AFPC



Due Date:Friday, February 2nd at 5:00pmTurn in to: athayer@tamu.edu Late Assignments: -10 points per day lateTotal Points Possible: 100Format: Please turn in written answers as well as the Excel workbook with separate sheets for each answer. Excel workbooks are necessary to receive credit for the problem.1. (40 pts) The ABC lumber company has three sources of wood and five markets where wood is demanded. The annual quantity of wood available in the supply source 1, 2, and 3 are 15, 20, and 15 million board feet, respectively. The amount that can be sold at market 1, 2, 3, 4, and 5 is 11, 12, 9, 10, and 8 million board feet, respectively. The company currently transports all of the wood by truck. It wishes to evaluate its transportation schedule, possibly shifting some or all of its transportation to trains. The unit cost of shipment (in $10,000) on the various routes using both methods is described in the table below. Table A. Cost per unit of truck transportationSupply sourceMarket 1Market 2Market 3Market 4Market 5Supply 15162354556Supply 25968503946Supply 34956535137Table B. Cost per unit of train transportationSupply sourceMarket 1Market 2Market 3Market 4Market 5Supply 1486848none54Supply 26675554957Supply 3None61645950The management needs to decide to what extent to continue to rely on truck transportation. Formulate a linear programming model for the ABC lumber company for the case that this company uses only truck transport. Write out the tableau with all of the relevant components.S1/M1S1/M2S1/M3S1/M4S1/M5S2/M1S2/M2S2/M3S2/M4S2/M5S3/M1S3/M2S3/M3S3/M4S3/M5Min516235455659685039464956535137Min Demand 1111>=11Min Demand 2111>=12Min Demand 3111>=9Min Demand 4111>=10Min Demand 5111>=8Max Supply 111111<=15Max Supply 211111<=20Max Supply 311111<=15b) Solve the problem in (a) using Excel. Write down and interpret the optimal values for:i. objective functionThe objective of the firm is minimizing cost. Given the parameters and constraints, the minimum amount of money the firm can pay for transportation is $2316 (in $10,000).ii. decision variablesThe firm will send 6 units (million board feet) from S1/M1 to minimize its objective function value.iii. shadow priceIf the firm relaxes the maximum supply #3 constraint (one more unit is available at supply 3), the value of the objective function would decrease by $10 (in $10,000). iv. reduced costIf the firm forces one unit (in million board feet) to be transported from S3/M4 then the value of the objective function would increase by $22 (in $10,000). How much does it cost if the company uses only train transport? Write down and interpret the optimal values for:NOTE: There are many ways to ensure that the transportation lines listed as “none” do not exist. Simply indicating a cost of $0 is not sufficient. You need to remove the variable, indicate it is not allowable in the constraints or, as I have shown here, made the cost so high that it essentially does not exist.i. objective functionThe objective of the firm is minimizing cost. Given the parameters and constraints, the minimum amount of money the firm can pay for transportation is $2652 (in $10,000).ii. decision variablesThe firm will send 11 units (million board feet) from S1/M1 to minimize its objective function value.iii. shadow priceIf the firm relaxes the maximum supply #3 constraint (has one more unit available at supply point 3), the value of the objective function would decrease by $7(in $10,000). iv. reduced costIf the firm forces one unit (in million board feet) to be transported from S3/M4 then the value of the objective function would increase by $17 (in $10,000). d) If the company has to commit to transporting the lumber by either train or truck, which method should they choose? The firm should choose to transport the material via truck. 2. (40 points) In a corn production operation, two products are created: corn and corn stover. Water, pesticides, fertilizer, and seeds are required. There are five production possibilities for producing corn as follows:Outputs and Inputs per Acre?Process??12345Corn Yield in Bushels6048769671Corn Stover Yield in Bales2329514849Water Use in Gallons113797Pesticide Use in Kilograms65887Fertilizer Use in Kilograms58768Seeds in pounds1010101010Land11111Corn sells for $4/bushel and corn stover sells for $3/bale. Water costs $0.6/gallon, pesticides cost $9/Kg, fertilizers cost $2/Kg, seeds cost $7/pound, and there is a $3 production cost for each process. There are 575 acres of land available for production.Write out the mathematical formulation of the problem. Explain your choice of minimization or maximization of the objective function.The objective function is maximized because the problem is a joint products problem which seeks to find the combination of inputs, production processes, and output that will earn the most money for the firm. Mathematical notation (specific):Full credit was also given for generic notation as seen in slidesMax 4Xwheat+3Xwheatstover-3XP1-3XP2-3XP3-3XP4-3XP5-.6Xwater-9XPest-2XFert-7XSeedS.t.1Xwheat-60XP1-48XP2-76XP3-96XP4-71XP5<=01Xwheatstover-23XP1-29XP2-51XP3-48XP4-49XP5<=011XP1+3XP2+7XP3+9XP4+7XP5-1Xwater<=06XP1+5XP2+8XP3+8XP4+7XP5-1XPest<=0 5XP1+8XP2+7XP3+6XP4+8XP5-1XFert<=010XP1+10XP2+10XP3+10XP4+10XP5-1XSeed<=01XP1+1XP2+1XP3+1XP4+1XP5<=575Xwheat, Xwheatstover, XP1, XP2, XP3, XP4, XP5, Xwater, XPest, XFert, XSeed >=0Numerically list the 3 types of constraints.Demand and supply balance (output)Demand and supply balance (input)Resource availability ** and non-negativity3. (30 points) You are a dietician at a summer camp. You want to make sure that you provide the campers an exciting, yet nutritious diet. You can provide the cooks the following food items: mixed vegetables, hotdogs, potato chips, breakfast cereal, chocolate chip cookies, and spaghetti. Each of the food items you purchase by the serving. The ingredient breakdown and nutritional minimum and maximum values are listed below:Mixed Vegetables (1 serving)Hotdog (1 serving)Potato ChipsBreakfast Cereal Chocolate Chip CookiesSpaghettiCost/lb..5111.532.5Protein (grams)2.215272.320Sugar (grams)5.7511121510Energy (calorie)118110160175221379Fat (grams).27101031113 Max requirementMin requirementProtein (grams)5020Sugar (grams)4825Energy (calorie)18001500Fats (grams)8040Write out the mathematical formulation of the problem. Be sure to include all relevant components. Min .5XMV+1XHD+1XPC+1.5XC+3XCCC+2.5XSS.t.2.21 XMV +5 XHD +2 XPC +7 XC +2.3 XCCC+20 XS >=205.75 XMV +1 XHD +1 XPC +12 XC +15 XCCC+10 XS >=25118 XMV +110 XHD +160 XPC +175 XC +221 XCCC+379 XS >=15000.27 XMV +10 XHD +10 XPC +3 XC +11 XCCC+13 XS >=402.21 XMV +5 XHD +2 XPC +7 XC +2.3 XCCC+20 XS <=505.75 XMV +1 XHD +1 XPC +12 XC +15 XCCC+10 XS <=48118 XMV +110 XHD +160 XPC +175 XC +221 XCCC+379 XS <=18000.27 XMV +10 XHD +10 XPC +3 XC +11 XCCC+13 XS <=80XMV, XHD, XPC, 1.5XC, XCCC, XS >=0You solve the problem and find the following values for the decision variables. Note: these are not the “correct” values. Mixed Vegetables0Hot Dog0Potato Chips3.465Breakfast Cereal.5Chocolate Chip Cookies1.264Spaghetti1.758Solve the mathematical formulation you indicated in part a and provide:The value of the objective function.5(0) +1 (0) +1(3.465) +1.5(.5) +3(1.264)+2.5(1.758) == 12.402The objective of the dietician is to minimize cost. Given the parameters and constraints, the minimum amount of money the dietician can pay to feed each camper is: $12.402.The value of each constraint. Interpret the slack for each constraint and if each constraint is binding. 2.21 (0) +5 (0) +2 (3.465) +7 (.5) +2.3 (1.264)+20 (1.758) >=205.75 (0) +1 (0) +1 (3.465) +12 (.5) +15 (1.264)+10 (1.758) >=25118 (0) +110 (0) +160 (3.465) +175 (.5) +221(1.264)+379 (1.758) >=15000.27 (0) +10 (0) +10 (3.465) +3 (.5) +11 (1.264)+13 (1.758) >=402.21 (0) +5 (0) +2 (3.465) +7 (.5) +2.3 (1.264)+20 (1.758) <=505.75 (0) +1 (0) +1 (3.465) +12 (.5) +15 (1.264)+10 (1.758) <=48118 (0) +110 (0) +160 (3.465) +175 (.5) +221 (1.264)+379 (1.758) <=18000.27 (0) +10 (0) +10 (3.465) +3 (.5) +11 (1.264)+13 (1.758) <=80Min Protein48.5>=20Not BindingSlack=28.5Min Sugar46>=25Not BindingSlack=21Min Energy1587.5>=1500Not BindingSlack=87.5Min Fat72.90564>=40Not BindingSlack=32.906Max Protein48.5<=50Not BindingSlack=-1.5Max Sugar46<=48Not BindingSlack=-2Max Energy1587.5<=1800Not BindingSlack=-212.5Max Fat72.90564<=80Not BindingSlack= -7.1Given the information above, none of the constraints are binding, the complementary slack associated with each constraint is not 0 indicating resources are underutilized.For the minimum nutrient constraints, the slacks are 28.5, 21, 87.5, and 32.906 for the minimum protein, sugar, energy and fat, respectively. This indicates that the mix of food is more than meeting the minimum requirements. For the maximum nutrient constraints, the slacks are -1.5, -2, -212.5, and -7.1 for the maximum protein, sugar, energy and fat, respectively. This indicates that the mix of food is not being constrained by the maximum constraints on each nutrient. 4. (0 pts) Suppose you are consulting with an investor to determine how much of 4 stocks to buy. From previous years' e xperience, the investor has observed the following data on returns per five hundred dollars invested from each of the four stocks: ????????ObsStock 1Stock 2Stock 3Stock 41?55?45?35?202?60?70?28?203?18?28?30?224?-30?42?12?195?50?15?40?216?60?40?22?187?41?30?35?208?16?19?36?20The investor has 500,000 to invest.Formulate the investor's problem using the EV criterion. Report the explicit formula associated with the problem. Full credit was given for specific or general notation as given below. Be able to write specific notation for a given problem. Solve the problem for the following five risk aversion coefficients: 0.01, 0.001, 0.0001, 0.00001, 0.00RAP0.010.0010.00010.000010X100044.430160X230.2347774.30893360.846955.56981000X3082.57085639.15400X4969.7652843.1202000Obj Func10049.1719967.3328563.0233494.1636125**Notice how the objective function value and choice of stocks changes as the RAP changes. Be able to explain this relationship. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download