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

Abstract

The aims of this paper are to consider cost and energy consumption to solve overhead crane systems’ setting issues. The main issue is that cranes use higher frequency handling equipment in heavy industry. In the overhead crane system setting issues, the determining factors, such as cost, load and energy consumption, are fuzzy. Moreover, decision makers must simultaneously consider real-world conflicting multi-objectives. In summary the question involves a fuzzy multi-objectives problem. Therefore, this paper adopted fuzzy multi-objective programming to construct a mathematical model aimed at minimizing cost and energy consumption with reference to the crane load, recovery period and budget, to carry out a crane system configuration. The model is used for a real problem to verify its correctness. Finally, this study provides a reference for decision makers to purchase crane device configurations.

Keywords

Fuzzy multi-objective linear programming overhead crane systems energy consumption and cost equipment assignment

1. Introduction

The overhead crane system is transportation equipment used frequently in heavy industry, such as the steel industry where overhead crane systems are mostly used in factories for transportation operation [1]. In an overhead crane system, the main equipment includes the main crane, bridge and trolley. Generally, the main crane is the core equipment of an overhead transportation system. Since equipment for different payloads consumes power differently, e.g. small payload consumes lower power; while big payload consumes more power. Therefore, when implementing an overhead traveling crane, power consumption calculations are required for the motor of main equipment, i.e. the main crane, bridges and small trolleys. The calculation of motor power consumption will differ according to the corresponding driving systems, which must consider the combination of motor and driving system. Current overhead crane systems can be roughly divided into old type and new type. The old type of overhead crane driving system uses full fixed-frequency motors and driving systems as its configuration. Under such configuration, though the initial cost for purchase is lower, the annual power consumption thereafter is relatively higher; on the contrary, the new type of overhead crane driving system uses variable-frequency motors and driving systems or recovery type variable-frequency motors, which not only has the advantage of energy saving, but also can charge back the motor with the power saved from terrain conditions. Therefore, an important issue in planning an overhead crane system is determining how to come up with the best combination of motor and driving system while achieving the targets of low cost and low power consumption [2].

Past researches related to equipment mostly focused on single objects and pursued overall cost minimization [3 –5]; few considered both goals: low cost and low power consumption. Therefore, this study utilizes fuzzy multi-objective linear programming (FMOLP) to solve the foregoing problems and offers reference to decision makers for the planning of overhead crane systems.

This study addresses a new equipment selection problem, which is distinguished by its inclusion of two salient features—energy consumption and cost-effectiveness for overhead crane drive systems in uncertain environments. However, this problem has received very little research attention. The proposed FMOLP model simultaneously optimizes both total energy consumption and costs considering the crane load, recovery period and budget constraints. That is, the model simultaneously achieves the targets of low energy consumption and low cost.

The remainder of this paper is organized as follows: Section 2 presents a literature review; Section 3 formulates the fuzzy multi-objective linear programming model; Section 4 assesses the feasibility of the proposed model, referring to the case of the steel reel factory. Finally, Section 5 draws conclusions and makes recommendations for future research.

2. Literature review

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. The proposed model

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.

4. Notations

d: Load, d = 1, 2 …, D

i: Equipment type of overhead crane system, i = 1, 2, 3, where i = 1 for signifies the bridge; i = 2 signifies the trolley; and i = 3 signifies the main hoist

j: Driving system type of overhead crane system, j = 1, 2, 3, where j = 1 signifies the fixed frequency driving system; j = 2 signifies the variable frequency driving system; and j = 3 signifies the recovery type variable frequency driving system

k: Brand of overhead crane system, k = 1, 2 …, K

t: Planning duration of overhead crane system (year), t = 0, 1 …, T, where t = 0 signifies the initial phase, and t = T signifies the overall planning phase

OC_dijk: Initial purchase cost of equipment type i of brand k with driving system j and under load d (NTD)

N_dijk: Initial installation cost of equipment type i of brand k with driving system j and under load d (NTD)

NC_dijk: Maintenance cost of equipment type i of brand k with driving system j and under load d (NTD)

M_t: Times of overhead crane system maintenance in the t-th year (times/year)

OT_dijk: Daily operation time of equipment type i of brand k with driving system j and under load d (h/day)

OY: Annual operation days of the overhead crane system (days/year)

m_dijk: Maintenance of equipment type i of brand k with driving system j and under load d after running for a certain time (times/h)

EC_dijk: Hourly electricity consumption of equipment type i of brand k with driving system j and under load d (kW/h)

Ce: Cost of electricity used by overhead crane system per kilowatt (NTD/kW)

Ce_t: Cost of electricity used by overhead crane system in the t-th year (NTD)

FD_tdijk: Power consumption of equipment type i of brand k with driving system j and under load d in the t-th year (kW/year)

CD_t: Electricity consumed by overhead crane system in the t-th year (kW/year)

CS_t: Electricity recovered by overhead crane system in the t-th year (kW/year)

ST_dijk: Startup power consumption of equipment type i of brand k with driving system j and under load d (kW/time)

FO_dijk: Cruising power consumption of equipment type i of brand k with driving system j and under load d (kW/time)

AC_dijk: Acceleration power consumption of equipment type i of brand k with driving system j and under load d (kW/time)

MS_j: Rotational speed of motor j (rpm)

LW: Hoisting weight of each time (tons)

g: Gravitational acceleration (m/s²)

HU: Hoisting height of load on main hoist (meter)

EF_dijk: Operation efficiency of main hoist

HO_d: Hook weight of main hoist under load d

HE: Ascending height of main hoist with no load (meter)

HB: Descending height of main hoist with load (meter)

HL: Descending height of main hoist with no load (meter)

PS_t: Power saving of main hoist with load in the t-th year (kW/year)

PE_t: Power saving of main hoist with no load in the t-th year (kW/year)

P_d: Load borne by overhead crane system under load d (tons)

D: Maximum hoisting weight of overhead crane (tons)

PD_dijk: Load borne by equipment type i of brand k with driving system j and under load d (kW/time)

Tr_d: Trolley weight under load d (tons)

Br_d: Bridge weight under load d (tons)

Y: Payback time of additional investment increased by the configuration suggested by this study (year)

B: Total equipment budget (NTD)

r_j: Energy saving rate using driving system j

$Y_{dijk} : {\begin{matrix} 1, \\ 0, \end{matrix} \begin{matrix} Equipment chosen in initial phase of equipment type i \\ and brand k with driving system j and under load d \\ Otherwise . \end{matrix}$

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

3.2.1. Objective functions

1. Minimizing total cost

The overall cost for equipment configuration includes the purchase cost in initial phase, and maintenance cost and operation cost every year, as shown in Equations (1–3).

$Min Z_{1} = \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [({OC}_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + {Ce}_{t}) \times Y_{dijk}]$ (2) $M_{t} = \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [\frac{O T_{dijk} \times OY}{m_{dijk}}]$ (3)

$C e_{t} = \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [O T_{dijk} \times OY \times E C_{dijk} \times Ce]$ (4)

2. Minimizing total power consumption

The goal of Equation (4) is to minimize the total power consumption of the equipment configuration in this study by deducting the power saved in the case of an installed recovery type variable frequency driving system, where CD_t is the equipment power consumption of the configuration in this study, as represented in Equation (5), and CS_t is the electricity recovered by the recovery type variable frequency driving system, as represented in Equation (6). $Min Z_{2} = \sum_{t = 0}^{T} (C D_{t} - C S_{t})$ (5)

$\begin{matrix} C D_{t} = \sum_{d}^{D} \sum_{i}^{3} \sum_{j}^{3} \sum_{k}^{K} [(S T_{d 1 jk} + F O_{d 1 jk} + A C_{d 1 jk}) \times {MS}_{j} \times Y_{d 1 jk} + (S T_{dijk} + F O_{dijk} + A C_{dijk}) \times \\ M S_{j} \times Y_{d 2 jk} + (LW \times g \times HU / E F_{d 3 jk}) + (H O_{d} \times g \times HE / E F_{d 3 jk}) + (LW \times g \times HB / E F_{d 3 jk}) \\ + (H O_{d} \times g \times HL / E F_{d 3 jk}) \times Y_{d 3 jk}] \end{matrix}$ (6) ${CS}_{t} = \sum_{t}^{T} \sum_{d}^{D} \sum_{k}^{K} [P S_{t} + P E_{t}] \times Y_{d 33 k}$ (7)

(1) Power saved from the main hoist descending with the load: The amount of power saved derived from the weight of the load is represented in Equation (7). $P S_{t} = \sum_{d}^{D} \sum_{k}^{K} (LW \times g \times HB \times E F_{d 33 k}) \times Y_{d 33 k}$ (8)

(2) Power saved from the main hoist descending with no load: The amount of power saved derived from the weight of the hook during the main moist descending with no load is represented in Equation (8).

$P E_{t} = \sum_{d}^{D} \sum_{k}^{K} (H O_{d} \times g \times HL \times E F_{d 33 k}) \times Y_{d 33 k}$ (9)

3.2.2. Constraints

1. Requirement constraints. $\sum_{d = 1}^{D} P_{d} \geq D + \sum_{d = 1}^{D} H O_{d}$ (10)

(1) Load constraint of the main hoist. $\sum_{d = 1}^{D} \sum_{j = 1}^{3} \sum_{k = 1}^{K} (P D_{d 3 jk} + Y_{td 3 jk}) \geq \sum_{d = 1}^{D} P_{d}$ (11)

(2) Load constraint of the trolley. $\sum_{d = 1}^{D} \sum_{j = 1}^{3} \sum_{k = 1}^{K} (P D_{d 2 jk} + Y_{td 2 jk}) \geq \sum_{d = 1}^{D} (P_{d} + T r_{d})$ (12)

(3) Load constraint of the bridge.

$\sum_{d = 1}^{D} \sum_{j = 1}^{3} \sum_{k = 1}^{K} (P D_{d 1 jk} + Y_{td 1 jk}) \geq \sum_{d = 1}^{D} (P_{d} + T r_{d} + B r_{d})$ (13)

2. Return on investment (ROI) constraint in a year for variable frequency and recovery type equipment.

$\begin{matrix} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(O C_{dijk} + N_{dijk}) \times Y_{dijk}] \\ - \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{k = 1}^{K} [(O C_{di 1 k} + N_{di 1 k}) \times Y_{di 1 k}] \\ \div \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{k = 1}^{K} [O T_{di 1 k} \times OY \times E C_{di 1 k} \times Ce \times Y_{di 1 k}] \\ - \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [O T_{dijk} \times OY \\ \times E C_{dijk} \times Ce \times Y_{dijk}] \leq Y \end{matrix}$ (14)

3. Budget constraint. $\sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [({OC}_{dijk} + N_{dijk}) \times Y_{dijk}] \leq B$ (15)

4. Non-negativity constraint.

$\begin{matrix} Y_{dijk} \geq 0 and integer, d = 1, 2, \dots, D, \\ i = 1, 2, \dots, I, j = 1, 2, \dots, J, k = 1, 2, \dots, K \end{matrix}$ (16)

3.3. Solving with the 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).

Table 1
Membership function f_k (z_k)

z ₁ > X₁₀ X ₁₀ X ₁₁ … X _1N X _1,N+1 < X_1,N+1

f₁ (z₁) 0 0 q ₁₁ … q _1N 1.0 1.0

z ₂ > X₂₀ X ₂₀ X ₂₁ … X _2N X _2,N+1 < X_2,N+1

f₂ (z₂) 0 0 q ₂₁ … q _2N 1.0 1.0

where, 0 ≤ q_kn ≤ 1.0 and q_kn ≤ q_k,n+1 for k = 1, 2; n = 1, 2, …, N.

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:

$\begin{matrix} f_{k} (z_{k}) = \sum_{n = 1}^{N} α_{kn} | z_{k} - X_{kn} | + β_{k} z_{k} + γ_{k}, \\ k = 1, 2, \dots, G, n = 1, 2, \dots, N \end{matrix}$ (17)

where $\begin{matrix} α_{kn} = - \frac{t_{k, n + 1} - t_{kn}}{2}, β_{k} = \frac{t_{k, N + 1} + t_{k 1}}{2}, \\ γ_{k} = \frac{S_{k, N + 1} + S_{k 1}}{2} . \end{matrix}$

For each segment X_k,r-1 ≤ L ≤ X_kr), assume that f_k (z_k) = t_krz_k + S_kr, where t_kr denotes the slope, and assume that S_kr represents the y-intercept of the line segment at [X_k,r-1, X_kr] in the piecewise linear membership function. Therefore,

$\begin{matrix} f_{1} (z_{1}) = - (\frac{t_{12} - t_{11}}{2}) | z_{1} - X_{11} | - (\frac{t_{13} - t_{12}}{2}) | z_{1} - X_{12} | - \dots - (\frac{t_{1, N + 1} - t_{1 N}}{2}) | z_{1} - X_{1 N} | \\ + (\frac{t_{1, N + 1} + t_{11}}{2}) z_{1} + \frac{S_{1, N + 1} + S_{11}}{2} (\frac{t_{1, n + 1} - t_{1 n}}{2}) \neq 0, n = 1, 2, \dots, N \end{matrix}$ (19)

where $t_{11} = (\frac{q_{11} - 0}{X_{11} - X_{10}}), t_{12} = (\frac{q_{12} - q_{11}}{X_{12} - X_{11}}), \dots, t_{1, N + 1} = (\frac{1.0 - q_{1 N}}{X_{1, N + 1} - X_{1 N}})$

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.

$\begin{matrix} f_{2} (z_{2}) = - (\frac{t_{22} - t_{21}}{2}) | z_{2} - X_{21} | - (\frac{t_{23} - t_{22}}{2}) | z_{2} - X_{22} | - \dots - (\frac{t_{2, N + 1} - t_{2 N}}{2}) | z_{2} - X_{2 N} | \\ + (\frac{t_{2, N + 1} + t_{21}}{2}) z_{2} + \frac{S_{2, N + 1} + S_{21}}{2}, (\frac{t_{2, n + 1} - t_{2 n}}{2}) \neq 0, n = 1, 2, \dots, N \end{matrix}$ (21)

where $t_{21} = (\frac{q_{21} - 0}{X_{21} - X_{20}}), t_{22} = (\frac{q_{22} - q_{21}}{X_{22} - X_{21}}), \dots, t_{2, N + 1} = (\frac{1.0 - q_{2 N}}{X_{2, N + 1} - X_{2 N}}),$

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 $d_{kn}^{-}$

$\begin{matrix} X_{1 n} = \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(O C_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + {Ce}_{t}) \times Y_{dijk}] \\ + d_{1 n}^{-} - d_{1 n}^{+}, n = 1, 2, \dots, N \end{matrix}$ (23)

$X_{2 n} = \sum_{t = 0}^{T} (C D_{t} - C S_{t}) + d_{2 n}^{-} - d_{2 n}^{+}, n = 1, 2, 3, \dots, N$ (24)

where $d_{kn}^{-}$ and $d_{kn}^{+}$ denote the negative and positive deviational variables, respectively, at the jth point, and X_kn 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:

$\begin{matrix} f_{1} (Z_{1}) = - (\frac{t_{12} - t_{11}}{2}) (d_{11}^{-} + d_{11}^{+}) - (\frac{t_{13} - t_{12}}{2}) (d_{12}^{-} + d_{12}^{+}) - \dots - (\frac{t_{1, N + 1} - t_{1 N}}{2}) (d_{1 N}^{-} + d_{1 N}^{+}) \\ + (\frac{t_{1, N} + t_{11}}{2}) \times {\sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(O C_{dijk} + N_{dijk}) \times Y_{dijk}] \\ + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(({NC}_{t} \times M_{t}) + C e_{t}) \times Y_{dijk}]} + \frac{S_{1, N + 1} + S_{11}}{2} \end{matrix}$ (25)

$\begin{matrix} f_{2} (Z_{2}) = - (\frac{t_{22} - t_{21}}{2}) (d_{21}^{-} + d_{21}^{+}) - (\frac{t_{23} - t_{22}}{2}) (d_{22}^{-} + d_{22}^{+}) - \dots - (\frac{t_{2, N + 1} - t_{2 N}}{2}) (d_{2 N}^{-} + d_{2 N}^{+}) \\ + (\frac{t_{2, N + 1} + t_{21}}{2}) \times \sum_{t = 0}^{T} (C D_{t} - C S_{t}) + \frac{S_{2, N + 1} + S_{21}}{2} \end{matrix}$ (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:

$\begin{matrix} Max L \\ s . t . \\ L \leq - (\frac{t_{12} - t_{11}}{2}) (d_{11}^{-} + d_{11}^{+}) - (\frac{t_{13} - t_{12}}{2}) (d_{12}^{-} + d_{12}^{+}) - \dots - (\frac{t_{1, N + 1} - t_{1 N}}{2}) (d_{1 N}^{-} + d_{1 N}^{+}) \\ + (\frac{t_{1, N + 1} + t_{11}}{2}) \times {\sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(O C_{dijk} + N_{dijk}) \times Y_{dijk}] \\ + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + C e_{t}) \times Y_{dijk}]} + \frac{S_{1, N + 1} + S_{11}}{2} \end{matrix}$ (27)

$\begin{matrix} L \leq - (\frac{t_{22} - t_{21}}{2}) (d_{21}^{-} + d_{21}^{+}) - (\frac{t_{23} - t_{22}}{2}) (d_{22}^{-} + d_{22}^{+}) - \dots - (\frac{t_{2, N + 1} - t_{2 N}}{2}) (d_{2 N}^{-} + d_{2 N}^{+}) \\ + (\frac{t_{2, N + 1} + t_{21}}{2}) \times {\sum_{t}^{T} (C D_{t} - C S_{t})} + \frac{S_{2, N + 1} + S_{21}}{2} \end{matrix}$ (28) $\begin{matrix} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [({OC}_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + {Ce}_{t}) \times Y_{dijk}] \\ + d_{1 n}^{-} - d_{1 n}^{+} = X_{1 n} (25) \end{matrix}$

$\sum_{t = 0}^{T} (C D_{t} - C S_{t}) + d_{2 n}^{-} - d_{2 n}^{+} = X_{2 n}$ (30)

Equations (10–15) $L, d_{1 n}^{-}, d_{1 n}^{+}, d_{2 n}^{-}, d_{2 n}^{+}, y_{dijk} \geq 0, \forall d, i, j, k, n$

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. Model implementation

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.

Table 2
Optimum equipment configuration of minimum overall cost

Objective 1: Minimum overall cost

Equipment type Main hoist Bridge Trolley

Motor type Fixed frequency Variable frequency Fixed frequency Variable frequency Fixed frequency Variable frequency

Equipment type Common variable frequency Recovery type variable frequency Common variable frequency Recovery type variable frequency Common variable frequency Recovery type variable frequency

Payload (tons) 50 – – – – – – – – –

100 2 – – 2 – – 2 – –

150 – – – – – – – – –

Z₁ = 8,472,160(NTD)

Z₂ = 5,557,020 (kWh/year)

Objective 1: Minimum overall cost
Payload (tons)	50	–	–	–	–	–	–	–	–	–
	100	2	–	–	2	–	–	2	–	–
	150	–	–	–	–	–	–	–	–	–
Z₁ = 8,472,160(NTD)
Z₂ = 5,557,020 (kWh/year)

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.

Table 3

Optimum equipment configuration of minimum overall power consumption

Objective 2: Minimum overall power consumption
Equipment type		Main hoist			Bridge			Trolley
Motor type		Fixed frequency	Variable frequency		Fixed frequency	Variable frequency		Fixed frequency	Variable frequency
Equipment type			Common variable frequency	Recovery type variable frequency		Common variable frequency	Recovery type variable frequency		Common variable frequency	Recovery type variable frequency
Payload (tons)	50	–	–	–	–	–	–	–	–	–
	100	–	–	2	–	–	2	–	–	2
	150	–	–	–	–	–	–	–	–	–
Z₁ = 12,459,170 (NTD)
Z₂ = 2,604,748 (kWh/year)

Table 4

Lists the piecewise linear membership functions of the proposed model

Z₁	≥ 12, 400, 000	11,100,000	9,800,000	≤ 8, 500, 000
f₁ (Z₁)	0	0.5	0.8	1.0
Z₂	≥ 5, 100, 000	4,270,000	3,440,000	≤ 2, 610, 000
f₂ (Z₂)	0	0.5	0.8	1.0

4.3. Solution procedure for the case study

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.

Fig. 1.

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

Fig. 2.

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

Step 3. Express the piecewise linear membership functions in the following format: $\begin{matrix} f_{1} (Z_{1}) = - 0.000000 0769 | Z_{1} - 13100000 | - 0.000000 0385 | Z_{1} - 11000000 | \\ - 0.0000002692 \times Z_{1} + 3.53846 (27) \end{matrix}$ $\begin{matrix} f_{2} (Z_{2}) = - 0.000000 1205 | Z_{2} - 4270000 | - 0.000000 0602 | Z_{2} - 3440000 | \\ - 0.0000004217 \times Z_{2} + 2.3506 (28) \end{matrix}$

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

$\begin{matrix} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [(O C_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + {Ce}_{t}) \times Y_{dijk}] \\ + d_{11}^{-} - d_{11}^{+} = 11100000 \end{matrix}$ (34)

$\begin{matrix} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [({OC}_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + C e_{t}) \times Y_{dijk}] \\ + d_{12}^{-} - d_{12}^{+} = 9800000 \end{matrix}$ (35)

$\sum_{t = 0}^{T} (C D_{t} - C S_{t}) + d_{21}^{-} - d_{21}^{+} = 4270000$ (36)

$\sum_{t = 0}^{T} (C D_{t} - C S_{t}) + d_{22}^{-} - d_{22}^{+} = 3440000$ (37)

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

$\begin{matrix} f_{1} (Z_{1}) = - 0.000000 0769 (d_{11}^{-} - d_{11}^{+}) - 0.000000 0385 (d_{12}^{-} - d_{12}^{+}) - 0.000000 2692 \\ \times {\sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [({OC}_{dijk} + N_{dijk}) \times Y_{dijk}] + \sum_{t = 0}^{T} \sum_{d = 1}^{D} \sum_{i = 1}^{3} \sum_{j = 1}^{3} \sum_{k = 1}^{K} [((N C_{t} \times M_{t}) + C e_{t}) \times Y_{dijk}]} \\ + 3.53846 \end{matrix}$ (38) $\begin{matrix} f_{2} (Z_{2}) = - 0.0000001205 (d_{21}^{-} - d_{21}^{+}) - 0.0000000602 (d_{22}^{-} - d_{22}^{+}) + 0 . 0000004217 \\ \times {\sum_{t = 0}^{T} (C D_{t} - C S_{t})} + 2.3506 (34) \end{matrix}$

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%.

Table 5

Optimum equipment configuration by FMOLP

Equipment type		Main hoist			Bridge			Trolley
Motor type		Fixed frequency	Variable frequency		Fixed frequency	Variable frequency		Fixed frequency	Variable frequency
Equipment type			Common variable frequency	Recovery type variable frequency		Common variable	Recovery type variable frequency		Common variable frequency	Recovery type variable frequency
Payload (tons)	50	–	–	–	–	–	–	–	–	–
	100	–	–	2	2	–	–	2	–	–
	150	–	–	–	–	–	–	–	–	–
Z₁ = 9,319,184 (NTD)
Z₂ = 3,116,623 (kWh/year)
L = 87.42%

Table 6

Equipment specifications by FMOLP

Objective 2: Minimum overall power consumption
Equipment type	Payload (tons)	Motor capacity	Motor type	Driving system	Brand	Equipment quantity
Main hoist	100	400 kW	Variable frequency	Recovery type variable frequency	B	2
Bridge	100	160 kW	Fixed frequency	Fixed frequency	C	2
Trolley	100	80 kW	Fixed frequency	Fixed frequency	E	2

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.

Table 7

Result comparison between single objective programming and FMOLP

Model Objective	LP-1	LP-2	FMOLP
Z₁ (NTD)	8,472,160	12,459,170	9,319,184
Z₂ (kWh)	5,557,020	2,604,748	3,116,623
L (%)	100%	100%	87.42%

5. Conclusion

This study proposes novel fuzzy multi-objective equipment planning decision problem for overhead crane drive systems distinguished by the inclusion of two salient features: energy consumption and cost-effectiveness in uncertain environments. The novel feature of this study is the introduction of a new recovery type variable frequency driving system, which was compared with current driving systems with respect to power consumption and cost, and used as one condition for the decision-making in purchasing an overhead crane system, regarding which an equipment configuration model complying with practical decision-making was established. The proposed formulation of equipment planning decisions for overhead crane drive systems, involving the integrated energy consumption and cost-effective equipment planning problems in a linear programming model with fuzzy multiple goals, is unique in the literature.

The study formulates the equipment planning decision problem as an FMOLP model. The proposed FMOLP model simultaneously optimizes total costs and total energy consumption with the crane load, recovery period and budget constraints. Models that consider economic and energy consumption issues as well as flexibility are needed in today’s increasingly industrialized environment. Moreover, the proposed FMOLP model provides a systematic procedure that assists the equipment planner, enabling a decision-maker to interactively modify the membership functions of the objective until a satisfactory degree is acquired. The case study of an overhead crane system in a steel reel manufacturer demonstrates the feasibility of applying the proposed model. Results of this study significantly contribute to the efforts of satisfying practical managerial requirements, and confirm that the FMOLP model can greatly facilitate decision-makers during equipment planning in fuzzy environments.

Further studies may apply the proposed model to the evaluation of equipment planning decisions involving different cases of payload uncertainty. Additionally, although this study implements a systematic decision-making procedure and obtains feasible solutions, additional data analysis is necessary to implement the proposed FMOLP model for fuzzy environments.

Footnotes

Acknowledgments

The author would like to thank the National Science Council of the Republic of China, Taiwan for financially/partially supporting this research under Contract No. MOST 106-2221-E-020-021.

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z ₁	> X₁₀	X ₁₀	X ₁₁	…	X _1N	X _1,N+1	< X_1,N+1
f₁ (z₁)	0	0	q ₁₁	…	q _1N	1.0	1.0
z ₂	> X₂₀	X ₂₀	X ₂₁	…	X _2N	X _2,N+1	< X_2,N+1
f₂ (z₂)	0	0	q ₂₁	…	q _2N	1.0	1.0