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# A Shipping Problem

## by Omar on Jul 13, 2011

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This topic deals on how to solve problems using linear programming. In the professional environment, this is used for selling projects to big companies. ...

This topic deals on how to solve problems using linear programming. In the professional environment, this is used for selling projects to big companies.
Linear programming is a modeling technique used to determine the optimal allocation of resources in business, the military, and other areasof human endeavor.
(More details: Stewart's Precalculus book)

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## A Shipping ProblemDocument Transcript

• -497205-544195<br />Instituto Tecnológico y de Estudios Superiores de Monterrey<br />Campus Ciudad de México<br />Juan José González OrtizA01212851<br />Omar Ulises Quiroz LimaA01213271<br />Professor Laura Tinoco<br />Mathematical Think<br />Group 30<br />March 30th, 2011<br />Final Project<br />Problem: An electronics discount sale has a sale on a certain brand of stereo. The chain has stores in Santa Monica and El Toro and warehouses in Long Beach and Pasadena. To satisfy rush orders, 15 sets must be shipped from the warehouses to the Santa Monica store, and 19 must be shipped to El Toro store. The cost of shipping a set is \$5 from Long Beach to Santa Monica, \$6 from Long Beach to El Toro, \$4 from Pasadena to Santa Monica, and \$5.50 from Pasadena to El Toro. If the Long Beach warehouse has 24 sets and the Pasadena warehouse has 18 sets in stock, how many sets should be shipped from each warehouse to each store to fill the orders at a minimum shipping cost?<br />Abstract<br />The problem requires calculating the minimum cost of ship the electronic sets from one place to another. After all the proper calculations, we reached to the solution where the lowest cost is located at the point (0,16) of the graph in which the company should ship: 0 sets from Long Beach to Santa Monica, 16 sets from Long Beach to El Toro, 15 sets from Pasadena to Santa Monica, and 0 sets from Pasadena to El Toro to reach to the total lowest cost of all: \$172.50<br />Design Bases<br />The diagram shows us the amount of sets represented by four equations that will be sent into Santa Monica and El Toro stores. This is the flow of sets from the warehouses to the respective dealing stores. <br />The x represents the number of cars that will be sent from the Long Beach warehouse to the Santa Monica delivery store. And, as Santa Monica delivery store has a stock (capacity) of only 15 sets, then the amount sent from Pasadena warehouse doesn’t have to overpass the 15 sets and it is represented by 15 – x; the difference between the total capacity and the cars already sent from Long Beach warehouse. The same case applies for the y but with different amount of sets.<br />Long Beach WarehousePasadena WarehouseSanta Monica StoreEl Toro Store24 sets18 sets15 sets19 setsx15 - x19 - yy<br />It is easier to represent this flow of cars through a drawing rather than by tables or charts because allow us to relate the same type of data and to order it, in order to work with it and understanding better the behavior of the same data.<br />Feasible Region<br />As the variables x and y are the number of sets that will be sent to the stores, they can’t be negative. In fact, it doesn’t exist negative number of sets. So x is bigger or equal to zero and also y. The number of sets from all the warehouses to all the stores must be bigger or equal to zero.<br />x≥0y≥015-x≥019-y≥0<br />Also, the number of electronic sets sent from Long Beach and Pasadena must not exceed the total capacity of those warehouses (24 and 18 sets respectively).<br />x+y≤2415-x+ (19-y)≤1834-x-y≤18-x-y≤-16x+y≥16<br />If we unify all our inequalities into one system of inequalities we have that:<br />0≤x≤150≤y≤19x+y≤24x+y≥16<br />Feasible regionVertex: (5,19)Vertex: (15,9)Vertex: (15,1)Vertex: (0,19)Vertex: (0,16)211452095<br />Evaluation of vertex:<br />When x=15:x+y=1615+y=16y=16-15y=1When x=15:x+y=2415+y=24y=24-15y=9When y=19:x+y=24x+19=24x=24-19x=5<br />The graph shows the intersection of all the equations giving as result the feasible region in which the minimum costs are possible. The feasible region is the solution set of a system; in this case the minimum costs that are possible for transporting the electronic sets.<br />Objective function<br />Our objective is to solve this problem, and solving this problem implies to find the number of sets that should be shipped from each warehouse to each store filling the orders of the minimum shipping cost.<br />Costs<br />First, we state our equation for the costs; which is:<br />C=x+0.50y+164.50<br />This equation comes from adding all the shipments from each ware house to each store having the addition:<br />C=\$5x+\$6y+\$415-x+\$5.50(19-y)<br />Where you pay: \$5 from Long Beach to Santa Monica, \$6 from Long Beach to El Toro, \$4 from Pasadena to Santa Monica, and \$5.50 from Pasadena to El Toro.<br />The minimum cost<br />VertexC=x+0.50y+164.50(0,19)\$174(0,16)\$172.50(5,19)\$179(15,9)\$184(15,1)\$180<br />Analysis and Conclusions<br />The lowest cost of all is at the point (0,16) of the graph, which represents the cost of \$172.50. So the electronics company should ship: 0 sets from Long Beach to Santa Monica, 16 sets from Long Beach to El Toro, 15 sets from Pasadena to Santa Monica, and 0 sets from Pasadena to El Toro.<br />The results that we got are the ones that will lead the shipping to its lowest cost, obtaining these results from one of the vertices that are within the range of possible options. The shaded area represents the possible combinations of sending cars from one place to another without exceeding the maximum number of sets available in each warehouse, neither shipping the wrong number of sets to a shop.<br />