a) Using an illustrating example briefly explain what "operations research" is. (2 Marks) b) Differentiate between feasible solution and optimum solution (2 Marks) c) Explain the conditions that must be satisfied for an operations research to exist (4 Marks) d) Explain the components of an operations research problem (3 Marks) c) Explain the structure of the structure of an input-output table using an example (5 Marks) f) Green valley producers produce yoghurt and fresh milk. One ton of fresh milk requires 1/3 hour of processing and 1/3 hour of packaging while one ton of yoghurt requires one hour of mixing and 2/3 hour of packaging. Ench of the three departments can be run for only 8 hours due to power rationing. One ton of yoghurt contributes Shs. 30 to profit while a ton of fresh milk contributes She 40 to profit. i) Formulate the linear programme stating all the necessary conditions. (2 marks) ii) Using simplex method , find the optimal values of fresh milk and yoghurt that must be produced to maximize profit and the maximum profit attainable . What is game theory? (12

a) Operations research is a discipline that deals with the application of advanced analytical methods to make better decisions. For example, a company might use operations research to optimize its production schedule by analyzing data on production rates, labor availability, and material costs.<br /><br />b) A feasible solution is a solution that satisfies all the constraints of a problem, while an optimum solution is the best possible solution that maximizes or minimizes the objective function. For instance, in a transportation problem, a feasible solution could be any allocation of resources that meets the supply and demand constraints, whereas the optimum solution would be the allocation that minimizes the total cost.<br /><br />c) For an operations research problem to exist, there must be a clear objective to optimize (such as maximizing profit or minimizing cost), constraints that the solution must satisfy (like limited resources or production capacity), and a set of decision variables (such as the quantity of products to produce).<br /><br />d) The components of an operations research problem include the objective function (what you are trying to maximize or minimize), constraints (limitations or requirements that must be met), and decision variables (the choices you can make).<br /><br />e) An input-output table shows the relationship between inputs and outputs of a system. For example, in a manufacturing process, the inputs could be raw materials, labor, and capital, while the outputs could be finished products. The table would show how much of each input is required to produce a certain number of outputs.<br /><br />f) <br />i) Let x be the amount of fresh milk (in tons) and y be the amount of yoghurt (in tons) produced. The linear program is:<br /> Maximize: 40x + 30y (profit)<br /> Subject to:<br /> 1/3x + 1/3y ≤ 8 (processing time constraint)<br /> 1/3x + 2/3y ≤ 8 (packaging time constraint)<br /> x ≤ 8 (power constraint for fresh milk)<br /> y ≤ 8 (power constraint for yoghurt)<br /> x ≥ 0, y ≥ 0 (non-negativity constraints)<br /><br />ii) Using the simplex method, you would solve the linear program to find the optimal values of x and y that maximize profit. The maximum profit attainable would be the value of the objective function at these optimal values.<br /><br />Game theory is a study of strategic decision-making in situations where the outcome depends on the actions of multiple agents. It involves analyzing the best responses of each agent to the actions of others. For example, in a game of chess, both players must consider the possible moves of the other to determine their own best move.