transportation
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
TRANSPORTATION
Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137
CHAPTER 7
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
Chapter 6 Organization
• A brief history of the transportation problem• An engineering modeling approach to problem-
solving using mathematics• Modeling and engineering• A transportation problem from an engineering
modeling perspective• 7.1 Example Transportation Problem• 7.2 Integer Solutions• 7.4 Validating Models• 7.5 Sensitivity Analysis
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
• We need to transport oranges from California (CA) and Florida (FL) to Michigan (MI), Missouri (MO), and North Carolina (NC)
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
• Supply of oranges▪ CA – no more than 50.3 tons of oranges▪ FL – no more than 49.7 tons of oranges
• Demand of oranges▪ MI – at least 33.6 tons of oranges▪ MO – at least 36.3 tons of oranges▪ NC – at least 30.1 tons of oranges
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
• Since the oranges are shipped by truck, costs to consider include
• Shipping costs per tons of oranges
▪ diesel fuel ▪ wear and tear
▪drivers’ wage ▪ insurance
From/To MI MO NC
CA $10/ton $9/ton $12/ton
FL $8/ton $8/ton $6/ton
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
CA(50.3)
$12×NofTCA, NC
$6×NofTFL, NC
$8×NofTFL, MO
$9×NofTCA, MO
$10×NofTCA, MI
Shipping CostsNofT => Number of Tons
Demand in tonsSupply in tons
$9×NofTFL, MI
FL(49.7)
MI(33.6)
MO(36.3) NC
(30.1)
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
CA(50.3)
$12×NofTCA, NC
$6×NofTFL, NC$8×NofTFL, MO
$9×NofTCA, MO
$10×NofTCA, MI
Shipping Costs NofT => Number of Tons
Demand in tonsSupply in tons
$9×NofTFL, MI
FL(49.7)
MI(33.6)
MO(36.3) NC
(30.1)
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 Example Transportation Problem
CA(50.3)
$12×NofTCA, NC
$6×NofTFL, NC
$8×NofTFL, MO
$9×NofTCA, MO
$10×NofTCA, MI
Shipping CostsDemand in tons
Supply in tons
$9×NofTFL, MI
FL(49.7)
MI(33.6)
MO(36.3)
NC(30.1)
NofT => Number of Tons
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – LP Formulation
• Define ▪ Decision Variables, Objective Function,
Constraints• Create Excel Spreadsheet
▪ Input decision variables, objective function, constraints, RHS constraint values
• Use Solver to find the solution▪ Output final values, objective function value,
LHS constraint values• Analyze results using Answer and Sensitivity Reports
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Decision Variables
• Let xij be the number of tons of oranges shipped from supply state i to demand state j
▪ i = 1 (CA) or 2 (FL)▪ j = 1 (MI), 2 (MO), 3 (NC)
• Much better than the notation NofTCA, MI
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Objective Function
• Total Shipping Costs per route
• Minimize Shipping Cost
▪ z = $10x11 + $9x12 + $12x13 +
$9x21 + $8x22 + $6x23
From/To MI MO NC
CA $10x11 $9x12 $12x13
FL $9x21 $8x22 $6x23
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Supply Constraints
• CA and FL have a limited number of tons of oranges to distribute to demand states
▪ CA has a maximum supply of 50.3 tons
x11 + x12 + x13 ≤ 50.3 tons
▪ FL has maximum supply of 49.7 tons
x21 + x22 + x23 ≤ 49.7 tons
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Demand Constraints
• MI, MO, and NC have a minimum number of tons of oranges to receive from the supply states
▪ MI has minimum demand of 33.6 tons
x11 + x21 ≥ 33.6 tons
▪ MO has minimum demand of 36.3 tons
x12 + x22 ≥ 36.3 tons
▪ NC has minimum demand of 30.1 tons
x13 + x23 ≥ 30.1 tons
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – NonNegativity Constraints
• Cannot have a negative amount of oranges
▪ xij ≥ 0 where i = 1, 2 and j = 1, 2, 3
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Excel Formulas
= SUMPRODUCT($B$7:$G$7,B9:G9)
= SUMPRODUCT($B$7:$G$7,B12:G12)
= SUMPRODUCT($B$7:$G$7,B16:G16)
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Excel Solver
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.1 EX Transportation Problem – Solution
• The minimum cost for shipping 100 tons of oranges from the supply states to the demand states is $823.70
• More specifically
From/To MI MO NC
CA 14 tons 36.3 tons 0 tons
FL 19.6 tons 0 tons 30.1 tons
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.2 Integer Solutions
• What if we wanted our number in tons to be an integer value?
• How would our LP formulation change?
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.2 Integer Solutions - Objective Function
• Total Shipping Costs per route
• Minimize Shipping Cost
▪ z = $10x11 + $7x12 + $15x13
+ $8x21 + $10x22 + $5x23
From/To MI MO NC
CA $10x11 $7x12 $15x13
FL $8x21 $10x22 $5x23
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.2 Integer Solutions - Solution
• The minimum cost for shipping 100 tons of oranges from the supply states to the demand states is $790
• More specifically
From/To MI MO NC
CA 10 tons 7 tons 15 tons
FL 8 tons 10 tons 5 tons
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
7.4 Validate Our Model
• Use data from government websites to validate the shipping costs per ton
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
In Summary
• Transportation problems are specific types of linear programming problems that address moving products from supply to demand
• The problems in chapter 7 are all balanced transportation problems
MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
www.MINDSETproject.org
Practicing Transportation
• Optional – Read pp. 1 – 5 for background information
• Work on problems 7.17 – 7.23 (pp. 6-10 – 6-12), Cape Fear River Basin Water Supply Plan, individually or in pairs
▪ In the HW solutions, the answer to 7.21 is incorrect. The optimal solution should be $32,000 - not $36,000.
▪ When solving these problems, make sure to increase your precision (add more zeros)
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