Transshipment Problem Solving With Linear Programming Essay Example For College

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.

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:

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

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.

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.

Kierkegaard’s Philosophy In “Fear And Trembling”

Thesis: Kierkegaard claims about absolute choice, which on being a realization of freedom, means a choice of not this or that, but self in the eternal meaning.

In Fear and Trembling Kierkegaard wrote: “If faith cannot make it a holy act to be willing to murder his son, then let the same judgment be passed on Abraham as on everyone else. If a person lacks the courage to think his thought all the way through and say that Abraham was a murderer, then it is certainly better to attain this courage than to waste time on unmerited eulogies. The ethical expression for what Abraham did is that he meant to murder Isaac; the religious expression is that he meant to sacrifice Isaac-but precisely in this contradiction is the anxiety that can make a person sleepless, and yet without this anxiety, Abraham is not who he is” (pg. 30).

To Kierkegaard’s opinion, there are three basic types of existence, or three stages of human existence: aesthetic, ethical, and religious, and they go in such sequence so first two types as a matter of fact act as preliminary stages on a way to religious existence. The focus of the third, religious stage is an instant of the “leap of faith”, which opens the true sense of existence consisting in the absolute relation to God, i.e. paradoxical contact of time and eternal, that in its turn is the existential repetition of the absolute Paradox: existing (= temporal) and eternal when the God existed in an image of a person. Davenport et al. state: “in a sense, the Christian life is essentially natural, for although life can only be understood retrospectively, it has to be lived forwards. Either way, it is a matter of relating the temporal to the eternal, but whilst

“[t]he speculative principle is that I arrive at the eternal retrogressively… an existing individual can have a relationship to the eternal only as something perspective, as something in the future” (CUP 380);

hence the need to relate to the eternal through the repetition of the encounter with the eternal-in-time rather than through time itself, in the philosophical or historical investigation”1.

Emphasizing on the personal character of God-relation, Kierkegaard rejects the mediated communication with God, recognizing absolute inexpressiveness of experience of belief, acting in that way as a successor of that line in interpretation of Christianity, which goes from epistles of Apostle Paul, through the philosophy of Tertullian, Augustine, medieval mysticism and Pascal to well-known “Sola Fide” of Luther. Any existential experience finds, in Kierkegaard’s opinion, true sense and concerns to the sphere of true existence so far as it promotes comprehension by the person of the religious value of own individuality (as opposed to non-true existence, connected with the dispersion of subjectivity and hence, taking away from the God).

Special attention thus Kierkegaard pay to the fear connected with the experience of the person of own existence as life “face to face with death”2, and also to despair as to an initial point for the achievement of absolute. Existence, according to Kierkegaard, demands constant spiritual pressure and suffering (in particular at a religious stage). The basic existential concepts, called to describe non-cognizable and inconceivable in its secret existence, are not deduced consistently one of another, but mutually conditioned in such a manner that each concept already encloses all the others.

The true existence has ethic-individual character. Thus an individual as the concrete according to Kierkegaard acts as a condition of realization of ethical as universal, i.e. has ethical (debt) not outside of self, but inside of self. The ethical essence of existence concentrates on the concept of choice. Kierkegaard claims about absolute choice, which is the realization of freedom, means a choice of not this or that, but self in the eternal meaning. “The notion of choice in Kierkegaard cannot be separated from his notion of ‘freedom’, for it is the freedom of the existing individual to make choices, together with the demand that he makes choices, which define the ‘paradoxical’ nature of ethical existence, that is, the continuous confrontation of the individual with alternative and exclusive possibilities of action”3.

Ethical existence has a set of quite steady, reliable correlations with the world. Reality challenges individuals, reality constantly rises before them as a train of tasks. Solving these tasks, an ethicist proceeds from moral continuity of own “self”. Any choice is an additional step in the construction of a personality. However, unlike an aesthete for whose own personality breaks up to a number of potential opportunities, an ethicist considers that realization of self is a task, which is quite realizable. We choose our own self quite freely, however, the concrete targets, which create a selection field, are put forward by circumstances of life. Kierkegaard claims:

“A person who aesthetically considers a whole range of life-tasks… is more likely to arrive at a multiplicity than an either/or, because here the factor of self-determination in the choice is not given an ethical emphasis, and because, if one does not choose absolutely, one chooses for that moment only and can, for that reason, choose something else the next instant. The ethical choice is therefore in a certain sense far easier, far simpler, but in another sense infinitely more difficult”4.

For Kierkegaard ethical existence as a whole is more worthy than an aesthetic one. Ethical existence is regarded as a state of maturity, responsibility, ability to make the riskiest and difficult decisions. Perhaps the only thing that it is necessary to leave at the previous stage is creativity, spontaneous impulse inducing us to art creativity.

But even consistently conducted ethical direction cannot relieve a person of despair. We can be as much as moral, we can persistently see to it that every instant of choice has been ethically colored, and however even the highest principles cannot relieve a person of fear of death, of comprehension of instability of own being. Nevertheless, individuals, who already have learned to choose resolutely, are already able to take the next step bringing them into the sphere of religion.

At the religious stage, the opportunity of overcoming despair appears connected with absolutely doubtful and absurd from the reasonable point of view acceptance of being of God. Kierkegaard quite realizes that the nature of Christian religiousness is all the way through paradoxical: human’s self only then recovers and may release from despair when it, by virtue of all of the same despair, begins to see self in God.

Footnotes

  1. Davenport, John J., Anthony Rudd, Alasdair C. MacIntyre, and Philip L. Quinn. Kierkegaard After MacIntyre Essays on Freedom, Narrative, and Virtue. Chicago: Open Court, 2001.
  2. In Fragments, Climacus confides that death is his lovely dancing partner, the partner, we might say, who gives him life. And in a story from Stages on LifeWay, a glance of recognition puts a walker face to face with death, which paradoxically starts him on toward life.
  3. Solomon, Robert C. From Rationalism to Existentialism; The Existentialists and Their Nineteenth-Century Backgrounds. New York: Harper & Row, 1972.
  4. Kierkegaard, Søren, and V. Eremita. Either/or. A Fragment of Life. Garden City: [s.n.], 1959.

The Golden Arches Theory Of Conflict Prevention

Introduction

The chapter “The Golden Arches Theory of Conflict Prevention” of “The Lexus and The Olive Tree” by Thomas Friedman is devoted to a unique and rather insightful finding. This finding is exclusively the author’s own and has been challenged by some. While McDonald’s is one of the restaurants around the world, the author was hit by the realization that no two countries with McDonald’s have gone to war after getting McDonald’s in their country.

Influences of globalization

The author is not only highlighting the influence of McDonald’s but is focusing on the economic side of war. When countries know that they have an economic stake in a country, they rarely go to war with that country. The author cites several examples of how countries after getting McDonald’s did not go to war with each other. Friedman explains: “It both increases the incentives for not making war and it increases the costs of going to war in more ways than in any previous era in modern history.” It’s not liked those countries will never fight each other, but now if they do, they have more to lose than before. This is the basic argument in the chapter and one that has been supported by ample examples.

The author clearly understands that his theory has, “a limited shelf life” which means that “sooner or later virtually every country would have McDonald’s, and sooner or later two of them would go to war.” But now the war will not only cost more, but it will also be rather damaging to the world on the whole.

Another important argument in the chapter is that regardless of how rapidly globalization is taking place, it is unable to change geopolitics. In other words, while the world may have become a village, people are still sentimental about their own culture and countries. Now the question is: what has globalization done then if it has not removed boundaries. The author replies that globalization may have been unable to alter geopolitics but it has certainly affected it. In other words, while people were ready to go to war before at the drop of a hat if they felt their country was in danger, the same is not true anymore. People will think twice before waging a war because now governments have a lot to lose economics-wise.

Conclusion

After presenting these important arguments, the author moves back to his main premise of the book i.e., how technology and boundary-less behavior can help countries succeed. He makes it clear that when countries open their doors to new brains, it automatically makes them self-reliant and strong. In probably one of the most important lines in the chapter, the author says, “…if you close your country off in any way to either the best brains in the world or the best technologies in the world, you will fall behind faster and faster.

That’s why the most open-minded, tolerant, creative, and diverse societies will have the easiest time with globalization, while the most closed, rigid, uptight, self-absorbed, and traditional companies and countries, which are just not comfortable with openness will struggle.” The whole idea seems to make sense because even we are witness to the fact that countries, where people are allowed to enter and use their brains freely, are the ones who prosper the most.

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