transportation lp-de guzman
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Linear Programming Problems- SolvedTRANSCRIPT
Problem SetDexby P. de GuzmanAMS 511 (Linear Programming)Prof. Rhoda A. NamocoFebrurary 16, 2015Problem SetGeneral Instructions:1. Solve the following transportation problems using NWC, MC and Vogel's methods.2. Formulate the following as transportation problems.3. Solve the resulting model using a computer solver.4. Email solver setup to [email protected] with subject "AMS 511 Trabsportation Problem".1. A transportation problem involves the following costs, supply, and demand:To (cost)From1234Supply1$500$750$300$450122$650$800$400$600173$400$700$500$55011Demand10101010402. Given a transportation problem with the following costs, supply, and demand, find the optimal solution by using the computer:To (cost)From123SupplyA$6$7$4100B$5$3$6180C$8$5$7200Demand1351751704803. Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno. These warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply, and demand, Because of an agreement between distributors, shipments are prohibited from Miami to Chicago:To (cost)FromNew YorkPhiladelphiaChicagoBostonSupplyTampa$9$14$12$17200Miami$11$10$6$10200Fresno$12$8$15$7200Demand130170100150550
#1Northwest Corner MethodLP MODEL:To (cost)OBJECTIVE FUNCTION:From1234Supplymin Z = 500X11 + 750X12 + 300X13 + 450X141$500$750$300$45012, 2, X+ 650X21 + 800X22 + 400X23 + 600X24102XX+ 400X31 + 700X32 + 500X33 + 550X34 $650$800$400$60017, 9, XX89Xsubject to3$400$700$500$55011, 10, XSUPPLY CONSTRAINTS:XX110X11 + X12 + X13 + X14 12Demand10, X10, 8, X10, 1, X10, XX21 + X22 + X23 + X24 17min Z=$22,500X31 + X32 + X33 + X34 11
Minimum Cost MethodDEMAND CONSTRAINTS:To (cost)X11 + X21 + X31 10From1234SupplyX12 + X22 + X32 101$500$750$300$45012, 2, XX13 + X23 + X33 10XX102X14 + X24 + X34 102$650$800$400$60017, 10, XX10X7NONNEGATIVITY CONSTRAINTS:3$400$700$500$55011, 1, XXij 0 (i=1,2,3; j= 1,2,3,4)10XX1Demand10, X10, X10, X10, 8, 7, Xmin Z=$20,650Vogel's MethodTo (cost)From1234SupplyPenalty1$500$750$300$45012, 2, X15050300X10X22$650$800$400$60017, 7, X20050200XX1073$400$700$500$55011, 1, X10015015010XX1Demand10, X10, X10, X10, 8, 7, XPenalty100501001001005010050100min Z=$21,150Using Excel SolverTo (cost)From1234Supply1$500$750$300$450122$650$800$400$600173$400$700$500$55011Demand1010101040123410.00.0$2$1020.0$9$80.03$10$10.00.0min Z=$20,200s.t.S11212S21717S31111D11010D21010D31010D41010
#2Northwest Corner MethodTo (cost)LP MODEL:From123SupplyOBJECTIVE FUNCTION:1$6$7$4100, Xmin Z = 6X11 +7X12 + 4X13100XX+ 5X21 + 3X22 + 6X232$5$3$6180, 145, X+ 8X31 + 5X32 + 7X3335145X3$8$5$7200, 170, Xsubject toX30170SUPPLY CONSTRAINTS:Demand135, 35175, 30, X170, XX11 + X12 + X13 100min Z=$2,550X21 + X22 + X23 180X31 + X32 + X33 200Minimum Cost MethodTo (cost)DEMAND CONSTRAINTS:From123SupplyX11 + X21 + X31 1351$6$7$4100, XX12 + X22 + X32 175XX100X13 + X23 + X33 1702$5$3$6180, 5, X5175XNONNEGATIVITY CONSTRAINTS:3$8$5$7200, 130Xij 0 (i=1,2,3; j= 1,2,3)130X70Demand135, 130, X175, X170, 70, Xmin Z=$2,480Vogel's MethodTo (cost)From123SupplyPenalty1$6$7$4100, X22XX1002$5$3$6180, 52115175X3$8$5$7200, 70, X211130X70Demand135, 130, X175, X170, 70, XPenalty1221231min Z=$2,480Using Excel SolverTo (cost)From123SupplyA$6$7$4100B$5$3$6180C$8$5$7200Demand13517517012310.00.0$1002$135$450.030.0$130$70min Z=$2,350s.t.S1100100S2180180S3200200D1135135D2175175D3170170
#3Northwest Corner MethodLP MODEL:To (cost)OBJECTIVE FUNCTION:FromNew YorkPhiladelphiaChicagoBostonDummySupplymin Z = 9X11 + 14X12 + 12X13 + 17X14Tampa$9$14$12$170.0200, 70, X+ 11X21 + 10X22 + 6X23 + 10X2413070XXX+ 12X31 + 8X32 + 15X33 + 7X34 Miami$11$10M$100.0200, 100, XX100X100Xsubject toFresno$12$8$15$70.0200, 100, XSUPPLY CONSTRAINTS:XX1005050X11 + X12 + X13 + X14 +X15 200Demand130, X170, 100, X100, X150, 50, X50, XX21 + X22 + X23 + X24 + X25 200min Z=$5,000X31 + X32 + X33 + X34 + X35 200
Minimum Cost MethodDEMAND CONSTRAINTS:To (cost)X11 + X21 + X31 130FromNew YorkPhiladelphiaChicagoBostonDummySupplyX12 + X22 + X32 170Tampa$9$14$12$170.0200, 150, 50, XX13 + X23 + X33 10050X100X50X14 + X24 + X34 150Miami$11$10M$100.0200, 120, XX15 + X25 + X35 5080120XXXFresno$12$8$15$70.0200, 50, XNONNEGATIVITY CONSTRAINTS:X50X150XXij 0 (i=1,2,3; j= 1,2,3,4)Demand130, 80, X170, 120, X100, X150, X50, Xmin Z=$5,180OTHERS:X23 = 0Vogel's MethodTo (cost)FromNew YorkPhiladelphiaChicagoBostonDummySupplyPenaltyTampa$9$14$12$170.0200, 70, X932130X70XXMiami$11$10M$100.0200, 150, X101X150XX50Fresno$12$8$15$70.0200, 170, 207111X2030150XDemand130170, 150, X100, 30, X150, X50, XPenalty22330223323323min Z=$5,170Using Excel SolverTo (cost)FromNew YorkPhiladelphiaChicagoBostonDummySupplyTampa$9$14$12$170.0200Miami$11$10$6$100.0200Fresno$12$8$15$70.0200Demand13017010015050New YorkPhiladelphiaChicagoBostonDummyTampa$1000.0$1000.00.0Miami$30$1200.00.0$50Fresno0.0$500.0$1500.0min Z=$5,080s.t.S1200200S2200200S3200200D1130130D2170170D3100100D4150150Dummy5050O100
Sheet1Dexby P. de GuzmanAMS 511 (Linear Programming)Prof. Rhoda A. NamocoFebrurary 16, 2015
Problem SetGeneral Instructions:1. Solve the following transportation problems using NWC, MC and Vogel's methods.2. Formulate the following as transportation problems.3. Solve the resulting model using a computer solver.4. Email solver setup to [email protected] with subject "AMS 511 Trabsportation Problem".
1. A transportation problem involves the following costs, supply, and demand:To (cost)From12341$500$750$300$4502$650$800$400$6003$400$700$500$550Demand10101010
2. Given a transportation problem with the following costs, supply, and demand, find the optimal solution by using the computer:To (cost)From123SupplyA$6$7$4100B$5$3$6180C$8$5$7200Demand135175170480
3. Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno. These warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply, and demand, Because of an agreement between distributors, shipments are prohibited from Miami to Chicago:To (cost)FromNew YorkPhiladelphiaChicagoBostonTampa$9$14$12$17Miami$11$10$6$10Fresno$12$8$15$7Demand130170100150
#1Northwest Corner MethodLP MODEL:To (cost)OBJECTIVE FUNCTION:From1234Supplymin Z = 500X11 + 750X12 + 300X13 + 450X141$500$750$300$45012, 2, X+ 650X21 + 800X22 + 400X23 + 600X24102XX+ 400X31 + 700X32 + 500X33 + 550X34 $650$800$400$60017, 9, XX89Xsubject to3$400$700$500$55011, 10, XSUPPLY CONSTRAINTS:XX110X11 + X12 + X13 + X14 12Demand10, X10, 8, X10, 1, X10, XX21 + X22 + X23 + X24 17min Z=$22,500X31 + X32 + X33 + X34 11
Minimum Cost MethodDEMAND CONSTRAINTS:To (cost)X11 + X21 + X31 10From1234SupplyX12 + X22 + X32 101$500$750$300$45012, 2, XX13 + X23 + X33 10XX102X14 + X24 + X34 102$650$800$400$60017, 10, XX10X7NONNEGATIVITY CONSTRAINTS:3$400$700$500$55011, 1, XXij 0 (i=1,2,3; j= 1,2,3,4)10XX1Demand10, X10, X10, X10, 8, 7, Xmin Z=$20,650
Vogel's MethodTo (cost)From1234SupplyPenalty1$500$750$300$45012, 2, X15050300X10X22$650$800$400$60017, 7, X20050200XX1073$400$700$500$55011, 1, X10015015010XX1Demand10, X10, X10, X10, 8, 7, XPenalty100501001001005010050100min Z=$21,150
Using Excel SolverTo (cost)From1234Supply1$500$750$300$450122$650$800$400$600173$400$700$500$55011Demand1010101040
123410.00.0$2$1020.0$9$80.03$10$10.00.0
min Z=$20,200s.t.S11212S21717S31111D11010D21010D31010D41010#2Northwest Corner MethodTo (cost)LP MODEL:From123SupplyOBJECTIVE FUNCTION:1$6$7$4100, Xmin Z = 6X11 +7X12 + 4X13100XX+ 5X21 + 3X22 + 6X232$5$3$6180, 145, X+ 8X31 + 5X32 + 7X3335145X3$8$5$7200, 170, Xsubject toX30170SUPPLY CONSTRAINTS:Demand135, 35175, 30, X170, XX11 + X12 + X13 100min Z=$2,550X21 + X22 + X23 180X31 + X32 + X33 200Minimum Cost MethodTo (cost)DEMAND CONSTRAINTS:From123SupplyX11 + X21 + X31 1351$6$7$4100, XX12 + X22 + X32 175XX100X13 + X23 + X33 1702$5$3$6180, 5, X5175XNONNEGATIVITY CONSTRAINTS:3$8$5$7200, 130Xij 0 (i=1,2,3; j= 1,2,3)130X70Demand135, 130, X175, X170, 70, Xmin Z=$2,480
Vogel's MethodTo (cost)From123SupplyPenalty1$6$7$4100, X22XX1002$5$3$6180, 52115175X3$8$5$7200, 70, X211130X70Demand135, 130, X175, X170, 70, XPenalty1221231min Z=$2,480
Using Excel SolverTo (cost)From123SupplyA$6$7$4100B$5$3$6180C$8$5$7200Demand135175170
12310.00.0$1002$135$450.030.0$130$70
min Z=$2,350s.t.S1100100S2180180S3200200D1135135D2175175D3170170
#3Northwest Corner MethodLP MODEL:To (cost)OBJECTIVE FUNCTION:FromNew YorkPhiladelphiaChicagoBostonDummySupplymin Z = 9X11 + 14X12 + 12X13 + 17X14Tampa$9$14$12$170.0200, 70, X+ 11X21 + 10X22 + 6X23 + 10X2413070XXX+ 12X31 + 8X32 + 15X33 + 7X34 Miami$11$10M$100.0200, 100, XX100X100Xsubject toFresno$12$8$15$70.0200, 100, XSUPPLY CONSTRAINTS:XX1005050X11 + X12 + X13 + X14 +X15 200Demand130, X170, 100, X100, X150, 50, X50, XX21 + X22 + X23 + X24 + X25 200min Z=$5,000X31 + X32 + X33 + X34 + X35 200
Minimum Cost MethodDEMAND CONSTRAINTS:To (cost)X11 + X21 + X31 130FromNew YorkPhiladelphiaChicagoBostonDummySupplyX12 + X22 + X32 170Tampa$9$14$12$170.0200, 150, 50, XX13 + X23 + X33 10050X100X50X14 + X24 + X34 150Miami$11$10M$100.0200, 120, XX15 + X25 + X35 5080120XXXFresno$12$8$15$70.0200, 50, XNONNEGATIVITY CONSTRAINTS:X50X150XXij 0 (i=1,2,3; j= 1,2,3,4,5)Demand130, 80, X170, 120, X100, X150, X50, Xmin Z=$5,180OTHERS:X23 = 0Vogel's MethodTo (cost)FromNew YorkPhiladelphiaChicagoBostonDummySupplyPenaltyTampa$9$14$12$170.0200, 70, X932130X70XXMiami$11$10M$100.0200, 150, X101X150XX50Fresno$12$8$15$70.0200, 170, 20711X2030150XDemand130170, 150, X100, 30, X150, X50, XPenalty22330223323323min Z=$5,170
Using Excel SolverTo (cost)FromNew YorkPhiladelphiaChicagoBostonDummySupplyTampa$9$14$12$170.0200Miami$11$10$6$100.0200Fresno$12$8$15$70.0200Demand13017010015050
New YorkPhiladelphiaChicagoBostonDummyTampa$1000.0$1000.00.0Miami$30$1200.00.0$50Fresno0.0$500.0$1500.0
min Z=$5,080s.t.S1200200S2200200S3200200D1130130D2170170D3100100D4150150Dummy5050O100