An analysis of energy consumption and cost-effectiveness for overhead crane drive systems by using fuzzy multi-objective linear programming

2.1. Equipment selection decision-related literature

The so-called equipment configuration denotes a configuration made after an environmental evaluation is concluded according to the corporate production guidelines. Usually for equipment configuration, there are many aspects to consider, such as the regulation and evaluation of equipment performance, price and lifetime. However, in practical situations, decision makers make decisions based on multiple attributes when it comes to equipment selection, but these attributes may conflict with each other; therefore, multi-attribute decision making (MADM) is used to solve such dilemmas. Hwang and Yoon [6] consider that MADA can simultaneously evaluate multiple qualitative and quantitative guidelines and prioritize all feasible scenarios; decision makers can then choose the best one from among them. Later, many scholars began to use the analytic hierarchy process (AHP) as a tool for problem solving. Kim et al. [7] used Delphi Method and AHP in the electronics industry to evaluate the preference between scrapped electric appliances and electronic equipment. Caputo et al. [8] adopted AHP for relevant equipment selection and configuration regarding safety issues involving the use of industrial equipment. Kannan et al. [9] introduced Fuzzy TOPSIS for the evaluation and selection of green equipment suppliers in the electronics industry. García-Flores et al. [10] developed a mathematical model for solving the problem of selecting the optimal combination of whey process equipment facility location and the transportation routes problem. This model maximizes the total profit of operating the supply chain subject to production, network flow, production capacity, equipment availability and site setup budget. Izadikhah [11] developed a fuzzy goal programming method for the machine tool selection problem to determine the best machine tool satisfying the order demands for a manufacturing company in Iran. Angius et al. [12] proposed an integrated analytical approach to select equipment with a minimum-cost configuration, and assess the performance of an adaptable assembly line. Temiz and Calis [13] used the AHP and the Preference Ranking Organization Method for Enrichment of Evaluation (PROMETHEE) decision-making methods to select the optimum excavation machine for a construction company in Izmir, Turkey. Zhong et al. [14] proposed an equipment selection knowledge base system to estimate the design process of industrial styrene for a more effective, convenient and smart way to determine the best equipment. Song et al. [15] developed a distributed generation system programming model, and proposed the memorized-firefly algorithm to solve the model for the installation capacity of various power generation units in a distributed generation system. Pei et al. [16] proposed a multi-stage Wiener process-based prognostic model for equipment considering the influence of imperfect maintenance activities on the degradation level and degradation rate.

2.2. Application of fuzzy multi-objective linear programming (FMOLP)

The applicability of FMOLP has been extended by several studies, and subsequently applied to solve imprecise problems in several fields. For instance, Ezzati et al. [17] developed a particularly simple algorithm to find a lexicographic/preemptive fuzzy optimal solution of a fuzzy lexicographic multi-objective linear programming problem with symmetric trapezoidal fuzzy numbers. Liu and Shi [18] proposed a fuzzy stratified simplex method for solving a type of FMOLP problem involving symmetric trapezoidal fuzzy numbers, and used this method to obtain the fuzzy optimal solution. Akbaş and Dalkiliç [19] proposed a solution algorithm based on the weighting of objective functions by using trapezoidal fuzzy numbers for solving a fuzzy multi-objective linear programming problem. Aggarwal and Sharma [20] proposed a new algorithm to solve a fully FMOLP problem with all the constraints as fuzzy inequalities, and to find the fuzzy Pareto optimal solution. Nguyen [21] developed a portfolio selection model using the FMOLP which was able to capture uncertainty for portfolio selection taking higher moments into account. Tu and Chang [22] proposed a binary fuzzy goal programming method to solve an airport logistics center expansion plans problem for airport ground handling service companies in Taiwan under fuzzy environment. Chen et al. [23] proposed an approach based on fuzzy goal programming and quality function deployment for facilitating new product planning to maximize customer satisfaction in a fuzzy environment. Mokhtari and Hasani [24] developed a fuzzy multi-objective programming model, and then solved it using a simulated annealing heuristic for a cleaner production-transportation planning problem involving multiple products produced in manufacturing plants. Saxena et al. [25] proposed a fuzzy multi-objective integer programming model with carbon tax policy using an interactive programming approach to solve remanufacturing strategic supply chain planning problems. Shojaie and Raoofpanah [26] developed a multi-objective linear integer mathematical programming model with triangular fuzzy numbers for solving two-objective fuzzy transportation problems.

3.1. Problem description

The configuration of overhead crane systems needs to be analyzed and then to proceed according to two considerations: cost and power consumption. For cost, there are roughly two stages: purchases in earlier stage, and the repairing and maintenance in the later stage. In the purchase stage, the focus should be on the main components of the overhead crane system, which include the main hoist, bridge, trolley, and the fixed and variable frequency motors, which are analyzed in respect to purchase and installation cost along with the corresponding driving system; while for the repair and maintenance in the later stage, the costs cover the regular maintenance for the overhead crane system, including labor, maintenance and repair.

For energy consumption, the main focus is on the calculation of the amount of power consumed and saved during the operation of the overhead crane system. The main power consumption results from the fixed and variable frequency motors that energize the main hoist, bridge and trolley for moving operations, among which the motor’s power consumption is subject to the influence of the driving system.

With practical considerations, decision makers must simultaneously consider multiple goals for the establishment of an overhead crane system. For example, the goals of minimum overall cost and power consumption become unclear due to the uncertainty regarding relevant decisive parameters, including cost, operation time and payload. Furthermore, there are conflicting considerations between among these goals, e.g. the cost of a full fixed frequency driving system is lower but will entail higher electric cost; on the contrary, the cost of a variable frequency driving system is higher but the power consumption is lower; if a recovery type variable frequency driving system is introduced, as such a system is able to generate power spontaneously for the use of crane operation, the power consumption is much lower, but the purchase cost is relatively higher.

In conclusion, this study considered various configuration for establishing an overhead crane system, including fixed frequency and variable frequency motors for the crane system, and fixed frequency, variable frequency and recovery type variable frequency driving systems, and used fussy FMOLP to construct an overhead crane system configuration complying with the practical decision making that satisfies the goals of minimum overall cost and overall power consumption.

3.2. Fuzzy multi-objective linear programming (FMOLP) model

The original FMOLP model for solving the above problems can be applied in the piecewise linear membership function developed by Hannan [27]. According to Hannan, piecewise linear membership functions should be specified to represent the fuzzy scenario, including the fuzzy decision-making of Bellman and Zadeh [28]. To derive an auxiliary variable L, a minimum operator is used to integrate the fuzzy set and transform the original FMOLP model into a single objective LP model, and L (0 ≤ L ≤ 1) is used to measure the degree of satisfaction regarding the decision-making. The solution procedure of an FMOLP model is derived [27] as follows:

Assuming X₁0, X₁₁, …,  X_1,N+1, q₁₁, …,  q_1N represent objective values and f _k (z _k ) scales, then:

Step 1. Derive a membership function f _k (z _k ) for each objective function z _k (k = 1, 2). Table 1. represents the piecewise linear membership functions, f₁ (z₁) and f₂ (z₂).

Step 2. Connect the points in the discrete membership function with line segments: (z _k ,   (f _k (z _k )), for k = 1, 2.

Step 3. Formulate each piecewise linear membership function f _k (z _k ) as follows:

f k ( z k ) = ∑ n = 1 N α kn | z k - X kn | + β k z k + γ k , k = 1 , 2 , … , G , n = 1 , 2 , … , N(17)

where α kn = - t k , n + 1 - t kn 2 , β k = t k , N + 1 + t k 1 2 , γ k = S k , N + 1 + S k 1 2 .

For each segment X_k,r-1 ≤ L ≤ X _kr ), assume that f _k (z _k ) = t _kr z _k + S _kr , where t _k r denotes the slope, and assume that S _k r represents the y-intercept of the line segment at [X_k,r-1,  X _kr ] in the piecewise linear membership function. Therefore,

f 1 ( z 1 ) = - ( t 12 - t 11 2 ) | z 1 - X 11 | - ( t 13 - t 12 2 ) | z 1 - X 12 | - … - ( t 1 , N + 1 - t 1 N 2 ) | z 1 - X 1 N | + ( t 1 , N + 1 + t 11 2 ) z 1 + S 1 , N + 1 + S 11 2 ( t 1 , n + 1 - t 1 n 2 ) ≠ 0 , n = 1 , 2 , … , N(19)

where t 11 = ( q 11 - 0 X 11 - X 10 ) , t 12 = ( q 12 - q 11 X 12 - X 11 ) , … , t 1 , N + 1 = ( 1.0 - q 1 N X 1 , N + 1 - X 1 N )

and S_1,N+1 delineates the intercept between the line segment of X₁N and X_1,N+1 to the vertical line, which is derived by f₁ (z₁) = t_1rz₁ + S_1r.

f 2 ( z 2 ) = - ( t 22 - t 21 2 ) | z 2 - X 21 | - ( t 23 - t 22 2 ) | z 2 - X 22 | - … - ( t 2 , N + 1 - t 2 N 2 ) | z 2 - X 2 N | + ( t 2 , N + 1 + t 21 2 ) z 2 + S 2 , N + 1 + S 21 2 , ( t 2 , n + 1 - t 2 n 2 ) ≠ 0 , n = 1 , 2 , … , N(21)

where t 21 = ( q 21 - 0 X 21 - X 20 ) , t 22 = ( q 22 - q 21 X 22 - X 21 ) , … , t 2 , N + 1 = ( 1.0 - q 2 N X 2 , N + 1 - X 2 N ) ,

and S_2,N+1 delineates the intercept between the line segment of X₂N and X_2,N+1 to the vertical line, which is derived by f₂ (z₂) = t_2rz₂ + S_2r.

Step 4. Introduce the nonnegative deviational variables d kn + and d kn -

X 1 n = ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( O C dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + Ce t ) × Y dijk ] + d 1 n - - d 1 n + , n = 1 , 2 , … , N(23)

X 2 n = ∑ t = 0 T ( C D t - C S t ) + d 2 n - - d 2 n + , n = 1 , 2 , 3 , … , N(24)

where d kn - and d kn + denote the negative and positive deviational variables, respectively, at the jth point, and X _k n represents the values of the kth objective function at the nth point.

Step 5. Substitute Equations (19 and 20) into (21) and (22), respectively, which yields:

f 1 ( Z 1 ) = - ( t 12 - t 11 2 ) ( d 11 - + d 11 + ) - ( t 13 - t 12 2 ) ( d 12 - + d 12 + ) - … - ( t 1 , N + 1 - t 1 N 2 ) ( d 1 N - + d 1 N + ) + ( t 1 , N + t 11 2 ) × { ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( O C dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( NC t × M t ) + C e t ) × Y dijk ] } + S 1 , N + 1 + S 11 2(25)

f 2 ( Z 2 ) = - ( t 22 - t 21 2 ) ( d 21 - + d 21 + ) - ( t 23 - t 22 2 ) ( d 22 - + d 22 + ) - ⋯ - ( t 2 , N + 1 - t 2 N 2 ) ( d 2 N - + d 2 N + ) + ( t 2 , N + 1 + t 21 2 ) × ∑ t = 0 T ( C D t - C S t ) + S 2 , N + 1 + S 21 2(26)

Step 6. Introduce the auxiliary variable L (0 ≤ L ≤ 1), and transform the original FMOLP problems into an equivalent crisp LP form by using the minimum operator to aggregate fuzzy sets. The resulting equivalent crisp LP form for solving the fuzzy multiple objectives problem can be formulated as follows:

Max L s . t . L ≤ - ( t 12 - t 11 2 ) ( d 11 - + d 11 + ) - ( t 13 - t 12 2 ) ( d 12 - + d 12 + ) - … - ( t 1 , N + 1 - t 1 N 2 ) ( d 1 N - + d 1 N + ) + ( t 1 , N + 1 + t 11 2 ) × { ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( O C dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + C e t ) × Y dijk ] } + S 1 , N + 1 + S 11 2(27)

L ≤ - ( t 22 - t 21 2 ) ( d 21 - + d 21 + ) - ( t 23 - t 22 2 ) ( d 22 - + d 22 + ) - … - ( t 2 , N + 1 - t 2 N 2 ) ( d 2 N - + d 2 N + ) + ( t 2 , N + 1 + t 21 2 ) × { ∑ t T ( C D t - C S t ) } + S 2 , N + 1 + S 21 2(28) ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( OC dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + Ce t ) × Y dijk ] + d 1 n - - d 1 n + = X 1 n ( 25 )

∑ t = 0 T ( C D t - C S t ) + d 2 n - - d 2 n + = X 2 n(30)

Equations (10–15) L , d 1 n - , d 1 n + , d 2 n - , d 2 n + , y dijk ≥ 0 , ∀ d , i , j , k , n

Step 7. Repeat, implement and modify the above process until a satisfactory solution is obtained.

The proposed model provides a systematic decision-making solving procedure facilitating the fuzzy decision-making process, enabling decision makers to interactively adjust the search direction during the solution procedure in order to obtain the preferred satisfactory solutions for a decision maker.

4.1. Case descriptions

This study took the overhead crane system in a steel reel factory as the object for model testing and verification. The main business of the steel reel factory is selling steel reels, and the factory size is about 6000 m². The maximum payload of the overhead crane system (D) is 200 tons, the budget upper limit (B) is 15 million NTD and the ROI (Y) is 10 years. The overhead crane system’s annual operation time (OY) is 250 days and daily operation time (OT _dijk ) is 24 hours. The cost and energy consumption are two factors studied in this study with regard to the main hoist, bridge and trolley. The energy consumption of all three pieces of equipment is mainly based on electricity, which is calculated by assuming the unit cost of 3 NTD/kWh.

4.2. Initial solution

Under the goal of minimum overall cost and satisfying all constraints, the overall cost for the overhead crane system configuration Z₁ is 8,472,160 NTD, and annual power consumption Z₂ is 5,557,020 kWh, as shown in Table 2.

It is known from Table 3 that under the goal of minimum overall cost, full fixed frequency is selected for each piece of equipment in combination for the optimum equipment configuration. Under the goal of minimum power consumption and satisfying all constraints, the annual overall power consumption for the overhead crane system configuration Z₂ is 2,604,748 kWh, and overall cost Z₁ is 12,459,170 NTD, as shown in Table 3.

It is known from Table 4 that under the goal of minimum overall power consumption, recovery type variable frequency is selected for each piece of equipment in combination for the optimum equipment configuration.

In all of the experiments herein, a personal computer equipped with an Intel^® 1.7GHZ CPU and 6G RAM was used. The fuzzy multiple objectives problem can be solved by applying the procedure described in Section 3.3.

Step 1. Use the conventional LP model to obtain initial solutions for each objective function. The results obtained are Z₁= 8,472,160 NTD and Z₂= 2,604,748 kWh. Then, formulate the FMOLP model using initial solutions.

Step 2. Use Table 4 to plot the piecewise linear membership functions (z₁,  f₁ (z₁)) and (z₂,  f₂ (z₂)) (Figs. 1 and 2). The curves for the fuzzy objective should have almost linear associations with the Hannan membership function.

OPEN IN VIEWER

Fig. 1.

Curve for piecewise linear membership function (z₁, (f₁ (z₁))).

OPEN IN VIEWER

Fig. 2.

Curve for piecewise linear membership function (z₂, (f₂ (z₂))).

Step 3. Express the piecewise linear membership functions in the following format: f 1 ( Z 1 ) = - 0.000000 0769 | Z 1 - 13100000 | - 0.000000 0385 | Z 1 - 11000000 | - 0.0000002692 × Z 1 + 3.53846 ( 27 ) f 2 ( Z 2 ) = - 0.000000 1205 | Z 2 - 4270000 | - 0.000000 0602 | Z 2 - 3440000 | - 0.0000004217 × Z 2 + 2.3506 ( 28 )

Step 4. Introduce the nonnegative deviational variables, which yield:

∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( O C dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + Ce t ) × Y dijk ] + d 11 - - d 11 + = 11100000(34)

∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( OC dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + C e t ) × Y dijk ] + d 12 - - d 12 + = 9800000(35)

∑ t = 0 T ( C D t - C S t ) + d 21 - - d 21 + = 4270000(36)

∑ t = 0 T ( C D t - C S t ) + d 22 - - d 22 + = 3440000(37)

Step 5. Formulate the piecewise linear equations for each membership function where f _k (z _k ), k = 1, 2.

f 1 ( Z 1 ) = - 0.000000 0769 ( d 11 - - d 11 + ) - 0.000000 0385 ( d 12 - - d 12 + ) - 0.000000 2692 × { ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( OC dijk + N dijk ) × Y dijk ] + ∑ t = 0 T ∑ d = 1 D ∑ i = 1 3 ∑ j = 1 3 ∑ k = 1 K [ ( ( N C t × M t ) + C e t ) × Y dijk ] } + 3.53846(38) f 2 ( Z 2 ) = - 0.0000001205 ( d 21 - - d 21 + ) - 0.0000000602 ( d 22 - - d 22 + ) + 0 . 0000004217 × { ∑ t = 0 T ( C D t - C S t ) } + 2.3506 ( 34 )

Step 6. Introduce the auxiliary variable L (0 ≤ L ≤ 1), and transform the original FMOLP problems into an equivalent crisp LP form by using the minimum operator to aggregate the fuzzy sets. The resulting equivalent crisp LP form for solving the fuzzy multi-objective problem.

For the above equations, using the linear programming software program LINGO version 11.0 to run

the ordinary single-objective LP model for fuzzy multi-objectives problem obtained the overall cost Z₁

as 9,319,184 NTD and overall power consumption Z₂ as 3,116,623 kWh, as shown in Tables 5 and 6, respectively. The satisfaction of the decision maker reaches 87.42%.

The solutions of single objective programming and FMOLP are summarized in Table 7. The solutions derived by FMOLP have considered both overall cost and overall power consumption, among which Z₁= 9,319,184 NTD and Z₂= 3,116,623 kWh, which are between the results of objectives 1 and 2. The overall satisfaction of decision maker L is 87.42%. The configuration result complies with the actual needs of the planner.