The Shortest Path to Destruction A Floral Shop Operations Research I Linear Programming Project Kathedra Burton Mari Juno Casey Haley December 6 2004 On my honor as an Aggie I have neither given nor received unauthorized aid on this academic work Signature of Each Group Member 1 Contents 1 0 Executive Summary 2 2 0 Problem Description 4 2 1 Problem 1 Grummins Engine Truck Production 4 2 2 Problem 2 Wheat Stock 5 2 3 Problem 3 Floral Dilemma 7 3 0 Computational Results 9 3 1 Problem 1 Grummins Engine Truck Production 9 3 2 Problem 2 Wheat Stock 12 3 3 Problem 3 Floral Dilemma 15 4 0 Conclusions Recommendations 21 5 0 References 23 6 0 Appendix 24 2 1 Executive Summary The purpose of this project was to take 3 situational problems and solve each one using linear programming methods The problems addressed are 1 Multi period Inventory 2 Multi period Work Scheduling and 3 Floral Buying and Scheduling All problems were solved using the LINDO optimization software In the Multi period inventory problem Grummins Engine must decide how many trucks to produce in a three year period to both maximize profit and satisfy a government imposed pollution constraint Grummins will produce according to the following schedule and realize a profit of 3 600 000 Table 1 Optimal Solution for Grummins Year 1 2 3 Truck Type Produced Sold Inventoried 1 100 100 0 2 200 200 0 1 180 180 0 2 100 100 0 1 170 170 0 2 150 150 0 Recommendations for Grummins 1 Study customers to see if they will accept an increase in selling price 2 Look into better manufacturing practices to reduce manufacturing cost In the multi period work scheduling problem a wheat warehouse must decide on how much wheat to sell and purchase according to an initial monthly inventory and a storage constraint of 20 000 bushels The maximum profit is 162 The optimal solution is shown below and is in 1000 bushels Table 2 Optimal Solution for Wheat Warehouse Month Initial Stock Wheat Sold Wheat Bought 1 6 0 0 2 6 0 0 3 6 6 20 3 4 20 0 0 5 20 0 0 6 20 20 20 7 20 20 0 8 0 0 20 9 20 20 0 10 0 0 0 Recommendations 1 Try to increase selling price 2 Look for a different retailer to buy wheat at a lower price 3 If 1 and 2 can be done look into increasing storage space The third problem looks at a floral shop The florist buys five different colors of flowers between the months of February and September She has to decide on what color to buy each month and how many while minimizing her total cost LINDO provided an optimal solution with a total cost of 20 500 Table 3 Optimal Solution for Floral Shop Color Feb Mar Apr May June 500 Red July Aug 500 500 White 500 500 Pink Sep 500 500 Yellow Purple 500 Recommendations 1 Look for other suppliers 2 Study customer demand to see if cheaper alternatives can be found such as underused colors that can be bought at reduced prices 4 2 1 Problem 1 Grummins Engine Diesel Truck Production Grummins Engine produces diesel trucks New government emission standards have dictated that the average pollution emissions of all trucks produced in the next three years cannot exceed 10 grams per truck Grummins produces two types of trucks Each type 1 truck sells for 20 000 cost 15 000 to manufacture and emits 15 grams of pollution Each type 2 sells for 17 000 cost 14 000 to manufacture and emits 5 grams of pollution Production capacity limits total truck production during each year to at most 320 trucks Grummins knows that the maximum number of ach truck type that can be sold during each of the next years is given in Table 62 Demand can be met from previous production or the current year s production It costs 2 000 to hold 1 truck of any type in inventory for one year 1 p117 Assumptions 1 There will be no trucks left over from year three 2 The divisibility assumption is not valid 3 All variables are nonnegative Table 62 1 Year Type 1 Type 2 1 100 200 2 200 100 3 300 150 Decision Variables Xij number of trucks made Yij number of trucks sold Zij number of trucks in inventory and carried over where i year made from 1to 3 and j type of truck 1 and 2 5 Objective Function Maximize 20000y11 20000y21 20000y31 17000y12 17000y22 17000y32 15000x11 15000x21 15000x31 14000x12 14000x22 14000x32 2000z11 2000z12 2000z21 2000z22 s t y11 100 y12 200 x11 x12 320 x11 y11 z11 0 x12 y12 z12 0 x21 x22 320 z11 y21 200 z12 y22 100 x21 z11 y21 z21 0 x22 z12 y22 z22 0 x31 x32 320 z21 y31 300 z22 y32 150 x31 z21 y31 0 x32 z22 y32 0 5y11 5y21 5y31 5y12 5y22 5y32 0 2 2 Problem 2 Wheat Stock You own a wheat warehouse with a capacity of 20 000 bushels At the beginning of month 1 you have 6 000 bushels of wheat Each month wheat can be bought and sold at the price per 1000 bushels given in table 46 The sequence of events during each month is as follows 1 You observe your initial stock of wheat 2 You can sell any amount of wheat up to your initial stock at the current month s selling price 3 You can buy at the current month s buying price as much wheat as you want subject to the warehouse size limitation Everything shown in per thousand bushels 1 112 Assumptions 1 All variables are nonnegative Table 46 1 Month Selling Price Purchase Price 1 3 8 6 2 6 8 3 7 2 4 1 3 5 4 4 6 5 3 7 5 3 8 1 2 9 3 5 10 2 5 Decision Variables Xi initial stock of wheat Yi amount of wheat sold Zi amount of wheat bought Objective Function Maximize 3y1 6y2 7y3 y4 4y5 5y6 5y7 y8 3y9 2y10 8z1 8z2 2z3 3z4 4z5 3z6 3z7 2z8 5z9 5z10 s t x1 6 y1 x1 0 z1 x1 y1 20 x2 z1 x1 y1 0 y2 x2 0 z2 x2 y2 20 x3 z2 x2 y2 0 y3 x3 0 z3 x3 y3 20 x4 z3 x3 y3 0 y4 x4 0 z4 x4 y4 20 x5 z4 x4 y4 0 y5 x5 0 z5 x5 y5 20 x6 z5 x5 y5 0 y6 x6 0 z6 x6 y6 20 x7 z6 x6 y6 0 y7 x7 0 z7 x7 y7 20 x8 z7 x7 y7 0 7 y8 x8 0 z8 x8 y8 20 x9 z8 x8 y8 0 y9 x9 0 z9 x9 y9 20 x10 z9 x9 y9 0 y10 x10 0 z10 x10 y10 20 2 3 Problem 3 Florist Dilemma A flower shop owner must buy bundles of flowers to make bouquets She buys five different colors of flowers between the months of February …