive maintenance of the equipment during its service lifetime. For modeling and computational reasons, it is difficult to integrate the decisions faced by the product designer and the maintenance manager; a collaborative, game-theoretic approach is more appropriate. Introductory information is offered in Section 4.1, followed by implementation of the five-step, game-theoretic approach in Sections 4.2 to Problem Statement Absorption chillers include four basic components: (1) an evaporator, (2) an absorber, (3) a generator, and (4) a condenser, as shown in Figure 4. Usually, the absorber and evaporator are combined in a single shell as illustrated in Figure 5. An absorber, a condenser, an evaporator and a heat exchanger are required for each of the four different chillers in the industrial complex. There are 16 subsystems in total, referenced with the subscript m ¼ 1;...; 16. The product designer is interested in selecting tubes ðq m Þ and determining their length ðl qm Þ, number ðn qm Þ, and number of passes ðn pm Þ for each subsystem m. The required capacity of refrigeration, temperature of condensing water entering the absorber ðt 1 Þ, and temperature of cooling water entering and exiting the evaporator (T 2 and T 3 ) are listed in Table I. The product designer has 11 different commercial tubes available for selection. The primary characteristics of these tubes are given in Table II. All of these tubes are made of copper with a thickness of 1:27 mm and can be purchased in lengths from 1:0 m to 15 m. For the finned tubes, the fin height is 1:42 mm and the fin width is 0:278 mm. Once the pipes are designed, the design of the shells is straightforward. The product designer is interested in two objectives: (1) minimizing the required capital cost, DC, as a function of the cost of the pipes and shells, and (2) minimizing the power, _W, required to pump water and solution through the various subsystems of the four chillers. More detailed information on the cost models, and the geometric and thermal constraints can be found in Ref. [7]. Thermal and fluid models for absorption chillers are developed elsewhere [1]. After the chillers are designed, the maintenance manager must specify the required number of spare tubes to maintain in inventory for replacement purposes. This problem determination of the number of spare units to carry in inventory is one of the most impor-

9 DESIGN FOR MAINTENANCE 569 FIGURE 4 Components of the absorption refrigeration cycle. tant decisions for preventive maintenance. Clearly, if the number of spare parts is less than necessary during the preventive maintenance cycle, the system to be repaired will experience unnecessary downtime until the spare parts are available. On the other hand, if more spares are carried in inventory than required, an excessive carrying cost will be incurred. Ideally, the number of spares in inventory should equal the number of repairs at the time of replacement. However, the number of replacements is a random variable; thus, determination of the appropriate number of spare parts is a difficult task [3]. Assume for simplicity that both preventive inspection of equipment and replenishment of inventory occur only once a year. The maintenance manager seeks to maintain an inventory of spare parts that is sufficient to supply any required replacements without delay. The maintenance manager requires a (probabilistic) confidence of at least 90% that the inventory of spare parts is larger than the number of parts to be replaced after inspection; hence: PfNF qmy I qmy g0:9 Y ¼ 1;...; 30 ð5þ FIGURE 5 The absorber evaporator module.

10 570 G. HERNANDEZ et al. TABLE I Nominal Chiller Requirements. Capacity [Tons] T 1,nom C T 2,nom C T 3,nom C where I qmy represents the number of tubes of type q and length L qm in inventory during the year Y and NF qmy represents the number of these tubes that failed inspection and need to be replaced that year. The expected lifetime of the equipment is 30 years. The probabilistic model for estimating the minimum inventory of spare parts required for satisfying Eq. (5) is described in Ref. [7]. This estimation is not trivial, and the model is solved using a numerical algorithm. Once the minimum number of spare parts is estimated, the total maintenance cost, MC, can be calculated as the sum of the maintenance costs for all tube types employed in the four chillers: MC ¼ XNDT MC i i¼1 ð6þ where NDT represents the number of different tubes. Depending on the choices made by the product designer, NDT can have a value between 1 (if there is a common tube and tube length for all 16 subsystems) and 16 (if all subsystems have a different tube and tube length). MC in Eq. (6) is in turn modeled as the sum of the cost of the spare parts, TC, and the holding cost of inventory, HC: MC ¼ XNDT MC i ¼ XNDT ðtc i þ HC i Þ i¼1 i¼1 ð7þ where TC i is the cost of the tube i (the unit cost per length multiplied by the length) and HC i is calculated for this example as: HC i ¼ X30 Y ¼1 irði iy TC i Þð30 Y Þ ð8þ where ir is an annual interest rate (considered here to be 10%), I i is the number of parts in inventory for tube i, andy represents the year for equipment with an expected life of 30 years. TABLE II Tubes Available for Selection. q Type Outer diameter (mm) Number of fins per meter Unit cost (dollars per meter) 1 Smooth Smooth Smooth Smooth Finned Finned Finned Finned Finned Finned Finned

11 DESIGN FOR MAINTENANCE 571 TABLE III Target Values for the Goals. Level k Who? 1. Design cost DC [dollars] Product designer 2. Power consumption _W [kw] Maintenance manager 3. Maintenance cost MC [dollars] 1 Ideal 160, ,000 2 Desirable 250, ,000 3 Tolerable 500, ,000 4 Undesirable 700, ,000 5 Highly Undesirable 1,000,000 12,000 1,000,000 6 Unacceptable 1,200,000 15,000 1,200,000 MC is the expected value of the maintenance cost, because maintenance cost is a random variable and the optimal purchasing quantity for each year can be determined only at the time of purchase. MC is affected by the decisions of the product designer in a number of ways: 1. The reliability of the tubes and their expected life depends on the selection of tubes by the product designer and the operating conditions. More reliable tubes will result in less frequent replenishments and smaller inventory costs. 2. The inventory level increases with the number of different tubes utilized (NDT), a parameter controlled by the product designer. Hence, from a maintenance standpoint, the product designer should select the most reliable tubes, minimize the number of tubes for each subsystem, and commonalize tubes as much as possible. These three objectives may conflict with one another and with the productdesigner s goals of minimizing initial capital cost and energy consumption. However, the mathematical problems of both design and maintenance are too complex to be integrated easily into one optimization problem. Therefore, the five-step method described in Section 3 is employed to find satisfactory solutions by modeling interactions between the product designer and the maintenance manager. 4.2 Step 1: Define a Common Objective Function Z to be Minimized Step 1.1 Identify the Relevant Goals for the Product Designer and Maintenance Manager As explained in Section 4.1, the product designer is interested in minimizing the capital cost (DC) of material and the power consumption of the equipment ð _W Þ. The maintenance manager is interested in minimizing the life-cycle costs associated with the inventory of spare parts (MC). TABLE IV Weights for the Goals in a Common Deviation Function. Product design Maintenance k 1. Design cost w 1,k 2. Power consumption w 2,k 3. Maintenance cost w 3,k

12 572 G. HERNANDEZ et al. FIGURE 6 Compromise DSP for maintenance. Step 1.2 Define Six Levels of Desirability for Each of the Goals from Step 1 The product designer and maintenance manager identify six levels of desirability for the goals, DC, _W, and MC. These values are shown in Table III. These values can be specified by each party independently, by a project manager, by a group of stakeholders in a meeting, or by other means suitable for the management style of the enterprise. FIGURE 7 Compromise DSP for the product designer.

13 DESIGN FOR MAINTENANCE 573 Step 1.3 Define the Common Objective Function defined as follows: Next, a common deviation function, Z, is Z ¼ X3 X 5 i¼1 k¼1 w i;k d þ i;k ð9þ The weights, w i;k, are determined with the algorithm described in Ref. [5]. The weights are listed in Table IV. The deviation variables, di;k þ, are associated with the formulation of goals as follows: DC maxðdc DC Target kþ1 ; 0Þ DC Target k þ d 1;k dþ 1;k ¼ 1 ð10þ _W maxð _W _W Target kþ1 ; 0Þ _W Target k þ d 2;k dþ 2;k ¼ 1 ð11þ MC maxðmc MC Target kþ1 ; 0Þ MC Target k þ d 3;k dþ 3;k ¼ 1 ð12þ 4.3 Step 2: Formulate a Compromise DSP for Maintenance The maintenance manager formulates a Compromise DSP (Fig. 6), using the portion of the deviation function, Z, associated with maintenance. In this case, since the maintenance manager has only one objective, there is no need for a weighted sum of deviation variables, and MC may be minimized directly. The Compromise DSP in Figure 6 is solved with a two-step process as follows: 1. Given the length, selection of tubes and the conditions of operation of equipment, a probability model is solved to estimate the expected lifetime for each pipe utilized. 2. Given the expected lifetime and purchasing cost of each pipe, the optimal inventory of spare parts is found using linear programming. Note that this problem can have from 30 variables (for one common tube for all subsystems) to ¼ 480 variables (if all subsystems have a different tube). Solving this problem with nonlinear programming would be very difficult; therefore, it is desirable to solve this portion of the problem independently. This would not be possible if design and maintenance decisions were fully integrated. 4.4 Step 3: Using the Compromise DSP from Step 2, Approximate the BRC of Maintenance Using a Response Surface Next, a parametric model of the best reply correspondence of the maintenance manager as a function of the product design variables is developed using the Compromise DSP in Figure 6. Note that the information needed by the leader (product designer) in this case is only the best reply value of MC as a function of the product design variables. This value is approximated here with a response surface by solving the Compromise DSP of Figure 6 for a number of combinations of the relevant design variables and parameters. For brevity, details on this approach are not included here. The resulting response surface model for maintenance cost follows: MCðX D Þ¼ XNDT MC i ðx D Þ i¼1 ð13þ

14 574 G. HERNANDEZ et al. TABLE V Solution for the Design Variables. m Equipment Tube q Length L qm Passes n tp Tubes N qm 1 Abs Abs Abs Abs Cond Cond Cond Cond Evap Evap Evap Evap HE HE HE HE with MC i ðx D Þ¼162;640 þ 53;134x 1i þ 53;454x 2i þ 37;370x 3i 5132x 2 1i 943x2 2i þ 14;394x 2 3i þ 17;525x 1ix 2i þ 14;468x 1i x 3i þ 10;667x 2i x 3i ð14þ x 1 ; x 2 and x 3 are normalized values for the number of tubes ðn i Þ, the cost of the pipe selected ðc i Þ and a friction factor ð f i Þ: x 1i ¼ 2: ðn i 8000Þ x 2i ¼ 1:4313ðc i 2:175Þ x 3i ¼ 1:461 lnð f i Þþ5:045 ð15þ ð16þ ð17þ The friction factor f i is estimated from the diameter of the tube and the conditions of flow. In Eqs. (15) to (17) the values of x 1i ; x 2i and x 3i range between 1:682 and Given Eq. (13) as an approximation of the expected maintenance cost, a compromise DSP for the product designer is formulated in Step Step 4: Formulate and Solve a Compromise DSP for the Product Designer A compromise DSP for the product designer is formulated as shown in Figure 7. Expected maintenance cost, MC, is included as a goal of product design and the common deviation function defined in Step 1 is the function to be minimized. The resulting compromise TABLE VI Comparison of Solutions. DC Power MC NDT GT 316, ,990 8 Seq 253, ,232, Min DC 164,381 11, , Min _W 998, , Min MC 280, ,494 6

15 DESIGN FOR MAINTENANCE 575 FIGURE 8 Comparison of design cost DC. DSP is a non-linear problem with 64 continuous and discrete design variables; combining this portion of the problem with the compromise DSP for maintenance (Fig. 6) would be difficult computationally. The compromise DSP in Figure 7 is solved using a simulated annealing algorithm with the commercial optimization software OptdesX with 500 cycles and starting from three different initial points. The solution is shown in Table V. 4.6 Step 5: Using the Product Design Solution Found in the Previous Step, Find the Solution to the Maintenance Problem The values for design cost, power consumption, and maintenance cost obtained with this game-theoretic approach are shown in the first row of Table VI, labeled GT. The game-theoretic solution can be compared with a solution obtained by solving compro- FIGURE 9 Comparison of power consumption.

16 576 G. HERNANDEZ et al. FIGURE 10 Comparison of maintenance costs. mise DSP s for design (without maintenance cost) and maintenance sequentially (labeled Seq in Tab. VI), and with three solutions obtained by minimizing DC, _W, and MC, respectively. In Table VI, NDT refers to the number of different types of tubes (combinations of tube selection and length) that are utilized and kept in inventory. All these solutions are compared graphically in Figures 8 to 10. The solution achieved with the proposed game theoretic approach yields the best balance between the three objectives. In this example, there is no motivation for commonalizing tubes from the perspective of the product designer; however, there is a large penalty in terms of maintenance costs associated with component variety. In summary, by modeling the dynamics between maintenance and product design using game theory, it is possible to achieve large savings in maintenance costs while obtaining acceptable levels of achievement for the product design objectives. This result has been achieved without integrating the models of product design and maintenance into a single, large, intractable design problem. 5 CLOSURE Incorporating maintenance considerations during product design usually involves collaboration between various agents within an enterprise. A practical and effective approach for collaborative design is described in this paper, based upon a formal and rigorous mathematical foundation, namely, game theory. To implement this approach, the compromise DSP, with a deviation function adapted from linear physical programming, is employed to model mathematically the decisions of multiple designers. System-level objectives and approximate models of the information and strategies of decision-makers are utilized to facilitate cooperation. This approach is intended for complex systems design for which modeling, computational, or organizational challenges make it difficult to integrate the entire design process into a single decision. The proposed approach has been demonstrated via a design problem involving selection and design of components to facilitate maintenance. This approach is shown to yield satisfactory solutions without requiring complete integration of complex and difficult individual decisions. The knowledge and expertise of each decisionmaker are utilized while keeping computational problems at a tractable level.

17 DESIGN FOR MAINTENANCE 577 Acknowledgements Gabriel Hernandez was sponsored by the National Council of Science and Technology of Mexico (CONACYT) during his doctoral studies. Carolyn Conner Seepersad has been supported by a National Science Foundation Graduate Research Fellowship and is currently supported by the Fannie and John Hertz Foundation. Financial support from NSF grant DMI and from Lutron Electronics is gratefully acknowledged. The cost of computer time was underwritten by the Systems Realization Laboratory of the Georgia Institute of Technology. References [1] ASHRAE (2001). ASHRAE Fundamentals Handbook, ASHRAE Handbook Series. [2] Chen, W., Allen, J. K. and Mistree, F. (1993). Hierarchical selection in gas turbine maintenance management. ASME Advances in Design Automation, ASME DE, 65, [3] Elsayed, E. A. (1996). Reliability Engineering. Addison-Wesley Publishing Company, Reading, Massachusetts. [4] Fudenburg, D. and Tirole, J. (1993). Game Theory. The MIT Press, Cambridge, MA. [5] Hernandez, G., Allen, J. K. and Mistree, F. (2001). The compromise decision support problem: Modeling the deviation function as in physical programming. Engineering Optimization, 33(4), [6] Hernandez, G. and Mistree, F. (2000). Integrating product design and manufacturing: A game theoretic approach. Engineering Optimization, 32(6), [7] Hernandez, G., Seepersad, C. C. and Mistree, F. (2000). Commonalizing subsystem components to facilitate maintenance: A game theoretic approach. 8th AIAA=USAF=NASA=ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, AIAA [8] Korilis, Y., Lazar, A. and Orda, A. (1997). Achieving network optima using Stackelberg routing strategies. IEEE=ACME Transactions on Networking, 5(1), [9] Kreps, D. M. (1990). Game Theory and Economic Modeling. Oxford University Press, Oxford. [10] Lewis, K. and Mistree, F. (1997). Modeling interaction in multidisciplinary design: A game theoretic approach. AIAA Journal, 35(8), [11] Marston, M. (2000). Game based design: A game theory based approach to engineering design. PhD dissertation, G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA. [12] Messac, A. (1996). Physical programming: Effective optimization for computational design. AIAA Journal, 34(1), [13] Messac, A., Gupta, S. M. and Akbulut, B. (1996). Linear physical programming: A new approach to multiple objective optimization. Transactions in Operations Research, 8, [14] Mistree, F., Hughes, O. F. and Bras, B. A. (1993). The compromise decision support problem and the adaptive linear programming algorithm. In: Kamat, M. P. (Ed.), Structural Optimization: Status and Promise. AIAA, Washington, D.C., [15] Myers, R. H. and Montgomery, D. C. (1995). Response Surface Methodology: Process and Product Optimization Using Designed Experiments. John Wiley and Sons, New York. [16] Myerson, R. B. (1991). Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, MA. [17] Rao, J. R. J., Badhrinath, K., Pakala, R. and Mistree, F. (1997). A study of optimal design under conflict using models of multi-player games. Engineering Optimization, 28(1), [18] Rao, S. S. and Freihet, T. I. (1991). A modified game theory approach to multiobjective optimization. ASME Journal of Mechanical Design, 113(3), [19] Vincent, T. L. (1983). Game theory as a design tool. ASME Journal of Mechanisms, Transmissions and Automation in Design, 105(2),

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