# Transshipment Problem Solving with Linear Programming

## Abstract

This term paper showcases the Application of Quantitative Methods for the exploration and analysis of a transshipment problem through a linear programming model for Lij Systems. An electronic company would like to determine an optimal transshipment plan that minimizes total transportation cost while meeting demands in each retail outlet and not exceeding the capacity at each production facility.

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The linear model reflects the production capacity at each of the 3 facilities; Atlanta, Boston, and Chicago, transportation cost per unit going to their regional warehouses at Edison and Fargo, and costs per unit in transportation to their supply retail outlets located in Houston, Indianapolis, and Jacksonville.

The researcher formulates the transshipment problem as a linear programming model and determines the optimal transportation costs from the production facilities through the warehouses and from the warehouses to the supply retail outlets taking into account the units per production costs for transportation.

The paper uses an analysis of polynomial equations arising from the maximum production capability at Lij system facilities, cost per unit of transportation to the regional warehouses, cost per unit from the warehouses to the supply retail outlets, and demand at supply retail outlets.

The research identifies the minimal transportation cost while meeting the demand in each retail outlet while not also exceeding the production capacity at each facility.

## Introduction

Integer programming, which is a quantitative technique for optimizing some objective subject to certain constraints, is quite useful in solving some problems. Capital rationing problems represent situations of constrained maximization since the objective is, to select the group of projects that maximizes cash inflows subject to a budget (or financial) constraint. Integer programming is used instead of linear programming so that the results will all be in terms of whole projects. It would be quite difficult to implement 6 of a project. Computer programs are available for solving integer programming problems.

The basic integer programming problem can be stated as follows:

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Maximize b1x1 + b2x2+……. + bnxn

When

C1x1+c2x2+……. +cnxn< c x 1 =0, 1(for all i=1,n)

Where

Bi (for i =1,n) =the present value of the cash inflows

Xi (for i = 1,n) = a decision variable which can have a value of either 0 or 1 depending on whether the project is accepted (if xi = 1) or rejected (if xi =0),

Ci (for = 1,n) = the net investment required for project i

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C = the cost constraint, and

n = the number of projects considered.

Using certain integer programming algorithms, the acceptable projects (those for which, xi = 1) can be determined.

Lij Systems has commissioned a research task to determine the optimal transportation costs from their production facilities to their regional warehouses and from their regional warehouses to their supply retail outlets ensuring that the demand at their supply retail outlets is met within their production constraints. This is to be achieved by an analysis of the polynomial equations arising out of the variables in the problem. The variables will arise out of cost per unit of transportation, production capacity, and demand. To achieve this and present it as a linear programming model, the researcher formulates the polynomial equations and determines the minimum possible transportation cost and the maximum possible transportation cost from the Costs of Transportation per unit that Lij Systems currently experience. The minimum and maximum costs are calculated from the production facilities to the warehouses and from the warehouses to the supply retail outlets.

Lij Systems has three production facilities located in Atlanta, Boston, and Chicago with a production capacity of 800, 500, and 700 respectively. The firm has to meet demands at each of their supply retail outlets in Houston, Indianapolis, and Jacksonville of 900, 600, and 500 respectively.

The following are transportation costs from the production facilities to the warehouse; The transportation cost per unit from Atlanta to Edison is 6, from Boston to Edison is 1 and from Chicago to Edison is 3 respectively. The transportation per unit from Atlanta to Fargo is 4, from Boston to Fargo is 8 and from Chicago to Fargo is 1.

The following are transportation costs from the warehouse to the supply retail outlets;

The transportation cost per unit from Edison to Houston is 8, from Edison to Indianapolis is 3 and from Edison to Jacksonville is 4 respectively. The transportation cost from Fargo to Houston is 2, from Fargo to Indianapolis is 3 and from Fargo to Jacksonville is 8. These costs per unit are to be used within the polynomial equations to determine the lowest and highest possible values of transportation that can be achieved within the constraints of production and distribution.

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## The problem Statement

The research task is to be bounded by the production capability of Lij Systems, and the supply retail outlets demand. The problem calls for an analysis of transportation costs to these two destinations, warehouses, and supply outlets while taking into account production at the facilities and demand at the outlets. A solution, presented as a linear programming model is to be designed that seeks to find the optimal transshipment plan that minimizes total transportation cost while meeting the demands in each retail outlet and not exceeding the capacity at each production facility.

## The Goal of the Solution

The solution requires a formulation of a solution as a linear programming model paradigm which determines the optimal transshipment plan. The solution is to take into account the production capability of Lij Systems, account for optimal transport costs, and reflect the demands at the supply retail outlets owned by the company. The solution aims to transform the polynomial equations into a linear programming model.

## Prior Research

In previous research located in Microsoft Student Encarta 2007, linear programming systems are outlined as series of equations representing the model problem to be solved as variables within the relative maximum values of production and demand.

In the text on Quantitative Methods from an Internet search on Linear programming models and Quantitative methods, the problem is presented as a series of polynomial equations of the different sets of conditions to be accounted for.

## The Methodology

The problem is broken down from a Casual flow chart into a Linear programming model. The casual flow chart represents production facilities capacity, costs per unit of transportation, Warehouse, and supply retail outlets demand.

Polynomial equations are derived out of the flow chart to represent the relationships between production capacities, cost per unit in transportation, and supply retail outlet demands.

## Definitions of LP Variables

Variables will be representing production capacity, transportation to the warehouse, and transportation from warehouse to supply retail outlets.

A1 is Chicago to Fargo route

A2 is Boston to Fargo route

A3 is Atlanta to Fargo route

B1 is Chicago to Edison route

B2 is Boston to Edison route

B3 is Atlanta to Edison route

C1 is Fargo to Jacksonville route

C2 is Fargo to Indianapolis route

C3 is Fargo to Houston route

D1 is Edison to Jacksonville route

D2 is Edison to Indianapolis route

D3 is Edison to Houston route

Atlanta total production capacity 800

Boston total production capacity 500

Chicago total production capacity 700

Houston total demand is 900

Indianapolis total demand is 600

Jacksonville total demand is 500

## The Objective Function

• To establish the most optimal transportation plan that utilizes the production capacity at each facility while meeting the supply retail demands.
• Movement of goods from the production facility to warehouses
• Movement of goods from warehouses to supply retail outlets
• Installing a linear programming model in the transportation of good

## The Constraints

The first constraint is production capacity about cost per unit in transportation.

6b3+4a3=800

b2+4a2=500

3b1+a1=700

The second constraint is maximum demand at supply retail outlets that may require a higher

Cost per unit in transportation to meet demand.

8d3+2c3<=900.

3d2+3c2<=600.

4d1+8c1<=500.

## The Results

Maximum possible shipment from Atlanta to Edison b3 + a3 = 800.

Maximum possible shipment from Boston to Edison b2 +a2 = 500.

Maximum possible shipment from Chicago to Edison b1 + a1 = 700.

Maximum demand in Houston d3 +c3 = 900.

Maximum demand in Indianapolis is d2+ c2 = 600.

Maximum demand in Jacksonville is d1 + c1 = 500.

## Linear Programming

12 needed variables to be used:

• A1 = Chicago to Fargo route ((a1*800)-(b1*800)).
• A2 = Boston to Fargo route ((a2*500-(b2*700)).
• A3 = Atlanta to Fargo route ((a3*700)-(b3*700)).
• B1 = Chicago to Edison route ((b1*800)-(a1*800)).
• B2 = Boston to Edison route ((b2*500)-(a2*500)).
• B3 = Atlanta to Edison ((b3*500)-(b2*500)).
• C1 = Fargo to Jacksonville ((c1*500)-(c2*500)-(c3*500)).
• C2 = Fargo to Indianapolis is ((c2*600)-(c1*600)-(c3*600)).
• C3 = Fargo to Houston ((c3*900)-(c1*900)-(c2*900).
• D1 = Edison to Jacksonville ((d1*500)-(d2*500)-(d3*500)).
• D2 = Edison to Indianapolis is ((d2*600)-(d1*600)-(d3*600)).
• D3 = Edison to Houston ((d3*900)-(d1*900)-(d2*900).

## Managerial interpretation of Results

The following equations represent all the variables presented within their relationships to transport to warehouses and production capability, transportation costs from the warehouses to the retail outlets, and their relationship to transportation unit cost within the points between the warehouses and the retail outlets. The minimal cost per unit of transportation cost is represented below with the variable values found in the introductory paragraph.

• b3 + a3 <= 800.
• b2 +a2 <= 500.
• b1 + a1 <= 700.
• d3 +c3 <= 900.
• d2+ c2 <= 600.
• d1 + c1< = 500.

The section below represents the individual costs to be isolated by substituting the units costs into the equations to derive the least costly form of transportation and the maximum possible cost of transportation for each route from production facilities to retail outlet points.

• Chicago to Fargo route ((a1*800)-(b1*800)).
• Boston to Fargo route ((a2*500-(b2*700)).
• Atlanta to Fargo route ((a3*700)-(b3*700)).
• Chicago to Edison route ((b1*800)-(a1*800)).
• Boston to Edison route ((b2*500)-(a2*500)).
• Atlanta to Edison ((b3*500)-(b2*500)).
• Fargo to Jacksonville ((c1*500)-(c2*500)-(c3*500)).
• Fargo to Indianapolis is ((c2*600)-(c1*600)-(c3*600)).
• Fargo to Houston ((c3*900)-(c1*900)-(c2*900).
• Edison to Jacksonville ((d1*500)-(d2*500)-(d3*500).
• Edison to Indianapolis is ((d2*600)-(d1*600)-(d3*600)).
• Edison to Houston ((d3*900)-(d1*900)-(d2*900).

## Conclusion

When the cost per unit for transportation variables are substituted in the equations making up the linear programming model presented above, an optimal transshipment that minimizes total transportation while meeting the demand in each retail outlet not exceeding the capacity at each production facility is achieved. The values for cost per unit of transportation from the point of production and the values for cost per unit of transportation from the warehouses are substituted to yield within the polynomial equations the desired maximum transportation cost that still falls within the bounds of production ceiling and maximum demand.

Jacksonville presents the most maximal transportation route from the warehouses at Edison and Fargo while Jacksonville presents the most expensive transportation line from the two warehouses. Indianapolis comes second inefficiency.

On the other hand, production facilities at Atlanta and Chicago are the most robust with Atlanta leading on transportation cost per unit of the three facilities. This production capability is followed by facilities at Chicago and Boston.

## References

Bernhard, Richard D., and George L. Castler,(1973); capital Investment Analysis Columbus, Ohio: Grid,.

Fogler, H.Russell, (1972); “ Ranking Techniques and capital rationing.” pp.134 -143.

Russell, B (1987); Qualitative and Quantitative methods.

Schwab, Bernard, and peter rusting; (1969); A comparative analysis of the net present value and the benefit- cost value as a measure of the economic desirability of investments, journal of Finance.

Teichroew, Daniel, Alexander A. Robichek, and Michael Monalbaano,(1969); analysis for criterion Investment and financing decisions under certainty”, management science.

Weigather H. Maartin (1963);. Mathematical programming and the analysis of capital budgeting problems, (Englewood Cliffs. N.J:Prenticel hall.

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