Mathematics for Management Science
Mathematics for Management Science MT 235
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Date Created: 10/03/15
Mathematics for Management Science Notes 05 prepared by Professor Jenny Baglivo Jenny A Baglivo 2002 All rights reserved Transportation and assignment problems Transportationassignment problems arise frequently in planning for the distribution of goods and services from several supply locations origins to several demand locations destinations Transportation assignment is to be made at minimum cost Origins are indexed using i1 2 etc Destinations are indexed using j1 2 etc Typical notation is as follows 39 For the decision variables we let Xij equal the number of units shipped from origin i to destination j 39 The supply at origin i is denoted by Si 39 The demand at destination j is denoted by If the total supply is greater than or equal to the total demand then 0 For each origin the constraint XH S Si is added to the model and 0 For each destination the constraint x Z is added to the model Not all products are shipped All demands are satisfied If the total supply is less than the total demand then 0 For each origin the constraint XH 2 Si is added to the model and 0 For each destination the constraint x S is added to the model All products are shipped Not all demands are satisfied page 1 of 19 Exercise 1 Tropicsun is a leading grower and distributor of fresh citrus products with three large citrus groves scattered around central Florida in the cities of Mt Dora Eustis and Clermont Tropicsun currently has 275000 bushels of citrus at the grove in Mt Dora 400000 bushels at the grove in Eustis and 300000 at the grove in Clermont Tropicsun has citrus processing plants in Ocala Orlando and Lessburg with processing capacities to handle 200000 600000 and 225000 bushels respectively Tropicsun contracts with a local trucking company to transport its fruit from the groves to the processing plants The trucking company charges a at rate for every mile that each bushel of fruit must be transported Each mile a bushel of fruit travels is known as a bushel mile The following table summarizes the distances in miles between the groves and processing plants Mt Dora Tropicsun wants to determine how many bushels to ship from each grove to each processing plant in order to minimize the total number of bushel miles Note The problem can be visualized using a transportation network where each source and destination is represented as a node and each route from source to destination as a link page 2 of 19 0 De ne the decision variables precisely Completely specify the LP model page 3 of 19 0 Clearly state the optimal solution 0 How would the total cost change if Leesburg could only process 200000 bushels Interpret the reduced cost for shipping from Eustis to Ocdld page 4 of 19 Exercise 1 solution sheet 250000 Changing B18DZO F18F20 H18H20 BZZD22 lt B24D24 linear model nonnegative page 5 of 19 Sensitivity and formulas sheets Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price RH Side Increase Decrease F18 from Mt Dora LHS 275000 48 275000 50000 75000 F19 from Eustis LHS 400000 30 400000 50000 250000 F20 from Clermont LHS 300000 20 300000 50000 300000 B22 LHS to Ocala 200000 27 200000 75000 50000 C22 LHS to Orlando 550000 0 600000 1 E30 50000 D22 LHS to Leesburg 225000 8 225000 250000 50000 page 6 of 19 Assignment problems are special cases of transportation problems where Si 2 dl Z 1 Exercise 2 Fowle Marketing Research Company has just received requests for market research studies from three new clients The company faces the task of assigning a project leader to each client Currently four individuals have no other commitments and are available for the project leader assignments Fowle s management realizes however that the time required to complete each study will depend on the experience and ability of the project leader assigned Estimated project completion times in days are given in the following table The three projects have approximately the same priority and the company wants to assign project leaders to minimize the total number of days required to complete all three projects If a project leader is to be assigned to one client only what assignments should be made This problem could be solved by enumerating all 24 possible assignments of three of the four project leaders to the clients page 7 of 19 0 De ne the decision variables precisely Completely specify the LP model 0 Clearly state the optimal solution page 8 of 19 Solution and sensitivity sheets Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable page 9 of 19 Transshipment problems Transshipment problems have the same basic goals as transportation problems in particular there is a need to ship goods from origins to destinations at minimum cost but 39 Goods may travel from a source through an intermediate location known as a transshipment node to a destination 39 Some destinations may also serve as transshipment points to other destinations 39 etcetera The locations sources destinations and transshipment points are visualized as nodes in a network and are numbered consecutively If nodes i and j are connected by a direct route in the network a link then the decision variable Xij is used to represent the number of items shipped from node i to node j For example if a company has 1 two plants sources 2 a warehouse serving as a pure transshipment point and 3 two retail outlets destinations a transportation network could look like the following 1 Plant 1 4 Outlet 1 2 Plant 2 gt 5 OutletZ The links correspond to i j equal to 13 14 15 23 24 25 34 35 page 10 of 19 For each node in the network there is a flow constraint where the left and right hand sides are as follows 1 LHS OUTFLOW INFLOW 2 RHS 2 supply with a plus sign g demand with a negative sign g zero if it is a pure transshipment node The outflow corresponds to the number of units leaving a given node The in ow corresponds to the number of units entering a given node You can use the same relational symbol s 2 for all flow constraints I Total Supply 2 Total Demand I LHS RHS I I Total Supply gt Total Demand I LHS s RHS I I Total Supply lt Total Demand I LHS 2 RHS I Exercise 3 A company has two plants at nodes 1 and 2 one regional warehouse at node 3 and two retail outlets at nodes 4 and 5 In the next production cycle 1 Plants 1 and 2 can produce 400 and 600 units of product respectively 2 Outlets 1 and 2 have a demand for 750 and 250 units of product respectively and 3 The costs for shipping each unit of the product are as follows At most 500 units can be shipped between the warehouse and the outlet at node 4 The company wants to determine the least costly way to ship the units page ll of 19 0 Construct the transshipment network including supplies and demands with appropriate signs and costs along each route 0 Define the decision variables precisely 0 Completely specify the LP model page 12 of 19 0 Clearly State the optimal solution 0 Interpret the Shadow price and range of feasibility for the last constraint page 13 of 19 Exercise 3 solution sheet Changing D14D21 t0 D28D32 F22F32 D34 lt F34 Assume linear model nonnegative page 14 of 19 Sensitivity and formulas sheets Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable page 15 of 19 Exercise 4 The Bavarian Motor Company BMC manufactures expensive luxury cars in Hamburg Germany and exports cars to sell in the United States The exported cars are shipped from Hamburg to ports in Newark New Jersey and Jacksonville Florida From these ports the cars are transported by rail or truck to distributors located in Boston Columbus Atlanta Richmond and Mobile 1 There are 200 cars in Newark and 300 in Jacksonville The numbers of cars needed in Boston Columbus Richmond Atlanta and Mobile are 100 60 80 170 and 70 respectively Per car shipping costs are as follows see the graph for numbering 9 BMC would like to determine the least costly way of transporting the cars from the ports to the locations where they are needed 0 Complete thefollowing transshipment network by filling in the supplies and demands with appropriate signs and the unit costs along each link 3 Columbus 4 Richmond 5 Atlanta page 16 of 19 0 Define the decision variables precisely 0 Completely specify the LP nwdel page 17 of 19 0 Clearly State the optimal solution 0 Interpret the Shadow price of the first constraint and its range offeasibility page 18 of 19 Exercise 4 solution and sensitivitV renorts Adjustable Cells Final Reduced Objective Allowable Allowable Constraints Final Shadow Constraint Allowable Allowable page 19 of 19
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