سیاست های موجودی برای کمک های بشردوستانه در طوفان

In this work we present a stochastic programming model minimizing costs, to support the decision process of inventory policy which best satisfies the demand for food in shelters when hurricane winds are about to impact a town. In this model we consider perishable products as well as the first in first out (FIFO) system for their consumption. In order to make the model closer to reality ordering cost is time-varying and we add a penalty cost in case the shortage exceeds a known limit for two days in a row. Finally the cost to dispose of expired food is greater than the purchase cost of the product since throwing away food has ethical implications. Starting from a stochastic programming model, we present a procedure to transform it to a deterministic mixed integer programming model (MIP) with non-convex objective function over its entire domain, which closely states the situation in reality. Preliminary computational results and discussion are presented.

The objective of this work is to propose an inventory policy to support the special conditions faced by those who manage shelters when a natural event like a hurricane, source of potential damage, puts the population at risk. In the case of such event, government or non-profit organizations operate shelters for the people in the affected areas. In Mexico, in many cases, the intervention of the Army is required because of the magnitude of the disaster and because of their expertise in handling of these kinds of situations. When that happens, Officers from the Army, evaluate in rough numbers the situation, estimate the supplies needed as a first approach, take them from their own inventory and then move to the target. All this procedure can take around six hours to be ready to start supporting operations. Once they are in the affected region, they organize all the workers and shelters. The specialists evaluate the real situation: count all the people affected as well as their needs and ask for supplies from Headquarters on a weekly basis. When the Army takes control or when the local authorities take the relief efforts under their control, in both cases they follow special procedures to ask for funds from the Federal Government to support all these special works. From all the products needed for shelters’ operation we will focus on food. The inventory policy will be the result of a stochastic programming model that will represent the situation when the shelter’s manager needs to place food orders – under probabilistic scenarios – that a distribution center must supply. Other works, e.g. Lodree and Taskins (2009) [10] was presented for general inventory problems for supplies such as flashlights, batteries, etc. However, to the best of our knowledge, there is no prior work on the food inventory problem based on the dynamic hurricane information. The measure of performance for the problem is to minimize all the expected costs incurred during the inventory management as it is usually done in inventory control problems. However, the ordering cost is not a fixed number over the planning horizon; as the hurricane approaches, the ordering cost is much higher compared to the time periods after the hurricane strikes. In this model, the shortage cost is much greater than the purchase cost because a shortage will result in not being able to feed the refugees. We also include a disposal cost because the products have an expiration date. Finally, because the same shortage of food for two or more days in a row causes much bigger problems compared to the shortage for two days or more apart from each other, besides the shortage cost we include a penalty cost for shortage in excess for two days in a row; the underlying assumption is that the refugees can probably not withstand two days without food and if that happens, medical equipment should be mobilized to help them. Those two days in the penalization process can be easily modified to three, four or any other number of days as needed. The other difference given by the nature of the products is that our scenario refers to lost sales, which means the product in shortage will not be supplied later. It is important to remark that we will try to avoid the nonlinear representations, to keep the model relatively simple and solvable with readily available commercial software. Some background information about the scenario developed in our work is described in the following paragraphs. Tropical cyclones are natural phenomena originated in the tropical zones of the planet, integrated by great masses of winds moving clockwise or counterclockwise. Besides their rotational movement, they present a translation movement. Along their way toward the earth’s surface, weather conditions produce changes in their speed. When the rotational speed reaches 118 km/h or more, the phenomenon is then called hurricane. Despite the benefit of the rain, a hurricane is a potential source of destruction that grows according to its intensity and to the vulnerability of the region that it is likely to hit. Although all of us had heard about hurricanes and their devastation, it was not until Katrina’s arrival in August 2005 that more people became aware of the importance of supply chain vulnerability and response; more than the events themselves, it was amazing to realize the lack of preparation authorities have to face the damage caused by hurricanes every year. The good news is that these events could be probabilistically predicted; scientists forecast the expected number and type of hurricanes that could arise every year, as well as their intensity and trajectory once every one of them appears. Regarding food supply, in most cases this activity is carried on “whenever needed” basis; there is no previous or specific inventory decision support. Moreover, as Wassenhove (2006) [14] points out, this process is extremely complex. To understand the magnitudes, it is enough to take a look at the number of unsolicited donations, excess in emergency orders, saturation of terrestrial transportation and so on when an emergency situation occurs. As we will see in this paper, there are some attempts to improve this situation but nothing with the scope we are presenting. In the next section previous works related to this topic are discussed; then we present the stochastic programming models we used to solve the problem. The models are presented according to the modifications they went through until we considered one model that accurately reflects the situation we wanted to represent; at the end we discuss the feasibility of the programming model and the convexity of the total cost function.

The properties for the conditional expectation were very useful to better handle the stochastic programming problems since we could write the expected value for inventories and shortages at any time as a function of their expected values associated with each scenario at any time and the probabilities of occurrence of each scenario – a parameter – that produces a more manageable model. ○ The introduction of the perishable characteristic of the product made the model closer to reality, and increased a lot the already big size of the problem. ○ Parameters as for example the shortage cost should be very carefully defined, because it could be estimated as an emergency order supplied by airplane but when the airplane is not an option, experts must give a number that represents the cost of having disturbances or other kind of situation. ○ About the k and k0 parameters, they should be determined by the people in charge of the situation that have experience with them. ○ It is easy to see all the variety of extensions that could be made to the model presented. For example, the introduction of different modes of transportation, the availability of them in terms of the distance to the hurricane; work with a variable purchasing cost instead of a fixed one; introduce capacity to the shelters, multi product, make the lead time variable, etc. ○ The problem is easily handled by the software we have now.