transportation

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



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



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)




• 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




• 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




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)




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)




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



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



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



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



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



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



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



7.1 EX Transportation Problem – Excel Formulas

= SUMPRODUCT($B$7:$G$7,B9:G9)





7.1 EX Transportation Problem – Excel Solver



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



7.2 Integer Solutions

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

• How would our LP formulation change?



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



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



7.4 Validate Our Model

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



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



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)

transportation

Documents

decision science

engineering tools

tons of orangesnc

tons of orangesmo

tons of orangesdemand

tons of orangesfl

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