possibilistic linear programming approach to the multi...

Int. J. Pure Appl. Sci. Technol., 7(2) (2011), pp. 117-131

International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in

Research Paper

Possibilistic Linear Programming Approach to the Multi-item Aggregate Production Planning Seema Sarkar (Mondal)1 and Savita Pathak2,* 1, 2 Department of Mathematics, National Institute of Technology, Durgapur, West Bengal, India * Corresponding author, e-mail: ([email protected]) (Received: 25-8-11; Accepted: 12-10-11)

Abstract: This paper presents an application of fuzzy mathematical programming model to solve aggregate production planning (APP). APP goals to maximize the profit through optimal levels of production, subcontracting, inventory, backorder and work levels to meet the demand over a time period. Fuzzy logic was applied to solve the uncertain production, demand, capital and warehouse spaces. All costs are taken as triangular fuzzy numbers. Model is developed such that the system takes minimum subcontracted units in each period and no inventories at the end of the planning horizon. Possibilistic Linear Programming approach is taken to solve the APP problem with imprecise costs, demands, warehouse spaces and capacities for optimization of profit function of the multi-item under material requirements constraints. A numerical example is presented to verify the model. Keywords: multi-item, uncertain production, uncertain demand, uncertain capital, uncertain warehouse space, fuzzy model.

Introduction: Many manufacturing company do not perform with appropriate production planning even though it plays an important role for them. The term” aggregate” represents the planning made for two or more production categories. The purpose of aggregate production planning (APP) is to determine production levels in all categories for matching recent certain demands. In order to achieve this aim, APP considers hiring, firing, overtime, backordering, subcontracting, inventory levels and the other elements of system to be modeled. It also determines appropriate sources for production (Paksoy and Atak, 2005). APP is a medium range capacity planning method that typically encompasses a time horizon from 2 to 12 months. A planner must make decisions regarding output rates, employment levels and

Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 117-131. 118

changes, inventory levels as well as subcontracting to optimize the production plan. Many researchers have developed mathematical optimization models including APP problems (Elion, 1975; Masud and Hwang, 1980). The aim of a manufacturing company for making multiproduct aggregate production planning is to obtain the maximum profit or minimum cost by determining the product quantity, subcontracting quantity, labor level, etc.,to meet the market demand in long term. A lot of researchers have developed various types of models and approached to solve APP decision making problems (Holt et al, 1955; Bergstrom and Smith, 1970; Hasman and Mcclain, 1971 and others). Fung et al, (2003) developed a fuzzy multiproduct aggregate production planning model whose solutions were introduced to cater to different scenarios under various decision making preferences by using parametric programming, best balance and interactive techniques. It is revealed that linear programming (LP) is a conventionally used technique (Tingley, 1987). The managerial decisions in material management are essentially conditioned by product stock out costs and inventory holding costs. In general, the profit rate is the decisive factor for the former, while material price and the inventory holding rate influence for the letter. For capacity analysis, the idleness of key machines has a vial impact upon investment utilization. Because of price fluctuation in a dynamic market, material obsolescence and the time value of capital, assigning a set of crisp values for parameters is no longer appropriate for dealing with such ambiguous decision problems. So, different multi-criteria decision making (MCDM) methods can be effectively used to solve such type of problem. Athawale et. al. [2011] presented a paper at a comparative study on the ranking performance of some multi-criteria decision – making methods for industrial robot selection. In which they considered ten most popular MCDM methods and compared their relative importance with respect to the ranking the alternative robots as engaged in some industrial pick-n-place operation. Fortunately, possibility distribution offers an effectual alternative for proceeding with inherent ambiguous phenomena in determining cost parameters [Guiffrida et. al., (1998), Inuiguchi, (1994), Luhandjula, (1989), Rommelfanger, (1996), Zadeh, (1978), Zimmermann, (1978)]. Therefore in this study we constructed a possibilistic linear programming model into a crisp multiple objective linear programming model and applied Zimmermann’s fuzzy programming technique [1978] to obtain a composite single objective goal. In real world APP problems, input data or related parameters are imprecise due to some information being incomplete. Traditional mathematical programming techniques cannot solve all fuzzy programming problems. Lai and Hwang (1992) developed an auxiliary multiple objective linear programming (MOLP) model for solving a possibilistic linear programming (PLP) problem with imprecise objective and/or constraint coefficients. Wang and Liang (2005) presented a novel ‘ interactive Possibilistic Linear Programming (PLP) approach’ for solving the multi-product APP problem with imprecise forecast demand. Wang and Fang (2001) presented a novel ‘Fuzzy Linear Programming (FLP) method’ for solving the APP problem with multi objectives where the product price, unit cost to subcontracting, work force level, production capacity and market demands are fuzzy in nature. Liang (2008) developed a fuzzy multi objective linear programming (FMOLP) model with piecewise linear membership function to solve integrated multi product and multi time period production/distribution planning decisions (PDPD) problems.


There are few combination of aggregate production planning: (1) the production rate can be altered by affecting changes in the workforce through hiring/firing of workers, (2) the production rate can also be altered by maintaining a constant labor force but introducing overtime or idle time, (3) the production rate may be kept on constant level and the fluctuation in demand can be met by altering the level of subcontracting, (4) the production rate may be kept constant and changes in demand can be absorbed by changes in the inventory level. Any combination of these strategies or alternative is possible. The concern of the APP is to select the strategy with the least cost to the manufacturing companies. This problem has been under an extensive discussion and several alternative methods for finding an optimal solution have been suggested in the literature (Leung et al, 2004; Aliev et al, 2007 and others). In this paper an application of fuzzy mathematical programming model is presented to solve aggregate production planning (APP) problem. APP goals to maximize the profit through optimal levels of production, subcontracting, inventory, backorder and work levels to meet the demand over a time period. Fuzzy logic was applied to solve the uncertain production, demand, capital and warehouse spaces. The proposed model accomplishes forecasting adjustment, items management. Because of price fluctuations, material obsolescence and the time value of capital, the ambiguity of cost is considered in the objective function of the model. All costs are taken as triangular fuzzy numbers. Model is developed such that the system takes minimum subcontracted units in each period and no inventories at the end of the planning horizon. Possibilistic Linear Programming approach is taken to solve the APP problem with imprecise costs, demands, warehouse spaces and capacities for optimization of profit function of the multi-item under material requirements constraints. Here, the fuzzy objective function is substituted with three crisp objectives: (1) minimizing the most possible cost, (2) maximizing the possibility of obtaining lower cost and (3) minimizing the risk of obtaining higher cost. Zimmermann’s fuzzy programming method is then applied for achieving an overall satisfactory compromise solution. Finally, a numerical example is presented to verify the model. Model for APP in fuzzy environments Assumptions and Notations: Assume that a company produces N types of products to meet the market demands over a planning horizon T. Following notations are used in the model: N types of items T planning horizon Z profit function over the planning horizon T

imprecise demand for the n-th item in period t (units) production in regular time of the n-th item by permanent workers in period t (units) production in over time of the n-th item by permanent workers in period t (units) production in regular time of the n-th item by temporary workers in period t (units) production in over time of the n-th item by temporary workers in period t (units) imprecise total over time produced units of the n-th item in period t


subcontracting unit of n-th item in period t (units) inventory level of the n-th item in period t (units) backorder level of the n-th item in period t (units) permanent workers in period t (numbers)

temporary workers in period t (numbers) workers hired in period t (numbers) workers fired in period t (numbers) imprecise warehouse spaces in period t (/unit)

conversion factor in hours of permanent worker per unit of production conversion factor in hours of temporary worker per unit of production total regular time per permanent worker (hours) total regular time per temporary worker (hours) total over time per permanent worker (hours) total over time per temporary worker (hours) space taken by n-th item ( /unit)

imprecise cost to hire one worker in period t ($/man-month) imprecise cost to fire one worker in period t ($/man-month) imprecise subcontracting cost per unit of the n-th item in period t ($/unit) imprecise inventory carrying cost per unit of the n-th product in period t ($/unit) imprecise backorder cost per unit of the n-th product in period t ($/unit) imprecise permanent workers wage in regular time in period t ($/worker)

imprecise temporary workers wage in regular time in period t ($/worker) imprecise permanent workers wage in overtime in period t ($/hour)

imprecise temporary workers wage in over time in period t ($/hour) imprecise sales revenue per unit of the n-th item in period t ($/unit)

All type of production costs are in $/unit and imprecise,

production cost by permanent workers in regular time per unit of the n-th item in period t

production cost by permanent workers in overtime per unit of the n-th item in period t production cost by temporary workers in regular time per unit of the n-th item in period t

production cost by temporary workers in overtime per unit of the n-th item in period t total production cost of the n-th item in period t

Fuzzy Model Formulation: In an APP decision making problem, the objective function can be defined for n-th item in t-th period as follows: Max

=


(1)

Subject to,

, (2)

,

(3)

(4)

, (5)

,

(6)

,

(7)

(8)

(9)

(10)

(11)

(12)

(13)

= ,

(14)

, (15)

, (16)

,

(17)

, , ,


(18)

,

(19)

Here, in equation (1), the terms

,

and represent the total revenue, the total production cost and the cost of managing labor level respectively. All costs are triangular fuzzy numbers. Then the profit function is also a triangular fuzzy number. Equation (2) shows the inventory level of n-th item in t-th period. Equation (3) shows the maximum fuzzy warehouse space and equation (4) is for minimum and maximum level of permanent workers for n-th item in t-th period. Equation (5) shows the maximum level of subcontracting units of n-th item in t-th period. Equation (6) represents that inventory of n-th item in t-th period is greater than the subcontracted units. Equation (7) indicates the maximum level of back order units, and backorder units in the last period T is greater than the inventory level in that period. Equations (8), (9), (10) and (11) represent that the time taken in production by the total produced units at overtime and regular time by permanent and temporary workers should be less than the available labor capacities for n-th item in t-th period respectively. Equation (12) shows the maximum level of temporary workers for n-th item in t-th period. Equation (13) represents that the total number of temporary workers in t-th period is the sum of the numbers of workers hired at the beginning of the (t-1)-th period and the t-th period for n-th item. Equation (14) shows that the number of the temporary workers hired at the beginning of the (t-2)-th period will be fired at the beginning of the t-th present period (due to The Labor Protection Act). Equation (15), (16) and (17) show the over time machine fuzzy capacity in units, the maximum fuzzy demand level and the maximum fuzzy capital level for production only for n-th item in t-th period respectively. Equation (18) indicates that all variables are non- negative and have continuous values. Equation (19) represents the initial conditions. Possibilistic Linear Programming Approach to Optimize the APP Problem of Profit Function of Multi-Item with Imprecise Costs, Demands, Warehouse Spaces and Capacities: With respect to the techniques for solving a linear programming problem with imprecise coefficients in the objective function, Rommelfanger, [1996] states that a fuzzy objective function should be interpreted as a multi objective demand. In general, an ideal solution to this problem does not exist. The first method to obtain a compromise solution was proposed by Tanaka et. al. [1984 b]. They adopt a weighted average as a substitute for the fuzzy


objective with a special crisp compromise objective. The extreme values of the parameters

will have an impact on the effect of the weighting sum. An - Pareto – optimal solution

proposed by Sakawa and Yano [1989] restricts the fuzzy coefficient to - level – sets.

Luhandjula’s [1989] - possibility effect solution is a similar concept. As these authors do

not explain the specifications of the levels and and use several restrictive assumptions, the application of these approaches may be limited in practice. Besides, when the goal of the objective can be given, Tanaka and Asai [1984] considered the objective function as a fuzzy constraint. However, a given goal for the objective function is always difficult for managers to decide. We solved the APP problem with imprecise over time productions, imprecise capital for production, imprecise warehouse spaces, imprecise demands, and imprecise costs by the Wang and Liang (2005)’s PLP approach. Here, we adapted the triangular fuzzy number to the APP problem under fuzzy material requirement constraints. The main advantages of the triangular fuzzy number are simplicity and flexibility of the fuzzy arithmetic operations. The

distribution of a triangular fuzzy number is shown in figure 1. In this study, weighted average method is used to convert triangular fuzzy number into a crisp number. If the minimum acceptable membership level α is given, the corresponding

auxiliary crisp inequality of a triangular fuzzy number can be expressed as:

(20) where, represent the corresponding weight of the most pessimistic, most likely and most optimistic values, respectively. In practice, the weights and the membership level α can be determined by Decision Maker (DM)’s experience (Liang, 2009). Lia and Hwang [1992] referred to portfolio theory and converted the fuzzy objective with a triangular possibility distribution into three crisp objectives. According to their model, equation (1) of our model is presented as Maximize = ( , x), where x is a feasible solution for the proposed PLP model. In geometrical representation, the three critical points ( ), ( ) and ( x, 1) in figure 2 are shown as a fuzzy objective. We therefore proceed to minimize the fuzzy objective by pushing the three points towards the right. Solving the fuzzy objective becomes the process of maximizing and x simultaneously. However, there may become a conflict in the simultaneous optimization process. The imprecise goal function of the PLP model has a triangular possibility distribution. The PLP approach simultaneously involves maximizing which is the most possible value of the imprecise total cost. Meanwhile, we maximize ( ) which is the possibility of obtaining higher total cost and minimize ( ) which is the risk of obtaining lower total cost (Wang and Liang, 2005). The last two objectives are actually relative measures from . The three replaced objective functions still guarantee the above declaration of pushing the possibility distribution toward the right in figure 2. In this way our problem can be transformed into a multiple objective linear programming (MOLP) as follows:


1

Crisp Number

0

Figure 1: The possibility distribution of a triangular fuzzy number

=

, (21)

=

=

1

Crisp Number

0

Figure 2: The possibility distribution of a triangular fuzzy number

There are many MOLP approaches to solve the above problem, such as goal programming, utility theory and so on. Since it is quite difficult for managers to determine the requisite objective goals or establish their utility functions, we suggest using Zimmermann’s fuzzy programming method with the normalization process. The solution procedure of this PLP method is given as:

Mem

ber

ship

Deg

ree

Mem

bers

hip

Deg

ree


Step 1: The three new crisp objective functions of the auxiliary MOLP problem are developed as shown in equations (21) to (23). Step 2: Given the minimum acceptable possibility for example α = 0.5, the imprecise over time productions, imprecise capital for production, imprecise warehouse spaces and imprecise demands constraints are converted to the crisp ones by the weighted average method as follows:

,

(24)

,

(25)

,

(26)

,

(27)

Step 3: The PIS (positive ideal solution) and the NIS (negative ideal solution) of the new three objective functions can be specified as:

, ;

, ;

, ; (28)

The corresponding linear membership functions of the three objective functions are defined by:

,

(29)


,

(30)

,

(31)

Step 4: Finally, we apply Zimmermann’s equivalent single-objective L P model (preference-based membership functions of the objective function) to obtain the overall satisfaction compromise solution. The single- objective ordinary L P model for solving the APP problem is formulated as follows: Max λ Subject to,

λ , i = 1,2,3; 0 , and equations (2), (4) to (14) and (24) to (27), (32) where, λ represents the overall DM satisfaction under the proposed strategy of maximizing the most possible value. If λ = 1, then each goal is fully satisfied. If λ = 0, then none of the goals are satisfied. The PLP approach provides the overall degree of DM satisfaction. Numerical Illustration Here, a numerical example is presented to illustrate the model with input data given in TABLE 1 to TABLE 6 with four term planning horizon for three items. First, all the fuzzy inequality constraints are converted to crisp ones using weighted average method at α = 0.5. The original single objective LP model for solving APP problem can be formulated as a multi-objective LP problem with equations (21), (22) and (23). Now, the problem is solved using the single objective LP problem to obtain the PIS and NIS of the objectives using LP solver for the parameters given in TABLES 1 to 6. The PIS and NIS values are given in TABLE 7. The problem is solved according to the equation (32) using membership functions (29), (30) and (31) with obtained PIS and NIS values.


TABLE-1 (Input data)

t=1,2,3,4 n

1 185,250,325 75,90.105 80,95,110 80,95,110 85,100,115 160,200,240

2 235,300,375 80,95,110 85,100,115 85,100,115 90,105,120 170,210,250

3 285,350,425 85,100,115 90,105,120 90,105,120 95,110,125 180,220,260


t=1,2,3,4 n

t=1 t=2 t=3 t=4

1 40,50,60 75,90,105 536,670,804 5 40, 44, 48 60, 70, 80 69, 74, 79 70, 80, 90

2 45,55,65 80,95,110 540,675,810 4 35, 40, 45 60, 65, 70 60, 70, 80 69, 74, 79

3 50,60,70 85,100,115 544,680,816 3 40, 45, 50 50, 60, 70 60, 65, 70 60, 70, 80


Min , Max n

t = 1 t = 2,3,4 t=1 t=2 t=3 t=4

1 30, 40 30, 35 250,255,260 255,260,265 260,265,270 265,270,275

2 30, 35 30, 32 240,245,250 245,250,255 250,255,360 255,260,265

3 30, 40 30, 35 235,240,245 240,245,250 245,250,255 250,255,260


and

40, 50, 60 40, 45,50 320, 400, 480 400, 500, 600 480,600,720 144 150



Max Max and n

t=2 t=3 t=4 t=1 t=2,3,4 t=1 t=2 t=3 t=4

1 10 15 10 40 30 60 50 40 35

2 10 15 15 30 30 70 75 40 35

3 7 10 7 30 30 40 40 30 30

150 165 600 660 460, 480, 500

TABLE-6 Input data TABLE-7 (using LP Solver)

n

t=1 t=2, 3, 4

1 27000, 28000, 29000 28000, 29000, 30000

PIS=

PIS=

PIS=

2 27500, 28500, 29500 31000, 32000, 33000

3 28000, 29000,30000 32000, 33000, 34000

NIS=

NIS=

NIS=

TABLE- 8 Optimum Results (using LP Solver)

n t

1 40 46.67 166.67 40 41.67 40 - - - - -

2 30 29.37 145.83 30 36.46 35 10 11 44 10 0

3 30.42 0 145.83 30 36.46 35 15 16.5 70 5 0 1

4 9.72 9.72 145.83 9.72 36.46 35 10 11 44 5 10

1 30 62.71 145.83 30 36.46 35 - - - - -

2 30 61.04 133.33 30 33.33 32 10 11 44 10 0

3 30 36.88 133.33 30 33.33 32 15 16.5 66 5 0 2

4 30 30 133.33 30 33.33 32 15 16.5 66 10 10

1 30 31.67 166.66 30 41.67 40 - - - - -

2 30 25.88 145.83 30 36.46 35 7 7.7 30.8 7 0

3

3 30 8.58 145.83 30 36.46 35 10 11 44 3 0


4 30 30 139.00 30 36.46 35 7 7.7 30.8 4 7

TABLE- 9 Optimum Results (using LP Solver)

( λ= 0.6522)

i =1 i =2 i =3

137465.11 90396.57 41098.96

(96366.15, 137465.11, 227861.68)

The feasible solutions of the problems are given in TABLE 8. The optimum solutions of three objectives are given in TABLE 9 as follows: Max ,

, . Therefore, = (96366.15, 137465.11, 227861.68). The DM specified the most likely value of the triangular distribution of each fuzzy number as the precise value. Conclusion The APP is concerned with the determination of production, the inventory level, the backorder level, the work force level, subcontracting units, hiring and firing of the workers on a finite planning horizon. The objective is maximizing the profit for three items under the consideration of the ambiguous costs and the uncertain increasing demands with minimum subcontracting units in each period and zero inventories at the end of the last period. The proposed model integrates forecasting activities, item management and production planning in APP. Although solving possibilistic mathematical models remains a concern, this research gives evidence that the proposed model is competent in dealing with vague and imprecise data to solve the decision making problems in APP to determine the managerial decisions. The model is verified with numerical example solved by using Possibilistic Linear Programming approach. The proposed model can be modified to make it better suited to practical applications. Future scope may also explore with fuzzy random variables, case study, using various simulation techniques such as genetic algorithm or heuristics, various hybridizations of harmony search algorithm, ant colony optimization, and particle swarm optimization. References [1] A. Aliev, B. Fazloliahi and B. Guirimov, Fuzzy-genetic approach to aggregate production

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