Int. J. Production Economics 134 (2011) 28–42
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty S.M.J. Mirzapour Al-e-hashem a,n, H. Malekly b, M.B. Aryanezhad a a b
Department of Industrial Engineering, Iran University of Science and Technology, P.C.: 16846113114, Tehran, Iran School of Industrial Engineering, Islamic Azad University-South Tehran Branch, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 1 January 2010 Accepted 28 January 2011 Available online 22 February 2011
Manufacturers need to satisfy consumer demands in order to compete in the real world. This requires the efficient operation of a supply chain planning. In this research we consider a supply chain including multiple suppliers, multiple manufacturers and multiple customers, addressing a multi-site, multiperiod, multi-product aggregate production planning (APP) problem under uncertainty. First a new robust multi-objective mixed integer nonlinear programming model is proposed to deal with APP considering two conflicting objectives simultaneously, as well as the uncertain nature of the supply chain. Cost parameters of the supply chain and demand fluctuations are subject to uncertainty. Then the problem transformed into a multi-objective linear one. The first objective function aims to minimize total losses of supply chain including production cost, hiring, firing and training cost, raw material and end product inventory holding cost, transportation and shortage cost. The second objective function considers customer satisfaction through minimizing sum of the maximum amount of shortages among the customers’ zones in all periods. Working levels, workers productivity, overtime, subcontracting, storage capacity and lead time are also considered. Finally, the proposed model is solved as a single-objective mixed integer programming model applying the LP-metrics method. The practicability of the proposed model is demonstrated through its application in solving an APP problem in an industrial case study. The results indicate that the proposed model can provide a promising approach to fulfill an efficient production planning in a supply chain. & 2011 Elsevier B.V. All rights reserved.
Keywords: Aggregate production planning Robust multi-objective optimization Uncertainty Supply chain
1. Introduction Nowadays, supply chain management (SCM) which covers production planning for entire supply chain from the raw material supplier to the end customer has become the foundation for operations management. Since SCM has become the core of the enterprise management in the 21st century, there is a high interest to exploit the full potential of SCM in enhancing organizational competitiveness. SCM has a tremendous impact on organizational performance in terms of competing based on price, quality, dependability, responsiveness, and flexibility in the global market and it is becoming a more matured discipline. Hence, this requires a more defined organizational structure, performance measures, etc. One of the problems that should be addressed in this scope is aggregate production planning (APP), which is focused in this paper along with the broader topics of SCM. The SCM has made managers and analysts to shift their focuses from
n
Corresponding author. E-mail addresses:
[email protected] (S.M.J. Mirzapour Al-e-hashem),
[email protected] (H. Malekly),
[email protected] (M.B. Aryanezhad). 0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2011.01.027
only manufacturing plant to entities plants interact with; for example, suppliers, warehouses, and customers. As a result SCM therefore, has recently received much attention (Dolgui and Ould¨ Louly, 2002; Wang and Liang, 2005; Gunnarsson and Ronnqvist, 2008; Lodree and Uzochukwu, 2008; Gebennini et al., 2009). Baykasoglu (2001) has defined APP as medium-term capacity planning over 3–18 months planning horizon and it determines the optimum production, workforce and inventory levels for each period of planning horizon for a given set of production resources and constraints. Such planning usually involves one product or a family of similar products with small differences so that considering the problem is justified from an aggregated viewpoint, and planners in the process of APP make decisions regarding overall production levels for each product category to meet the fluctuating demands in near future, also make policies and decisions relating to the issues of hiring, laying off, overtime, backorder and inventory. APP is an important technical level planning in a production management system. In the field of planning it falls between the broad decisions of long-range planning and the highly specific and detailed short-range planning decisions. Other forms of family disaggregation plans such as master production schedule, capacity requirements planning and
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
material requirements planning all depend on APP in a hierarchical way (Ozdamar et al., 1998). APP has attracted considerable attention from both practitioners and academia (Shi and Haase, 1996). Numerous APP models with varying degrees of sophistication have been introduced in the last decades. Since Holt et al. (1955) proposed the approach for the first time; scholars have developed numerous models to help solving the APP problems, each with their own supporters and detractors. As a comprehensive remark, Nam and Logendran (1992) reviewed APP models from 140 journal articles and 14 books and categorized the models into optimal and near-optimal classifications. Hanssman and Hess (1960) developed a model based on the linear programming approach using a linear cost structure of the decision variables. Haehling (1970) extended the Hanssman and Hess (1960) model for multi-product, multi-stage production systems in which optimal disaggregation decisions can be made under capacity constraints. Masud and Hwang (1980) presented three MCDM methods, which were goal programming (GP), the step method and sequential multi-objective problem. These methods were applied to solve APP problem with maximizing profit, minimizing changes in workforce level, minimizing inventory investment and minimizing backorders. A set of data consisting of two products, a single production plant and eight planning periods was generated to compare the results. Goodman (1974) developed a GP model which approximates the original nonlinear cost terms of the Holt’s model by linear terms and solves it using a variant of the simplex method. Baykasoglu (2001) extended Masud and Hwang’s model with additional constraints such as subcontractor selection and setup decisions. A tabu search algorithm was designed to solve the pre-emptive GP model. The APP can be considered as the combination of several classical production planning problems in the literature which has been modeled by some kinds of mathematical programming such as scheduling problems (Buxey, 1993; Foote et al., 1998), workforce planning problems (Mazzola et al., 1998) and long set up time problems (Porkka et al., 2003). However, if decisions are made based on the deterministic model, there is a risk that demand might not be met with the right products. It is an unfortunate reality that some critical parameters such as customer demand, price and manufacturing capacity are not known with certainty. If the supply chain designed by the decision makers is not robust with respect to the uncertain environment, the impact of performance inefficiency (e.g. delay) could be devastating for all kinds of enterprises. Since they cannot usually protect themselves completely against the risk, they have to manage it. Risk management can be used as a tool for greater rewards, not just control against loss. There are lots of papers to deal with enterprise risk management (see Wu and Olson, 2008, 2009a, 2009b, 2010a, 2010b; Wu et. al., 2010). APP in many manufacturing environments is based on some parameters with uncertain values. Uncertainties might arise in product demand, etc. Thus, the robustness of a production plan in terms of fulfillment of product demand depends on incorporating the uncertain parameters in production planning models. Our research considers a supply chain problem under uncertainty of demands and various cost parameters. This implies that the problem is more realistic since demand and cost forecasts are seldom precise and in advanced forecasting systems they are usually given as more than a single value. Furthermore, due to demand uncertainty it may not be possible to meet all demands with their available capacity. The problem is introduced in details further. Research that considers uncertainty can be categorized according to the four primary approaches (Sahinidis, 2004): (1) Stochastic programming approach, (2) fuzzy programming approach, (3)
29
stochastic dynamic programming approach, and (4) robust optimization approach. In the first approach, some parameters are regarded as random variables with known probability distributions. The second one seeks the solution considering some variables as fuzzy numbers. The third one includes applications of random variables in dynamic programming which can be found essentially in all areas of multi-stage decision making. In comparison, the last one represents uncertainty through setting up different scenarios which demonstrate realizations of uncertain parameters. The aim of this approach is to find a robust solution which ensures that all specified scenarios are ‘‘close’’ to the optimum in response to changing input data. As mentioned above, to deal with the real-world planning problems involving noisy, incomplete or erroneous data, the methods were employed in some cases such as stochastic programming (Kall and Wallace, 1994; Birge and Louveaux, 1997; Kall and Mayer, 2005); fuzzy set theory (Wang and Fang, 2000); robust optimization (Bertsimas and Sim, 2003, 2004, 2006; BenTal and Nemirovski, 1998, 1999, 2000) and stochastic dynamic programming. The last one was used mostly in the past to obtain closed-form solutions of analytically tractable models and numerical solutions to relatively small problem instances. With the recent developments in approximations, especially neurodynamic programming, this methodology offers the potential of dealing with problems that for a long time were considered intractable due to either a large state space or the lack of an accurate model. Some applications have included: production planning (Cheng et al., 2003), and supply chain management (Bitran et al., 1998). Bakir and Byrne (1998) developed a stochastic LP model based on the two-stage deterministic equivalent problem to incorporate demand uncertainty in a multi-period multi-product (MPMP) production planning model. In Escudero et al. (1993) a multi-stage stochastic programming approach was proposed for solving an MPMP production planning model with random demand. It is important to note that stochastic programming approach focuses on optimizing the expected performance (e.g. cost) over a range of possible scenarios for the random parameters. We can expect that the system would behave optimally in the mean sense. However, the system might perform poorly at a particular realization of scenarios such as the worst case scenario. More precisely, unacceptable inventory and backorder size for some scenarios might be observed by implementing the solution of two-stage stochastic model. The solutions in the form of fuzzy number provide different conditions to the production management in an uncertain environment. Tang et al. (2000) introduced a novel approach to modeling multi-product APP problems with fuzzy demand and fuzzy capacities. The objective of the considered problem was to minimize the total costs of quadratic production costs and linear inventory holding costs. Tang et al. (2003) proposed an approach that focuses on a formulation and simulation analysis for multi-product APP problems with fuzzy demands and fuzzy capacities. The fuzzy multiproduct APP model was transformed into a parametric programming model. A simulation of a practical instance was conducted to illustrate the model and demonstrate the performance and effect of various parameters on the optimal APP. Wang and Fang (2001) extended a four-objective APP model defined by Masud and Hwang (1980) with fuzzy parameters such as fuzzy demand, fuzzy machine time, fuzzy machine capacity and fuzzy relevant costs. Wang and Liang (2004) developed a fuzzy multi-objective linear programming model for solving a multi-product APP decision problem in a fuzzy environment. Also Aliev et al. (2007) developed a fuzzy integrated multi-period and multiproduct aggregate production and distribution model in supply chain. The model was formulated in terms of fuzzy programming and the solution was provided by genetic algorithm.
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To handle the trade-off associated with the expected cost and its variability in stochastic programs, Mulvey et al. (1995) proposed the concept of stochastic robust optimization. Leung and Wu (2004) proposed a robust optimization model for stochastic APP. In Leung et al. (2007) a robust optimization model was developed to address a multi-site APP problem in an uncertain environment. In Kazemi-Zanjani et al. (2010) robust optimization approach was proposed as one of the potential methodologies to address MPMP production planning in a manufacturing environment with random yield. Feng and Rakesh (2010) considered an integrated optimization of logistics and production costs associated with the supply chain members based on the scenario approach to handle the uncertainty of demand. The formulation was a robust optimization model with expected total costs, cost variability due to demand uncertainty, and expected penalty. In this paper we are to develop a robust multi-objective aggregate production planning (RMAPP) model to solve the multi-period multiproduct multi-site APP problem for a medium-term planning horizon based on existing conflict between the total losses of supply chain and the customer satisfaction level over the planning horizon under different scenarios. Cost parameters of the supply chain as well as demand fluctuations are subject to uncertainty. Working levels, workers productivity, overtime, subcontracting, storage capacity and lead time are also considered. This makes modeling more complicated. While there is a vast literature devoted on this type of problem, to the best of our knowledge, the rich deal of such researches has been dedicated to some of these aspects individually or to specific combinations of them. Going through the literature indicates that the prior approaches on robust optimization models have a limitation inside which cannot generate a solution that optimizes all the important and conflicting criteria like total losses and customer satisfaction simultaneously. So there is a gap between the theoretical and practical contributions responsible for developing APP models. The purpose of our research is to bridge this gap by introducing a framework that is a realistic alternative to more sophisticated complex APP models. Additionally, the model may serve as a basis for the study of APP problems. The simplicity of the implementation and operation of the model enables the decision makers to manage production in terms of production loading capacity and workforce arrangement without having to learn complex operations and programming procedures. The rest of the paper is organized as follows: In Section 2, the background of robust optimization formulation is described. The third Section presents the problem description and formulation. Then the solution procedure is presented in Section 4. Next, the robustness and effectiveness of the proposed model are demonstrated by a case analysis in Section 5. Finally, conclusions are presented in Section 6.
2. Robust optimization Mulvey et al. (1995) introduce a framework for robust optimization that involves two types of robustness: ‘‘solution robustness’’ (the solution is nearly optimal in all scenarios) and ‘‘model robustness’’ (the solution is nearly feasible in all scenarios). The definition of ‘‘nearly’’ is left up to the modeler; their objective function has general penalty function for both model and solution robustness, weighted by a parameter intended to capture the modeler’s preference between the two. The robust optimization method developed by Mulvey et al. (1995), in fact, extends stochastic programming through replacing traditional expected cost minimization objective by one that explicitly addresses cost variability. In the following, the framework of robust optimization is briefly described (Feng and Rakesh, 2010). Consider the
following LP model that includes random parameters: MincT x þ dT y
ð1Þ
subject to Ax ¼ b,
ð2Þ
Bxþ Cy ¼ e,
ð3Þ
x,yZ 0,
ð4Þ
where x denotes the vector of decision variables that should be determined under the uncertainty of model parameters. B, C and e represent random technological coefficient matrix and righthand side vector, respectively. Assume a finite set of scenarios O ¼{1, 2, y, z} to model the uncertain parameters; with each scenario xAO we associate the subset {dx; Bx; Cx; ex } and the P probability of the scenario px ( xpx ¼1). Note that a scenario is a series of data realizations over the planning horizon. We use O to denote these 1 to z scenarios. Uncertain coefficients B, C, e can be denoted as Bx, Cx, and ex for each scenario xAO. Also, control variable y, which is subject to adjustment when one scenario is realized, can be denoted as yx for scenario x. Because of parameter uncertainty, the model may be infeasible for some scenarios. Therefore, dx presents the infeasibility of the model under scenario x. If the model is feasible, dx will be equal to 0. Otherwise; dx will be assigned a positive value according to Eq. (7). A robust optimization model is formulated as follows: Min sðx,y1 ,y2 ,. . .,yx Þ þ orðd1 , d2 ,. . ., dx Þ
ð5Þ
subject to Ax ¼ b,
ð6Þ
Bx x þ Cx yx þ dx ¼ ex
8x A O,
x Z0, yx Z 0, dx Z 0
ð7Þ
8x A O,
ð8Þ
The first term represents solution robustness, capturing the firm’s desire for low costs and its degree of risk aversion, while the second term represents model robustness, penalizing solutions that fail to meet demand in a scenario or violate other physical constraints like capacity. We use c to represent f (x, y), which is a cost or benefit function. cx ¼ f (x, yx) for scenario x. A high variance for cx ¼f (x, yx) means that the solution is a highrisk decision. In other words, a small change of the value of uncertain parameters can cause a big change of the value of the measure function. Mulvey et al. (1995) used:
sð3Þ ¼
X xAO
px cx þ l
X
0 px @cx
xAO
X
12 px0 cx0 A
ð9Þ
x0 A O
to represent solution robustness, where l denotes the weight placed on solution variance in which the solution is less sensitive to change in the data under all scenarios as l increases. As can be seen, there is a quadratic term in Eq. (9). Yu and Li (2000) discussed the computational effort required due to the quadratic term and proposed an absolute deviation instead of the quadratic term, which is shown as follows: X X X sð3Þ ¼ px cx þ l px 9cx px0 cx0 j: ð10Þ xAO
xAO
x0 A O
Although objective (10) is a nonlinear function, it can be optimized by converting the problem into a linear programming model having a linear objective function with linear constraints by introducing two non-negative deviational variables (Wagner, 1975). The justification for the equivalence is a simple change of
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
variables. Based on Leung et al. (2007), instead of minimizing the sum of absolute deviations in (10), two deviational variables are minimized subject to original constrains and additional soft constraints which give positive values of the difference inside the absolute functions. However, Yu and Li (2000) stated that this direct linearization approach is largely restricted due to many non-negative deviational variables and constraints introduced. Eq. (9) is transformed to the following linear programming problem: 20 1 3 X X X 4 @ A Min ð11Þ px cx þ l px cx px0 cx0 þ2yx 5 xAO
xAO
subject to X cx px cx þ yx Z 0,
sites
customers
x0 A O
8 x A O,
ð12Þ
xAO
Fig. 1. General schema for supply chain configuration.
yx Z0, 8 x A O:
ð13Þ
It can be interpreted that as the amount by which cx is greater P P than xAO pxcx, then yx ¼0, while as the amount by which xAO P pxcx is greater than cx, then yx ¼ xAO pxcx cx. The second term in the objective function, r(d1, d2, y, dx), is a feasibility penalty function, which is used to penalize violations of the control constraints under some of the scenarios. The violation of control constraints means that the infeasible solution to a problem under some of the scenarios is obtained. Using the weight o, the trade-off between solution robustness measured from the first term s(1) and model robustness measured from the penalty term r(1) can be modeled under the MCDM process. Based on this discussion, the objective function can be formulated as follows: 20 1 3 X X X X 4 @ A 0 0 Min px cx þ l px cx px cx þ2yx 5 þ o px dx : xAO
suppliers
31
xAO
x0 A O
xAO
ð14Þ
3. Model description According to Gallego (2001), aggregate production planning is a traditional problem which firms are dealing with. APP is concerned with the determination of production, inventory, and workforce levels to meet varying demand requirements over a medium-term planning horizon that ranges from 3 to 18 months. The planning horizon is often divided into some periods. Typically, the physical resources of the firm are assumed to be fixed during the planning horizon of interest. Given the external demand requirements, the planning effort is oriented toward the best utilization of those resources. The plan must take into account the various ways; a firm can cope with demand fluctuations as well as the cost associated with them. Generally these fluctuations deal with: i. Changing the size of the workforce by firing and hiring via allowing alterations in the production rate. Excessive use of firing and hiring may be limited by union regulations and may create severe labor problems. Workforce training may be a good strategy for increasing the workers productivity to compensate the needed extra production rate in demand’s peaks. ii. Varying the production rate by introducing overtime or outside subcontracting. iii. Accumulating seasonal inventories. The trade-off between the cost incurred in changing the production rate and holding the inventory is the basic question to be resolved in most practical situations. iv. Planning backorders.
Costs relevant to APP in a supply chain are as follows: i. Basic production costs: Raw material purchasing costs, direct labor costs, and overhead costs. It is customary to divide these costs into variable and fixed costs. ii. Costs associated with changes in the production rate: costs involved in hiring, training, and laying off personnel, as well as overtime and subcontracting compensations. iii. Inventory and backorder related costs. iv. Transportation cost: cost involved in transporting raw material from suppliers to factories and from factories to customers. The proposed multi-objective multi-product multi-site APP problem in a supply chain can be described as follows: There are J sites, S suppliers and C customers (see Fig. 1). Each site produces several product items assembled from some parts supplied by suppliers, regarding to consumption rates. Production cost of a certain item at different sites and raw material cost in different suppliers can be different. Each site characterized by its own raw material and end product inventory capacities and the available time for its production which is limited to its number of k-level workers beside the allowed amount of regular and overtime. Every site could subcontract an allowed proportion of its product to subcontractors. All sites, suppliers and customers’ zones are spread geographically, and then the transportation cost from suppliers to sites and from sites to customers’ zones can vary. Being aware of that the storing of end products in customers’ zones is impossible, the problem is to determine: (1) the quantity of product i manufactured at site j to fulfill stochastic demand of customer’s zone c in each period of time by k-level workers; (2) the quantity of raw material m provided by supplier s in all periods to fulfill the net requirements of site j regarding to the consumption rates and the lead times; (3) the number of klevel workers would be hired, fired or trained at each site in each period; (4) the quantity of raw material m and end product i stored at site j in each period; (5) the amount of demand in each customer’s zone is not met in each period, in a way that the total losses of supply chain will be minimized and customer satisfaction will be maximized, simultaneously.
3.1. Notations Parameters demand for product i (1, 2, y, I) in demand point c (1, Dxict 2, y, C) in period t (1, 2, y, T) in scenario x (1, 2, ..., z) x production cost per hour, in regular time (g¼1), Cjg overtime (g¼ 2), subcontracting (g¼ 3) at site j (1, 2, y, J) in scenario x
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S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
x Pict x
SLkjt aij x FCkjt x HCkjt x
TCkk0 jt x CIMmjt x
CIPijt
gim at TCSxsjt x TCCict x CMsmt
uk TCAPgjt
CAPMj CAPPj CAPSsmt LTsj LTjc UPkk0
pxict rx
sale price of product i in customer’s zone c in period t in scenario x labor cost of k-level (k¼1, 2, y, K) workers at site j in period t in scenario x production time of product i at site j firing cost of k-level worker at site j in period t in scenario x hiring cost of k-level worker at site j in period t in scenario x training cost for k-level worker trained to level k’ at site j in period t in scenario x inventory holding cost for raw material m (1, 2, y, M) at site j in period t in scenario x inventory holding cost for finished product i at site j in period t in scenario x number of units of raw material m required for each unit of product i fraction of the workforce variation allowed in period t transportation cost from supplier s (1, 2, y, S) to site j in period t in scenario x transportation cost from site j to demand point c in period t in scenario x cost of raw material m provided by supplier s in period t in scenario x productivity of k-level workers (0ruk r1) available regular time (g¼1), overtime (g ¼2) and capacity of subcontracting in terms of time unit (g ¼3) at site j in period t raw material storage capacity at site j end product storage capacity at site j maximum number of raw material m supplier s could provide in period t lead time required for shipping raw material from supplier s to site j lead time required for shipping end product from site j to demand point c 1 if training from skill level k to skill level k0 is possible; 0 otherwise shortage cost of product i in customer’s zone c in period t in scenario x Occurrence probability of scenario x
þ
X
x CIMmjt IMmjt þ
m,j,t
X
þ
i,j,t
x
TCCict CUSijct þ
i,j,c,t
Min
s,m,j,t
X
x
x
pict BDict
i,c,t
X x X Max BDict c
t
X X x CIPijt IPijt þ TCSxsjt SUPsmjt X
x Pict CUSijct ,
ð15Þ
i,j,c,t
! ð16Þ
i
subject to IPijt ¼ IPijðt1Þ þ
X X Xijgt CUSijct g
8i,j,t,
ð17Þ
c
IMmjt ¼ IMmjðt1Þ þ
X X SUPsmj½tLTsj Xijgt :gim s
8m,j,t,
ð18Þ
8k,j,t,
ð19Þ
g,i
Lkjt ¼ Lkjðt1Þ þHLkjt FLkjt þ
X X ULk0 kjt ULkk0 jt k0
k0
X X Lkjt uk ðTCAP1jt þ TCAP2jt Þ Z Xijgt axij
8j,t,
ð20Þ
i,g A f1,2g
k
X Xij3t axij r TCAP3jt
8j,t,
ð21Þ
i
X BDxict ¼ BDxicðt1Þ þ Dxict CUSijc½tLTjc
8i,c,t, x,
ð22Þ
j
X IMmjt rCAPMj
8j,t,
ð23Þ
m
X IPijt rCAPPj
8j,t,
ð24Þ
i
X X ðFLkjt þ HLkjt Þ r aðt1Þ Lkjðt1Þ k
8j,t,
ð25Þ
k
FLkjt þ
X ULkk0 jt r Lkjðt1Þ
8k,j,t,
ð26Þ
k0
X ULk0 kjt :FLkjt ¼ 0
8k,j,t,
ð27Þ
k0
ULk0 kjt rM UPkk0
8k,
X SUPsmjt rCAPSsmt
k0 ,j,t,
ð28Þ
8s,m,t,
ð29Þ
j
Variables Xijgt number of product i produced at site j using method g in period t Lkjt number of k-level workers at site j in period t FLkjt number of k-level workers at site j fired in period t HLkjt number of k-level workers at site j hired in period t ULkk’jt number of k-level workers at site j trained to level k0 in period t IMmjt inventory level of raw material m at site j at the end of period t IPijt inventory level of end product i at site j in period t SUPsmjt number of units of raw material m shipped from supplier s to site j CUSijct number of units of end product i provided by site j for demand point c in period t shortage of product i in demand point c in period t in BDxict scenario x
Xijgt ,SUPsmjt ,IMmjt ,IPijt ,CUSijct ,BDxict Z0 Lkjt ,FLkjt ,HLkjt ,ULkk0 jt Z0, and integer
8i,j,c,g,k,s,m,t:
ð30Þ
First objective function (15) aims to minimize total losses of supply chain including production cost, labor cost, hiring cost, firing cost, training cost, raw material inventory holding cost, end product inventory holding cost, transportation cost, raw material purchasing cost and shortage cost, from which the total sales is deducted. Second objective function (16) considers customer satisfaction through minimizing sum of the maximum shortages among the customers’ zones in all periods. This objective is explicitly nonlinear and the linear equivalent equations could be rewritten with the help of an auxiliary variable as follows: X x Min Wt ð31Þ t
3.2. Problem formulation X X x X x x aij cjg Xijgt þ CMsmt SUPsmjt þ SLkjt Lkjt Min i,j,g,t
s,m,j,t
k,j,t
X x X x X x þ FCkjt :FLkjt þ HCkjt :HLkjt þ TCkk0 j ULkk0 jt k,j,t
k,j,t
subject to X x BDict Wtx Z
k,k0 ,j,t
8c,t,
ð32Þ
i
Wtx Z 0
8t:
ð33Þ
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
Constraint (17) is a balance equation for the end product inventory at site j. Constraint (18) is an inventory balance equation for the raw material level at site j. Constraint (19) is also a balance equation for workforce level and ensures that the available k-skill level workers equals the workforce with the same skill level in previous period in addition to the change of workforce level in current period. Constraint (20) limits the available production time to available workforce regular and overtime, considering their productivity. Constraint (21) restricts the amount of products manufactured by subcontractor. Constraint (22) is a balance equation for shortage in demand point c. Constraints (23) and (24) limit the raw material and end product inventory levels to the related inventory storage capacities. Constraint (25) guarantees that the change in workforce level cannot exceed the proportion of workers in previous period. Constraint (26) ensures that the number of k-level workers who are fired or trained for upper skill levels in current period cannot exceed the available k-level workforce in previous period. Constraint (27) denotes that the workers who are trained for skill level k should not be fired in the same period. This constraint has an explicit nonlinear term which can be transformed to a linear one with the help of a binary variable and equivalent linear equations which are as follows: X ULk0 kjt rM Ykjt 8k,j,t, ð34Þ k0
there is no limitation for the number of considered uncertain parameters. However, the applicability of this approach is limited by the fact that it requires anticipating all possible consequences. The latter approach is applied where only a continuous range of potential future outcomes can be anticipated. The advantage of this methodology is that by assigning a probability distribution function to the continuous range of possible consequences, the need to forecast exact scenarios is eliminated. However, complexity of applying distribution function limits the number of considered uncertain parameters. In this paper a novel multi-objective stochastic robust optimization approach based on Mulvey’s model is presented in which uncertainty is represented by a set of discrete scenarios (x). X x X x X x LC x ðlabor costÞ : SLkjt Lkjt þ FCkjt FLkjt þ HCkjt HLkjt þ
k,j,t
k,j,t
8k,j,t,
8k,j,t,
TCkk0 j ULkk0 jt ,
IC x ðinventory costÞ :
X
The need for considering uncertainty in the production planning arises from this fact that the midterm planning models is to allocate resources for the future according to current information and future circumstances. On the other hand, the uncertain nature of the environment makes this issue more complicated. The first step to incorporate uncertainty into the planning decisions is the determination of the appropriate approach to deal with the uncertain parameters. Approaches to optimization under uncertainty have followed a variety of modeling philosophies, including expectation minimization, minimization of deviation from goals and minimization of maximum costs. According to Section 1, the main approaches to deal with uncertainty are stochastic programming (recourse models, robust stochastic programming and probabilities models), fuzzy programming (flexible and possibilistic programming), stochastic dynamic programming (Sahinidis, 2004) and robust optimization (Bertsimas and Sim, 2003, 2004, 2006; Ben-Tal and Nemirovski, 1998, 1999, 2000). The next step is the determination of the appropriate representation of the uncertain parameters. In stochastic programming approach, for representing uncertainty two different methodologies can be applied. These are the scenario-based approach and the distribution-based approach. In the former approach, the uncertainty is described by a set of discrete scenarios forecasting that how the uncertainty might take effect in the future. Each scenario is associated with a probability level representing the decision makers’ expectation of the occurrence of a specific scenario. The advantage of this methodology is that
x CIMmjt IMmjt þ
X x CIPijt IPijt ,
ð38Þ
i,j,t
TC x ðtransportation costÞ :
X
TCSxsjt SUPsmjt þ
s,m,j,t
X
x TCCict CUSijct
i,j,c,t
ð39Þ
X
X
x aij cjg Xijgt þ
i,j,g,t
X
x CMsmt SUPsmjt
s,m,j,t
x Pict CUSijct ,
ð40Þ
i,j,c,t
ð36Þ
3.3. Robust optimization formulation
ð37Þ
m,j,t
ð35Þ
where M is an arbitrary big number. Constraint (28) guarantees that training workers from skill level k to level k0 is possible, once this training program exists. Constraint (29) ensures that the amount of shipments from supplier s cannot exceed the supplier capacity. To conclude the formulation, variables are defined in (30).
k,j,t
x
t,j,k,k0
Ykjt A f0,1g
X
PC x ðproduction costÞ
FLkjt r Mð1Ykjt Þ
33
SC x ðshortage costÞ :
X
pxict BDxict ,
ð41Þ
i,c,t
SMSx ðsum of maximum shortageÞ :
X x Wt
ð42Þ
t
We present the above discussion in the following RMAPP formulation: X rx ðLC x þ IC x þ TC x þ PC x þ SC x Þ Min Z1 ¼ þ l1
X
x
rx ½ðLC x þ IC x þ TC x þ PC x þ SC x Þ
x
X X x rx dx1ict , rx’ ðLC x0 þ IC x0 þ TC x0 þ PC x0 þSC x0 Þ þ 2y1 þ o x0
i,c,t, x
ð43Þ Min Z2 ¼
X
X
x
x
rx SMSx þ l2
X rx ½SMSx rx0 SMSx0 þ2yx2
ð44Þ
x0
subject to ðLC x þ IC x þTC x þ PC x þ SC x Þ
X
rx ðLC x þ IC x þ TC x þ PC x þ SC x Þ
x x
þ y1 Z 0
8x,
ð45Þ
X x SMSx rx SMSx þ y2 Z0
8x,
ð46Þ
x
X x BDxict ¼ BDxicðt1Þ þ Dxict CUSijc½tLTjc þ dict
8i,c,t, x,
ð47Þ
j
yx1 , yx2 , dxict Z0 8i,c,t, x:
ð48Þ
Constraints (17)–(21), (23)–(26), (28)–(30) and (32)–(42) where rx is the probability of scenario x. Eqs. (37)–(42) are defined for formulation convenience. The first and second terms in Eqs. (43) and (44) are mean value and variance of the objective functions, respectively. The last term in (43) measures the model robustness with respect to infeasibility associated with control
34
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
constraint (47) under scenario x. Constraints (45) and (46) are auxiliary constraints for linearization defined in (14). Constraint (47) is a control constraint that is used to determine the amount of products transferred to customers’ zones and the amount of shortage in each period. It should be noted that if the demand of each customer’s zone in period t plus its previous shortage at period t 1 are smaller than the total quantity of each product transferred to that customer’s zone, then the shortage at P period t will be equal to BDxict ¼ BDxicðt1Þ þ Dxict j CUSijc½tLTjc and under
minimization,
the
x
deviation d1ict ¼ 0; whereas, if P x is greater than j CUSijc½tLTjc , then BDict ¼ 0 and
BDxicðt1Þ þ Dxict P dx1ict ¼ j CUSijc½tLTjc BDxicðt1Þ Dxict , indicating products inventory in customers’ zones, which is impossible based on the problem assumptions, thus an infeasible solution is obtained. Note that, end products storage in customers’ zones is impossible either due to the large amount of space needed there to store products or because of the high level of deterioration rate of products. Constraint (48) specifies non-negative variables.
4. Solution procedure Since RMAPP model is a multi-objective, mixed integer linear programming model whose objective functions are completely inconsistent, we used the LP-metrics method which is one of the famous MCDM methods for solving multi-objective problems with conflicting objectives simultaneously. According to this method, a multi-objective problem is solved by considering each objective function separately and then a single objective is reformulated which aims to minimize the summation of normalized differences between each objective and the optimal values of
them. In our proposed model, just you can assume that two objective functions are named as Z1, Z2. Based on LP-metrics method, RAPPM should be solved for each one of these two objectives separately. Assume that the optimal values for these two problems are Zn1, Zn2. Now, the LP-metrics objective function can be formulated as follows: Z1 Z1 Z2 Z2 , ð49Þ þ ð1 $ Þ MinZ3 ¼ $ Z1 Z2 where 0 r $ r1 is the relative weight of components of the objective function (49) which given by the decision maker(s). Using LP-metrics objective function and considering RMAPP model constraints, we have a single objective, mixed integer programming model, which can be efficiently solved by linear programming solvers.
5. Computational results In this section, as a real-world industrial case a data set is provided from CHOUKA Co. in Iran to illustrate the applicability of proposed model to practical problems. 5.1. Case description In 1973, with the cooperation of the Ministry of Agriculture, Organization of Natural Resources and Industrial Development and Renovation Organization, Iran Wood and Paper Industries Company (CHOUKA) consisting of some forest and industry sectors was established in GILAN province of Iran with the purpose of creating a wood and paper industry. In 1986, due to the specialization of the tasks and in accordance with the ratified
Fig. 2. The scatter of CHOUKA’s supply chain components.
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
35
Table 1 Production time- raw material and end product inventory cost. Site j
Scenario x
End product inventory holding cost ($/unit period)
Raw material inventory holding cost ($/unit period)
Production time (min)
Product i
Raw material m
Product i
1
2
3
4
5
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
1
1 2 3 4
5 6 7 8
7 8 9 10
9 10 11 12
11 12 13 14
13 14 15 16
4 5 6 7
4 5 6 7
4 5 6 7
4 5 6 7
4 5 6 7
5 6 7 8
5 6 7 8
5 6 7 8
6 7 8 9
6 7 8 9
35
48
40
45
62
2
1 2 3 4
8 9 10 11
10 11 12 13
12 13 14 15
14 15 16 17
16 17 18 19
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
5 6 7 8
36
55
45
35
72
3
1 2 3 4
9 10 11 12
9 10 11 12
9 10 11 12
9 10 11 12
9 10 11 12
5 6 7 8
6 7 8 9
7 8 9 10
8 9 10 11
8 9 10 11
9 10 11 12
9 10 11 12
7 8 9 10
7 8 9 10
6 7 8 9
33
45
37
47
82
8 20 15 10
9 15 0 0
10 20 0 20
Initial end product inventory
Initial raw material inventory
Product i
Raw material m
1 2 2 0
1 2 3
2 1 2 0
3 5 0 20
4 10 0 1
5 2 1 10
1 10 20 10
Table 2 Labor cost.
2 15 20 0
3 20 20 0
4 12 0 10
5 15 0 0
6 20 15 0
7 20 15 0
Table 3 Training cost at site 1 (10 $/manpower). Firing cost (10 $/ manpower)
Site j Scenario x Hiring cost (10 $/ manpower) Worker type k
Worker type k
1 2 3 4
1
4 5 6 7
5
2
3
Salary cost (10 $/ manpower)
Worker type k
4
5
1
2
3
4
5
1
1 2 3 4
4 5 6 7
4 5 6 7
4 5 6 7
4 7 8 9 5 8 9 10 6 9 10 11 7 10 11 12
10 11 12 13
11 12 13 14
18 19 20 21
20 21 22 23
22 23 24 25
24 25 26 27
26 27 28 29
2
1 2 3 4
4 5 6 7
5 6 7 8
6 7 8 8 8 9 7 8 9 9 9 10 8 9 10 10 10 11 9 10 11 11 11 12
10 11 12 13
12 13 14 15
21 22 23 24
23 24 25 26
25 26 27 28
27 28 30 32
29 30 32 34
3
1 2 3 4
4 5 6 7
4 5 6 7
5 6 7 8
8 9 16 17 5 10 17 18 6 11 18 19 7 12 19 20
19 21 22 24
20 22 24 25
22 24 26 27
5 6 7 8
5 6 7 8
5 4 5 6
6 5 6 7
7 5 6 7
Scenario x
Worker type k 1
2
3
4
5
1
1 2 3 4
– – – –
10 – – –
15 10 – –
20 15 10 –
25 20 15 10
2
1 2 3 4
– – – –
11 – – –
16 11 – –
21 16 11 –
26 21 16 11
3
1 2 3 4
– – – –
12 – – –
17 12 – –
22 17 12 –
27 22 17 12
4
1 2 3 4
– – – –
13 – – –
18 13 – –
23 18 13 –
28 23 18 13
0.65
0.70
0.75
0.85
0.95
Workers’ productivity
law of the (presidential) cabinet, the wood sector was separated and transferred to the other provinces, SEMNAN and KHUSASTAN. At the moment, the area of the company’s factories is about 500,000 hectares altogether in the form of 73 comprehensive forest and farm management plans. Among the most important tasks of the company the protection, reforestation, development and exploitation by considering the social-economic conditions can be mentioned. Additionally, providing and executing comprehensive forest and farm management plans, establishing thousands of kilometers of ramp networks in order to access natural resources, silviculture and reforestation operations and finally, proper and substantial exploitation of the forests with the goal of continuous output, wood production increment and providing a part of the country’s cellulose industrial needs are some of the other tasks seek company supervision.
Worker type k
Note that these costs for site 2 and 3 are about 10 and 20% more expensive, respectively.
Creation of the huge CHOUKA facility in the recent past, existence of other factories for production of pressed wooden board and wood fiber and many kinds of papers processing workshops have pushed up daily demand for various types of raw materials that should be provided from different locations nearby the relevant suppliers and customers. Fig. 2 shows the relative locations of the supply chain of CHOUKA Co. There are three types of product which CHOUKA has sold during the recent years: i. Floating Paper: this product is delivered according to customers’ requisition and market prices. The orders can be customized up to 440 cm paper width.
36
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
Table 4 Market demands for Scenario 1. Customer’s zone c
Product i
Period t 1
2
3
4
5
6
7
8
9
10
11
12
1
1 2 3 4 5
100 200 150 250 150
250 250 200 100 200
350 300 250 300 200
300 350 300 250 400
100 200 100 200 300
200 200 50 100 350
250 200 0 200 100
0 350 100 300 100
100 400 200 400 150
150 450 250 400 100
100 500 300 400 100
100 350 400 300 100
2
1 2 3 4 5
190 280 210 300 290
350 330 370 180 400
540 320 490 370 220
590 570 400 410 690
120 370 150 310 420
320 330 70 130 380
380 290 100 270 170
200 690 160 460 190
180 670 330 770 190
190 650 380 780 120
130 950 400 520 170
110 430 620 590 140
3
1 2 3 4 5
90 60 90 190 80
190 250 70 130 170
30 530 140 230 150
80 140 400 40 290
40 150 10 160 280
300 80 60 20 300
140 160 80 100 80
100 190 100 180 20
130 330 160 540 240
50 290 260 510 50
60 560 200 300 120
20 450 610 20 110
4
1 2 3 4 5
170 460 200 710 400
580 620 500 240 310
750 470 300 530 490
880 710 830 810 600
290 680 160 620 630
350 540 90 180 1110
560 570 0 260 320
0 920 140 520 200
230 830 620 980 170
310 660 540 460 180
250 1260 550 810 250
330 810 850 710 190
For scenarios 2, 3 and 4, the estimations are multiplied by 1.1, 1.2 and 1.3, respectively.
Table 5 Sites’ data. Site j
Storage capacity
Initial workforce
Raw material m
1 2 3
End product i
10000 12000 10000
15000 10000 10000
Workforce change rate
Worker type k 1
2
3
4
5
6 5 10
6 10 20
6 15 20
6 5 0
6 10 0
0.2 0.2 0.2
Table 6 Available time. Period t
Regular time (hour/period)
Overtime (hour/period)
Subcontracting (hour/period)
Site 1
Site 2
Site 3
Site 1
Site 2
Site 3
Site 1
Site 2
Site 3
144 160 168 176 120 192 200 200 192 176 184 160
144 160 168 176 200 120 200 200 192 176 184 152
144 160 168 176 200 120 200 200 200 176 184 152
50 50 50 60 40 60 60 60 60 60 60 50
50 50 50 60 60 40 60 60 60 60 60 50
50 50 50 60 60 40 60 60 60 60 60 50
200 220 230 240 170 270 280 280 270 240 260 220
200 220 230 240 280 170 280 280 270 240 260 210
200 220 230 240 280 170 280 280 280 240 260 210
Production cost ($/min) in scenario x 1 0.5 0.55 2 0.55 0.60 3 0.6 0.65 4 0.65 0.70
0.4 0.45 0.50 0.55
0.9 0.95 1 1.05
0.95 0.1 1.05 1.10
1 1.05 1.10 1.15
1.25 1.30 1.35 1.40
1.3 1.35 1.40 1.45
1.20 1.25 1.30 1.35
1 2 3 4 5 6 7 8 9 10 11 12
ii. Newspaper Printing Paper: this product is delivered under authorizations issued by Ministry of Culture and Islamic Guidance. This product, which is mainly used for newspaper printing, can be delivered up to 614 cm width
and covers the entire requirements by newspaper publications. iii. Printing and Writing Paper: this product is also sold under authorizations issued by Ministry of Commerce.
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
Including five end products provided for customers: i. ii. iii. iv. v.
Printing and writing typical specifications 50 g/m2 Newsprint typical specifications (HB) Fluting typical specifications 113–127 g/m2 Newsprint typical specifications 48.8 g/m2 Printing and writing typical specifications 70 g/m2
There are ten substances that supplied as the combinations of following cellulosic sources: i. Bombast stem, Hempseed and Cotton which have a long yarn (about 1.2–6 mm). ii. Plant stems like Wheat, Grain, Cane, Hemp, etc. iii. Pine trees (with long yarn) or plane tree (with short yarn about 0.5–1.2 mm). iv. Types of discarded papers or scrap cartons. The planning horizon of monitored time is assumed to be 12 periods, from March 2009 to March 2010. According to Fig. 2, there are 3 sites each located near the customer’s zone as well as suppliers at closest possible distance. Suppliers are spread geographically through entire of the country Table 7 Consumption rate. Product i
Raw material m
1 2 3 4 5
1
2
3
4
5
6
7
8
9
10
2 2 1 0 0
3 3 0 0 1
0 1 1 0 2
4 2 2 0 0
0 2 0 2 1
0 2 0 3 0
1 0 1 2 0
2 0 0 3 0
3 0 0 2 1
0 0 2 3 2
Scenario x Site j Supplier s 2
Customer’s zone c 3
4
1
2
3
4
1
1 2 3
0.014 0.029 0.079 0.101 0.036 0.058 0.072 0.065 0.029 0.014 0.108 0.086 0.065 0.043 0.086 0.036 0.13 0.144 0.05 0.072 0.094 0.115 0.072 0.151
2
1 2 3
0.016 0.032 0.088 0.112 0.04 0.064 0.08 0.072 0.032 0.016 0.12 0.096 0.072 0.048 0.096 0.04 0.144 0.16 0.056 0.08 0.104 0.128 0.08 0.168
3
1 2 3
0.02 0.04 0.18
4
1 2 3
0.024 0.048 0.132 0.168 0.06 0.096 0.12 0.108 0.048 0.024 0.18 0.144 0.108 0.072 0.144 0.06 0.216 0.24 0.084 0.12 0.156 0.192 0.12 0.252
0.04 0.02 0.20
0.11 0.15 0.07
0.14 0.12 0.10
0.05 0.09 0.13
0.08 0.06 0.16
0.10 0.12 0.10
0.09 0.05 0.21
Table 9 Lead time (period). Site j
1 2 3
and also near the woods. To describe the proposed model, the data included in Tables 1–11 gathered form CHOUKA Co. As mentioned before the number of sites, demand points, suppliers, products, raw materials, periods and scenarios are equal to 3, 4, 4, 5, 10, 12 and 4, respectively. Table 1 shows the production time for each product type at each site. Also the inventory holding cost and the initial inventory for the raw material and the end product are presented for every site. Human related costs for any singular site are reported in Tables 2 and 3. Forecasted market demand is presented in Table 4. In Table 5, initial workforce level and storage capacity for each site are given. Available regular time, overtime and subcontracting in terms of time unit are shown in Table 6. Consumption rate of raw materials is presented in Table 7. Tables 8 and 9 show the transportation cost and lead time between suppliers and sites and between sites and customers’ zones. Cost and capacity of raw materials provided by suppliers are stated in Table 10. Finally, Table 11 presents the shortage cost and the sales price of products for each customer’s zone. According to the above-mentioned data and considering the four scenarios like optimistic with associated probability of 0.25, mostly expected with associated probability of 0.5, pessimistic with associated probability of 0.15 and extremely pessimistic with associated probability of 0.1, RMAPP model is optimally solved three times, each time with one of the objective functions Z1, Z2 and Z3. First objective function (Z1) aims to minimize expected value in addition to the weighted of variance and the infeasibility penalty of total losses of supply chain production planning. Second one (Z2) is to minimize expected value plus the weighted of variance of sum of the maximum shortage amongst periods. Third one (Z3) is the LP-metrics objective function in which the best values of above mentioned objective functions (Zn1, Zn2) are imbedded and lead us to make a trade-off between cost and service level and also gives the planning manger a chance of making robust such plans to deal with fluctuations of cost and demand significantly.
5.2. Results
Table 8 Transportation cost ($/unit).
1
37
Supplier s
Customer’s zone c
1
2
3
4
1
2
3
4
0 0 2
0 0 3
1 2 0
2 1 1
0 1 0
0 0 0
0 0 0
1 0 1
All computations were run using the branch and bound algorithm accessed via LINGO 8.0 on a PC Pentium IV-3 GHz and 2 GB RAM DDR under win XP SP3. The presented hereunder are the resulted solution for which we have relied on a set of abovementioned records in respect of projection reported from CHOUKA Co. Tables 12–14 provide an insight into the output data characteristics by setting the relative weight ($) of each objective function component to 0.5 and the model robustness (o) to 5000. Hereafter we denote any pair of supplier-site or sitecustomer’s zone linkage as facility-demand link. In Table 12 the blank cells similar to other unreported relevant data are equal to 0, which means the end products are not delivered to the customers in the same period as ordered. Moreover the first column indicates the product type, the second column shows the active sites for each product and the third one determines the production state in regular time (g ¼1), overtime (g¼2) and subcontracting (g¼3). For example, the amount of product type 5 is produced at site (2) in period 5 in regular time is equal to 280 and the amount of subcontracted product type 2 at the same site and at the same period is equal to 367. Furthermore, similar considerations can be drawn for workforce plan (see Table 13), interestingly during the entire planning horizon no hiring were engaged. Just within a couple of periods upgrading of the staff (level 4) were made. Table 14 presents the interactions of supply chain components. As expected most of demands (sites and customers’ zones)
38
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
Table 10 Cost and capacity of raw material m provided by supplier s in period 1 in scenario x Scenario x
Supplier s
Purchasing cost ($) Raw material m 1
2
3
4
5
6
7
8
9
10
1
1 2 3 4
1 1 1.5 1.5
2 2 1 1.5
1 1 1 1
3 3 2 2
2 2 1.5 2
1 1 2 1
2 2 1.5 1
1 1 1 1
2 2 1.5 1.5
1 1 1 2
2
1 2 3 4
1.1 1.1 1.65 1.65
2.2 2.2 1.1 1.65
1.1 1.1 1.1 1.1
3.3 3.3 2.2 2.2
2.2 2.2 1.65 2.2
1.1 1.1 2.2 1.1
2.2 2.2 1.65 1.1
1.1 1.1 1.1 1.1
2.2 2.2 1.65 1.65
1.1 1.1 1.1 2.2
3
1 2 3 4
1.2 1.2 1.8 1.8
2.4 2.4 1.2 1.8
1.2 1.2 1.2 1.2
3.6 3.6 2.4 2.4
2.4 2.4 1.8 2.4
1.2 1.2 2.4 1.2
2.4 2.4 1.8 1.2
1.2 1.2 1.2 1.2
2.4 2.4 1.8 1.8
1.2 1.2 1.2 2.4
4
1 2 3 4
1.3 1.3 1.95 1.95
2.6 2.6 1.3 1.95
1.3 1.3 1.3 1.3
3.9 3.9 2.6 2.6
2.6 2.6 1.95 2.6
1.3 1.3 2.6 1.3
2.6 2.6 1.95 1.3
1.3 1.3 1.3 1.3
2.6 2.6 1.95 1.95
1.3 1.3 1.3 2.6
Production capacity Raw material m Supplier s
1
2
3
4
5
6
7
8
9
10
1 2 3 4
3500 3000 3500 3000
3500 3000 3000 3500
3500 3000 4500 3500
3500 3500 4000 3000
3500 3000 4000 3000
2500 3000 3500 3500
4000 3500 3500 3500
3500 3500 4500 3500
3000 3500 3500 3500
3500 3500 3000 3500
It is assumed that their capacity remains fixed in all periods.
Table 11 Shortage cost, sales price. Scenario x
Customer’s zone c
Shortage cost ($/period. unit)
Sales price ($/unit)
Product i
Product i
1
2
3
4
5
1
2
3
4
5
1
1 2 3 4
2 3 2 2
2 4 2 2
2 4 2 3
3 4 2 2
1 2 2 2
25 30 26 25
37 40 37 38
46 50 45 48
28 30 29 30
33 35 33 35
2
1 2 3 4
2.25 3.25 2.25 2.25
2.25 4.25 2.25 2.25
2.25 4.25 2.25 3.25
3.25 4.25 2.25 2.25
1.25 2.25 2.25 2.25
26 31 27 26
38 41 38 39
47 51 46 49
29 31 30 31
34 36 34 36
3
1 2 3 4
2.5 3.5 2.5 2.5
2.5 4.5 2.5 2.5
2.5 4.5 2.5 3.5
3.5 4.5 2.5 2.5
1.5 2.5 2.5 2.5
26.5 31.5 27.5 26.5
38.5 41.5 38.5 39.5
47.5 51.5 46.5 49.5
29.5 31.5 30.5 31.5
34.5 36.5 34.5 36.5
4
1 2 3 4
2.75 3.75 2.75 2.75
2.75 4.75 2.75 2.75
2.75 4.75 2.75 3.75
3.75 4.75 2.75 2.75
1.75 2.75 2.75 2.75
37 42 38 37
49 52 49 50
58 62 57 60
40 42 41 42
45 47 45 47
are served by their closest facilities, however, in exceptional cases, shortage come into existence and the savings in other costs do not offset charges for disconnecting the supply chain as much as possible. Nevertheless, downstream side of the chain cannot be served which were caused in two following instances; site (2)customer’s zone (1) and site (2)-customer’s zone (3). It is implicitly because of the readjustments due to supply restrictions
partially and partially by reason of the varying replenishment of site (2) among different scenarios. Table 15 represents the state of the upgraded staff concerning different values of o. For instance, when o ¼1500, 12 workers are upgraded from working level 2–3 at site (3) in period 1, and 2 workers are upgraded from working level 3–5 at site (1) in period 2. This implies that in many industrial cases it is possible
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
39
Table 12 Aggregate production plan obtained from solving the proposed model. Product i
1
2
3
4
5
Site j
Method g
Period t 1
2
3
4
5
6
7
8
9
1
1
19.5
359.2
308.4
2
1
204.6
820.7
750
653
330
435
200
3
1
231.8
496
590
120
320
263
1
1
904.7
1136.7
656.4
994
223.3
200
583.6
2
1
458
194.2
211.5
561.2
680
540
860
920
3
1
40
570
306.6
1
1
295.3
235.1
248.9
2
1
200
970.1
768.2
830
160
3
1
39.6
101.6
780
250
1
1
481.6
283.2
729
407.2
2
1
767.3
516
654.5
625.8
930
3
1
226
200
1
1 3
742.1
1090
1047.4
990.9 170
752.7
483.4
183.2 276.7
2
1 3
399
5.8
73.4
366.8 280
164.3 170
320
66.5
3
1
70
170
332.5
597.9
277.2
120
716.5 230
150
100
90
10
11
90.3
936.3
670
559.6
100
530
326.9
400
300
620
593
550
130
236.6
353.3
280
426.6
386.6
12
833
106.6
20
353
370
348.9
Table 13 Workforce plan obtained from solving the proposed model. Level k
Labor
2
3
4
1 2 3
6 5 8
2 5 8
1 5 8
2 3
1 2 3
6 10 22
4 10 17
1 4 10
3
2 3
15 5
8
4
1
3
2
5
1
6
1
1
2
3
4
5
Upgrading labor
Period t 1
2
Firing labor
Site j
4-5
5
6
7
1 2 10
1 2 10
1 2 10
1
8
88
8
7
9
9
6
1 2 3
4
1
1 3 5
2 3
1 2 3
2 5
1 2 3
6 15
7 5
1 2
6 3
2
1 2
10
1
3 6 7
8
9
10
11
12
10
7
7
5
4
4
8
8
8
8
8
8
8
4
4
4
3
3
3
3
2
1
1 2
2 3
3
3
1
2
2
1
40
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
Table 14 Interactions of supply chain components. Supplier s
Site j
No. of periods shipped from supplier s
Average raw materials shipped from supplier s
Site j
Customer’s zone c
No. of periods customer’s zone c delivered
Average products customer’s zone c delivered
1
1 2 3
11 4 8
1848.6 509.7 1138.4
1 2 3
1
8 0 6
188.7 0 141.6
2
1 2 3
2 11 1
180.2 2045.6 1767.1
1 2 3
2
11 9 9
245.3 198.4 307.8
3
1 2 3
10 5 10
1462 1143.9 1188.6
1 2 3
3
2 0 8
51.3 0 188.4
4
1 2 3
9 10 8
453.8 2389.7 855.4
1 2 3
4
8 11 1
303.1 405.7 246.3
Table 15 State of the staff upgrading versus the multiplier of model robustness.
o
Upgraded level
Site j
Period t 1
500
1500
2000
3500
2
2-5 3-5
2 2
2-3 2-3 3-5
2 3 1
2-3 2-4
3 1
12
1-3 2-3
3 3
3 12
3
4
5
6
7
8
9
10
11
12
5 8 1 12 2
1
(×105) 5
ϖ=1
Cumulative shortage
4.5 4 3.5
ϖ = 0.9 ϖ = 0.8
3
ϖ = 0.7 ϖ = 0.6
2.5 2
ϖ = 0.5 ϖ = 0.3 ϖ = 0.4
ϖ = 0.2
ϖ = 0.1
1.5
ϖ=0
1 0.5 2.31 2.32 2.34
2.36 2.38 2.39 2.43 2.46 Total losses
2.58 2.99
28.30 (×107)
Fig. 3. Trade-off between Z1 and Z2 for RMAPP model.
to raise workers productivity as well as production rate through providing training courses instead of hiring new workers. As introduced in previous, to present the importance of considering total losses and customer satisfaction simultaneously three models are extracted for further analysis as follows: 1. Model1: consists of total losses of supply chain (Z1) subject to the relevant constraints. 2. Model2: consists of sum of maximum shortage through entire periods (Z2) subject to the relevant constraints.
3. LP-metrics model: combination of Model1 and Model2 which is calculated by Z3 subject to the relevant constraints. Thus a series of multi-objective solutions for the model are obtained by varying$. Fig. 3 graphically illustrates the trade-off between Z1 and Z2 value of these solutions for $ ¼1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 0.1, 0, respectively. Note that when $ ¼1, LP-metrics model is equivalent to Model1, and when $ ¼0, LPmetrics model is equivalent to Model2. The best value of Model1 (Zn1) is obtained for $ ¼1 while the worst value of Z2 is achieved.
S.M.J. Mirzapour Al-e-hashem et al. / Int. J. Production Economics 134 (2011) 28–42
41
Fig. 4. Trade-off between model robustness and (a) Z1, (b) Z2.
Fig. 5. Trade-off between model robustness and Z1 value of LP-metrics model.
Conversely, the best value of Model2 (Zn2) is come by $ ¼ 0 but the inferior value of Z1 is derived. In other words, considering merely one objective may sacrifice the other. Comparison of results shows that the LP-metrics model makes a trade-off between these two objective functions. Fig. 3 is an efficient frontier curve and gives this chance to the decision maker to select a suitable $ from his/her perspective. In Fig. 4 a sensitivity analysis is done for the model robustness (o) versus Z1 and Z2 obtained via solving Model1, Model2 and LPmetrics model. As Fig. 4(a) demonstrates, in Model2 the value of Z1 is enhanced exponentially by increasing o but in the case of Model1 and LP-metrics model, this growing is not considerable compared to Model2. The fact behind this result is that in Model2 penalty of infeasibility is not considered. Fig. 4(b) shows that the best and worst values of Z2 are obtained from solving Model2 and Model1, respectively, and this objective function is not sensitive
to the value of model robustness (o). Fig. 4 illustrates that the proposed LP-metrics model behaves in a way that the values of Z1 and Z2 are close to their best values Zn1, Zn2 as much as possible. A sensitivity analysis for Z1 which is obtained from solving LPmetrics model versus model robustness is separately depicted in Fig. 5. As expected when the amount of o enlarges, the value of Z1 increases but the slope of this increasing gradually decreases.
6. Conclusion In this paper a new robust multi-objective aggregate production planning (RMAPP) model was presented. Some of the features of proposed model are as follows: (i) Considering the majority of supply chain cost parameters such as transportation cost, inventory holding cost, shortage cost, production cost and
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human related cost; (ii) considering some respects as employment, dismissal and workers productivity; (iii) considering the working levels and possibility of staff training and upgrading; (iv) considering the lead time between suppliers and sites and between sites and customers’ zones; (v) Cost parameters and demand fluctuations are subject to uncertainty. First this problem was formulated as a multi-objective mixed integer nonlinear programming and then transformed into a linear one and reformulated as a robust multi-objective linear programming in which the risk of solution was measured through absolute deviation method instead of sum of square error to remain model linearity. Finally this robust multi-objective model was solved as a singleobjective problem by applying LP-metrics method. The practicability of the proposed model was demonstrated through its application in solving an APP problem of CHOUKA Co. The results indicate that the proposed model can provide a promising approach to fulfill an efficient production planning in the supply chain. At the end, sensitivity analysis was done for ranges of LP-metrics weights, solution robustness and model robustness. The RMAPP model has not come to an end and the path is still open for researchers to extend some combinations of robust optimization and multi-objective programming approaches to take advantages of them, simultaneously. References Aliev, R.A., Fazlollahi, B., Guirimov, B.G., Aliev, R.R., 2007. Fuzzy-genetic approach to aggregate production-distribution planning in supply chain management. Information Sciences 177 (20), 4241–4255. Bakir, M.A., Byrne, M.D., 1998. Stochastic linear optimization of an MPMP production planning model. International Journal of Production Economics 55 (1), 87–96. Baykasoglu, A., 2001. MOAPPS 1.0: aggregate production planning using the multiple-objective tabu search. International Journal of Production Research 39 (16), 3685–3702. Ben-Tal, A., Nemirovski, A., 1998. Robust convex optimization. Mathematics of Operations Research 23 (4), 769–805. Ben-Tal, A., Nemirovski, A., 1999. Robust solutions to uncertain programs. Mathematics of Operations Research 25 (1), 1–13. Ben-Tal, A., Nemirovski, A., 2000. Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming 88 (3), 411–424. Bertsimas, D., Sim, M., 2003. Robust discrete optimization and network flows. Mathematical Programming 98 (1-3), 49–71. Bertsimas, D., Sim, M., 2004. The price of robustness. Operations Research 52 (1), 35–53. Bertsimas, D., Sim, M., 2006. Tractable approximations to robust conic optimization problems. Mathematical Programming 107 (1), 5–36. Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic Programming. Springer, New York. Bitran, G.R., Caldentey, R., Mondschein, S., 1998. Coordinating clearance markdown sales of seasonal products in retail chains. Operations Research 46 (5), 609–624. Buxey, G., 1993. Production planning and scheduling for seasonal demand. International Journal of Operations and Production Management 13 (7), 4–21. Cheng, L., Subrahmanian, E., Westerberg, A.W., 2003. Design and planning under uncertainty: issues on problem formulation and solution. Computers & Chemical Engineering 27 (6), 781–801. Dolgui, A., Ould-Louly, M.A., 2002. A model for supply planning under lead time uncertainty. International Journal of Production Economics 78 (2), 145–152. Escudero, L.F., Kamesam, P.V., King, A.J., Wets, R.J.-B., 1993. Production planning via scenario modeling. Annals of Operations Research 43 (6), 309–335. Feng, P., Rakesh, N., 2010. Robust supply chain design under uncertain demand in agile manufacturing. Computers & Operations Research 37 (4), 668–683. Foote, B.L., Ravindran, A., Lashine, S., 1998. Production planning and scheduling: computational feasibility of multi-criteria models of production, planning and scheduling. Computers and Industrial Engineering 15 (1), 129–138. Gallego, G., 2001, lecture note, IEOR 4000: Production Management, Lecture 5 /http://columbia.edu/ gmg2/4000/pdf/lect_05.pdfS. Gebennini, E., Gamberini, R., Manzini, R., 2009. An integrated production– distribution model for the dynamic location and allocation problem with safety stock optimization. International Journal of Production Economics 122 (1), 286–304. Goodman, D.A., 1974. A goal programming approach to aggregate planning of production and work force. Management Science 20 (12), 1569–1575.
¨ Gunnarsson, H., Ronnqvist, M., 2008. Solving a multi-period supply chain problem ¨ for a pulp company using heuristics—an application to Sodra Cell AB. International Journal of Production Economics 116 (1), 75–94. Haehling, L.C., 1970. Production and employment scheduling in multi-stage production systems. Naval Research Logistics Quarterly 17 (2), 193–198. Hanssman, F., Hess, S., 1960. A linear programming approach to production and employment scheduling. Management Technology 1 (1), 46–51. Holt, C, Modigliani, F., Simon, H.A., 1955. Linear decision rule for production and employment scheduling. Management Science 2 (1), 1–30. Kall, P., Mayer, J., 2005. Stochastic Linear Programming. Springer’s International Series, New York. Kall, P., Wallace, S.W., 1994. Stochastic Programming. John Wiley and Sons, New York. Kazemi-Zanjani, M., Ait-Kadi, D., Nourelfath, M., 2010. Robust production planning in a manufacturing environment with random yield: a case in sawmill production planning. European Journal of Operational Research 201 (3), 882–891. Leung, S.C.H., Wu, Y., 2004. A robust optimization model for stochastic aggregate production planning. Production Planning and Control 15 (5), 502–514. Leung, S.C.H., Tsang, S.O.S., Ng, W.L., Wu, Y., 2007. A robust optimization model for multi-site production planning problem in an uncertain environment. European Journal of Operational Research 181 (1), 224–238. Lodree Jr., E.J., Uzochukwu, B.M., 2008. Production planning for a deteriorating item with stochastic demand and consumer choice. International Journal of Production Economics 116 (2), 219–232. Masud, A.S.M., Hwang, C.L., 1980. An aggregate production planning model and application of three multiple objective decision methods. International Journal of Production Research 18 (6), 741–752. Mazzola, J.B., Neebe, A.W., Rump, C.M., 1998. Multiproduct production planning in the presence of work-force learning. European Journal of Operational Research 10 (2–3), 336–356. Mulvey, J.M., Vanderbei, R.J., Zenios, S.A., 1995. Robust optimization of large-scale systems. Operations Research 43 (2), 264–281. Nam, S.J., Logendran, R., 1992. Aggregate production planning—a survey of models and methodologies. European Journal of Operational Research 61 (3), 255–272. Ozdamar, L., Bozyel, M.A., Birbil, S.L., 1998. A hierarchical decision support system for production planning (with case study). European Journal of Operational Research 104 (3), 403–422. Porkka, P., Vepsalainen, A.P.J., Kuula, M., 2003. Multiperiod production planning carrying over set-up time. International Journal of Production Research 41 (6), 1113–1148. Sahinidis, N.V., 2004. Optimization under uncertainty: state-of-the-art and opportunities. Computers and Chemical Engineering 28 (6–7), 971–983. Shi, Y., Haase, C., 1996. Optimal trade-offs of aggregate production planning with multiple objective and multi-capacity demand levels. International Journal of Operations and Quantitative Management 2 (2), 127–143. Tang, J., Fung, R.Y.K., Yung, K.L., 2003. Fuzzy modelling and simulation for aggregate production planning. International Journal of Systems Science 34 (1), 661–673. Tang, J., Wang, D., Fang, R.Y.K., 2000. Fuzzy formulation for multi-product aggregate production planning. Production Planning and Control 11 (7), 670–676. Wagner, H.M., 1975. Principles of Operations Research, second ed. Prentice Hall, New Jersey. Wang, R.C., Fang, H.H., 2000. Aggregate production planning in a fuzzy environment. International Journal of Industrial Engineering—Theory, Applications, and Practice 7 (1), 5–14. Wang, R.C., Fang, H.H., 2001. Aggregate production planning with multiple objectives in a fuzzy environment. European Journal of Operational Research 133 (3), 521–536. Wang, R.C., Liang, T.F., 2004. Application of fuzzy multi-objective linear programming to aggregate production planning. Computers & Industrial Engineering 46 (1), 17–41. Wang, R.-C., Liang, T.F., 2005. Applying possibilistic linear programming to aggregate production planning. International Journal of Production Economics 98 (3), 328–341. Wu, D., Olson, D.L., 2008. Supply chain risk, simulation, and vendor selection. International Journal of Production Economics 114 (2), 646–655. Wu, D., Olson, D.L., 2009a. Enterprise risk management: small business scorecard analysis. Production Planning & Control 120 (4), 362–369. Wu, D., Olson, D.L., 2009b. Introduction to the special issue on ‘‘Optimizing Risk Management: Methods and Tools’’. Human and Ecological Risk Assessment 15 (2), 220–226. Wu, D., Olson, D.L., 2010a. Enterprise risk management: coping with model risk in a large bank. Journal of the Operational Research Society 61 (2), 179–190. Wu, D., Olson, D.L., 2010b. Enterprise risk management: a DEA VaR approach in vendor selection. International Journal of Production Research 48 (16), 4919–4932. Wu, D., Xie, K., Chen, G., et al., 2010. A risk analysis model in concurrent engineering product development. Risk Analysis 30 (9), 1440–1453. Yu, C.S., Li, H.L., 2000. A robust optimization model for stochastic logistic problems. International Journal of Production Economics 64 (1–3), 385–397.
