(44g) Allocation and Transportation Optimization of Products to Contracts Under Multiple up to Second Order Statistics Conditions on a Variable List of Product Characteristics

Given a general product with many characteristics and a list of contracts where given quantities of the product need to be delivered at certain destinations, and where contracts could freely impose conditions on some or all of the characteristics such as the average, or coefficient of variation of a dimension to be within certain limits for the overall contract, and assuming that various quantities of the product are spread geographically in many warehouses, the problem is to optimize the assignment of which individual product item from which warehouse should be given to which contract, such that the overall cost of both the intrinsic value of the products and their transportation be minimized.

Many products, particularly in the agricultural field can be described by the methodology presented here. To simplify things, a “product” will be assumed to be already packaged in a container which is economically unfeasible to break down. Therefore, when referring to the characteristics of a product, we will actually refer to the average values of the constitutive elements of that product in the package. As an example, let’s assume that one of the characteristics of the product is length. If the product is a grain of some type, we can measure every single grain in a given container and determine the average length of the grains in that container. In short, we will refer to this as the length of the product (in that container). Details on the sampling techniques and methodology on determining, for every characteristic the corresponding value in each container, are beyond the scope of this work.

Customers are limited in their need of products in several ways. They depend on the particular end product they manufacture which may require specific ranges of the input product. They are also restricted by the machinery they have available and which may not be able to cope with a wide variability of one or another characteristic. For example, while the end product requirements might be satisfied by a group of products with an average length between 1 and 4 units, the machines available may not be able to process the product effectively unless the length range is reduce to between 2 and 3 units.

Characteristics of a product are assumed to be open ended and generic. Some taken in consideration are: length, height, weight, colors (3 numbers), as well as very specific ones. Some characteristics might be of a nominal type, such as the name of the location where the product was harvested. The list of products is open ended because, even if so far a particular one was not taken in consideration, such as humidity, the methodology should be flexible to allow the use of this new variable without change.

The contracts can specify for a given characteristic only a limited set of condition types. As an example, a contract can specify the length of the product to be between a min and a max, the average length to be between a min and a max, and the coefficient of variation of the length of the product to be within certain limits. Not all types of conditions are present in a given contract for every single characteristic, and in that regard, the contracts can be assume to be of “free form”.

The actual example for which an optimization solution was built deals with approximately 3,000,000 products, each product has about 15 characteristics, and the products are distributed over several hundred warehouses. The number of contracts were also in the hundreds. For practical purposes, every single product can be considered to be unique because of the individual resolution of each of the characteristics measurements. Similarly, no two contracts are alike, even from the same customer – probably because even if they manufacture the same end product as before, their internal optimization needs require more restrictive ranges of the product.

Additional restrictions in the assignment process apply because as products are assigned to a contract, deliveries in batches start to happen and once on the move, deliveries cannot be stopped. Even more, at a given moment in time assignments can be made to a contract based on the available products at that time. However, at a later time, when some of the previously assigned products have not yet shipped, and are therefore still in play, and furthermore, if products better suited for that contract (meeting the conditions and also be cheaper), a reassignment can take place.

Another restriction is dictated by transportation needs to group products in containers or truck loads.

The solution proposed for the optimization starts with the intuitive observation that this is an iterative process, where at every individual step additional products need to be assigned to a contract such that at the end, the contract conditions are met, and also the value of the assigned products together with the transportation cost to destination is minimized. To take the example of length as one of the characteristics, we can assume a contract condition specifying the average length to be between min_length and MAX_length. If we face a shortage of long products which are also cheap at the beginning of the assignment process we will naturally be forced to assign short products, some of which might even be delivered. However, after a while, because we assigned many short products, we can still satisfy the overall contract condition intuitively by assigning, from this point on, even longer products, which are even cheaper. In this way, we can regard the process of assigning products to a contract as a dynamic process, where, after each assignment of a product or group of products to the contract the individual conditions on each of the characteristics may change and still guarantee that in the end, the overall contract conditions will be met.

The paper presents in detail the recurrent mathematical relationships for a given characteristic linking the summary of what was assigned so far (average, coefficient of variation) with the overall contract conditions to determine the relative conditions to be imposed on the products still to be assigned. It is like at each step of the assignment we are dealing with a dynamic contract rather than an absolute one, where the conditions are adapted based on what was assigned before.

The overall algorithm is also presented, where the transportation cost is included. In brief, candidate groups of product are determined from each warehouse then the one with the overall minimum value after including the transportation cost is selected.

Additional practical IT considerations are also discussed. For example, the entire data repository was available in an Oracle database and rather than extracting it for processing in a more convenient computing platform, the optimization algorithm was developed in PL/SQL, which is the Oracle’s procedural language, thus keeping the data in place and bringing the algorithm to it.