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1、MultiPlant Production and Transportation Planning Based on DataMultiPlant Production and Transportation Planning Based on Data Abstract This paper proposes a methodology for developing a coordinated aggregate production plan for manufacturers producing multiple products at multiple plants simultaneo
2、usly， in a centralized environment via data envelopment analysis （DEA）. Based on demand forecast of the planning horizon， the central decision maker （DM） specifies the optimal combination of input resources required by the optimal output targets for each plant to keep the supply and demand in balanc
3、e， and the accompanying transportation trips and volumes among distribution centers （DCs） or warehouse facilities. In this paper， we focus on an integrated production-transportation problem since production and transportation are two fundamental ingredients in the whole operation chain. We deal with
4、 multiple products manufactured in multiple plants. The proposed mixed integer DEA models minimize both production costs and transportation costs. The capacity constraint for each plant is enforced by using the production possibility set theory. Finally， we validate our models by a numerical example
5、 and sensitivity analysis. Key words： Integrated production-transportation planning； Data envelopment analysis Liu. F.， Bi， G. B.， & Ding， J. J. （2014）. Multi-Plant Production and Transportation Planning Based on Data Envelopment Analysis. Canadian Social Science， 10（3）， -0. Available from： http：/ D
6、OI： http：/dx.doi.org/10.3968/4536 INTRODUCTION We consider an integrated production and transportation planning problem： how to optimally determine aggregate production planning and transportation trips among distribution centers （DCs） and the corresponding transportation volumes， where production a
7、nd transportation plan are considered simultaneously. The production decisions concerns how to allocate input resources and set output targets among different production units， while the transportation decisions work out how to transport superfluous outputs for one DC to other under-supply DCs when
8、all these DCs are accommodated by the corresponding production unit. We are interested in making an integral decision to minimize the aggregate costs including production costs， here mainly referring to the costs of input resources， and transportation costs to satisfy each DCs market demand. In supp
9、ly chain management， it concerns efficient policies related to purchasing raw materials from suppliers according to order or market forecast， transforming them into finished goods considering production capacity， and delivering them to end customer. Traditionally， the activities are optimized separa
10、tely due to the intractability of large model. It is obvious that such pattern neglects the internal relation in the chain compared with optimizing these steps simultaneously since optimization of each step separately does not necessarily lead to the optimization of all steps in an integrated manner
11、. That is especially true when we deal with multi-plant and multi-DC under a centralized environment， where mutual cooperation is permitted and often required as long as such decision is cost-efficient for each DC to meet its demand. Consequently， the coordinated operations of the main stages will l
12、ead to remarkable cost reductions for the company. For example， in a research of Libbey-Owens-Ford Company （Martin et al.， 1993）， integrated approach saves nearly $2，000，000 compared with separated operations in annual cost. Another production-distribution study for Procter&Gamble （P&G） company （Cam
13、m et al.， 1997） shows that integrated planning cuts down almost 20% of total cost. Integrated production and planning has become a new branch of supply chain management （Hugos， 2011； Papageorgiu， 2009）. Our model differs from previous works in the technique to characterize production function. We as
14、sume no a priori information on production technology. In particular， this paper introduces data envelopment analysis （DEA）， a nonparametric method to describe production process， into integrated production-transportation problem， which is a different approach compared to the previous works in this
15、field. There have been many papers covering the integrated production-transportation problem in a tactical level， some of which include the management of inventory especially in multi-period situations. However， most of them link the production process with a priori production relationship. For exam
16、ple Zuo et al.， （1991）， Barbaroso?lu and ?zgr （1999）， Jayaraman and Pirkul （2001）， Jain and Palekar （2005）， Kanyalkar and Adil （2007） etc. propose models with production capacity or capacity expansion as consistent constraints； Tuy et al.， （1993）， Hochbaum and Hong （1996）， Tuy et al.， （1996）， Kuno a
17、nd Utsunomiya （1997； 2000）， etc. explicitly draw on exogenous production functions. In fact， such valuable a priori information is not always available， which reduces the applicability of their models. DEA is the one of the best modeling tools for providing a satisfactory solution. By using “satisfa
18、ctory solution”， we imply that our model is based on limited information about production process that the decision maker （DM） could be able to secure. The characterization of functional dependency between inputs and outputs in a production process is not an easy undertaking in some applications. Th
19、is becomes more severe when the dimensions of inputs and outputs increase as exemplifying the features of the modern manufacturing， which partially motivate the research of this paper. Besides， DEA technique helps to identify whether the production process is efficient or not. The rest of the paper
20、is organized as follows. Section 1 reviews the current literature on DEA-based production planning and integrated production-transportation problem. An integrated model of DEA-based production and transportation planning is proposed in section 2. An illustration of the model is given in section 3. S
21、ensitivity analyses on the inputs and transportation prices order of magnitude in section 4. Conclusions are drawn in the last section. 1. LITERATURE REVIEW Our paper relates to two bodies of research： The literature on integrated production-transportation and the literature on production planning b
22、ased on DEA. Dhaenens-Flipo and Finke （2001） deals with a multi-facility and multi-product planning problem， where production costs and transportation costs are regarded simultaneously. Simchi-Levi et al.， （2004） gives a comprehensive review on the explicit production-distribution （EPD） problems. Va
23、rious EPD problems are classified by three criteria： decision level， integration structure and problem parameters. In this paper we focus on the production-transportation problems， one class of great attention. Kanyalkar and Adil （2007） present a linear programming model to overcome the weaknesses o
24、f sequential planning approaches in a multi-site environment， where specific factors， are considered for a consumer goods enterprise. Alemany et al.， （2010） proposes a mixed-integer linear programming （MILP） model under a centralized ceramic tile sector. The objective function is to maximize total n
25、et profit while the master planning is determined in multi-period and multi-item. Kopanos et al.， （2012） develop a discrete/continuous-time MILP model in real-life semi-continuous food industries. They take alternative transportation modes， for example different kinds of trucks， into account. First
26、put forward by Charnes et al.， （1978） as a nonparametric method for estimating the relative efficiency of a group of homogenous decision making units （DMUs）， DEA now has been widely applied to the public sector. Recently DEA has been applied to make production planning. This approach bases on histor
27、y inputs and outputs data. Golany （1988） first presents an interactive multi-objective linear programming procedure to help the central DM decide realistic performance goals. Beasley （2003） puts forward an approach to maximizing average DMU efficiency while simultaneously deciding for all DMUs more
28、acceptable results. Korhonen and Syrj?nen （2004）， like Golany， suggest a method to maximize the total amount of outputs of all DMUs by a multi-objective linear programming to find the most preferred allocating plan. Du et al.， （2010） recommend two planning ideas for arranging new input-output mix. O
29、ne is to optimize the average production efficiency， and the other is to maximizing total outputs while simultaneously minimizing the total inputs. As far as we are aware， there is no DEA-based work regarding integrated production-transportation problem in the literature. Thus， the current paper sug
30、gests a new direction to address this problem. 2. THE MODEL The problem we study can be graphically illustrated in Figure 1. There are several production plants， each of which is directly connected to a large-scale DC by a solid line. Each production plant has its own production plan. These solid li
31、nes indicate that all the goods are transported to the corresponding DC once finished. In addition， these DCs are inter-connected by the dotted lines， which indicate possible transportation trips. Here “possible” means transportation trips are needed depends on whether they are cost-effective. In Fi
32、gure 1， each DC is surrounded by some people. This indicates the predicted market demand in next period. Note that the transportation volume depends on the production plan we make and how is the DCs market demand satisfied by the corresponding production plants. As a result， all production plants ar
33、e connected together indirectly. Figure 1 Graphical Representation of the Problem Our objective is to minimize total costs， including production costs taking place in production plants and transportation costs among DCs， while the transportation costs between production plants and DCs are neglected
34、to highlight the other two kinds of costs. For modeling purpose， we assume there are n production plants （denoted as DMUj （ j=1，n）. History inputs and outputs data are stated as xhij （i=1，2，.m） and yhri （r=1，2，.s） for i-th input and r-th output of j-th DMU respectively. Here the data are non-negativ
35、e and the superscript h stands for history data. xij （i=1，2，.m） and yrj （r=1，2，.srepresent i-th input and target r-th output of j-th DMU for the next period. They， as a whole， are the production plans for all DMUs in our integrated production-transportation problem. We use the notation drj for the j
36、-th DCs r-th market demand which can be predicted in advance， and t（r）jk for the transportation volume of the r-th product from the j-th DC to the k-th DC. According to the problem description above， model （1） for the production-transportation planning is given. （1） （2） where ci stands for the unit
37、price of the i-th input， ejk for the unit transportation price from the j-th DC to the k-th DC. The objective function is the total costs. The first-group and second-group constraints represent that new input-output combination must be enveloped by the production possibility set （PPS）. Various retru
38、ns-to-scale （VRS） assumption （Banker et al.， 1984） is indicated by the third-group constraints. Note that if the convexity constraints are deleted， then we get the model concerning constant return-to-scale （CRS） assumption （Charnes et al.， 1978）， which corresponds to model （2）. The fourth-group cons
39、traints in model （1） consider the outcome after transportation. They imply that the sum of each DCs volume of products which are equal to the products produced locally and the net volume of products of transportations are able to satisfy the corresponding market demand， where means the amount of r-t
40、h output to be transported from the j-th DC to other DC and implies the amount of r-th output to be transported from other DC to the j-th DC. The fifth-group constraints in model （1） impose restrictions on the DC to ensure this DCs state of transportation for a certain kind of output. That means the
41、 state only belong to one of the three states， namely inward freight， outward freight and neither inward nor outward freight. Here， inward freight indicates transporting this certain kind of output from other DC to the under-considering DC， outward freight on the contrary， and the last state indicat
42、es zero transportation. As a consequence， this will prevent redundant conveyance， avoiding the appearance of both inward and outward freight at the same time. Since model （1） and （2） are so much alike except the convexity constraints， in the following we mainly deal with model （1） while similar resu
43、lts about model （2） can be achieved in analogous way. Note that the product of decision variables makes model （1） a non-linear programming. By introducing 0-1 variables， model （1） is equivalently transformed to the following model （3）， where （i）rj are binary variables and M is a sufficiently large p
44、ositive number. After the transformation， model （3）， equivalent to model （1）， is a MILP. Model （2） can be transformed to a similar model except the convexity constraints； we omit it for simplicity in the following paper. Table 4 shows the detailed arrangements about the transportation plan. Allowing
45、 for the unit transportation price matrix in Appendix， the top 5 lowest unit transportation price are between DC5 and DC7， DC1 and DC8， DC3 and DC6， DC3 and DC4， and DC1 and DC6 respectively. Take the transportation plan in CCR as an example. When the transportation price is too high in comparison w
46、ith input price， （i.e. in scene1， scene2， and scene3）， there is no transportation trips among all the DCs. When the transportation trips first appear in scene4， all of them belong to the top 5 lowest unit transportation price. In scene5， 4 trips belong to the top 5， accounting for 80% trips. As tran
47、sportation price declines， trips which are outside of the top 5 come out. Similar conclusions can be found in BCC. Note that transportation trips first appear in scene1 in BCC although transportation price is relatively high. This is because the strict requirements regarding variable returns to scal
48、e， which makes transportation necessary to meet market demand if the DMU cant produce enough outputs under current production technology. Excluding this particular situation， it is worth mentioning that in all the two kinds of PPSs， transportation trips in scene1， scene2， scene3， and scene4 all belo
49、ng to the top 5. Whats more， the amount related with such trips is relatively larger than other trips. This makes sense since the objective is to minimize total costs both in production and in transportation. CONCLUSIONS This paper presents a series of models to deal with integrated production-transportation problem based on DEA. It offers an alternative approach to investigate the product
