transportation

MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)

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TRANSPORTATION

Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137

CHAPTER 7

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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

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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)

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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

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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

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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)

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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)

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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

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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

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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

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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

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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

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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

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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

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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)

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7.1 EX Transportation Problem – Excel Solver

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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

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7.2 Integer Solutions

• What if we wanted our number in tons to be an integer value?

• How would our LP formulation change?

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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

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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

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7.4 Validate Our Model

• Use data from government websites to validate the shipping costs per ton

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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

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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)