OptimizationA Transportation Example

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NINE MILEM a n a g e m e n t C o n s u l t i n g

Elliott, G., Elliott, A. & Brar, H.

Copyright © 2012. All Rights Reserved. The Nine Mile Management Consulting Group

Table 1.1 Market 1 (M1)

Market 2 (M2)

Market 3 (M3)

Warehouse Supplies

Warehouse 1 (W1)

$2/ton (c11)

$1/ton (c12)

$2/ton (c13)

40 tons (s1)

Warehouse 2 (W2)

$9/ton (c21)

$4/ton (c22)

$7/ton (c23)

60 tons (s2)

Warehouse 3 (W3)

$1/ton (c31)

$2/ton (c32)

$9/ton (c33)

10 tons (s3)

Market Demands

40 tons (d1)

50 tons (d2)

20 tons (d3)

110 tons Total Demand

Total Supply

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Optimization: A Transportation Example

What are some of the reasons businesses end up failing? Is it because businesses have difficulty keeping up with rapidly evolving times – or is it something deeper? According to a recent Harvard Business Review article – “of 750 of the most significant U.S. business failures of the past quarter century… nearly half could have been avoided” (Carroll & Mui, 2008). From taking the wrong strategy, to consolidation, to staying the course even as conditions worsen – businesses in today’s economy have a unique set of challenges – and near the top of the list is transportation.

The transportation optimization example that follows exemplifies the many areas of optimization that can always occur within a business. The footnote here is that many small-to-medium sized businesses do not have the luxury of conducting complex analysis – either due to a lack of man-power or a lack of inherent knowledge. However, this does not mean that it cannot be done. And this brings us to the next topic of this article – math!

Whether its liked or not, the importance of math in today’s business environment cannot be underesti-mated. With more large-scale businesses (and medium too) moving towards the collection of vast amounts of sales, consumer, shipping and transportation information – this information is useless until processed. The term “Big Data” was coined to refer to the real-time collection and ultimate analysis of this data. At the end of the day, this means getting down to the nitty-gritty; creating programs, algorithms, and sequences to extract the right information, i.e. information that can be used in order to determine what business strategy to take.

The rest of this short article demonstrates the use of applied mathematical frameworks to support strategic decision-making practices within organizations.

According to studies carried out by several Ivy League American institutions, companies that utilize applied mathematical and statistical models to justify, optimize, and streamline decision-making processes can significantly improve the performance of operations related sub-activities within the business – and in this case, the optimization of transportation routes, i.e. whether to make only right-handed turns, how to optimize for traffic, length of routes, and gas mileage.

The transportation problem is a classical example of a linear programming problem whose solution is concerned with the optimization of product unit distribution from several points of origin to several destination points. This example also showcases how transportation optimization and strategy can help companies cut costs, maximize profits, and efficiency.

Example:

Suppose Company A is a manufacturer of bricks. Company A owns 3 warehouses and sells its product to 3 markets. The supply of each warehouse (si), the demand (dj) of each market, and the shipping costs (cij) per ton are presented in the matrix below (Table 1.1).

Question?

How should Company A transport bricks in order to minimize overall transportation cost? Thankfully advanced linear programming techniques allow us to answer this problem in a reasonable fashion.

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Answer

Note: Before we proceed it should be made quite clear that the purpose of this article is to merely showcase the importance of utilizing mathematical frameworks to solve complex business problems, not to provide rigorous solutions. Instead, the algorithm will be partitioned into segments, where each part will highlight the algorithms' function as well as what it is trying to reconcile. However, all rigor will be omitted in an effort to negate confusion experienced by the reader.

The question explicitly outlines the objective, which is to minimize the overall transportation cost function:

The above summation can be expressed in the following form:

Of course the cost function itself is subject to several supply as well as demand constraints. Table 1.1 indicates that the supply of all 3 warehouses is 110 tons, and that the total market demand for bricks is 110 tons. In order to satisfy market demand each warehouse must exhaust its existing supply of brick. In addition, the product being shipped from Warehouse "i" must equal (not exceed) the demand of Market "j". Therefore, the following equality constraints are as follows:

VAM calculates the lowest feasible transportation cost while simultaneously satisfying market demand, and warehouse supply constraints. Imagine utilizing a completely different transportation structure. Such a transition, according to any feasible estimation, would translate into tens of thousands of dollars in inefficiency related losses, and this is only one operating area of one's business. Thousands of companies across Canada go bankrupt because they simply don't understand the value of optimizing internal process related inefficiencies. Growing economies of scale can decrease your per unit cost, but the journey to achieving efficiency can be much costlier, unless you receive the proper guidance. Let Nine Mile Management Consulting work with you so that you can enact strategies that will yield the sustainable long and short term growth you're looking for.

Carroll, P. B. & C. Mui. (2008). Seven Ways to Fail Big. Harvard Business Review - The Magazine, September. J. K. Strayer. Linear Porgramming and its Applications. New York: Springer-Verlag New York Inc., pp. 141-150.

Formally the problem asks us to: Minimize Transportation Cost = ∑ cij ∑ xij Define variables as follows:

cij represents the unit shipping cost from Warehouse "i" to Market "j" ie c11=$2 xij represents the # of units (in tons) to be shipped from Warehouse "i" to Market "j" dj for j=1,...,n represents the demand corresponding to Market "j" si for i=1,...,m represents the supply of Warehouse "i"

C = ∑ cij ∑ xij

C = 2x11 + x12 + 2x13 + 9x21 + 4x22 + 7x23 + x31 + 2x32 + 9x33

x11 + x12 + x13 = 40 (Warehouse 1 supply constraint) x21 + x22 + x23 = 60 (Warehouse 2 supply constraint) x31 + x32 + x33 = 10 (Warehouse 3 supply constraint) x11 + x21 + x31 = 40 (Market 1 demand constraint) x12 + x22 + x32 = 50 (Market 2 demand constraint) x13 + x23 + x33 = 20 (Market 3 demand constraint)

Where xij≥0 for all combinations of i,j Another condition must be met: Total Supply=Total Demand=110

Once all parameters and conditions have been properly outlined, the Vogel Advanced-Start Method (VAM) for solving "Balanced Transportation Problems" can be applied. VAM application yields the following feasible & optimal solution. Note - all steps have been omitted for the sake of brevity: x11 = 20, x12 = 0, x13 = 20, x21 = 10, x22 = 50, x23 = 0, x31 = 10, x32 = 0, x33 = 0 YIELDS Optimized Cost Function Below: Costoptimized = $2(20) + $2(20) + $9(10) + $4(50) + $1(10) = $380

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