Estudios de la infraestructura del sector de la exhibición

heuristic solution methods

Having discussed the various deterministic multi-level FLPs under exact and heuristic solution methods, we now present the stochastic models and their solution approaches. In FLPs, plants, DCs, transportation network and other facilities can work for several years or decades, during which time the environment in which they operate may change significantly [76]. The parameters such as costs, demands, travel times, and other inputs to hierarchical facility location models, may be highly uncertain. Thus, the development of models for multi-level facility location under uncertainty are of great importance [69, 74, 76]. There are a large number of approaches that have been proposed for optimization under uncertainty in general, which have also been applied to hierarchical facility location problems [76].

As defined by Snyder [76], risk and uncertainty are situations where randomness occurs. The problems with risk situations are the one with known probability distributions to the decision maker, and such problems are known as stochastic optimization problems. Under uncertainty conditions, parameters are uncertain, and probabilities are not known. These problems are termed as robust optimization problems [10, 76]. This part of the literature study discusses the various stochastic and robust problems which appeared in the context of hierarchical facility location problems.

A study carried out by Min and Melachrinoudis [60] is a three-level hierarchical location- allocation application problem based on the banking industry. It addresses the internal dynamics and functional dependence of different hierarchies of banking services. Three layers considered for banking services are automatic teller machines (ATMs), branch bank offices, and main banks [60]. “In the banking industry, banking services are often rendered to the clients through successive levels of banking facilities” [59]. In the lower level, ATMs or drive-in banks allow clients to deposit or to receive cash, and get a statement of current- account balance. Branch bank offices at the next level of hierarchy, provide a variety of larger order services such as opening accounts and maintaining safe deposit boxes. These are in addition to basic services provided by ATMs. At the highest level of the hierarchy, the main bank offers the extended services such as corporate loan financing, credit approvals, and long-term investment consultation [60].

The banking facilities location-allocation decisions, should comparatively be evaluated according to the following conflicting criteria: the maximization of the market profitability of open banks, the maximization of the customer drawing power of open banks, and the minimization of all the risks associated with resource commitments made to open banks [60]. Through the planners guidelines for evaluating the profitability, accessibility and risk of bank location-allocation, a chance-constrained goal programming (CCGP) model is developed [60]. However, due to the stochastic nature of risk, a chance-constrained (probabilistic constraint) risk goal is developed. The objective function is a deterministic nonlinear integer goal programming model computed optimally using LINGO’s (a software) modelling language. A similar study in this area by Hochreiter and Pflug [31] based on heuristic algorithms, can be consulted.

topology. They used the expected value approach to formulate the problem’s objective. The model considered coordinates for the three layers namely; facility, transfer point and demand points. The coordinates from the three layers are used in computing distances. The coordinates of demand points are independent random variables (stochastic), with a bi-variate uniform distribution. Thus the problem is to find the optimal location of the transfer point, such that the maximum expected weighted distance from the fixed facility to all demand points through the transfer point is minimized. In this problem, only the single facility is considered, one transfer point, and several demand points. The model is computed numerically using “fminimax” in Matlab software package [32]. The problem can be applied to a situation where a city and its dwellers are uniformly distributed in the square region, and a transfer point is to be located. This transfer point (e.g. helicopter pad) is to be located so as to serve accidents such as earthquakes, floods, medical emergencies, etc [32].

A study by Tadei et al. [77] addressed the problem of locating transshipment facilities for freight transportation from origin to destination through transshipment facility for maximization of the total net utility. This is done by taking the expected total shipping utility minus the total fixed cost of the facilities [77]. The problem considers the handling utilities (costs) at the transshipment facilities as stochastic variables. The handling operations are organized in alternative scenarios, and finite capacity and congestion effects make costs to be stochastic variables with unknown probability distributions [77]. This process can be termed as the robust optimization problem as defined by Snyder [76]. The problem is computed heuristically using Lagrangian relaxation. A similar problem by Tadei et al. [78] is presented with general transportation costs from origin to destination as a stochastic variable.

A stochastic supply chain network model under risk, with three tiers of suppliers, distribution centres (DCs) and customers is studied by Azad & Davoudpour [8]. They considered the

customers’ demands as stochastic variables using a financial risk measure (conditional value at-risk (CVaR) measure). The problem is formulated as a convex mixed integer programming and a heuristic method is developed to solve the problem. In the model, different DC capacity levels were used as in Amiri [7]. The authors also considered the routing design between DCs and customers [8].

A multi-period study by You et al. [89] is a global multi-product chemical supply chain with demand and freight rate as stochastic variables. It is a case study where the multi- period planning model takes into consideration the production and inventory levels, the transportation models, the times of shipments and customer service levels [89]. This real world application study originates from the Dow Chemical Company, which supplies multiple products to world-wide customers [89]. The company has several global business units (DCs) to supply to its customers, and even customers can be supplied directly from manufacturing plants (multi-sourcing). In the solution methods, the authors incorporated the Monte Carlo sampling in a stochastic programming. They also proposed a simulation framework based on an iteration method for solving deterministic and stochastic problems [89]. The study considers a planning horizon as one year, and a month as a planning period.

A robust optimization model by Butler et al. [15] focuses on the strategic-production and distribution planning for a new product in the market environment. There is no historical data for the new product, and hence the probability distribution is not known. The study is a supply chain based on a new product having uncertainties in the demand, as well as the cost and changes in the market conditions over time to be addressed [15]. The model is implemented as the robust Lagrangian model using the mixed integer programming solver of CPLEX 7.5 (ILOG, Inc., 2001) [15].

customer demand, travel times, number of customers and other input data in the context of VRPs and LRPs [19, 47, 51, 70, 71, 74, 82]. The stochasticity due to travel times have been considered in VRPs where there is a delay to the destination due to several factors. Factors revealed from literature associated with the delay are: traffic jams due to road capacity, road blocks as a result of traffic authorities, car accidents; and weather and floods [33, 47, 51, 70, 82]. The consideration for these factors are mostly on a daily basis. So there will be the increase of travel times and hence travel costs that need to be considered prior to planning decisions.

Generally, the stochastic multi-level FLPs discussed in the literature have considered various random variables or parameters. The research in this thesis, considers the stochastic transportation links between the DCs and customer points (CPs) during the rainy season in Tanzanian maize crop transportation network. On the other hand, the transportation links between production centres (PCs) to DCs are reliable since inter-regional roads are paved, and it usually takes place during the summer season. It is a multi-period planning horizon (rainy season), where the period of time is 17 weeks. Each week’s shipping of goods is required to meet the known demands at CPs. The weekly actual amount of rainfall data over 4 years will be used in our study.