Game Theory and Effectiveness of Logistic Processes Under Uncertainty and Risk Conditions 2

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1 Michał Tymiński 1 Jan Kochanowski University Kielce Branch in Piotrków Trybunalski (Poland) Game Theory and Effectiveness of Logistic Processes Under Uncertainty and Risk Conditions 2 Introduction The article presents the use of game theory for production scale in a company producing equipment for the army. Taking logistic decisions, which originated in the military, is connected with the process of management of the so called supply chains. This area of logistics is responsible for supplying materials and equipment and providing a range of social and economic services, as well as logistic services for troops taking part in military operations 3 in multinational space. As the very nature of economic phenomena is non-deterministic, it seems justifiable to use game theory to support uncertain and conflict processes of taking decisions. Operational decisions as well as strategic decisions taken under such conditions usually pose a serious risk and therefore they should include elements of risk and uncertainty 4. Conditions of uncertainty are mainly characterized by a lack of significant information and the randomness of the analysed processes. In case of logistic supply chains this may be, for instance, an unknown and variable primary demand for a given product in a given period of time. On the other hand, conflict situations are related to conditions under which decisions result from a changing environment or counter-decisions of other players involved in economic and market processes. The paper presents a possibility of applying game theory as a method supporting the process of making strategic decisions by a non-military company integrated with the military environment by the logistic supply chain. Our considerations are carried out with the use of a case of a manufacturing company which is a link of the logistic chain supporting military units dispatched on peace missions. Alfa Company is a manufacturer of military goods and materials. Participation in the supply chains gives the company an opportunity to implement the strategy of strengthening its market position. In order to illustrate the decision- making procedure some contractual data was used in the numerical part of the paper. ELEMENTS OF CONTEMPORARY MILITARY LOGISTICS Contemporary conditions of carrying out military operations enforce taking a new approach towards logistic support of troops which frequently perform their tasks under extremely difficult climate and field conditions and in areas which are distant from their home countries. One of the basic forms of contemporary logistic support of troops is the formula of multinational logistics developed by NATO. One of the key aspects of the formula is the integration of logistic processes while taking into account the criterion of hierarchy and functional level what is illustrated in Figure 1. The new concept of NATO logistic approach is aimed primarily at optimizing the size of national units of logistic support and, consequently, reducing logistic costs. It should be added that this approach may lead to an increase in effectiveness and flexibility of operations and synergy between units taking part in logistic support 5. An increase in integrity in various areas of logistics is based on the premise that globalization leads to integrated logistic security of military actions. One of the essential elements of contemporary multi-national logistics is participation of foreign partners seen as supporting and supplementing their military logistic potential. In both theoretical and practical terms globalization is understood as cooperation and mutual connections between countries, societies and economies which form a global network of connections. This is seen as a manifestation of systematic modification of various connections and relations with an emphasis put on economization of actions and economic effectiveness. The latter one is fostered by accuracy of the decisions taken therefore, to support decision making processes, companies should make use of specialist tools such as game theory. 1 M.Sc. M. Tymiński Ph.D. Jan Kochanowski University, Kielce; Branch in Piotrków Trybunalski, The Chair of National Security 2 Article reviewed. 3 K. Ficoń, Logistyka kryzysowa, Wyd. BEL Studio Sp. z o.o., the uncertainty situation is characterized by the fact that not all conditions and factors having impact on the decision- making process are fully known (Ficoń, 2007). 5 T. Smal (ed.): Zabezpieczenie logistyczne PKW w operacjach poza granicami kraju, WSOW Ląd Wrocław 2007, p

2 Fig. 1. Strategic and tactical areas of military logistics. Source: Ficoń K., Logistyka ekonomiczna. Procesy logistyczne, p. 23. USE OF GAME THEORY IN TAKING DECISIONS ON PRODUCTION SCALE UNDER UNCERTAINTY CASE OF ALFA COMPANY In essence, game theory deals with conflict situations between players, where despite compatible interests it is difficult to find a uniform mode of operation. The peculiar quality of game theory in stochastic games consists in minimizing loss functions. It amounts to probabilistic modelling in which one cause and various results can be determined or vice versa. The game theory is a tool which can be applied to provide a description of economic phenomena which are of conflict character. Mathematical algorithms are seen as key procedures in game theory as they allow to search for optimal decisions. Forecasting future events is particularly difficult in the process of decision-taking because of the occurrence of risk. One of many possible categories of risk is economic risk which results from the fact that all data is applicable to a past period and not to future 6. ABSOLUTE CERTAINTY RISK UNCERTAINTY Fig. 2. Types of decision- making situations. Source: Author s own elaboration. 6 K. Ficoń, Inżynieria zarządzania kryzysowego, Bel Studio sp. z o.o. Warszawa 2007, p

3 Due to the fact that Poland is involved in peace missions the army may signal a changeable demand for products manufactured by the Alfa company. The Alfa company, which manufactures military products and materials plans to increase its production scale. Defining a quantitative production plan for the years to come is impossible. This follows from the situation of uncertainty which accompanies quantitative production plans, as the army does not provide definite demand for products at that stage. On the other hand, the company has to support its economic potential, and simultaneously take into account the risk of failure (low profits or even losses). This kind of situation calls for variance analysis of decision on the size of production taking into account uncertainty conditions. In order to expand a production range it is necessary for the company to purchase the necessary tools and machines and increase its production- warehouse space. Decision- making possibilities Let us take into consideration three possibilities of increasing the company s economic potential: a a purchase enabling a large- scale production, b a purchase enabling a medium- scale production, c a purchase enabling a small- scale production. Decision-making criterion The choice of a decision should lead to company s obtaining maximum profit in T (threshold) period of time. However, the profit depends on market behaviour i.e. the army s demand or, in other words, demand for the produced materials and military goods. Decision- making situations Two possible situations are considered- high (s1) or low (s2) level of acceptance of products offered by clients (army). Each pair corresponds with a certain amount of profit d which is made by the company in T period of time. The managing board of the company assumes that clients will not act against the company s interest. Also, the company itself is not interested in offering low quality products at high prices. Table 1. Estimated company profits (in million PLN). Decision Production scale High acceptance Low acceptance A Large B Medium C Small Source: Author s own elaboration. The example illustrates a certain game of coincidence in which the following players take part: Company having three possibilities as a decision- maker. Market (in this case the army) having two possibilities as the nature. When operating under conditions of uncertainty the following criteria can be used: Maxi max: behavior of a risk-taker,an optimist, Maxi mini: behavior of a risk-averse person, Hurwitz s: behavior which is balanced between a risk-taker s and a risk-averse person s behavior, Savage s: behavior minimizing a loss which results from a wrong decision (also known as Mini max criterion, regret and convenient loss), Laplace s: behavior maximizing an expected profit. Maxi max criterion In this type of criterion for each d a maximum profit is determined and then some decisions are indicated for which the maximum profit is the highest. In case of the analyzed company, under the assumption on risk inclination, profits for each decision amount to, respectively: z1 = max (360, 40) = z2 = max (270, 40) = z3 = max (180, 85) = The largest maximum profit is obtained in case of decision a (360 million PLN). It means that taking this criterion into account the best decision is to purchase space and machines which will allow to produce on a large scale. Maxi mini criterion This criterion consists in determining a minimum guaranteed profit for each decision and then pointing at the decision for which the guaranteed profit is the highest. In the analyzed example the minimum guaranteed profit can be presented as follows: g1 = min (360, 40) = 40.0 g2 = min (270, 40) = 40.0 g1 = min (180, 85) =

4 The highest guaranteed profit is obtained for decision g3. This shows that, taking this criterion into account, the best decision is to get ready for the production on a small scale. It has been assumed in this example that a small-scale production does not require additional investments. Hurwitz criterion Hurwitz criterion requires determining for each decision an average weighted profit using the extreme approaches of the risktaker and the risk-averse person. This is an average weighted from maximum and guaranteed profits while the two weights are risk inclinations. The next step is to determine a decision for which the average weighted profit is the highest. The Hurwitz criterion formula is as follows: L = s * z max + (1 s) * g min where: L average weighted profit, s risk inclination coefficient, z max maximum risk level, g min minimum guaranteed profit level. In order to analyse the presented case using the Hurwitz criterion it is necessary to assume certain risk inclinations which are characteristic of the managing board of the company: 32% for purchase which enables a large-scale production, 64% for purchase which enables a medium-scale production, 79 % for purchase which enables a small-scale production. For these decisions the average weighted profits are equal to, respectively: L1 = 0.32 * (1 0.32) * ( 40) = 88.0 L2 = 0.64 * (1 0.64) * 40 = L3 = 0.79 * (1 0.79) * 85 = As the highest profit weighted by risk inclination is obtained for decision b (187.2), then taking this criterion into account, the best decision seems to be the purchase of production space and machines which will allow to start production on a medium scale. Savage criterion For every state of nature there is an optimal decision which should produce the maximum profit. With differences in nature every decision can be characterized by an amount which is the difference between the maximum profit and the profit obtained for a given decision. This type of difference is called regret. In the beginning the regret matrix is constructed. To determine the matrix one should start with establishing optimal profits separately for two states of nature: s1 : a1 = max (360, 270, 180) = s2 : a2 = max ( 40, 40, 85) = 85.0 For state s1 the best decision is decision a, as it should cause maximum profit (360) while for state s2 the best decision is decision c. The regret matrix is calculated in the following way: ( 40) Next the maximum regret is determined for each decision, and then the decision for which this regret is the smallest is selected. In the analyzed problem the maximum losses for particular decisions amount to: r1 = max (0, 125) = r2 = max (90, 45) = 90.0 r3 = max (180, 0) = The smallest of the biggest losses occurs for decision b. min (125, 90, 180) = 90 which means that, taking into account the Savage criterion, it is best to purchase production space and equipment which will allow to start production on a medium scale. LaPlace criterion In this approach an assumption is made that probability of occurrence of each of the states of nature n is the same and it is equal to 1 : n. The first step involves determining an expected profit value for each decision using probability of the states of nature, and then selecting the decision for which the expected profit is the highest. It follows from calculations that the highest of the expected profit value occurs for decision a i.e.. max {160, 155; 132.5} = 160. If this approach is adopted the best decision is the purchase of production space and machines that will enable the company to produce on a large scale. 480

5 Under the assumption that there is the same chance of occurrence of each of the two market states, the expected profits for subsequent decisions are as follows: h1 = (360 + ( 40) : 2 = h2 = ( ) : 2 = h3 = ( : 2 = SUMMING UP In the example we deal with the process decision- making under conditions of uncertainty. Uncertainty is characterized by the fact that each decision corresponds to more than one result and probability distribution of occurrence of particular states is unknown. Decision-making evaluation of games strategy: According to criteria 1 and 5 it is necessary to choose decision a.i.e. prepare to start production on a large scale. Decision b is recommended when criteria 3 and 4 are applied. Decision c recommends to start production according to criterion 2. The choice of decision made by the company s managing board depends on the assumed criterion of the game with nature as well as prospective information obtained from the army. At that stage of decision- making evaluation we need to conduct risk analysis in the area of investment program of selected decisions as it will allow to evaluate economic effectiveness of investment decisions. Abstract The paper deals with the process of taking logistic decisions in supply chain for the army. The armaments company which secures supplies for the army is considering three forecasting variants of extra supplies. Producing the supplies requires an increased industrial efficiency which is obtained through investments in fixed assets ( production plants, warehouses). However, these variants cannot be implemented in the normal course of action ( i.e. in deterministic situations) as they deal with future orders. They may only be implemented in future under uncertainty conditions to determine extra production profitability. An analysis of economic effectiveness is conducted by the company so that the optimum variant is selected. The analysis includes some elements of game theory which proves most useful in uncertainty situations. Key words: logistic decisions, uncertainty, game theory, conflict situations. LITERATURA / BIBLIOGRAPHY [1]. Ficoń K., Logistyka kryzysowa, Wyd. BEL Studio Sp. z o.o., Warszawa [2]. Ficoń K., Inżynieria zarządzania kryzysowego. Podejście systemowe, Wyd. BEL Studio Sp. z o.o., Warszawa [3]. Coyle J. I., Bardi E. I, Langley C. J. Jr., Zarządzanie logistyczne, PWE, Warszawa [4]. Kisperska-Moroń D., Changing Paradigm for Inventory Management in a Supply Chain Context, Wyd. Uniwersytet Ekonomiczny w Katowicach, [5]. Kisperska-Moroń D., Płaczek E., Liniecki R., Zarządzanie logistyczne w firmach usługowych, Wyd. AE w Katowicach,