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Optimization Techniques

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  • Optimization Techniques


    It is important that modern management be trained in the various decision-making processes. Judgment by itself will not always provide the best answer. The various decision-making processes are based on theory of probability, statistical analysis, and engineering economy. Decision making under certainty assumes the states of the product or market are known at a given time. Usually the decision-making strategy under certainty would be based on that alternative that maximizes if we are seeking quality, profit, etc., and that minimizes if we are studying scrap, customer complaints, etc. The several alternatives are compared as to the results for a particular state (quantity, hours of service, anticipated life, etc.). Usually decisions are not made under an assumed certainty. Often risk is involved in providing a future state of the market or product. If several possible states of a market prevail, a probability value is assigned to each state. Then a logical decision-making strategy would be to calculate the expected return under each decision alternative and to select the largest value if we are maximizing or the smallest value if we are minimizing. Here

    where E(a) 5 expected value of the alternative; pj 5 probability of each state of product or market occurring; cij 5 outcome for particular alternative i at a state of product or market j.

    A different decision-making strategy would be to consider the state of the market that has the greatest chance of occurring. Then the alternative, based upon the most probable future, would be that one that is either a maximum or minimum for that particular state.

    A third decision-making strategy under risk would be based upon a level of aspiration. Here the decision maker assigns an outcome value cij which represents the consequence she or he is willing to settle for if it is reasonably certain that this consequence or better will be achieved most of the time. This assigned value may be referred to as representing a level of aspiration which can be identified as A. Now the probability for each aj where the cij (each decision alternative) is equal or greater to A is determined. The alternative with the greatest p(cij $ A) is selected if we are maximizing.

    There are other decision-making strategies based on decision under risk and uncertainty. The above examples provide the reader the desirability of considering several alternatives with respect to the different states of the product or market.

    Linear Programming

    At the heart of management’s responsibility is the best or optimum use of limited resources including money, personnel, materials, facilities, and time. Linear programming, a mathematical technique, permits determination of the best use which can be made of available resources. It provides a systematic and efficient procedure which can be used as a guide in decision making.
    As an example, imagine the simple problem of a small machine shop that manufactures two models, standard and deluxe. Each standard model requires 2 h of grinding and 4 h of polishing. Each deluxe model requires 5 h of grinding and 2 h of polishing. The manufacturer has three grinders and two polishers; therefore, in a 40-h week there are 120 h of grinding capacity and 80 h of polishing capacity. There is a profit of $3 on each standard model and $4 on each deluxe model and a ready market for both models. The management must decide on: (1) the allocation of the available production capacity to standard and deluxe models and (2) the number of units of each model in order to maximize profit.

    To solve this linear programming problem, the symbol X is assigned to the number of standard models and Y to the number of deluxe models. The profit from making X standard models and Y deluxe models is 3X 1 4Y dollars. The term profit refers to the profit contribution, also referred to as contribution margin or marginal income.
    The profit contribution per unit is the selling price per unit less the unit variable cost. Total contribution is the per-unit contribution multiplied by the number of units. The restrictions on machine capacity are expressed in this manner: To manufacture one standard unit requires 2 h of grinding time, so that making X standard models uses 2X h. Similarly, the production of Y deluxe models uses 5Y h of grinding time. With 120 h of grinding time available, the grinding capacity is written as follows: 2X 1 5Y # 120 h of grinding capacity per week. The limitation on polishing capacity is expressed as follows: 4X 1 2Y # 80 h per week. In summary, the basic information is

    Two basic linear programming techniques, the graphic method and the simplex method, are described and illustrated using the above capacity-allocation–profit-contribution maximization data. Graphic Method

    The lowest number in each of the two columns at the extreme right measures the impact of the hours limitations. The company can produce 20 standard models with a profit contribution of $60 (20 x $3) or 24 deluxe models at a profit contribution of $96 (24 x $4). Is there a better solution?
    To determine production levels in order to maximize the profit contribution of $3X + $4Y when:

    a graph (Fig. 17.1.5) is drawn with the constraints shown. The two-dimensional graphic technique is limited to problems having only two variables—in this example, standard and deluxe models. However, more than two constraints can be considered, although this case uses only two, grinding and polishing.
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