Networks: A Management Science Solutions Approach

Background

The term 'networks' is widely used in contexts such as roads, telephone systems, electricity distribution systems, towns / cities in a region or state, manufacturing facilities, supermarkets, warehouses or outlets across some geographical area to mention but a few. Our example will draw on a network of towns across Uganda with a view to find a cost-effective delivery of goods, since the 'end-user' price tag also includes the cost of shipment, which depending on circumstances, can be a big percentage.

Transport cost components can be significant, with such examples to be found in provision of lake sand for construction. While the sand itself, its 'mining' and loading on a delivery truck may be low, the cost of transport can be significant.

In our example of in the coffee industry, transportation is a major aspect to reckon with. Farmers need to move plants from 'nurseries' for planting, tools and implements need to be brought to the farm, dried coffee needs to be taken to coffee factories, and 'beans' and the 'instant coffee' ready for export need to be transported thousands of miles to respective markets.

Transportation is a major activity in many business operations, be it for the public to commute to work; school textbooks to be distributed across the country; foods and beverages to be distributed to supermarkets and shops countrywide; raw materials to manufacturing facilities or finished goods to market outlets.

Network Contexts

These can be classified as:

Such situations can be tackled by a general solution approach known as the 'simplex method', or better still, by specially designed 'transportation method' or its variants.

Supplying to Meet Demand Optimally

'Optimisation' is derived from the word 'optimum', which is defined in an 'adjective' context as 'most conducive to a favourable outcome' in the New Oxford English dictionary.

Demand can be met from 'inventories' of available supplies in store, carried forward from a previous period, and, manufactured goods in the period. At the end of a period, any surplus goods can then be added to inventories, if any, in stores for future periods. A good example is the fuel in a car tank, where one may drive in to 'top-up' for a journey, at the end of which there may be surplus fuel. In this context, fuel in the tank can be viewed as 'opening inventory' that bought as 'supplied', and that remaining at the end of the journey and 'closing inventory'.

The stage is now set for our example, set in Uganda, with the centres listed below constituting interconnected 'nodes' in our network via our existing road network. It is going to be assumed that any deficits may be met by imports ordered at the 'right' time, and any surplus is carried forward as inventory into subsequent periods. The data for the example may be presented by the tables below:

Supply/Demand Matrix

There is a frequent term used, namely, 'feasibility', requiring that it is important to have validity in assumptions made. In this case, transportation problems require that 'demand can be met from inventories at hand, production during the period, and, imports if need be'. These three categories are lumped under supply.

In the figure above, any of the three supply points can contribute to meeting demand of any of the five nodes. For each combination of 'supply and demand' in the 15 (3 rows x 5 columns) box table above, there is an associated 'shipping' or 'transportation' cost, with the 8 nodes (3 supply + 5 demand) possibly being any of the twelve (12) cited in the earlier table above.

We shall assume that the cost of manufacture for any good at any supply point is the same, and that total supply is equal to total demand. We therefore only need to look at shipping cost as the cost of manufacture is 'independent', with the objective of 'minimising the total shipping cost'.

We will put some figures in the table, with 'unit shipping cost' from a supply point to a demand node being designated as in the table below:

In conventional solution methods, costs are entered in a small box in the 'top left hand' corner, with other corners used for other purposes. Also, supplies and demands use 'counters' 'i' and 'j' respectively.

Solution Approach

The solution approach uses a repetitive or 'iterative' method, starting with an 'initial feasible' solution, designed by some 'convention', and, proceeds with checks to see if there are indications of potential improvement in the objective function, until the 'best' solution is found. The said solution approach is applied to the example above hereinafter:

Note: Technical 'jargon' may be ignored, and, solution methods 'need' not be of concern as they can always be undertaken for your business.

Iteration 1: Initial Basic Feasible Solution (IBFS)

IBFS can be determined using the 'North-West Corner' Rule.

Cost of Starting Solution =(6x8+11x6+13x5+7x5+8x1+14x9+3x14) = 390

Total cost is sum of all 'route costs', listed above as (route cost x route flow). In this example, the 5 demands are 8, 11, 6, 9 and 14, while the 3 supplies are 14, 10 and 24, whose totals are equal (48). Applying the solution method further, gives the following route flows, with corresponding total costs shown:

Iteration 2: Total Cost = 335

Iteration 3: Total Cost = 271

Iteration 4: Total Cost = 247

Iteration 5: Total Cost = 231

Iteration 6: Total Cost = 215

Iteration 7: Total Cost = 212

Notes:

Such a solution approach can be applied to several situations, sometimes in unique ways, including but not limited to the following:

Potential Savings

It should be said that, as the cost can be progressively reduced in this simple example, starting from 390, down through 335, 271, 247, 231 and 215 up to 212, representing a 46% reduction, so can the impact be as great for large operations in manufacturing. Yet, typical problem sizes can be for hundreds of variables, sizes that could not be tackled in the old styles of work without 'risking' incurring substantial losses.

Examples can be found for capital budgeting in any organisation, transportation for such big companies as those in beverages, agriculture, transport and several others.

Solutions can be of great benefit for all, irrespective of size of company, nature and scope of work, or other considerations. Currently, these are areas that are not being taken advantage of, to great loss of many enterprises.

Variation to Multi-Period Model

This model can be extended to cover multiple periods, ie one can look at many consecutive periods, incorporating the capacity to consider inventories 'brought forward' from previous periods, or those 'carried forward' to meet demands in subsequent periods. This is also known as 'dynamic inventory models', incorporating production plans, along with carrying inventories, a typical situation in a business concern.