Simplex Method Paper Rough Draft

Simplex Method Paper Rough Draft

Business executives have to make crucial decisions everyday regarding costs and profits when buying and selling their products or services. When faced with the dilemma how to maximize costs while simultaneously minimizing costs, one can use the simplex method, created by George b. Dantzig; the simplex method allows one to "determine allocations of resources that maximize profit, and we determine production schedules that minimize cost" (Barrett, Byleen, and Ziegler, 2011). In this following paper, Team B will evaluate a current situation with ClueCross ClueShield and use the simplex method to determine the most cost efficient solutions to their issue. The team will also identify constraints as well as evaluate the results of the solutions determined by the simplex method.

The selected issue

ClueCross ClueChield's (CCCC) Workstation Support (WSS) has a work order requesting 600 Personal computers (PCs) for the Bronx Health Club and 400 PCs for the Long Island Companion Hospital. WSS has 700 PCs currently in storage in White Plains and 800 PCs in Storage in Manhattan. UPS will charge five dollars per PC for a shipment from White Plains to the Bronx Health Club; however UPS charges ten dollars for shipments to Long Island. Furthermore, UPS Charges fifteen dollars per PC for a shipment from Manhattan to the Bronx, and four dollars for shipments from Manhattan to Long Island Companion Hospital. The team will use the simplex method to determine the total number of PCs that CCCC can ship out from the individual deployment warehouse to the Bronx Health Club and the Long Island Companion Hospital Island at minimal costs.

The Objective Function and Constraints

According to "Britannica'S Objective Function" (2012), Linear programming (LP) refers to a family of mathematical optimization techniques that have proved effective in solving resource allocation problems, such as the one Team A details below, particularly those found in industrial production systems (linear programming). These Linear programming methods are algebraic techniques based on a series of equations or inequalities that limit a problem and are used to optimize a mathematical expression called an objective function ("The Free Dictionary", 2012). After analyzing the data above, CCCC is seeking to find the total number of PCs that they need to ship out from the individual deployment warehouse to each line of business. Below are the variables that are unidentified at this time

T = # of PCs from White Plains to Bronx Health Club

E = # of PCs from White Plains to Long Island Companion Hospital

A = # of PCs from Manhattan to Bronx Health Club

M = # of PCs from Manhattan to Long Island Companion Hospital.

The objective is to minimize the cost: C = 5T + 10E + 15A + 4M

The first two constraints have to do with the work orders.

Bronx Health Club: T + A = 600

Long Island Companion Hospital: E + M = 400

T + A = 600 and E + M = 400 are both known as equations. The Bronx Health Club put in an order for 600 PCs. They have stated that they do not want any more or any less than 600 PCs shipped to them. Likewise, Long Island Companion Hospital has to have no more or no less than 400 PCs. Therefore, the equations can be solved to express A and M in terms of T and E as shown in example one below:

Figure 1:

Now below, the following two constraints are dealing with supplies

Currently, there is 700 PCs in White Plains: T + E < 700

There is 800 PCs in Manhattan: A + M < 800

Substituting for A and M, allows us to express the constraint in variables of T and E as shown in the example below:

Figure 2:

The work orders total 1000 PCs. There are 800 PCs in Manhattan; this means that 200 PCs that would have to come from White Plains.

Below are the implicit constraints in Figure 3:

Figure 4:

Substituting yields

Now to recap the restrictions spoken applying just T and E as shown in the example below:

Figure 5:

Evaluating the solutions of the problem

Barnett, Ziegler, and Byleen (2011) states that, the points of intersection of the lines that form the boundary of a solution region will play a fundamental role in the solution of linear programming problems. Barnett, Ziegler, and Byleen (2011) go on to state that, A corner point of a solution region is a point in the solution region that is the intersection of two boundary lines. The following feasible points for our outlined issue are as followed:

Figure 6:

Now let us take a look at the objective function within each coordinate. The values above show us the number of PCs that are shipped through UPS from White Plains to Bronx Health Club and Long Island Companion Hospital. As soon as this is known, the next step would be to figure out the number of PCs UPS will ship from Manhattan. It is assumed that CCCC will have UPS ship a sufficient amount of PCs from Manhattan to satisfy the orders requested.

With the (0, 400) coordinates, CCCC would have UPS ship 400 PCs from White Plains to Long Island Companion Hospital. That would fill the Long Island Companion Hospital work order, so we would not need to ship any PCs from Manhattan to Long Island Companion Hospital, but Bronx Health Club would still need their 600 PCs. These will need to come from Manhattan. This is not the most practical answer because this solution would require the 600 PC's to be shipped at the highest rate.

With the (0, 200) coordinates, CCCC would ship