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MB0048 Q3. Explain how the profit maximization transportation problem can be converted to an equivalent cost minimization transportation problem.

Answer:The transportation problem in terms of minimization problems. But the situations may arise where we need to solve the transportation problem to maximize the profit and we have profit units given in the problem instead of unit cost of routes. This situations handled by converting the maximization problem into minimization problem by replacing each element of the transportation table by its difference from the maximum element of the transportation table. The transportation problems, with minimum cost as the
objective, and the traffic light control problem, with maximum flow per unit time as the objective, illustrate classical network flow problems. The best known flow-type problem in involves the flow of cash or funds between sub organizations of an firm and between the firm and other sources or uses of funds. For example, Crum and Nye designed general network flow models for three operations of a multiple-line property casualty firm: portfolio operations, investment portfolio operations, and the capital acquisition operations. We introduce the first network model and refer readers to Crum and Nye for the other two case studies. In network flow models, there are four types of nodes: the nodes representing the cash balance, the nodes designating the lines of business, the nodes representing existing claims, and the nodes representing new claims. A network model of the portfolio of an insurance company with two lines of business and spanning three time periods. The objective function is expressed to maximize the value of the firm after all incremental capital acquired by the model has been repaid. This can be shown to be equivalent to maximizing the value of the existing equity the appropriate objective for a public corporation. The investment portfolio problem can be similarly formulated as a minimum cost flow problem. Combined with the investment portfolio network, the whole network model represents flow of funds over time for the multiple line company. Given the maximum period premium volume, the initial asset amount, the investment choices, and some other parameters, the network flow model is able to find the optimal portfolios for the company. However, the decision of a major company depends upon how the company acquires external capital such the capital structure (for example, debt-equity composition) and how well the company manages its assets and liabilities. Hence, Crum and Nye used network models to formulate the capital acquisition problem and investment portfolio problem in addition to the above cited model for the portfolio problem. These three models combined with the objective function form the complete model for a major company. Such an insurer can either be a multilane property and casualty company or a life company including life products and annuities. To further illustrate the usefulness of network flow models in finance and , we briefly discuss how network optimization might be used to find arbitrage opportunities in currency exchanges for a multinational company. Suppose an American company is the target company and dollar is the target currency; that is, the company has an amount of excess cash that can be used for currency exchange. There exists a set of foreign currencies available for trading. A network model for this problem, as usual, consists of a set of nodes and a set of arcs. The nodes represent currencies, one node for each currency. The arcs represent the possible exchange between the two currencies with an exchange rate attached to the arc. The problem is simply to increment the amount of dollars by finding optimal exchanging paths and amounts. To implement the model, a source node and a sink node are artificially added. The source node has an initial excess of capital (the amount to be invested). An arc connects the source node to the dollar node. Another arc connects the dollar currency node to the sink node. The problem is then transformed into a maximum flow problem (that is, the company is trying to maximize its current dollar holdings by circulating currencies within the market). Clearly, if the exchange rates are spot rates and are fed into the model in real time, the model can be integrated into the company's whole financial management system. Under different specifications, the model can be extended to currency swap or interest swap problems.

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