Linear / Binary Integer Programming and Inventory Control

Linear Programming

Linear programming involves the planning of activities and is a resourceful problem solving tool that aids management in making decisions about how to allocate its resources to meet objectives. From its meaning alone, linear programming uses mathematical problems or models to represent the problem or activity involved and the linear function is either maximized or minimized, subject to constraints.

Linear programming was developed under the management science discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the WWII postwar period as many industries found valuable uses for linear programming. "The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel Prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich and the economist Tjalling Koopmans for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role" (Overton, 1997).

In general, linear programming can be divided into five key categories or applications that can be applied to the many types of businesses and industries. Out of the five, probably the most common is that of the transportation model, which involves optimizing a shipping plan for transporting goods. The resource-allocation model deals with limits that are set and, to make the most of those resources, one develops a model that involves the allocation of those resources to certain activities. Cost-benefit-trade-off models deal with benefits that must be achieved by undertaking certain activities without consideration of usage, however at the minimum cost. The mix of levels of various activities is chosen to achieve minimum levels at minimum cost. One common application of this type of model in businesses involves personnel scheduling. The last two models are mixed models, which involve combination of the other mentioned models and assignment models. Assignment models involve making assignments or otherwise, the assignment of people to jobs. This assignment may involve matching up personnel to job tasks or assigning machines or plants to tasks. No matter what type of linear programming model is used, the model is formulated, analyzed and used to serve and aid management in making business decisions.

Another type of linear programming model that can be utilized and deals with a different type of decision variable is the Binary Integer Programming model. With binary models, binary decision variables are used the decision to be made concerns yes or no decisions as opposed to the how-much decisions used in our original linear programming model. When a particular option is being considered, a yes or no decision is based on choosing yes if you want to go ahead with that option or choosing no if you decline the option. Each variable can only take on the value of 0 or 1 and this represents the selection or rejection of an option. Just like linear programming models, binary integer programming models can be formulated using a spreadsheet and these types of models arise quite frequently in a wide array of business applications.

Many industries use linear programming as a standard tool. In applying linear programming, management looks to allocate a finite set of resources in an optimal way. Examples of important application areas include staff scheduling, shipping and truck routing, distribution networks, product mix planning, petroleum refining, corporate restructuring and stock and bond portfolio selection. The petroleum refining industry has effectively embraced the software solutions of liner programming to optimize the business supply chain to maximize their profit margins and to optimize the entire supply chain from crude evaluation and selection, production planning and product logistic planning. The supply chain of a typical petroleum refining company involves a wide spectrum of activities, starting from crude purchase and crude transportation to refineries, refining operations, product transportation and finally delivering the product to the end consumer. "The whole process of selecting the right crude is linked not only to the transportation costs involved in delivering it to the refinery, but it must take into consideration the refinery configuration, capabilities and constraints in converting the crude into products, as well as the product volume and price fluctuations" (Jain...2008). Petroleum refinery planning and optimization is mainly addressed using linear programming like Refinery and Petrochemical Modeling System, which was introduced by Honeywell and is used today by more than 100 refineries worldwide. It is used in operations planning, crude oil evaluation and selection, inventory management, future investment analysis and to analyze "what if" scenarios. Ultimately linear programming is commonly used in refinery planning to identify with confidence the most profitable refinery-wide operating strategy and it provides a model that is an excellent economic evaluation tool to drive the entire petroleum refining supply chain toward higher profit.

An example of linear programming models is being utilized at Mike's place of employment, specifically a cost-benefit-trade-off model. Mike currently works for a call center that's main mode of business is resolving customer needs by taking inbound and making outbound telephone calls. The call center is a pure cost center and does not contribute any profits to the organization. As the business analyst for the location, it is Mike's job to minimize the operating costs of the business to ensure profitability for the business. Mike was tasked to create the requirements for a group of team members that would answer internal employee inquires. The group would be a subject matter expert on company standard operating procedures and would assist employees on making the best decisions for their customers. The current process has a Team Manager answering these questions. The goal is take this responsibility from the Team Managers and give it to a HELP desk group so that the Team Managers can concentrate on more important responsibilities. Minimizing of the salaries is also part of the scope of the project. With Mike's location being a pure cost center, 90% of the location's costs are from salaries and minimizing the salaries paid out to his group is crucial.

The requirements for the creation of the group were that they needed to answer all calls that come to their HELP desk line. Data was collected and analyzed from the current process to come up staffing requirements for HELP desk line. From historical