Development of web-based decision support system for business process reengineering in a health-care system.

Lee, Chang Won

ABSTRACT

A web-based decision support system (DSS) can provide management with a strategic plan for business process reengineering (BPR) in a health-care system. An attempt is made to develop a DSS for designing, evaluating, and implementing a strategic plan for BPR. A model is developed and analyzed to simulate a real hospital setting. Goal criteria and priorities are identified and established. The model results are evaluated and discussions are made along with web application in order to enhance the model applicability. The model provides the management with valuable information for planning and controlling hospital e-business activities.

INTRODUCTION

As the dynamics of the demanding marketplace and the requirement of competitive advantage have transformed, the need for decision support system (DSS) models for process reengineering in the health-care e-business system has been emphasized. Successful linkages of these planning processes play a critical role affecting business performance (Ackerman, Wall & Borman, 1999; Aldowaisan and Gaafar, 1999; Sheng, 2002; Short and Venkatraman, 1992; and Teng, Grover and Fiedler, 1996). Factors affecting business performance in a health-care e-business system are widely identified. Financial and non-financial factors should be considered together in the health-care decision process. The health-care e-business strategy needs to be based on compromise among the diverse stakeholders in the health-care system. Among DSS types, a model-driven DSS emphasizes access and control of a model that uses data and parameters provided by decision-makers to support them in analyzing a real-world e-business situation.

Due to the web technology and organizational paradigm shift, business operations in healthcare e-business systems may become more tightly coupled with primary business processes such as admissions, capacity, financing, manpower, and revenue planning. Web technology can deliver unprecedented opportunities to reengineer the business processes in health-care systems. Web technology is considered an emerging area for DSS and an important tool for DSS development. Web technology can enrich model-driven DSS. Web-based DSS has the advantages of reducing current technological obstacles of computerized systems, because of better sharing in decision information, and making more efficient decisions with less cost for model implementation (Courtney, 2001; Rao & Turoff, 2000; and Shim et al., 2002).

Strategic business process reengineering (BPR) in a health-care environment is a growing requirement for improving profitability and productivity. Subjective decision-making processes can be very critical in the multiple and complicated problems with trade-off relationships. When management considers several conflicting goals to achieve, a DSS model enables effective results in business processes and other operational environments in health-care e-business system (Sarker & Lee, 1999; and Stoddard & Jarvenpaa, 1995). However, previous DSS studies have rarely explored to develop an integrated model based DSS dealing with comprehensive core functions in health-care e-business system. Thus, an appropriate model development is essential to create a long-term opportunity for BPR in a health-care e-business system.

The purposes of this paper are (1) to develop a DSS model that aims at designing, evaluating, and implementing a strategic plan for BPR in a health-care system, and (2) to provide management with an insight on a web-enabled DSS model that can enhance business performance and refine operational strategy in health-care e-business activities.

Section 2 reviews a brief literature of decision support systems and business process reengineering. Section 3 presents a problem background with data modeling and goal decomposition. Section 4 presents model development dealing with decision variables, constraints and model formulation. Section 5 provides the developed model solution and discussion, followed by a conclusion.

LITERATURE REVIEW

Decision Support System

Decision support system (DSS) is an intelligent model dealing with semi- or ill-defined and structured decision-making problems in order to support better judgment amongst decision-makers. The concept of DSS has been defined as a system using management activities and decision-making types (Gory & Morton, 1971). Decision-making process stages in DSS consist of four stages: intelligence stage for recognizing appropriate problems in management environment, formulation stage for developing possible alternatives, choice stage for selecting a satisfying solution among potential alternatives, and implementation stage for analyzing and evaluating solutions with sensitivity analysis.

Information needs in DSS environment are characterized as requiring different types of information systems and technologies (Min, 1998). DSS has following characteristics: it is designed specifically for facilitating decision-making process and planning process; it is responded promptly for fulfilling decision-makers needs in short and long term; and it is supported intelligently for making better decision. With this context, DSS research has focused on four directions: intelligent computer system, model application, problem-solving model, and user-interface system. DSS applications have extended to collaborative DSS, negotiate DSS, knowledge-based DSS, and web-based DSS.

Web-based DSS is defined as a system that communicates decision support information or tools to decision-makers through a web environment. Web-based DSS is a DSS using web technology in order to provide decision-makers with business information through internets, extranets and intranets. The web technology is considered as an emerging area for development of DSS. Web-based DSS has advantages of reducing current technological obstacles of computerized systems, because of better sharing in decision information, and making more efficient decisions with less cost for model implementation.

DSS in health services sector has steadily appeared in the literature due to the quality of care, information technology, and financial significance (Collen, 1999; and Miller, 1994). Effective decision support systems in health-care systems rely on accurate patient data and health information, efficient decision-making models, and standardized data production mechanisms. Several issues in health-care decision support areas have changed the current research paradigm. Health-care DSS have recently focused on admissions planning (De Veries & Beekman, 1998; and Kurster & Groot, 1996), health-care financing (Mosmans, Praet & Dumont, 2002), information management (Lee & Kwak, 1999), information technology (Forgionne & Kohli, 1996), knowledge management (Pederson & Larsen, 2001), medical diagnosis (Mangiameli, West & Rampal, 2004); patient relationship (Kohli et al., 2001), and resource allocation (Vissers, 1998).

BUSINESS PROCESS REENGINEERING

Business process reengineering (BPR) is defined as fundamental rethinking and innovative redesign of business processes to achieve a dramatic improvement in critical and core measures of performance, such as cost, quality, speed, flexibility, and value-added service. Business process itself is a management philosophy or strategy that considers a collection of management activities taking input resources and deriving valuable outputs for a customer.

BPR characteristics take various forms. There are four types of characteristics based on scope and scale: functional integration, functional refinement, business redefinition, and process redesign. The efforts of BPR focus on (1) getting users of the outcome of the process, (2) merging information-processing into the origin of information production, (3) treating geographically dispersed resources, (4) linking parallel activities, (5) putting the decision point into the process, and (6) capturing necessary information when needed. The goals in BPR decisions are conflicting due to the existence of different goals in each sub-unit. It is difficult to meet current needs of multidimensional sub-units unless a systematic approach to evaluate potential future BPR decisions is undertaken (Davenport & Short, 1990; and Kettinger, Teng & Guha, 1997).

Many BPR issues in a health-care system have appeared in recent literature (Corlett, Maher & Sidman, 1998; Grimson, 2001; and Li, Benton & Leong, 2002). Efforts of BPR in a health-care system are called for within the organization. Most researchers and practitioners agree that BPR success relies on mission, leadership, new investments, process reengineering, resource allocation, and strategic alliance (Ho, Chan & Kidwell, 1999; Kohn, 1994; Newman, 1997; and Seymour & Guillett, 1997).

PROBLEM STATEMENT

Data Collection

The health-care system in this study is a comprehensive hospital, a leading patient-oriented provider of health services. The hospital's goal is to provide high quality and cost-effective health services while enriching a catholic-affiliated organization's mission. The organization has five different independent health-care systems located at different areas. The organization has built a web system with intelligent functions in supporting hospital e-business activities through web-based order communication system (OCS). Top decision-makers from the hospital and a consulting firm have participated in the overall review process. The related goals and criteria are justifies by the task force team. Data templates pertinent to the strategic proposal are derived. The task force team is responsible for all validated data sets from the hospital, examines the data, and acknowledges the validation for the collected data set. Based on the data set, an initial proposal of the DSS model development for process reengineering is established. Some data utilized in this study had been modified slightly to meet a software system requirement, even though the modified data have not distorted the original justification of input data.

The management wants to provide better services for patients in the health-care organization. Among 17 departments in the hospital, OB/GYN/pediatrics departments, five surgery departments, and an internal medicine department are selected for this study since they are the most core areas in the hospital. Characteristics of patients are divided by residency status (resident in the city or non-resident in the city) and visit type (first visits or revisits). Identifying these characteristics is very important to estimate the potential profitability of the hospital. Three major divisions have an admissions goal as well as hospital admissions status and system utilization rate.

GOAL JUSTIFICATION

Establishing goal decomposition and prioritization is completed for the proposed decision support model development. Synthesized priority is calculated for each goal in order to obtain the overall relative importance of the five goals using an expert opinion (Saaty, 1980).

Based on the above data, the goal priorities and the relevant information about business process redesign are established as follows: priority 1 ([P.sub.1])--financial goal ([G.sub.3]), priority 2 ([P.sub.2])--manpower goal ([G.sub.4]), priority 3 ([P.sub.3])--revenue goal ([G.sub.5]), priority 4 ([P.sub.4])--capacity goal ([G.sub.2]) and priority 5 ([P.sub.5])--admissions goal ([G.sub.1]).

[FIGURE 1 OMITTED]

There are sub-goals under five major goals. Financing planning goal has two sub-goals: service expenditure goal ([P.sub.11]) and information facility goal ([P.sub.12]). Manpower planning goal have two sub-goals: manpower utilization ([P.sub.21]) and payroll increase agreement ([P.sub.22]). Revenue planning goal has two sub-goals: total revenue increase ([P.sub.31]) and profitability fulfillment ([P.sub.32]). Capacity planning goal have three sub-goals: accommodation ([P.sub.41]), hospital utilization ([P.sub.42]), and hospital admission ([P.sub.43]). Admission planning goal has three goals: residential patient admission ([P.sub.51]), non-residential patient admission ([P.sub.52]), and revisit patient admission ([P.sub.53]). These sub-goals are prioritized based on internal agreements. Decision-makers such as the hospital president, a chief information officer (CIO), a medical unit director, and a financial unit director have justified the synthesized prioritization of the overall goals for the business process in the health-care organization under consideration.

MODEL DEVELOPMENT

Decision Variables

DSS models in business process reengineering have generally been limited to addressing financial goals, rather than other strategic policies of an organization. In this paper, a DSS model is formulated based on the following information. There are five different types of decision variables embracing 24 decision variables.

[X.sup.a.sub.ij] = admissions level in patient group i (i = 1, 2, 3, and 4) and department j (j=1,2, and 3)

[X.sup.f.sub.i] = financing level for services expenditure (i=1) and for information facilities (i=2)

[X.sup.m.sub.i] = manpower level in different types of work i (i = 1, 2,...,6)

[X.sup.p.sub.i] = payroll level in different types of work i (i = 1, 2,...,6)

[X.sup.r.sub.i] = revenue level in different types of work i (i = 1, 2,...,6)

where [X.sup.a.sub.ij], [X.sup.f.sub.i], [X.sup.m.sub.i], [X.sup.p.sub.i], and [X.sup.r.sub.i] [sup.3] 0

[X.sup.a] is number of admission in patient group i: first-visit resident (i=1), re-visit resident (i=2), first-visit non-resident (i=3), and re-visit non-resident (i=4); department j: OB/GYN/ pediatrics (j=1), surgery (j=2), and internal medicine (j=3). [X.sup.m] is number of manpower in physician (i=1), nurse (i=2), technician I (i=3), technician II (i=4), management I (i=5), and management II (i=6). [X.sup.p] and [X.sup.r] are payroll amounts and revenue amounts in physician (i=1), nurse (i=2), technician I (i=3), technician II (i=4), management I (i=5), and management II (i=6).

Systems Constraints

There are two different types of constraints: system constraints and goal constraints. System constraints (1-3): First-visit resident patient cannot exceed the maximum level of accommodation in each patient group of in OB/GYN/pediatrics (1,800 patients), surgery (900 patients), and internal medicine (850 patients). System constraints (4-6): Re-visit resident patient cannot exceed the maximum level of accommodation in each patient group of in OB/GYN/pediatrics (5,700 patients), surgery (1,900 patients), and internal medicine (2,100 patients).System constraints (7-9): First-visit non-resident patient cannot exceed the maximum level of accommodation in each patient group of in OB/GYN/pediatrics (1,500 patients), surgery (400 patients), and internal medicine (550 patients). System constraints (10-12): Re-visit non-resident patient cannot exceed the maximum level of accommodation in each patient group of in OB/GYN/pediatrics (2,500 patients), surgery (800 patients), and internal medicine (1,200 patients).

Goal Constraints

Financial Planning Goal Constraints have two sub-goals. Goal constraint (13) of sub-goal ([P.sub.11])--Prepare proper funds for service expenditure ($2,5200,000). Goal constraint (14) of sub-goal ([P.sub.12])--Supply an appropriate budget for information facilities ($2,088,000).

Manpower Planning Goal Constraints have two sub-goals. Goal constraints (15-20) of sub-goal ([P.sub.21])--Meet effective utilization of the required human resource level of physician group (37 persons), nurse group (166 persons), technician I (10 persons), technician II (39 persons), management I (53 persons), and management II (13 persons). Goal constraints (21-26) of sub-goal ([P.sub.22])--Achieve the payroll increase agreement by certain percentage points required from the current salary level of physician group ($53,860,000), nurse group ($11,090,000), technician I ($18,070,000), technician II ($13,30,000), management I ($13,250,000), and management II ($14,300,000).

Revenue Planning Goal Constraints have two sub-goals. Goal constraint (27) of sub-goal ([P.sub.31])--Do not allow an over-achievement of total revenue increase ($ 2,860,000) from the current level in terms of patient satisfaction. Goal constraint (28) of sub-goal ([P.sub.32])--Achieve the expected profitability level ($80,000) that is the difference between the expected revenue increase amount and the expected expenditure increase amount.

Capacity Planning Goal Constraints have three sub-goals. Goal constraint (29) of sub-goal ([P.sub.41])--Minimize the under-achievement of the accommodation goal of patients (1,380 persons) that is the sum of three divisions based on the residential status and patient type: capacity utilization percentage with first visit and resident (70%), revisit and resident (80%), first visit and non-resident (40%), revisit and non-resident (50%). Goal constraints (30-32) of sub-goal ([P.sub.42])--Meet the hospital resource utilization capacity to handle total admissions of 9,000 in OB/GYN/pediatrics, 3,500 in surgery, and 4,000 in internal medicine. Goal constraint (33) of sub-goal ([P.sub.43])--Meet the hospital admissions goal of 15,000 new patients in three divisions.

Admission Planning Goal Constraints have three sub-goals. Goal constraint (34) of sub-goal ([P.sub.51])--Minimize the under-achievement of the targeted admission with 70% of hospital admission capacity for residential patients. Goal constraint (35) of sub-goal ([P.sub.52])--Minimize the overachievement of the targeted admission with 30% of hospital admission capacity for non-residential patients. Goal constraint (36) of sub-goal ([P.sub.53])--Meet the targeted goal with 60% of hospital admission capacity for revisit patients.

MODEL FORMULATION

DSS for BPR in the health-care system is to minimize the value of the objective function subject to goal constraints (13)-(36), satisfying the preemptive priority rules, as shown in Table 1.

MODEL ANALYSIS AND DISCUSSION

Model Analysis

In this DSS model, a non-dominated solution has been sought. A non-dominated solution is defined in the following manner: a feasible solution to a multicriteria decision-making problem is non-inferior, if no other feasible solutions derive an improvement in one objective, without creating a trade-off in another objective. Regardless of the weighting structures and the goals, this model can lead to inferior, sub-optimal solutions. These solutions are not necessarily the optimal ones available to the decision-maker. Therefore, it is called a satisfying solution. Opportunity costs are given as well as the increases and decreases in the values of the coefficients and the right-hand-side elements. Management can determine in advance what will happen if the outcome deviates from overall objectives. In the example given, management can use the information from the solutions to alter their decision variables as any plan can come up with the new satisfying solution.

Web Applications

Web-based DSS is important for strategic decision-making process. More effective DSS can be implemented by web-based model dealing with organizational view in decision-making processes. Recent DSS applications take advantage of the opportunities in web technologies, along with other internet technologies. With this perspective, web-based DSS can be one of the most promising options, increasing core business competition in the new health-care market environment. However, simply making an existing DSS accessible by using a web technology to hospital managers, patients or other stakeholders will often lead to unsatisfactory results and less competitiveness within the market. Developing the user interface, modeling, data mining for web-based DSS remains hard and major tasks. Thus, developing web-based DSS model should be considered system's flexibility for the future expansions. Figure 1 shows brief web-based DSS components along with their relationships.

The hospital has launched its strategic DSS model with on-going base. The hospital top decision-makers have accepted the final results as valid and feasible outputs for implementing the DSS model in their web environment. The effects from this model outputs will be evaluated in the next two or three fiscal years, since any mistakes in medical and clinical management may result in serious damage for the patients and the hospital operations and performance. The future agenda will be arranged to compare with this proposed DSS model for hospital BPR planning. The BPR planning based on the model will provide the management with a significant insight to set an appropriate hospital resource planning and order communication system in their web environment, while meeting customer demand and enhancing competitive advantages. Thus, the hospital currently reviews all these planning strategies as the possible alternative operations.

CONCLUSION

In today's information technology age, rapid penetration of web technology into a business process enables more efficient and strategic management decisions. The health-care environment is not an exception to this trend. The growth of web technology can allow decision-makers to overcome many of the challenges confronting health-care systems. Health-care business environments are dramatically changing with multiple and complicated decision-making problems. The global health-care environment provides new business markets to management.

The DSS model for hospital business process planning is developed and analyzed to aid total resource planning. The health-care system in this study considers the proposed DSS model as the potential business strategies. Thus, this study provides both the satisfying solution and other important implications.

This study's contributions are as follows. This proposed DSS model enhances a practical way for planning the hospital business process planning considering tangible and intangible business aspects. Previous studies in DSS are limited to covering simultaneously comprehensive issues such as patient admission, hospital capacity, financing, manpower, and revenue. This study provides the management with insights improving overall performance through web-enabled hospital business process. This study utilizes an integrated multicriteria decision-making model that most previous studies in developing DSS models have not been explored in health-care area. The proposed results make better implication and more meaningful suggestions to the real decision-making settings.

The hospital decision-makers have accepted the final results as valid and feasible for implementing the hospital business process planning in real-situation. This proposed model's outputs will be evaluated during the next two or three fiscal years. In short, in situations where many complex e-business activities and conditions are involved, it can be much more practical to use the proposed DSS model to find a satisfying solution, especially when time and resources are limited.

ACKNOWLEDGMENT

This work was supported by the Korea Research Foundation Grant (KRF-2001-041-C00346).

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Chang Won Lee, Jinju National University

Table 1. Modeling for BPR in Health-Care System

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Subject to

[X.sup.a.sub.11] 1800 [pounds sterling] (1)

[X.sup.a.sub.12] 900 [pounds sterling] (2)

[X.sup.a.sub.13] 850 [pounds sterling] (3)

[X.sup.a.sub.21] 5700 [pounds sterling] (4)

[X.sup.a.sub.22] 1900 [pounds sterling] (5)

[X.sup.a.sub.23] 2100 [pounds sterling] (6)

[X.sup.a.sub.31] 1500 [pounds sterling] (7)

[X.sup.a.sub.32] 400 [pounds sterling] (8)

[X.sup.a.sub.33] 550 [pounds sterling] (9)

[X.sup.a.sub.41] 2500 [pounds sterling] (10)

[X.sup.a.sub.42] 800 [pounds sterling] (11)

[X.sup.a.sub.43] 1200 [pounds sterling] (12)

[X.sup.f.sub.1] + [d.sup.-.sub.1] - [d.sup.+.sub.1] = 2520 (13)

[X.sup.f.sub.2] + [d.sup.-.sub.2] - [d.sup.+.sub.2] = 2080 (14)

[X.sup.m.sub.1] + [d.sup.-.sub.3] - [d.sup.+.sub.3] = 37 (15)

[X.sup.m.sub.2] + [d.sup.-.sub.4] - [d.sup.+.sub.4] = 166 (16)

[X.sup.m.sub.3] + [d.sup.-.sub.5] - [d.sup.+.sub.5] = 10 (17)

[X.sup.m.sub.4] + [d.sup.-.sub.6] - [d.sup.+.sub.6] = 39 (18)

[X.sup.m.sub.5] + [d.sup.-.sub.7] - [d.sup.+.sub.7] = 53 (19)

[X.sup.m.sub.6] + [d.sup.-.sub.8] - [d.sup.+.sub.8] = 13 (20)

[X.sup.p.sub.1] + [d.sup.-.sub.9] - [d.sup.+.sub.9] = 59,400 (21)

[X.sup.p.sub.2] + [d.sup.-.sub.10] - [d.sup.+.sub.10] = 13,200 (22)

[X.sup.p.sub.3] + [d.sup.-.sub.11] - [d.sup.+.sub.11] = 19,800 (23)

[X.sup.p.sub.4] + [d.sup.-.sub.12] - [d.sup.+.sub.12] = 14,850 (24)

[X.sup.p.sub.5] + [d.sup.-.sub.13] - [d.sup.+.sub.13] = 14,850 (25)

[X.sup.p.sub.6] + [d.sup.-.sub.14] - [d.sup.+.sub.14] = 15,950 (26)

[X.sup.P.sub.1] + [X.sup.R.sub.2] + [X.sup.R.sub.3] + (27)
[X.sup.R.sub.4] + [X.sup.R.sub.5] + [X.sup.R.sub.6] -
[X.sup.+.sub.15] = 2860

[X.sup.R.sub.1] + [X.sup.R.sub.2] + [X.sup.R.sub.3] + (28)
[X.sup.R.sub.4] + [X.sup.R.sub.5] + [X.sup.R.sub.6] -
[X.sup.B.sub.1] + [X.sup.-.sub.16] - = [X.sup.+.sub.16] = 80

0.77[X.sup.a.sub.11] + 0.7[X.sup.a.sub.12] + (29)
0.7[X.sup.a.sub.13] + 0.8[X.sup.a.sub.21] +
0.8[X.sup.a.sub.22] + 0.8[X.sup.a.sub.23]
+ 0.4[X.sup.a.sub.31] + 0.4[X.sup.a.sub.32] +
0.4[X.sup.a.sub.33] + 0.5[X.sup.a.sub.41] +
0.5[X.sup.a.sub.42] + 0.4[X.sup.a.sub.33] +
[d.sup-.sub.17] = 1380

[X.sup.a.sub.11] + [X.sup.a.sub.21] + [X.sup.a.sub.31] + (30)
[X.sup.a.sub.41] + [d.sup.-.sub.18] -
[d.sup.+.sub.18] = 9000

[X.sup.a.sub.12] + [X.sup.a.sub.22] + [X.sup.a.sub.32] + (31)
[X.sup.a.sub.42] + [d.sup.-.sub.19] -
[d.sup.+.sub.19] = 3500

[X.sup.a.sub.13] + [X.sup.a.sub.23] + [X.sup.a.sub.33] + (32)
[X.sup.a.sub.43] + [d.sup.-.sub.20] -
[d.sup.+.sub.20] = 4000

[X.sup.a.sub.11] + [X.sup.a.sub.12] + [X.sup.a.sub.13] + (33)
[X.sup.a.sub.21] + [X.sup.a.sub.22] + [X.sup.a.sub.23] +
[X.sup.a.sub.31] + [X.sup.a.sub.32] + [X.sup.a.sub.33] +
[X.sup.a.sub.41] + [X.sup.a.sub.42] + [X.sup.a.sub.43] +
[d.sup.-.sub.21] + [d.sup.+.sub.21] + = 15,000

0.3[X.sup.a.sub.11] + 0.3[X.sup.a.sub.12] + (34)
0.3[X.sup.a.sub.13] + 0.3[X.sup.a.sub.21] +
0.3[X.sup.a.sub.22] + 0.3[X.sup.a.sub.23] -
0.7[X.sup.a.sub.31] - 0.7[X.sup.a.sub.32]
- 0.7[X.sup.a.sub.33] - 0.7[X.sup.a.sub.41] -
0.7[X.sup.a.sub.42] - 0.7[X.sup.a.sub.43] +
[d.sup.-.sub.22] = 0

-0.3[X.sup.a.sub.11] - 0.3[X.sup.a.sub.12] - (35)
0.3[X.sup.a.sub.13] - 0.3[X.sup.a.sub.21] -
0.3[X.sup.a.sub.22] - 0.3[X.sup.a.sub.23] +
0.7[X.sup.a.sub.31] + 0.7[X.sup.a.sub.32] +
0.7[X.sup.a.sub.33] + 0.7[X.sup.a.sub.41] +
0.7[X.sup.a.sub.42] + 0.7[X.sup.a.sub.43] -
[d.sup.-.sub.23] = 0

-0.6[X.sup.a.sub.11] - 0.6[X.sup.a.sub.12] - (36)
0.6[X.sup.a.sub.13] - 0.4[X.sup.a.sub.21] +
0.4[X.sup.a.sub.22] + 0.4[X.sup.a.sub.23] +
0.6[X.sup.a.sub.31] - 0.6[X.sup.a.sub.32] -
0.6[X.sup.a.sub.33] + 0.4[X.sup.a.sub.41] +
0.4[X.sup.a.sub.42] + 0.7[X.sup.a.sub.43] +
[d.sup.-.sub.24] + [d.sup.+.sub.24] = 0