Corporate optimal production planning with varying environmental costs: A grey compromise programming approach

European Journal of Operational Research 155 (2004) 68–95 www.elsevier.com/locate/dsw

Production, Manufacturing and Logistics

Corporate optimal production planning with varying environmental costs: A grey compromise programming approach Chia-Chin Wu a, Ni-Bin Chang b

b,*

a Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC Department of Environmental Engineering, Texas A&M University, Kingsville, TX 78363, USA

Received 13 March 2001; accepted 7 October 2002

Abstract Corporate environmental and resources management has become more strategically oriented when many pollution charges, environmental taxes, and resources conservation fees have been gradually imposing to the industry. The need for explicit consideration and incorporation of varying environmental costs within production-planning program is becoming critical to corporate management. This paper attempts to assess an optimal production-planning program in response to varying environmental costs in an uncertain environment. The optimal production strategy concerning numerous screening of possible production alternatives of dyeing cloth in a textile-dyeing firm in terms of market demand, resources availability, and impact of environmental costs is treated as an integral part of the multi-criteria decision-making framework based on the grey compromise programming approach. It covers not only the regular part of production costs and the direct income from product sales but also the emission/effluent charges and water resource fees reflecting part of the goals for internalization of external cost in a sustainable society. In particular, all the crucial variables in the model are addressed by interval expressions, the same as they are frequently applied in the grey systems theory, in support of a vital uncertainty assessment, which is much better suited for this particular study than other approaches. Research results demonstrate the applicability and significance of such an approach based on a case study. Industry looking for the competitive advantage of environmental management must be aware of the potential benefits from such an integrated production-planning program once the trend of increasing pollution charges, environmental taxes, and resources conservation fees remains. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Environmental management; Uncertainty analysis; Grey systems theory; Multiple objective programming; Optimal production planning

*

Corresponding author. Tel.: +361-593-3898455-3179; fax: +361-593-2069. E-mail address: [email protected] (N.-B. Chang).

0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00820-2

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

69

1. Introduction Understanding the environmental hazards and socio-economic impacts from production activities have been recognized as the initial effort towards a new social responsibility in industry. However, whether to implement sustainable corporate environmental and resource management practices or not is no longer a real choice for industrialists today, given the ultimate limitations of Nature––limitations that are close to being reached (Ulhøi, 1995). Yet it has been still far from being an operational concept for individual companies until the emergence of economic instruments from the public side and the environmental management system (EMS) in ISO 14001-accreditation program from the private side. The use of economic instruments, such as pollution charges or environmental taxes, as policy tools to internalize the external cost for the justification of possible market failure in the society, have gradually driven environmental management from the focus of ‘‘command and control’’ to the utilization of ‘‘economic incentives’’. In recent years, many planners and decision makers in corporate centers have started to recognize the role and significance of the integration of economic and environmental efforts in a single production-planning program. This trend has been initiating an increasing awareness of the importance of ‘‘green production planning’’ and ‘‘life cycle assessment’’ on the overall environment (Guide et al., 1996). Environmental performance is no longer classified as the use of add-on technologies to comply with environmental laws. The traditional ‘‘end-of-pipe’’ approach has proven to be costly and ultimately unsustainable (Higgins, 1990). Hence, the current industry must consider flexible planning and design to accommodate the rapid changes in market demands and respond effectively to new environmental policies, such as the emergence of various pollution charges, environmental taxes, resource conservation fees, and technology evolution. Production planning develops a series of projects concerning the production activities to achieve business targets of enterprises. Production planning programs in the late 1960s and earlier 1970s cover many fundamental aspects of production issues including the level of production, production scheduling, methods of production, or supply of raw material that are traditionally proposed as an issue with regards to cost minimization. They can be broadly classified as seven categories: (1) the linear decision rule (Holt et al., 1955); (2) transportation method (Bowman, 1956); (3) linear programming (Hanssmann and Hess, 1960; Eilon, 1975); (4) management coefficient method (Bowman, 1963); (5) production switching (Orr, 1962; Mellichamp and Love, 1978); (6) parametric production planning (Jones, 1967); (7) search decision rule (Taubert, 1968). In addition to the work above, later studies found that multiple conflicting objectives frequently characterize the complex production systems. Lee (1973) first applied the goal programming technique to cover the objective functions for the maximization of sales, the maximization of production capacity, the minimization of inventory, and the minimization of overtime working simultaneously. The others further examined the multi-objective programming scheme applied for the production planning (Hindelang and Hill, 1978; Arthur and Lawrence, 1982; Rakes et al., 1989; Tabucanon and Mukysnkoon, 1985). The ways for improving a cost-effective production-planning program in the last decade frequently focused on applying optimization approaches within the manufacturing stage. For example, Gupta and Manousiouthakis (1990) started to use an innovative approach––the mass exchange network synthesis (MEMS) for optimizing the removal of the multiple toxic substances and other species from plant streams, while achieving specified separation targets at minimum operating costs. Douglas (1992) proposed a hierarchical decision procedure that provides a simple way to identify potential pollution problems early in a design program. Some applications of such a hierarchical decision procedure have been examined and found useful in planning the replacement component in production (Krupp, 1992; Rossiter et al., 1993; Smith and Patella, 1994; Mizsey and Fonyo, 1995). Paluzzi and Greiner (1993) drafted a plan to reduce hazardous substance consumption and eliminate wastewater discharges by applying the principle of total quality environmental management (TQEM). Kleiner (1993), Clarke et al. (1994), and Walley and Whitehead (1994) have devoted considerable attention to the effect of increasing environmental costs of

70

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

production to both individual firms and to society as a whole. Ahmad and Subhas (1994) introduced a new methodology, based on a generic pollution balance equation, to minimize waste production in manufacturing processes. This goal could also be achieved by one of the most promising optimization methods–– the mixed-integer nonlinear programming model, which provides us with a simultaneous synthesis procedure for the overall manufacturing process through using a computer software package (PROSYN) (Kravanja and Grosmann, 1994). Sarkis and Rasheed (1995) and Gupta (1995) further described the environmentally conscious manufacturing strategies and environmental challenges for a modern firm in terms of operations objectives. In particular, Saydam and Cooper (1995) developed a computer-based information system (i.e., a linear programming model) to aid in scheduling of textile fabric rolls through the dyeing process without considering recycling impacts. Due to increasing environmental concerns associated with incorporating pollution prevention strategies into those industrial processes, various waste minimization programs were developed for coping with different industrial pollution problems (Crittenden and Kolaczkowski, 1995). Recent work have also been focusing on integrated environmental and resources management in terms of a holistic approach (Ulhøi, 1995; Spengler et al., 1997). Besides, Guide et al. (1996) and Russo and Fouts (1997) highlighted the possible linkage between environmental and economic performance. However, uncertainty plays an important role in decision-making. Interval analysis was explored by Moore that was recognized as the emergence of the idea of grey systems theory (Moore, 1979). Later on, the grey systems theory was formally introduced by Deng (1984a,b, 1986), in which all systems are divided into three categories, including the white, grey, and black parts. While the white part shows completely certain and clear messages in a system, the black part reveals totally unknown characteristics. The message released in the grey part is in between which can be easily described as an interval or grey number. Hence, the grey uncertainty covers both known and unknown messages. In an interval analysis, with limited samples during investigation, a parameter can better be defined as a closed interval with upper and lower limits while the vagueness of its intrinsic characteristics remains. With an iterative experimental process, however, the range of an interval number could differ over time and a probabilistic distribution could be identified eventually. To ease the real world applications, it is much easier to define an interval or a grey number in a grey system in terms of a closed interval with upper and lower limits. Huang and Moore (1993) and Huang et al. (1992, 1993, 1994, 1995a,b) developed a series of mathematical programming models using grey systems theory as a tool for illustrating the uncertainties in the environmental planning processes. The spectrum of applications rapidly covers various problems of systems analysis that include the use of grey linear programming (GLP), grey fuzzy linear programming (GFLP), grey fuzzy dynamic programming (GFDP), and grey integer programming (GIP) approaches (Huang et al., 1992, 1993, 1994, 1995a,b). Grey multi-objective programming was also applied for solving solid waste and watershed management issues (Chang et al., 1996a,b, 1997). The characterization of grey compromise programming and its stability analysis have been fully discussed by Chang et al. (1999). Previous studies of environmental costs in a production system rarely clarified the impacts from pollution charges, environmental taxes, and resource conservation fees that may result in a higher risk in production decision-making. Within the last decade, those pollution charges are normally incurred with respect to the official required levels in relation to air emission, wastewater discharge, and solid waste disposal. The consumption of surface or groundwater resources in the manufacturing process may be subject to pay for the water right gained on a regular basis. But assessing the interactions between the production planning, pollution control, and the impact of pollution charges and resource conservation fees in an integrated industrial production program by a multi-criteria decision-making framework remains unclear due to not only the conflicting objectives but also the embedded systematic uncertainties. This paper is designed to first address the policy evolution with regards to applying the economic instruments as policy tools for improving environmental resources conservation in Taiwan, and then concentrates on deriving the grey compromise-programming model for demonstrating an environmental and resources

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

71

management model in search of optimal production strategies in a multiple-production-line system with obtainable resource capacity. The case study would delineate the grey approach is better suited for this particular study than other approaches, such as deterministic goal programming approach.

2. Policy analysis of using the economic instruments for environmental resources conservation in Taiwan The conservation of environmental resources had never been so much emphasized in Taiwan until the late 1990s. To create substantial incentives for sustainable development, charge systems for pricing water resources consumption and water/air pollution impacts to the local ecosystem were intensively proposed, discussed, and designed between 1995 and 1998 in Taiwan. Emissions and effluent taxes or fees are conceptually regarded as ‘‘Pigouvian environmental taxes’’, which is identical to the ‘‘pollution charge’’. They are imposed directly on the pollutant and are set at rates equal to the incremental damage from an additional unit of pollutant at the optimum theoretical pollution level. Although those analyses in scholarsÕ work are of interest, practical implementation, however, must rely on a simplified estimation mechanism. After a long-term debate by the legislators, the pollution charge for air emissions in the initial stage has been formally putting into practice in Taiwan since July, 1995. However, it was originally performed as an indirect pollution charge being collected through the oil or gasoline consumption process. In order to be more consistent with the ‘‘polluters-pay-principle (PPP)’’, a new program based on the direct emissions of hazardous pollutants, such as SOx , NOx , and VOC, from point sources and construction sites has been initiated for possible further improvement of air quality since July 1997. The standards designed for air pollution charge based on various application levels of air pollution control device (APCD) in Taiwan are listed in Table 1. On the other hand, the standards for water pollution charge is described in terms of the ‘‘equivalent hazard’’ in relation to several designated pollutants in the effluents. The term ‘‘equivalent hazard’’ is defined below.

Table 1 The air pollution charge in Taiwan for stationary sources Charge NOx 12 NT$/kg 10 NT$/kg 8 NT$/kg 6 NT$/kg 0 NT$/kg SOx 10 NT$/kg 7.5 NT$/kg 5 NT$/kg 2.5 NT$/kg

Criterion Uncontrolled stationary sources Monthly average emission concentration is lower than three-fourth of the emission standard via the use of APCD Monthly average emission concentration is lower than one half of the emission standard via the use of APCD Monthly average emission concentration is lower than one-fifth of the emission standard via the use of APCD Using natural gas as fuel and low-NOx burner Uncontrolled stationary sources Monthly average emission concentration is lower than one-fifth of the emission standard via the use of APCD Monthly average emission concentration is lower than one-tenth of the emission standard via the use of APCD Monthly average emission concentration is lower than one-twentieth of the emission standard via the use of APCD

Note: The currency ratio is 31NT$/1US$ in 1999.

72

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

Table 2 The transfer coefficients of equivalent hazard used in Taiwan Pollutant

Hazard equivalent/kg

Suspended solids Chemical oxygen demand Mercuric compounds Cadmium Chromic compounds Lead Nickel Copper Arsenic Ammonia nitrogen Phosphate Cyanide Phenolic compounds

0.001 0.022 50 10 2 1 1 1 2 0.04 0.33 10 10

The equivalent hazard of pollutant i ¼ the flowrate of effluent (m3 )
103 (l/m3 )
the average concentration of the pollutant i (mg/l)
106 (kg/mg)
the transfer coefficient of equivalent hazard of pollutant i. Obviously, pollution charge due to the disposal of treated wastewater effluents is intimately related to both the quantity and the quality of the effluents simultaneously. The transfer coefficients prepared for illustrating the equivalent hazard with respect to those designated pollutants are listed in Table 2. Besides, in the earlier time period in Taiwan, water resources were regarded as a public good and could be distributed according to the pre-registered water rights by the public or private sectors, irrespective of any special charges. Due to variations of annual rainfall, however, river flows usually vary with location and time. Without sufficient number of reservoirs, the shortage of the water supply limited many regional development programs. Such impacts were especially critical in South Taiwan where the uneven stream flows has resulted in an extremely subtle condition of water resources consumption in the last few years. As a result, in order to fully utilize the economic instruments for preventing the possible abuse of water resources, pricing policy for water resources was seriously considered by the Ministry of Economics from 1995 to 1997. Like many European countries and the United States, the implied economic value of water resources from a sense of resource conservation should be considered in terms of many influential factors, such as the abundance of rainfall, seasonal effect, category of consumers, location of sources, and the availability and reliability of water supply (i.e., surface or groundwater). Accordingly, the possible fee of conserving water resources, as listed in Table 3, was formally proposed in 1998 in Taiwan. The standards mainly vary with the consumption locations, the categories of uses, and the sources of water. Overall, in addition to mandating compliance with the effluent standards, both ‘‘water pollution charge’’ and ‘‘water resources fee’’ are highly likely to be implemented in the year 2003. No matter in which formats the economic instruments will be implemented in the way to change the consumerÕs behavior, enterprises for compliance with the new policy will face a challenge of increasing production cost via the charge systems that can be taken into account as sort of ‘‘fees’’ or ‘‘taxes’’ practically in the production program. This may inevitably generate phenomenal cost impacts to those water-consuming industries, such as petrochemical, steel manufacturing, pulp, and textile industries. Advanced cost analysis in relation to production planning is essential for handling such issues. The following case study would demonstrate how could we apply the multi-criteria decision-making approach to figure out the optimal production-planning program in an uncertain environment.

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

73

Table 3 The proposal of water resource fees in Taiwan (NT$/m3 ) Area

Utilization

Surface water

Groundwater

Northern Taiwan

Water supply Agricultural use Hydraulic use Industrial use Other use

0.23 0.19 0.21 0.38 0.37

1.10 0.59 1.27 1.68 1.68

Central Taiwan

Water supply Agricultural use Hydraulic use Industrial use Other use

0.29 0.24 0.26 0.46 0.46

1.59 0.90 1.94 2.58 2.55

Southern Taiwan

Water supply Agricultural use Hydraulic use Industrial use Other use

0.38 0.32 0.34 0.62 0.62

1.44 0.86 1.85 2.46 2.45

Eastern Taiwan

Water supply Agricultural use Hydraulic use Industrial use Other use

0.28 0.23 0.25 0.42 0.45

0.64 0.36 0.77 1.03 1.02

Note: (1) The above chart only identifies the rate in wet season and the rate in the dry season is 30% higher. (2) If the average rainfall in the dry season in a year is higher than that in the wet season in the last year, the rate of both seasons will be the same. (3) The currency ratio is 31NT$/1US$ in 1999.

3. Case study Textile industry has played an important role in economic development since the early 1980s in Taiwan. In 1991, the textile industry became one of the largest industrial sectors in terms of the international trade in Taiwan, among which the printing, dyeing, and finishing industry (i.e., the textile dyeing industry) plays a significant role in the entire textile industry. However, as it becomes a major sector in export towards making tremendous international trade surplus, the generation of a lot of effluents with high organic strength and colors also causes serious environmental pollution. Recently, the government has promulgated or has been preparing several environmental laws and regulations regarding the use of economic instruments as policy tools for pollution prevention and control. The pricing of environmental resources with respect to water consumption and pollution impact due to the discharge of effluents/emissions into water and air media has been conducted since 1996. Imposing water/air pollution charges and water resource fees, commencing 2003 probably, however, will create obvious cost impacts to the printing, dyeing, and finishing industries. This practice describes how to assess such cost impacts within a textile-dying production system using the grey compromise-programming model to address the conflicting objectives and the embedded uncertainties simultaneously leading to provide a set of flexible production solutions. 3.1. The development of printing, dyeing, and finishing industry in Taiwan The textile dyeing industry provides a vital link between the upstream man-made fiber, spinning and knitting industries, and the downstream ready-made clothes and apparel industries. Furthermore, it bears the responsibilities of processing upstream raw material and preparing for downstream manufacturing of

74

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

clothes. According to the governmental investigation (Annual Statistical Report, 1999), there are 343 printing, dyeing, and finishing mills registered in Taiwan in which approximately two-third of them are organized as medium or small firms in size to retain their flexibility in business operation. They tend to process products for other textile firms with relatively diversified operational patterns in order to meet the rapidly changing requirements in the hierarchical production network. Competition in the international trade market, however, has resulted in less profitable condition in recent years. 3.2. The impacts of pollution charges and water resource fees The wastewater effluents, discharged from the printing, dyeing, and finishing processes, are often concerned by the public with regards to the potential impacts of metals, salt, color, and aquatic toxicity to the local ecosystem. Especially, the effluent from the printing operation usually imposes remarkable impacts due to the higher concentrations of BOD and COD, while the effluents from the finishing operation, such as softening and waterproofing, often generates additional impacts due to the higher concentration of total suspended solids (TSS). Besides, NOx and SOx emissions from the boiler system, that is required for producing vapor for use in printing, might result in air pollution impact in the atmospheric environment. Table 4 summarizes the sources of pollution produced when handling the printing, dyeing, and finishing processes. Although process difference in similar production programs always exists in various types of printing, dyeing and finishing factories, the wastewater treatment processes in these factories commonly involve typical unit operations of pH adjustment, color removal, and organics removal. Depending on the types of fibers in raw material, the kinds of chemicals required and the demands of products in the market, the characteristics of the wastewater effluents discharged from the textile dying industry may vary apparently. Tables 5 and 6 describe the typical characteristics of those wastewater streams before and after on-site treatment. In recent years, the environmental issues in relation to the printing, dyeing, and finishing industry in Taiwan have been growing in both quantity and complexity. First, due to the environmental rules and regulations that are increasingly stringent over time, existing wastewater treatment facilities and production equipment may need to be further retrofitted to meet the new governmental requirements. Second, the costs for pollution prevention and control are getting higher even though there is no large-scale retrofit program required. Third, the public tends to be better educated in terms of environmental knowledge, which inevitably generate stronger feeling for environmental protection. The industry may be further urged to follow governmental rules and regulations more closely. Any violation of the environmental regulation often leads to not only the penalty with monetary loss, but also the serious problems of public relationship. Table 4 Pollutants profile from printing, dyeing and finishing process Types of pollution

Sources of pollution

pH

Bases: souring, mercerizing, bleaching unit and vat dyes, sulfurized dyes, coloring from reactive dyes, reductive washing; Acids: acidic dyes, chromium dyes, acetic acid, hydrochloride bleaching agents Nap, scrap fibers, paste, auxiliaries Paste, surfactants, oils, auxiliaries Reductive bleaching agents, decolorants and detergents, paste, aldehyde compounds from the dyes Chromium dyes, direct dye fixation, soluble vat dyes, oxidation dyes, etc. Special bleaching agents, resist agents, decolorants, hydro-sulphites Iron and coal dyes, pigment from natural dyes, soluble vat dyes Fixing agents, metal complex dyes, direct dyes, post-treatment dyes, oxidation dyes, auxiliaries for natural dyes High temperature discharges from the dyeing and scouring units

Suspended solid BOD COD Trivalent chromium Manganese, Zinc Iron Copper Temperature

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

75

Table 5 The characteristics of wastewater streams generated from printing, dyeing and finishing mills before on-site treatment Items

Cotton fiber

Wool fiber

Synthetic fiber

Mixed fiber

Total

pH COD (mg/l) BOD (mg/l) SS (mg/l) Opacity (cm) Temperature (°C)

5.95–11.50 525–2800 250–1200 150–500 1.0–15.0 30–68

6.137–7.70 333–630 120–258 9–180 2.0–12.0 33–42

5.00–11.00 310–1400 65–450 27–360 1.0–10.0 37–53

6.00–11.00 350–2500 138–500 75–650 0.1–8.0 28–50

5.00–11.50 310–2800 65–1200 9–650 0.1–15.0 28–68

Table 6 The characteristics of wastewater streams discharged from the printing, dyeing and finishing mills after on-site treatment Items

Cotton fiber

Wool fiber

Synthetic fiber

Mixed fiber

Total

pH COD (mg/l) BOD (mg/l) SS (mg/l) Opacity (cm) Temperature (°C)

6.44–8.00 80–250 20–70 16–60 8.0–30.0 27–30

6.50–7.50 54–82 4–23 2–10 20.0–30.0 27–30

5.50–8.10 52–300 10–120 6–60 10.0–50.0 21–35

6.20–9.50 36–205 1–80 4–95 10.0–30.0 25–40

5.50–9.50 36–300 1–120 2–95 8.0–50.0 21–40

Finally, there are higher needs of technologies for pollution prevention and control in order for the industries to meet the new wastewater discharge standards. Hence, some of the printing, dyeing, and finishing firms in Taiwan are accelerating various waste minimization programs with the considerations of possible alternatives for wastewater recycling and energy saving in their production programs. 3.3. Formulation of optimization model Production planning with the consideration of environmental cost impacts has become a growing segment of the overall effort to gain the competitive excellence in the market. But many production planning and control may not be able to address the upcoming issues of potential costs for pollution charges and resource conservation fees. The following optimization model describes the method and procedure for optimizing a textile-dyeing production program with respect to the potential impact of pollution charges/ water resources fees and the limitations of available production resources simultaneously. Such application covers not only the production costs and direct incomes from production sale but also the emission/effluent charges and water resource fees from environmental perspectives, which notably reflects the basic requirement of internalizing the external cost in the short-run. To cover the uncertainties in the identification of some parameter values, grey variables are defined as long as they are needed in the formulation. Based on such variations of parameter values, the use of grey programming approach is absolutely better suited for this particular study than other approaches, such as deterministic goal programming approach. All grey variables, such as Xi;j that is defined as the yield of product ÔiÕ (dyeing cloth), are expressed as interval variables, such as Xi;j , in the following model. 3.3.1. Objective function Decision makers in charge of the corporate operation have to cogitate not only the maximization of business profit, but also additional considerations with respect to environmental impacts and green initiatives. In this study, to ensure maximum consensus, three broad-based planning objectives were proposed for such an optimal production planning: (1) minimization of total production cost; (2) maximization of total production capacity; (3) minimization of inventory cost. They are described below. Except for the

76

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

definitions of notation listed below each objective function and constraint, nomenclature is also prepared at the end of this paper as an appendix. (1) Minimization of total production cost. In the short-run, the annual production cost of a firm can be expressed as the summation of production cost incurred in the daytime and nighttime if there are two turns in a routine working day. However, the production cost should be figured out in terms of the cost impacts of imposed economic instruments so that it may provide the essential trade-off in the multiobjective evaluation of optimal production scheme. The objective function can be formulated as below: Min F1 ¼

12 X 2 X j¼1

   ðc ti
Xi;j þ dti
Yi;j Þ

ð1Þ

i¼1

in which the Xi;j is the yield of product ÔiÕ (dyeing cloth) in the daytime in the jth month (m); Yi;j is the yield of product ÔiÕ (dyeing cloth) in the nighttime in the jth month (m); cti is the unitary production cost of product ÔiÕ incurred in the daytime t (NT$/m); and dti is the unitary production cost of product ÔiÕ incurred in the nighttime t (NT$/m). (2) Maximization of total production capacity. This objective function is defined to maximize the utilization efficiency of production equipment in the production planning. The objective function can be formulated as below: !   12 X 2 X X Y i;j i;j Max F2 ¼ s þ  ð2Þ i  Si;j Si;j j¼1 i¼1 in which si is the required time of production equipment for producing unitary product ÔiÕ in the jth month (hours/m); and Si;j is the time of key equipment available for producing product ÔiÕ in the jth month (hours). (3) Minimization of inventory cost. Inventory of products inevitably causes additional opportunity cost. The inventory cost can be properly minimized with regard to the demand and production information in such a planning scheme. The objective function can be formulated as below: Min F3 ¼

12 X 2 X j¼1

 e i
Zi;j

ð3Þ

i¼1

in which Zi;j is the inventory of product ÔiÕ (dyeing cloth) in the jth month (m); and ei is the unitary inventory cost of product ÔiÕ (NT$/m). 3.3.2. Constraint set In general, if the market price and production cost are fixed, production profit should be proportional to the production capacity. However, there exist some restrictions when figuring out the optimal production program. They include certain amount of interior limitations with regard to the material, manpower, and equipment available for production and external limitations with respect to the market demand as well as the rising environmental costs, such as pollution charges and resource fees. The basic constraint set in this model therefore consists of the demand constraint, labor constraint, production capacity constraint, and inventory constraint. Monthly time period was used in this planning scheme. The configuration of the proposed constraint set in this model is illustrated as follows. (1) Demand constraint. This constraint ensures the flow balance in the production profile. It implies that the summation of production level in the present month and inventory of both products in last month must be equal to the market demand and the inventory in the present month:   Xi;j þ Yi;j þ Zi;k ¼ D i;j þ Zi;j

8i; 8j; 8k; k ¼ j  1;

in which D i;j is the grey demand level of product ÔiÕ in the jth month (m).

ð4Þ

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

77

(2) Labor constraint. Labor availability could be a limiting factor in performing the monthly production program. This constraint accounts for such a situation with respect to both products: 2 X

 ri;h
Xi;j 6 R 1;j;h

8j; 8h;

ð5Þ

 ri;h
Yi;j 6 R 2;j;h

8j; 8h;

ð6Þ

i¼1 2 X i¼1

in which R1;j;h is the limitation of available labor in the daytime subject to the use of ÔhÕ type of labor in the jth month (capita hours); R2;j;h is the limitation of available labor in the nighttime subject to the use of ÔhÕ type of labor in the jth month (capita hours); and ri;h is the required input of ÔhÕ type of labor or manpower for producing unitary product ÔiÕ in the jth month (capita hours/m). (3) Production capacity constraint. This constraint addresses the limitation of equipment available for production in each month. But only the key equipment, the dyeing machine, is included in this analysis:   s i
Xi;j 6 S1;i;j

8i; 8j;

ð7Þ

  s i
Yi;j 6 S2;i;j

8i; 8j;

ð8Þ

in which S1;i;j is the limitation of key equipment available for producing the product ÔiÕ in the daytime in the jth month (hours); and S2;i;j is the limitation of key equipment available for producing the product ÔiÕ in the nighttime in the jth month (hours). (4) Inventory constraint. This constraint ensures the possible range of inventory that can be accepted in the production program: 2 X

 Zi;j 6 Vi;j

8i; 8j;

ð9Þ

i¼1 2 X

 Zi;j P Ui;j

8i; 8j;

ð10Þ

i¼1

in which Ui;j is the minimum inventory level of product ÔiÕ in the jth month (m); and Vi;j is the maximum inventory level of the storage (m). (5) Non-negativity constraints Xij P 0

8i; 8j;

ð11Þ

Yij P 0

8i; 8j;

ð12Þ

Zij P 0

8i; 8j:

ð13Þ

3.3.3. Solution procedure of grey compromise programming Solving a compromise programming problem is equivalent to solving a dimensionless objective function (i.e., a distance-based function) that represents the relative measure of a decision makerÕs preference from

78

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

the ideal solution subject to a set of definitional constraints. The associated solution procedure is integrated with both solution techniques of compromise programming and grey programming. They are expressed as follows (Chang et al., 1999). Assume the original model is expressed as kt lt X X Max Zt ¼ ctj xj þ ctj xj t ¼ 1; . . . ; q; ð14Þ j¼kt þ1

j¼1

Min

Zt ¼

kt X

ctj xj þ

m X

ctj xj

t ¼ q þ 1; . . . ; r;

ð15Þ

j¼kt þ1

j¼1

s:t:

lt X

aij xj 6 bi

8i;

ð16Þ

j¼1

ð17Þ xj P 0 8j; where there are q maximization objective functions and ðr  qÞ minimization objective functions, totally r objective functions, and ctj P 0 for j ¼ 1; . . . ; kt 8t; ctj < 0 for j ¼ kt þ 1; . . . ; lt 8t; kt þ lt ¼ m 8t; m is the number of decision variables. The scaling function applied in the model is n t ¼

Zt  Zt ðx j Þ  Zt

ð18Þ

in which Zt is the grey ideal solution of the individual objective function, which can be obtained from a payoff table; and n t is a scaling variable in compromise programming. The distance-based formulation of the objective functions can be described as (
 s )1=s r X Zt  Zt ðx j Þ  s Min Ls ¼ Min xt ð19Þ Zt t¼1 s:t:

m X

  a ij xj 6 bi

8i;

ð20Þ

j¼1

8j; ð21Þ x j P0 s where ‘‘s’’ is the distance parameter; ‘‘r’’ represents the total number of objectives; ‘‘xt ’’ is the corresponding weight associated with each objective, which could also be defined as an interval number. In addition, the following restrictions should be satisfied as long as it is defined as a decision-making problem: r X 1 6 s 6 1; xst > 0; and xst ¼ 1: ð22Þ If we assume

t¼1

Pr

s t¼1 xt ¼ 1 for all t and s, the algorithm can couple three cases as below.

(I) Distance parameter s ¼ 1 The objective function is define as Manhattan distance, and the optimization problem turns out to be a GLP problem as below: r X Min L ¼ Max Zt ðx ð23Þ 1 j Þ t¼1

s:t:

m X

  a ij xj 6 bi

8i;

ð24Þ

j¼1

x j P0

8j:

ð25Þ

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

79

To find the grey optimal solution, the solution algorithm of GLP (Huang et al., 1993) and multiobjective linear programming (Zeleny, 2002) must be applied. (II) Distance parameter 1 < s < 1 For 1 < s < 1, the satisfactory solution will be the noninferior feasible solution which is closest to the ideal solution Zt in terms of a weighted geometric distance. Although the formulation turns out to be nonlinear, the grey information can still propagate through the objective function and constraint set. Four situations have to be discussed as follows. (i) Distance parameter s ¼ even number The generic distance-based formulation can be expressed as  L s ðxj Þ

¼

s r   X Zt  Zt ðx j Þ Zt

t¼1

¼

q X

1

s ðZt Þ t¼1

¼

s q   X Zt  Zt ðx j Þ Zt

t¼1

Q t þ

r X t¼qþ1

s r   X Zt  Zt ðx j Þ þ Zt t¼qþ1

1  sQ ðZt Þ t

ð26Þ

where Zt are the grey ideal solutions for the model. It can be recognized that Zt P Zt ðxÞ for t ¼ 1; . . . ; q    and Zt 6 Zt ðxÞ for t ¼ q þ 1; . . . ; r. Here Q  Zt ðxÞ for t ¼ 1; . . . ; q and Q for t ¼ Zt t ¼ Zt ðxÞ  Zt t ¼ q þ 1; . . . ; r.   Assume c j P 0 for j ¼ 1; . . . ; kt 8t; cj < 0 for j ¼ kt þ 1; . . . ; lt 8t; bi > 0 for i ¼ 1; . . . ; m, and let   þ þ    Lðaij Þ ¼ signðaij Þjaij j; H ðaij Þ ¼ signðaij Þjaij j, where sign ðaij Þ ¼ 1, if aij P 0; sign ða ij Þ ¼ 1, if aij < 0, then the two relevant submodels can be formulated as follows. Submodel 1 Min

s:t:

q X

1

s ðjZtþ jÞ t¼1

" Qt ¼ Zt ðxÞ  Zt ¼



Qt

s

r X

1   s þ s Qt ð Z j t jÞ t¼qþ1 " # kt lt X X  þ þ þ þ  ctj xj þ ctj xj ; Qt ¼ Zt  Zt ðxÞ ¼ Zt 

 L s ðxj Þ ¼

þ

j¼kt þ1

j¼1 kt X

 c tj xj þ

lt X

t ¼ 1; . . . q;

ð28Þ

t ¼ q þ 1; . . . r;

ð29Þ

#

þ c  Zt ; tj xj

j¼kt þ1

j¼1

kt lt X X Lðaij Þ þ H ðaij Þ  x þ xj 6 1 j þ b b i i j¼1 j¼kt þ1

Zt 6 Zt 6 Ztþ Qt P 0 8t;

ð27Þ

8t; 8i;

8t;

ð31Þ ð32Þ

 xþ 8j; j P 0; xj P 0

xþ j

P x j

if both

xþ j

ð30Þ

ð33Þ and

x j

exist in submodel 1 at the same time for all j:

ð34Þ

  Some of xþ jopt and xjopt can be obtained from this submodel, which correspond to Fsopt , if the global  optimal solution can be obtained. Qtopt for all t and Ztopt for all t can also be obtained from this submodel.

80

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

Submodel 2 r  s X  s 1 1  s Q þ  s Qþ þ t t Z   Z   t t t¼qþ1 t¼1 " # kt lt X X þ þ  þ    þ ctj xj þ ctj xj ; Qt ¼ Zt  Zt ðxÞ ¼ Zt 

 Min Lþ s ðxj Þ ¼

s:t:

q X

" ¼

Ztþ ðxÞ



Zt

¼

kt X

þ cþ tj xj

t ¼ 1; . . . ; q;

ð36Þ

t ¼ q þ 1; . . . ; r;

ð37Þ

j¼kt þ1

j¼1

Qþ t

ð35Þ

þ

lt X

#

 cþ tj xj

 Zt ;

j¼kt þ1

j¼1

kt lt X X H ðaij Þ  Lðaij Þ þ x þ xj 6 1 8t; 8i; j  bi bþ i j¼1 j¼kt þ1

ð38Þ

Qþ t P0

ð39Þ

8t;

 Qþ t P Qt

8t;

ð40Þ

xþ j

P 0 8j;

ð41Þ

P 0;

x j

þ xþ j ¼ ðxjopt Þ

if xþ j is exiting in submodel 1 for all j;

ð42Þ

 x j ¼ ðxjopt Þ

if x j is exiting in submodel 1 for all j;

ð43Þ

þ x if x j 6 ðxjopt Þ j is not exiting in submodel 1 for all j;

ð44Þ

þ xþ j P ðxjopt Þ

ð45Þ

if x j is not exiting in submodel 1 for all j:

 þ Some of the xþ jopt and xjopt can be obtained from this submodel in the way to find the optimal value Fsopt and þ Qtopt , if the global optimal solution exists.

(ii) Distance parameter s ¼ odd number The generic distance-based formulation can be expressed as  L s ðxj Þ

¼

s p   X Zt  Zt ðx j Þ Zt

t¼1

¼ ¼

q X

1

s ðZt Þ t¼1 u X

1

ðZt Þ t¼1



Q t

¼

s q   X Zt  Zt ðx j Þ t¼1

s

s



 s ðQt Þ þ

r X

1

s ðZt Þ t¼qþ1 q X

1

ðZt Þs t¼uþ1

Zt

  s r X Zt  Zt ðx j Þ þ Zt t¼qþ1

s ðQ t Þ

s

ðQ t Þ 

v X

1

ðZt Þs t¼qþ1

s

ðQ t Þ 

r X t¼vþ1

1 s ðQ Þ ðZt Þs t

ð46Þ

where Zt are the ideal solutions for the model. Hence, it can be deduced that Zt P Zt ðxÞ for t ¼ 1; . . . ; q;    Zt 6 Zt ðxÞ for t ¼ q þ 1; . . . ; r must hold. Let Qs  Zt ðxÞ for t ¼ 1; . . . ; q; and Qs t ¼ Zt t ¼ Zt ðxÞ  Zt   for t ¼ q þ 1; . . . ; r. Then, Eq. (46) is applicable for the following cases Zt > 0 for t ¼ 1; . . . ; u; Zt < 0 for t ¼ u þ 1; . . . ; q; Zt > 0 for t ¼ q þ 1; . . . ; v; and Zt < 0 for t ¼ v þ 1; . . . ; r.   Assume c tj P 0 for j ¼ 1; . . . ; kt 8t; ctj < 0 for j ¼ kt þ 1; . . . ; lt 8t; bi > 0 for i ¼ 1; . . . ; m, the two relevant submodels can be formulated as follows.

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

81

Submodel 1 u X

q v r 1   s X 1  þ s X 1  þ s X 1  s   þ Q  Q  s s s Qt s ðQt Þ t t þ  ðZt Þ ðZt Þ ðZt Þ t¼uþ1 t¼1 t¼qþ1 Zt t¼vþ1 " # kt lt X X   þ  þ þ þ  Qt ¼ Zt  Zt ðxÞ ¼ Zt  ctj xj þ ctj xj ; t ¼ 1; . . . ; u;  L s ðxj Þ ¼

Min s:t:

" Qþ t

¼

Ztþ



Zt ðxÞ

¼

Ztþ "

Qþ t

¼

Ztþ ðxÞ



Zt

¼ "

Q t

¼

Zt ðxÞ



Ztþ

¼



j¼kt þ1

j¼1 kt X

 c tj xj

þ cþ tj xj

 cþ tj xj

j¼kt þ1  c tj xj

# þ c tj xj

;

t ¼ u þ 1; . . . ; q;

ð49Þ

 Zt ;

t ¼ q þ 1; . . . ; v;

ð50Þ

 Ztþ ;

t ¼ v þ 1; . . . ; r;

ð51Þ

#

lt X

þ

j¼1 kt X

ð48Þ

j¼kt þ1

lt X

þ

# þ c tj xj

j¼kt þ1

j¼1 kt lt X X Lðaij Þ þ H ðaij Þ  xj 6 1 þ xj þ b b i i j¼1 j¼kt þ1

8t; 8i;

Zt 6 Zt 6 Ztþ ; t ¼ 1; . . . ; u and  8j; xþ j P 0; xj P 0  xþ j P xj

þ

j¼1 kt X

lt X

ð47Þ

ð52Þ

t ¼ v þ 1; . . . ; r;

ð53Þ ð54Þ

 if both xþ j and xj exist in submodel 1 at the same time for all j:

ð55Þ

  Some of xþ jopt and xjopt can be obtained from this submodel in the way to find Fsopt , if the global optimal þ  solution exist. In addition, the values of Qtopt for t ¼ u þ 1; . . . ; v; Qtopt for t ¼ 1; . . . ; u and t ¼ v þ 1; . . . ; r and Ztopt for t ¼ 1; . . . ; u and t ¼ v þ 1; . . . ; r can also be obtained from this submodel.

Submodel 2 Min

s:t:

v r 1   s X 1   s X 1 þ s Q  Q  s s t t þ s ðQt Þ ðZ ðZ Þ Þ Z ð Þ Z t t t t t¼uþ1 t¼1 t¼qþ1 t¼vþ1 " # kt lt X X þ  þ Qþ  Zt ðxÞ ¼ Ztþ  c c t ¼ 1; . . . ; u; t ¼ Zt tj xj þ tj xj ;  Lþ s ðxj Þ ¼

u X



q  s X s Qþ þ t 

1

"   Ztþ ðxÞ ¼ Zt  Q t ¼ Zt

" Q t

¼

Zt ðxÞ



Ztþ

¼ "

Qþ t

¼

Ztþ ðxÞ



Zt

¼

j¼kt þ1

j¼1 kt X

þ cþ tj xj þ

 c tj xj

þ

kt X

þ cþ tj xj

þ

j¼1 kt lt X X H ðaij Þ  Lðaij Þ þ xj þ xj 6 1  b bþ i i j¼1 j¼kt þ1

Zt 6 Zt 6 Ztþ ; Qþ t

P ðQ topt Þ;

lt X

#

 cþ tj xj ;

t ¼ u þ 1; . . . ; q;

ð58Þ

 Ztþ ;

t ¼ q þ 1; . . . ; v;

ð59Þ

 Zt ;

t ¼ v þ 1; . . . ; r;

ð60Þ

#

þ c tj xj

j¼kt þ1

j¼1

ð57Þ

j¼kt þ1

j¼1 kt X

lt X

ð56Þ

lt X

#  cþ tj xj

j¼kt þ1

8t; 8i;

t ¼ u þ 1; . . . ; v; t ¼ 1; . . . ; u and t ¼ v þ 1; . . . ; r;

ð61Þ ð62Þ ð63Þ

82

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95 þ Q t 6 ðQtopt Þ;

t ¼ u þ 1; . . . ; v;

 xþ j P 0; xj P 0

xþ j

¼

ðxþ jopt Þ

ð64Þ

8j; xþ j

ð65Þ

exits in submodel 1 for all j;

ð66Þ

 x if x j ¼ ðxjopt Þ j exits in submodel 1 for all j;

ð67Þ

þ x j 6 ðxjopt Þ

ð68Þ

if

if x j does not exit in submodel 1 for all j;

 xþ j P ðxjopt Þ

xþ jopt

if xþ j does not exit in submodel 1 for all j:

ð69Þ

x jopt

Fsþopt ,

and can be obtained from this submodel in the way to find if the global optimal Some of  solution exists. In addition, Qþ for t ¼ 1; . . . ; u and t ¼ v þ 1; . . . ; r, Q for t ¼ u þ 1; . . . ; v, and Ztopt (for topt topt t ¼ u þ 1; . . . ; v) can also be obtained from this submodel. (iii) Distance parameter s 6¼ integer number The generic distance-based formulation can be expressed as  L s ðxj Þ ¼

ðc=bÞ r   X Zt  Zt ðx j Þ Zt

t¼1

¼

ðc=bÞ q   X Zt  Zt ðx j Þ t¼1

Zt

þ

ðc=bÞ r   X Zt  Zt ðx j Þ Zt t¼qþ1

ð70Þ

where c=b ¼ s; b and c are both integer number; and if Zt are the ideal solutions for the model, then Zt P Zt ðx for t ¼ 1; . . . ; q; Zt 6 Zt ðx for t ¼ q þ 1; . . . ; r. If c is an odd number, j Þ j Þ  ðc=bÞ Pr Zt Zt ðx Þ j turns out to be a complex number, and therefore it is meaningless in real-life events. t¼qþ1 Z  t

If c is an even number, the solution procedure is similar to the case when distance parameter s is an even number. (iv) Distance parameter s ¼ 2 Since distance parameter s ¼ 2, objective functions of two submodels both are convex functions and all constraints are linear and concave functions such that both submodels can be classified as convex programming models. So we can obtain global optimum solutions from these two submodels, and get a set of grey interval for those global optimum solutions. (III) Distance parameter s ¼ 1 For the case of distance parameter s ¼ 1, the objective function is denoted as the Tchebycheff distance, and the problem can be transformed into a linear programming model in which the largest weighted deviation determines the satisfactory solution and the situation of trade-off mechanics turns out to be not only competitive but noncompensatory. The model formulation becomes Min L ¼ Min V  1 s:t: Zt  Zt ðx Þ 6V  j z X

  a ij xj 6 bi

8i;

ð71Þ 8t;

ð72Þ ð73Þ

j¼1

x j P0

8j:

ð74Þ

The min–max operation in the above model would ensure that the maximum deviation from the ideal solution could be minimized. Therefore, the traditional GLP algorithm can still be applied.

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

83

Fig. 1. The flowchart of solution procedure.

3.3.4. Data collection and analysis This case study emphasizes assessing the impacts of pollution charges and water resource fees to the production-planning program in a textile-dyeing firm in Central Taiwan. Fig. 1 provides the flowchart to explain the complexity of solution procedure involved using the grey compromise programming model as a tool. To build an effective model, information flows of internal input, external input, and the other resources input as well as products and effluents information in the firm must be gathered in the very beginning, which must rely on the managerial department in the textile-dyeing factory to provide the firsthand data in support of such an analysis. Then, formulation of the model based on required technical settings and planning scenarios becomes the central focus. Finally, emphasis will be placed upon deriving

84

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

two submodels in search of the grey optimal solution and performing the final impact assessment with regards to the inclusion of economic instruments in the production-planning program. Although the textile-dyeing industry in Taiwan mainly prints four types of fibers, consisting of the cotton fibers, wool fibers, synthetic fibers, and the mixed fibers, there are only two production processes being designed for producing synthetic fibers (product I) and mixed fibers (product II) in the target firm. This implies the subscript i in the model formulation can only be either 1 or 2 in this case study. But it may not be the case when applying for the other applications. The processing line I (i.e., the production sector I) is in charge of the production of synthetic fibers and has 18 dyeing machines. The process allows the raw cloth be fed into a wool burning machine, then shipped into a desizing unit. The cloth that has been well desized can experience bleaching, mercerizing, and scouring sequentially. Finally, the cloth are printed or scoured, as it is required by the demand side. The processing line II (i.e., the production sector II) is responsible for the production of mixed fibers and has 12 dyeing machines. The process allows the raw cloth be first fed into a drying machine, and then shipped into a finishing unit. The cloth that has been well dried requires a manual examination for quality assurance/quality control. Finally, the semi-product of cloth are packed and sent out to the demand side. There are a total of 170 workers in two production lines; they generally work in turns for 8 hours in the daytime (full time job) and 3 hours in the nighttime (part time job). Manpower can be distributed into four types of work, including pretreatment (like knitting and wool burning), core treatment (like mercerizing; scouring; and bleaching), final treatment (like printing, finishing, and drying), and quality assurance/quality control (like product check and packing), within these two production lines. Therefore, the parameter  values of ri;h to be applied in the following modeling analysis is an average of any one out of the four manpower requirements. The production capacity of the dyeing machine is 400 m per minute for product I and 350 m per minute for product II. Besides, there are capacity for inventory of 90,000 m of product I and 60,000 m of product II, and the unitary cost of inventory is 1.5 times as many as that of producing original products. This factory generally discharges a wastewater stream of 6200–6500 m3 per day and the flue gas of 1.9–2.5
105 Nm3 per day, and a total amount of water resources required is 6700–7000 m3 per day. In summary, Table 1 may be used as a basis to decide the air pollution charge. Tables 2 and 6 may help determine the variations of effluent quality and the possible ranges of water pollution charge to be incurred. Table 3 can be applied for the calculation of water resources fees in terms of the location of the firm, the source of water resources used, and the seasonal option. The aggregate wastewater effluents from two production lines (synthetic fibers and mixed fibers) were investigated for the determination of pollution Table 7 The concentrations of pollutants in the wastewater stream and flue gas Pollutant

Concentration (mg/l)

Suspended solids Chemical oxygen demand Mercuric compounds Cadmium Chromic compounds Lead Nickel Copper Arsenic Ammonia nitrogen Phosphate Cyanide Phenolic compounds SOx (ppm) NOx (ppm)

43–47 175–190 0 0 0 0 0 0 0 17–15 8–6 0 0 200–250 250–300

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

85

Table 8 The production cost of each product Product I (synthetic fibers)

Product II (mixed fibers)

Before imposing pollution charges and water resource fees Daytime ½2:4215; 2:6313 Nighttime ½2:5513; 2:7706

½2:4125; 2:6119 ½2:5368; 2:7431

After imposing pollution charges and water resource fees Daytime ½2:7291; 2:9896 Nighttime ½2:8589; 3:1262

½2:7796; 3:0366 ½2:9094; 3:1759

charge. To ease the practice, Table 7 summarizes the information being used directly in the model application. Table 8 shows the derivation of the production cost associated with each product with respect to two planning scenarios that concerned about the inclusion or exclusion of pollution charges and water resource fees. These values come from one of the previous studies using grey input/output analysis as a tool (Wu and Chang, 2003). Obviously, the inclusion of pollution charges and water resource fees would increase the level of product cost by about 15%. To fully apply this grey compromise programming model, some other grey parameter values related to the production program are needed. In Table 9(A), the information of material type, aid agent, and manpower group are organized. Part of them is drawn from confidential data reports so that the titles of them cannot be released in this paper. In Table 9(B), scheduled working days, demand of products, maximum input of manpower of all types, and the maximum production capacity of products associated with equipment are summarized in a 12-month time frame. 4. Results and discussions LINDOâ software package was used as a computer solver. When the value of ‘‘s’’ is equal to 1 and infinity, the program can be solved by the traditional linear programming scheme. However, when the value of ÔsÕ is equal to 2, it can be solved as a quadratic programming problem based on the Kuhn–Tucker (K–T) method. In such circumstances, it assures the findings of global optimal solution. Although there are six cases, consisting of s is 1, s is an even number, s is an odd number, s is not an integer, s is 2, and s is infinity, are covered in the solution procedure for ensuring the mathematical integrity in theory, only three cases are needed in real world decision-making. They include s is 1, s is 2, and s is infinity from a practical sense. If s ¼ 1, it implies the objectives considered is compensatory in decision-making. On the other hand, if s ¼ 2, it implies the objectives considered is competitive in decision-making. If s ¼ 1, the model can be transformed into a linear programming model in which the largest weighted deviation determines the preferred solution and the situation among trade-off mechanics turns out to be not only competitive but noncompensatory. To conduct a comparative study, the planning scenarios associated with the inclusion of economic instruments were separately analyzed so as to show the differences of each optimal production program due to the impacts of environmental costs. Table 10(A)–(C) stand for the optimal production scheme based on the scenario of single-objective optimization before the inclusion of any economic instrument. Considering the minimization of production cost independently would eliminate the possible production of product I in the nighttime. But considering the maximization of production capacity or the minimization of inventory cost independently would allow us to choose a more aggressive production program leading to propose a combined production activity in both daytime and nighttime. In all those three cases, none of the parameter values came out as interval numbers in the final solution, which implies that the grey message embedded in the real world system is completely diminished over the optimization procedure. Table 11 considers the case for the inclusion of the economic instruments, when the single object optimization scheme remains the same. The cost for producing product II would clearly become larger than that of

86

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

producing product I after levying those pollution charges and water resource fees, and the cost for unitary production would increase due to the impacts from the emergence of economic instruments. However, the inclusion of such impact would not result in a big final difference in optimal production program once the Table 9 Grey parameter values used in the production program Itemsa

Price (NT$/unit)b

Panel A Material type 1 Material type 2 Material type 3 Material type 4 Material type 5 Material type 6 Material type 7 Aid agent Manpower group Manpower group Manpower group Manpower group

4.2–4.5 3.3–3.6 5.3–5.7 6.1–6.5 5.1–5.3 6.1–6.6 5.9–6.2 2.1–2.4 112–115 116–119 120–124 115–118

1 2 3 4

Month 1

2

3

4

5

6

25

24

25

24

25

24

Demand of product I (10,000 m)

½253; 260

½187; 192

½236; 240

½246; 252

½267; 271

½247; 252

Demand of product II (10,000 m)

½165; 173

½118; 125

½135; 141

½155; 162

½126; 132

½175; 183

Maximum input of manpower of type I (capita hours/month)

½8950; 9000c ½3325; 3375d

½8590; 8640c ½3190; 3240d

½8950; 9000c ½3325; 3375d

½8590; 8640c ½3190; 3240d

½8950; 9000c ½3325; 3375d

½8590; 8640c ½3190; 3240d

Maximum input of manpower of type II (capita hours/month)

½10950; 11000c ½4075; 4125d

½10510; 10560 ½3910; 3960d

½10950; 11000 ½4075; 4125d

½10510; 10560 ½3910; 3960d

½10950; 11000 ½4075; 4125d

½10510; 10560c ½3910; 3960d

Maximum input of manpower of type III (capita hours/month)

½5950; 6000c ½2200; 2250d

½5710; 5760c ½2110; 2160d

½5950; 6000c ½2200; 2250d

½5710; 5760 ½2110; 2160d

½5950; 6000 ½2200; 2250d

½5710; 5760c ½2110; 2160d

Maximum input of manpower of type IV (capita hours/month)

½7950; 8000c ½2950; 3000d

½7630; 7680c ½2830; 2880d

½7950; 8000c ½2950; 3000d

½7630; 7680c ½2830; 2880d

½7950; 8000c ½2950; 3000d

½7630; 7680c ½2830; 2880d

Maximum production capacity of product I associated with the equipment (hours)

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3406; 3456c ½1246; 1296d ½4652; 4752e

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3406; 3456c ½1246; 1296d ½4652; 4752e

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3406; 3456c ½1246; 1296d ½4652; 4752e

Maximum production capacity of product II associated with the equipment (hours)

½2350; 2400c ½850; 900d ½3200; 3300e

½2254; 2304c ½814; 864d ½3068; 3168e

½2350; 2400c ½850; 900d ½3200; 3300e

½2254; 2304c ½814; 864d ½3068; 3168e

½2350; 2400c ½850; 900d ½3200; 3300e

½2254; 2304c ½814; 864d ½3068; 3168e

Panel B Scheduled working days (day/month)

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

87

Table 9 (continued) Itemsa

Price (NT$/unit)b Month 7

8

9

10

11

12

Scheduled working days (day/month)

25

25

24

25

24

25

Demand of product I (10,000 m)

½275; 280

½256; 260

½192; 196

½271; 275

½192; 196

½232; 238

Demand of product II (10,000 m)

½147; 155

½168; 172

½122; 128

½188; 192

½131; 135

½151; 155

Maximum input of manpower of type I (capita hours/month)

½8950; 9000c ½3325; 3375d

½8950; 9000c ½3325; 3375d

½8590; 8640c ½3190; 3240d

½8950; 9000c ½3325; 3375d

½8590; 8640c ½3190; 3240d

½8950; 9000c ½3325; 3375d

Maximum input of manpower of type II (capita hours/month)

½10950; 11000c ½4075; 4125d

½10950; 11000c ½4075; 4125d

½10510; 10560 ½3910; 3960d

½10950; 11000c ½4075; 4125d

½10510; 10560c ½3910; 3960d

½10950; 11000 ½4075; 4125d

Maximum input of manpower of type III (capita hours/month)

½5950; 6000c ½2200; 2250d

½5950; 6000c ½2200; 2250d

½5710; 5760 ½2110; 2160d

½5950; 6000c ½2200; 2250d

½5710; 5760c ½2110; 2160d

½5950; 6000c ½2200; 2250d

Maximum input of manpower of type IV (capita hours/month)

½7950; 8000c ½2950; 3000d

½7950; 8000c ½2950; 3000d

½7630; 7680c ½2830; 2880d

½7950; 8000c ½2950; 3000d

½7630; 7680c ½2830; 2880d

½7950; 8000c ½2950; 3000d

Maximum production capacity of product I associated with the equipment (hours)

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3406; 3456c ½1246; 1296d ½4652; 4752e

½3550; 3600c ½1300; 1350d ½4850; 4950e

½3406; 3456c ½1246; 1296d ½4652; 4752e

½3550; 3600c ½1300; 1350d ½4850; 4950e

Maximum production capacity of product II associated with the equipment (hours)

½2350; 2400c ½850; 900d ½3200; 3300e

½2350; 2400c ½850; 900d ½3200; 3300e

½2254; 2304c ½814; 864d ½3068; 3168e

½2350; 2400c ½850; 900d ½3200; 3300e

½2254; 2304c ½814; 864d ½3068; 3168e

½2350; 2400c ½850; 900d ½3200; 3300e

a

The dyeing materials and agent were referred as confidential data in a propriety report. The currency ratio was 31NT$/1US$ in 1999. c This represents the maximum input of manpower or equipment in daytime. d This represents the maximum input of manpower or equipment in nighttime. e This represents the total maximum input of manpower or equipment. b

trade-off in the multiobjective programming is not taken into account. Only the differences of optimal production of product I in the nighttime exist, as evidenced by the outputs in Tables 10(A) and 11(A). Findings indicate that the production of product I is in favored of due to relatively lower production cost. Part of the production program for producing product I must be scheduled in the nighttime due to the labor and equipment constraints. In the case of multicriteria decision-making, the concept of compensatory versus noncompensatory and competitive versus noncompetitive in the trade-off procedure are noticeably taken into account. Table 12(A)–(C) therefore list the optimal production scheme based on the scenario of multiobjective optimization without the inclusion of the impacts of implementing economic instruments temporarily. No matter which ‘‘s’’ value is applied in this case, the production of product I in the nighttime is always excluded. Another significant observation is that when ‘‘s’’ value is equal to 2 or infinity, almost no grey message appears in the final noninferior solution set. This implies that the inherent trade-off mechanism in

88

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

Table 10 The optimal solution for the objective function Month

The production level of product I  in the daytime X1;j (m)

Panel A: The optimal solution for economic instruments) 1 2607000 2 ½1870000; 1920000 3 ½2360000; 2400000 4 ½2460000; 2520000 5 ½2670000; 2710000 6 ½2470000; 2520000 7 ½2750000; 2800000 8 ½2560000; 2600000 9 ½1920000; 1960000 10 ½2710000; 2750000 11 ½1920000; 1960000 12 ½2320000; 2380000

The production level of product II  in the daytime X2;j (m)

The production level of product I in the nighttime  Y1;j (m)

The production level of product II in the nighttime  Y2;j (m)

The inventory level of product I  Z1;j (m)

The inventory level of product II  (m) Z2;j

the objective function of minimization of production cost independently (before implementing ½973000; 1053000 1303000 ½1220000; 1316000 ½976000; 1080000 ½910000; 970000 ½966000; 1046000 ½830000; 910000 ½1020000; 1060000 1343000 ½870000; 970000 1433000 ½1260000; 1340000

0 0 0 0 0 0 0 0 0 0 0 0

727000 0 130000 451000 350000 784000 640000 660000 0 887000 0 127000

77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000

50000 ½103000; 173000 ½139000; 173000 50000 50000 50000 50000 50000 ½113000; 173000 50000 ½133000; 173000 50000

Objective function value F1 ½112974500; 125979200 Panel B: The optimal solution for economic instruments) 1 1114500 2 420000 3 ½2342500; 2400000 4 2520000 5 ½2610000; 2710000 6 2520000 7 1237500 8 2600000 9 460000 10 2750000 11 1933000 12 ½2200000; 2052500

the objective function of maximization production capacity independently (before implementing ½1700000; 1780000 ½1355500; 1373000 0 ½150500; 243000 ½970000; 1040000 453000 ½1512500; 1550000 34500 1403000 ½217000; 234500 0 ½1510000; 1550000

½1475000; 1562500 ½1415000; 1500000 0 0 0 0 ½1475000; 1562500 0 ½1415000; 1500000 0 27000 180000

0 0 1287000 ½1420000; 1500000 1570000 ½1420000; 1500000 0 ½1480000; 1562500 0 ½1480000; 1562500 ½1310000; 1473000 0

77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000

50000 ½113000; 173000 50000 ½103000; 173000 50000 173000 ½139000; 173000 50000 173000 50000 173000 173000

Objective function value F2 ½0:50628; 0:5382 Panel C: The optimal solution for the objective function of minimization of inventory cost independently (before implementing economic instruments) 1 ½1044500; 2607000 ½220000; 1700000 ½0; 1562500 ½0; 1480000 77000 50000 2 1870000 ½0; 1180000 0 ½0; 1180000 77000 50000 3 ½797500; 885000 1350000 ½1475000; 1562500 0 77000 50000 4 ½2050000; 2460000 ½130000; 1550000 ½0; 410000 ½0; 1420000 77000 50000 5 ½2320000; 2367500 ½0; 1260000 ½302500; 350000 ½0; 1260000 77000 50000 6 ½970000; 2470000 ½330000; 1750000 ½0; 1500000 ½0; 1420000 77000 50000 7 ½2657500; 2750000 0 ½0; 92500 1470000 77000 50000 8 2560000 ½117500; 200000 0 ½1480000; 1562500 77000 50000 9 ½1640000; 1920000 0 ½0; 280000 1220000 77000 50000 10 2710000 ½317500; 400000 0 ½1480000; 1562500 77000 50000 11 ½420000; 600000 ½1210000; 1310000 ½1320000; 1500000 ½0; 100000 77000 50000 12 2320000 ½0; 30000 0 ½1480000; 1510000 77000 50000 Objective function value F3 ½5819268; 6308871

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

89

Table 11 The optimal solution with respect to the objective function Month

The production level of product I  in the daytime X1;j (m)

Panel A: The optimal solution economic instruments) 1 ½2607000; 2677000 2 1993000 3 ½2237000; 2337000 4 ½2009000; 2059000 5 ½2197000; 2237000 6 ½1809000; 1859000 7 ½2110000; 2160000 8 ½1900000; 1940000 9 2043000 10 ½1700000; 1780000 11 ½1920000; 1960000 12 ½2320000; 2380000

The production level of product II  in the daytime X2;j (m)

The production level of product I in the nighttime  Y1;j (m)

The production level of product II in the nighttime  Y2;j (m)

The inventory level of product  I Z1;j (m)

The inventory level of product II  Z2;j (m)

with respect to the objective function of minimization of production cost only (after implementing ½973000; 1053000 ½1180000; 1300000 1343000 ½1427000; 1507000 ½1383000; 1443000 ½1627000; 1707000 ½1470000; 1550000 ½1680000; 1720000 ½1220000; 1280000 ½1880000; 1960000 1433000 ½1260000; 1300000

0 0 0 451000 473000 661000 640000 660000 0 887000 0 0

727000 0 130000 0 0 0 0 0 0 0 0 127000

77000 ½150000; 200000 ½77000; 87000 77000 77000 77000 77000 77000 ½160000; 200000 77000 77000 77000

50000 ½50000; 100000 ½163000; 173000 50000 173000 50000 50000 50000 50000 ½50000; 90000 173000 50000

Objective function value F1 ½128359600; 144359700 Panel B: The optimal solution economic instruments) 1 1114500 2 420000 3 ½2342500; 2400000 4 2520000 5 ½2610000; 2710000 6 2520000 7 1237500 8 2600000 9 460000 10 2750000 11 1933000 12 ½2052500; 2200000

with respect to the objective function of maximization of production capacity only (after implementing ½1700000; 1780000 ½1355500; 1373000 0 ½150500; 243000 ½970000; 1040000 453000 ½1512500; 1550000 34500 1403000 ½217000; 234500 0 ½1510000; 1550000

½1475000; 1562500 ½1415000; 1500000 0 0 0 0 ½1475000; 1562500 0 ½1415000; 1500000 0 27000 180000

0 0 1287000 ½1420000; 1500000 1570000 ½1420000; 1500000 0 ½1480000; 1562500 0 ½1480000; 1562500 ½1310000; 1473000 0

77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000 77000

50000 ½113000; 173000 50000 ½103000; 173000 50000 173000 ½139000; 173000 50000 173000 50000 173000 173000

Objective function value F2 ½0:50628; 0:5382 Panel C: The optimal solution with respect to the objective function of minimization of inventory cost only (after implementing economic instruments) 1 ½1044500; 2607000 ½220000; 1700000 ½0; 1562500 ½0; 1480000 77000 50000 2 1870000 ½0; 1180000 0 ½0; 1180000 77000 50000 3 ½797500; 885000 1350000 ½1475000; 1562500 0 77000 50000 4 ½2050000; 2460000 ½130000; 1550000 ½0; 410000 ½0; 1420000 77000 50000 5 ½2320000; 2367500 ½0; 1260000 ½302500; 350000 ½0; 1260000 77000 50000 6 ½970000; 2470000 ½330000; 1750000 ½0; 1500000 ½0; 1420000 77000 50000 7 ½2657500; 2750000 0 ½0; 92500 1470000 77000 50000 8 2560000 ½117500; 200000 0 ½1480000; 1562500 77000 50000 9 ½1640000; 1920000 0 ½0; 280000 1220000 77000 50000 10 2710000 ½317500; 400000 0 ½1480000; 1562500 77000 50000 11 ½420000; 600000 ½1210000; 1310000 ½1320000; 1500000 ½0; 100000 77000 50000 12 2320000 ½0; 30000 0 ½1480000; 1510000 77000 50000 Objective function value F3 ½6580942; 7191253

90

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

Table 12 The optimal solution of grey compromise programming Month

The production level of product I in the  daytime X1;j (m)

The production level of product II in the  daytime X2;j (m)

The production level of product I in the night (m) time Y1;j

The production level of product II in the night time Y2;j (m)

Panel A: The optimal solution of grey compromise programming as s ¼ 1 (before implementing 1 ½2587000; 2677000 ½993000; 1073000 0 707000 2 1920000 1250000 0 0 3 ½2330000; 2400000 ½1250000; 1350000 0 60000 4 2520000 ½916000; 1080000 0 540000 5 ½2600000; 2710000 ½980000; 1040000 0 280000 6 ½2436000; 2520000 ½1000000; 1080000 0 750000 7 ½2710000; 2800000 ½870000; 950000 0 600000 8 ½2470000; 2600000 ½1110000; 1150000 0 570000 9 1960000 1280000 0 0 10 ½2680000; 2750000 ½900000; 1000000 0 920000 11 1960000 1350000 0 0 12 ½2280000; 2380000 ½1300000; 1370000 0 303000

The inventory level of product  I Z1;j (m)

The inventory level of product  II Z2;j (m)

economic instruments) 77000 50000 ½77000; 107000 ½50000; 120000 77000 ½50000; 80000 ½77000; 137000 50000 ½77000; 151000 50000 ½77000; 117000 50000 77000 50000 77000 50000 ½77000; 107000 ½50000; 110000 77000 50000 ½77000; 117000 ½50000; 90000 77000 173000

Objective function value L 1 ½375295900; 415125500 Panel B: The optimal solution of grey compromise programming as s ¼ 2 (before implementing 1 ½2607000; 2677000 ½973000; 1053000 0 727000 2 ½1870000; 1920000 ½1180000; 1250000 0 0 3 ½2360000; 2400000 ½1220000; 1280000 0 130000 4 ½2460000; 2520000 ½976000; 1046000 0 574000 5 ½2670000; 2710000 ½910000; 970000 0 350000 6 ½2470000; 2520000 ½966000; 1046000 0 784000 7 ½2750000; 2800000 ½830000; 910000 0 640000 8 ½2560000; 2600000 ½1020000; 1060000 0 660000 9 ½1920000; 1960000 ½1220000; 1280000 0 0 10 ½2710000; 2750000 ½870000; 910000 0 1010000 11 ½1920000; 1960000 ½1310000; 1350000 0 0 12 ½2320000; 2380000 ½1260000; 1300000 0 250000

economic instruments) 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000

Objective function value L 2 ½26:7635; 29:5891 Panel C: The optimal solution of grey compromise programming as s ¼ 1 (before implementing 1 ½2607000; 2677000 ½973000; 1053000 0 850000 2 ½1870000; 1920000 ½1057000; 1127000 0 0 3 ½2360000; 2400000 ½1220000; 1280000 0 253000 4 ½2460000; 2520000 ½976000; 1046000 0 451000 5 ½2670000; 2710000 ½910000; 970000 0 473000 6 ½2470000; 2520000 ½966000; 1046000 0 661000 7 ½2750000; 2800000 ½830000; 950000 0 640000 8 ½2560000; 2600000 1020000 0 783000 9 ½1920000; 1960000 ½1097000; 1157000 0 0 10 ½2710000; 2750000 ½870000; 910000 0 1133000 11 ½1920000; 1960000 ½1187000; 1227000 0 0 12 ½2320000; 2380000 ½1260000; 1300000 0 30000

economic instruments) 77000 173000 77000 50000 77000 173000 77000 50000 77000 173000 77000 50000 77000 ½50000; 90000 77000 173000 77000 50000 77000 173000 77000 50000 77000 50000

Objective function value L 1 ½118319600; 131814300

multiobjective evaluation framework would prevent the possible interference embedded in grey uncertainties from final decision-making. But this is not true when ‘‘s’’ is equal to 1, implying that relatively

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

91

Table 13 The optimal solution of grey compromise programming Month

The production level of product I  in the daytime X1;j (m)

Panel A: The optimal solution 1 ½2587000; 2677000 2 1920000 3 ½2270000; 2340000 4 1980000 5 ½2600000; 2710000 6 ½1646000; 1770000 7 ½2750000; 2800000 8 ½1900000; 2030000 9 1960000 10 ½2680000; 2750000 11 1960000 12 ½1967000; 2077000

The production level of product II  in the daytime X2;j (m)

The production level of product I  in the nighttime Y1;j (m)

The production level of product II in the night time Y2;j (m)

of grey compromise programming as s ¼ 1 (after implementing ½993000; 1073000 0 707000 1250000 0 0 ½1310000; 1410000 60000 0 ½1456000; 1620000 540000 0 ½980000; 1040000 0 280000 ½1790000; 1830000 750000 0 ½830000; 950000 0 600000 ½1680000; 1720000 570000 0 1280000 0 0 ½900000; 1000000 0 920000 1350000 0 0 ½1613000; 1673000 303000 0

The inventory level of product  I Z1;j (m)

The inventory level  of product II Z2;j (m)

economic instruments) 77000 50000 ½77000; 107000 ½50000; 120000 77000 ½50000; 80000 ½77000; 137000 50000 ½77000; 151000 50000 77000 ½50000; 90000 77000 50000 77000 50000 ½77000; 117000 ½50000; 110000 ½77000; 87000 50000 ½77000; 127000 ½50000; 70000 77000 173000

Objective function value L 1 ½356206600; 398422100 Panel B: The optimal solution 1 ½2607000; 2677000 2 ½1870000; 1920000 3 ½2360000; 2400000 4 ½1886000; 1946000 5 ½2320000; 2360000 6 ½1686000; 1736000 7 ½2110000; 2160000 8 ½2560000; 2600000 9 ½1920000; 1960000 10 ½2710000; 2750000 11 ½1920000; 1960000 12 ½2070000; 2130000

of grey compromise programming as s ¼ 2 (after implementing ½973000; 1053000 0 727000 ½1180000; 1250000 0 0 ½1220000; 1280000 0 130000 ½1550000; 1620000 574000 0 ½1550000; 1320000 350000 0 ½1750000; 1830000 784000 0 ½1470000; 1550000 640000 0 ½1020000; 1060000 0 660000 ½1220000; 1280000 0 0 ½870000; 910000 0 1010000 ½1310000; 1350000 0 0 ½1510000; 1550000 250000 0

economic instruments) 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000 77000 50000

Objective function value L 2 ½26:97207; 29:7911 Panel C: The optimal solution 1 ½1757000; 1827000 2 ½1870000; 1920000 3 ½2107000; 2147000 4 ½2460000; 2520000 5 ½2197000; 2237000 6 ½2470000; 2520000 7 ½2750000; 2800000 8 ½2560000; 2600000 9 ½1920000; 1960000 10 ½1577000; 1617000 11 ½1920000; 1960000 12 ½2320000; 2380000

of grey compromise programming as s ¼ 1 ½1823000; 1903000 850000 ½1057000; 1127000 0 ½1473000; 1533000 253000 ½976000; 1046000 0 ½1383000; 1443000 473000 ½966000; 1046000 0 ½830000; 950000 0 1020000 0 ½1097000; 1157000 0 ½2003000; 2043000 1133000 ½1187000; 1227000 0 ½1260000; 1300000 0

(after implementing economic instruments) 0 77000 173000 0 77000 50000 0 77000 173000 451000 77000 50000 0 77000 173000 661000 77000 50000 640000 77000 ½50000; 90000 783000 77000 173000 0 77000 50000 0 77000 173000 0 77000 50000 30000 77000 50000

Objective function value L 1 ½134468500; 151083400

strong compensatory effect in the trade-off mechanism may appear to retain those grey uncertainties throughout the final noninferior solution set.

92

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

On the other hand, Table 13(A)–(C) list the optimal production schemes based on the planning scenario with the inclusion of the economic instruments in the multiobjective optimization framework. Observation reveals that there is relatively strong compensatory effect in the trade-off mechanism when ‘‘s’’ is equal to 1 leading to retain those grey uncertainties in the final noninferior solution set. In all cases, the required inventory level is equivalent to those outputs gained simply from the minimization of inventory cost, as shown in Table 10(C), irrespective of which value of ÔsÕ is chosen. It appears that the objective for the minimization of inventory cost is relatively competitive to the others so that it would restrain other objectives in the trade-off process. Based on Tables 12(A) and 13(A), in which the value of ÕsÕ is fixed by 1, results clearly indicate that the optimal production program tends to be in favor of producing more product I for the reason of reducing resource consumption. In any circumstances, all of the cases in the proposed multiobjective evaluation framework, as listed in Tables 12 and 13, present effective production schemes leading to satisfy the changing demand with respect to uncertain resources constraints and varying environmental costs at the same time. 5. Conclusions Optimization skill has been usefully applied to a wide variety of production planning problem areas. Despite almost two decades of such research activity, challenging production planning problems still remain in many of these areas. Today, there is a growing awareness of the environmental costs, such as pollution charges and resource conservation fees, which have to be taken into account in the productionplanning scheme in an uncertain environment. This paper extends multiobjective production planning scheme in which grey uncertainties are particularly addressed. The results of this study clearly indicate that optimization model may help managers in making environmentally sound decisions without necessarily sacrificing the economic interests of the firm to any great extent. In addition, from a technical point of view, multiple objectives and criteria considered in the model formulation with the aid of interval expressions in the parameter values do present a unique configuration allowing the uncertain information for being propagated in the trade-off process. The case study successfully incorporates the environmental costs and production benefits into a production-planning system facing in a textile-dyeing firm. It demonstrates how uncertain messages in a production-planning system can be quantified through the use of interval numbers in a multiobjective evaluation framework, and thereby generates a set of more flexible solutions for decision makers. Such a grey multiobjective programming approach may be further applied in search of the integrated corporate operational strategies, management structures, and remanufacturing systems. Apart from the concern of conventional cost accounting, enterprises facing the corresponding challenges should experience such an assessment in the future. Acknowledgements The authors acknowledge the helpful comments provided by anonymous referees in the reviewing process. Appendix A. Notation Decision variables Xi;j the yield of product ÔiÕ (dyeing cloth) expressed by grey interval in the daytime in the jth month (m) Yi;j the yield of product ÔiÕ (dyeing cloth) expressed by grey interval in the nighttime in the jth month (m)

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95  Zi;j  nt

93

the inventory of product ÔiÕ (dyeing cloth) expressed by grey interval in the jth month (m) a scaling variable in compromise programming (unitless)

System parameters D the grey demand level of product ÔiÕ in the jth month (m) i;j R the grey limitation of available labor in the daytime subject to the use of ÔhÕ type of labor in the jth 1;j;h month (capita hours) R the grey limitation of available labor in the daytime subject to the use of ÔhÕ type of labor in the jth 2;j;h month (capita hours)  S1;i;j the grey limitation of available equipment for production of product ÔiÕ in the daytime in the jth month (hours)  S2;i;j the grey limitation of available equipment for production of product ÔiÕ in the nighttime in the jth month (hours)  Si;j the grey limitation of available equipment for production of product ÔiÕ in the jth month (hours) Ui;j Vi;j

the grey minimum inventory level of product ÔiÕ in the jth month (m) the grey maximum inventory level of product ÔiÕ in the jth month (m)

c ti dti

the grey unitary production cost of product ÔiÕ in the daytime t (NT$/m) the grey unitary production cost of product ÔiÕ in the nighttime t (NT$/m)

e i  ri;h

the grey inventory cost of product ÔiÕ (NT$/m) the grey required input of ÔhÕ type of labor for producing unitary product ÔiÕ in the jth month (capita hours/m) the grey required input of equipment for producing unitary product ÔiÕ in the jth month (hours/m).

s i

References Ahmad, K.H., Subhas, K.S., 1994. Pollution balance: A new methodology for minimization waste production in manufacturing processes. Journal of Air and Waste Management Association 44, 1303–1308. Annual Statistical Report, 1999. Administrative Yen, Taiwan, ROC. Arthur, J.L., Lawrence, K.D., 1982. Multiple goal production and logistics planning in a chemical and pharmaceutical company. Computer and Operations Research 9 (2), 127–137. Bowman, E.H., 1956. Production planning by the transportation method of linear programming. Journal of the Operational Research Society 4 (1), 100–103. Bowman, E.H., 1963. Consistency and optimality in managerial decision making. Management Science 9, 301–321. Chang, N.B., Chen, Y.L., Wang, S.F., 1997. A fuzzy interval multi-objective mixed integer programming approach for the optimal planning of metropolitan solid waste management system. Fuzzy Sets and Systems 89 (1), 35–60. Chang, N.B., Wen, C.G., Chen, Y.L., Yong, Y.C., 1996a. Optimal planning of the reservoir watershed by grey fuzzy multi-objective programming (I): Theory. Water Research 30 (10), 2329–2334. Chang, N.B., Wen, C.G., Chen, Y.L., Yong, Y.C., 1996b. Optimal planning of the reservoir watershed by grey fuzzy multi-objective programming (II): Application. Water Research 30 (10), 2335–2340. Chang, N.B., Yeh, S.C., Wu, G.C., 1999. Stability analysis of grey compromise programming and its applications. International Journal of Systems Science 30 (6), 571–589. Clarke, R.A., Stavins, R.N., Greeno, J.L., Bavaria, J.L., 1994. The challenge of going green. Havard Business Review 72 (4), 37– 50. Crittenden, B., Kolaczkowski, S., 1995. Waste minimization, a practical guide. Institution of Chemical Engineers, Rugby, UK. Deng, J., 1984a. The Theory and Methods of Socio-economic Grey Systems. Science Press, Beijing, China (in Chinese). Deng, J., 1984b. Contributions to Grey Systems and Agriculture. Science and Technology Press, Taiyuan, Shanxi, China (in Chinese). Deng, J., 1986. Grey Prediction and Decision. Huazhong Institute of Technology Press, Wuhan, China (in Chinese). Douglas, J.M., 1992. Process synthesis for waste minimization. Industrial Engineering Chemical Research 31, 238–243.

94

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

Eilon, S., 1975. Five approaches to aggregate production planning. AIIE Transactions 7 (2), 16–29. Guide Jr., V.D.R., Srivastava, R., Spencer, M.S., 1996. Are production systems ready for the green revolution? Production and Inventory Management Journal 4, 70–76. Gupta, A., Manousiouthakis, V., 1990. Waste reduction through multicomponent mass exchange network synthesis. Comparative Chemical Engineering 18 (Suppl), S585–S590. Gupta, M.C., 1995. Environmental management and its impact on the operations function. International Journal of Operations and Production Management 15 (8), 34–51. Hanssmann, F., Hess, W., 1960. A linear programming approach to production and employment scheduling. Management Technology Monograph 1, 46–51. Hindelang, T.J., Hill, J.L., 1978. A New Model for Aggregate Output Planning. Omega, 267–272. Higgins, H.M., 1990. Hazardous Waste Minimization Handbook. Lewis Publishers, New York. Holt, C.C., Modigliani, F., Dimon, H., 1955. A linear decision rule for production and employment scheduling. Management Science 2 (1), 46–51. Huang, G.H., Baetz, B.W., Patry, G.G., 1992. A grey linear programming approach for municipal solid waste management planning under uncertainty. Civil Engineering Systems 9, 319–335. Huang, G.H., Moore, R.D., 1993. Grey linear programming, its solving approach, and its application. International Journal of Systems Science 24 (1), 159–172. Huang, G.H., Baetz, B.W., Patry, G.G., 1993. A grey fuzzy linear programming approach for municipal solid waste management planning under uncertainty. Civil Engineering Systems 10, 123–146. Huang, G.H., Baetz, B.W., Patry, G.G., 1994. Grey dynamic programming for solid waste management planning under uncertainty. Civil Engineering Systems 11, 43–73. Huang, G.H., Baetz, B.W., Patry, G.G., 1995a. Grey integer programming: An application to waste management planning under uncertainty. European Journal of Operational Research 83, 594–620. Huang, G.H., Baetz, B.W., Patry, G.G., 1995b. A grey integer programming for solid waste management planning under uncertainty. European Journal of Operational Research 83, 594–620. Jones, C.H., 1967. Parametric production planning. Management Science, 22–34. Kleiner, J.A., 1993. What does it mean to be green? Harvard Business Review 69 (4), 38–47. Kravanja, Z., Grosmann, I.E., 1994. New topology and capabilities in prosyn––an automated topology and parameter process synthesizer. Comparative Chemical Engineering 8 (11/12), 1097–1114. Krupp, J.A.G., 1992. Core Obsolescence Forecasting in Remanufacturing. Production and Inventory Management Journal 33 (2), 12–17. Lee, S.M., 1973. Goal Programming for Decision Analysis. Averbach, Philadelphia, USA. Mellichamp, J.M., Love, R.M., 1978. Production switching heuristics for the aggregate planning problem. Management Science 24 (12), 1242–1251. Mizsey, P., Fonyo, Z., 1995. Waste reduction in the chemical industry. In: Freeman, H.M., Puskas, Z., Orbina, R. (Eds.), NATO ASI Series, Vol. 2, Cleaner Technologies and Cleaner Products for Sustainable Development. Springer, Berlin, pp. 141– 152. Moore, R.E., 1979. Method and Application of Interval Analysis. SIAM, Philadelphia, PA. Orr, D.A., 1962. Random walk production-inventory Policy: Rationale and Implementation. Management Science 9, 108– 122. Paluzzi, J.E., Greiner, T.J., 1993. Finding green in clean: Progressive pollution prevention. Total Quality Environmental Management 2 (3), 283–290. Rakes, T.R., Lori, S.F., Wynne, A.J., 1989. Aggregate production planning using chance constrained goal programming. International Journal Production Research 22 (4), 673–684. Rossiter, A.P., Spriggs, H.D., Klee Jr., H., 1993. Apply process integration to waste minimization. Chemical Engineering Progress 1, 30–36. Russo, M.V., Fouts, P.A., 1997. A resource-based perspective on corporate environmental performance and profitability. Academy of Management Journal 40 (3), 534–549. Sarkis, J., Rasheed, A., 1995. Greening the manufacturing function. Business Horizons 38 (5), 17–27. Saydam, C., Cooper, W.D., 1995. Dye machine scheduling and roll selection. Production and Inventory Management Journal 4, 64– 70. Smith, R., Patella, E.A., 1994. Waste minimization in the process industries. Chemical Engineering 5, 24–25. Spengler, Th., P€ uchert, H., Penkuhn, T., Rentz, O., 1997. Environmental integrated production and recycling management. European Journal of Operational Research 97 (2), 308–326. Tabucanon, M.T., Mukysnkoon, S., 1985. Multiple-objective micro-computer based interactive production. International Journal of Production Research 23 (5), 1001–1023. Taubert, W.H., 1968. A search decision rule for the aggregate scheduling problem. Management Science, 122–135.

C.-C. Wu, N.-B. Chang / European Journal of Operational Research 155 (2004) 68–95

95

Ulhøi, J.P., 1995. Corporate environmental and resource management: In search of a new managerial paradigm. European Journal of Operational Research 80 (1), 2–15. Walley, N., Whitehead, B., 1994. It is not easy being green. Harvard Business Review 72 (3), 46–52. Wu, C.C., Chang, N.B., 2003. Grey input–output analysis and its application for environmental cost allocation. European Journal of Operational Research 145 (1), 175–201. Zeleny, M., 1974. Linear Multi-objective Programming. Springer, New York, pp. 197–220.