OR

• Goal programming approach establishes a specific numeric goal for each of the objective and then attempts to achieve each goal sequentially up to a satisfactory level rather than an optimal level
• GP; Channes and Cooper (1961); Suggested a method for solving an infeasible LP

  problem arising from various conflicting resource constraints (Goals).
• Ijiri (1965) developed the concept of preemptive priority factors, assigning different

  priority levels to incommensurable goals and different weights to the goals at the

  same priority level
• A key idea in goal programming is that one goal is more important than another. Priorities are

  assigned to each deviational variable.
• Goal programming models are all minimization problems.
• In GP, there is no a single objective, but multiple goals to be attained.
• The deviation from the high-priority goal must be minimized to the greatest extent possible before the next- highest-priority goal is considered.
• System constraints may influence but are not directly related to goals
• Goal constraints that are directly related to goals.
• Goal Programming: Satisficing
• Firms often have more than one goal
• They may want to achieve several, sometime contradictory goals
• It is not possible for LP to have multiple goals unless they are all measured in the

  same units, and this is a highly un usual situation.
• In linear and integer programming methods the objective function is measured in one dimension only.
• An important techniques that has been developed to supplement LP is called goal programming
• In GP, instead of trying to minimize or maximize the objective function directly, as in

  case of an LP, the deviations from established goals within given set of constraints are Minimized
• In GP, Slack and Surplus variables are known as deviational variables (di – and di +)

  (means underachievement & overachivement ); These deviations from

  each goal or sub-goal.
• The idea of working backward by solving two well-known

  puzzles and then show how dynamic programming can be

  used to solve network, inventory, and resource-allocation

  problems.
• Dynamic programming computes its solution bottom up by

  synthesizing them from smaller sub solutions, and by trying

  many possibilities and choices before it arrives at the

  optimal set of choices.
• richard ernest bellman (1950), a mathematician

  that coined the dynamic

  programming when he

  was writing his book “Än

  Approximate Dynamic

  Programming”.
• Decision theory a

  discipline addressing

  important practices to

  assess a recommended

  course of action for the

  decision-make
• Properties of Dynamic Programming

  − The problem structure is divided into stages

  − Each stage has a number of states associated with it

  − Making decisions at one stage transforms one state of the current stage into a state in

  the next stage.

  − Given the current state, the optimal decision for each of the remaining states does not

  depend on the previous states or decisions. This is known as the principle of optimality

  for dynamic programming.

  − The principle of optimality allows to solve the problem stage by stage recursively.
• States

  › Each stage has a number of states associated with it. Depending what decisions are

  made in one stage, the system might end up in different states in the next stage.

  › If a geographical region corresponds to a stage then the states associated with it

  could be some particular locations (cities, warehouses, etc.) in that region.

  › In other situations a state might correspond to amounts of certain resources which

  are essential for optimizing the system.
• Decisions

  › Making decisions at one stage transforms one state of the current stage into a state

  in the next stage.

  › In a geographical example, it could be a decision to go from one city to another. In

  resource allocation problems, it might be a decision to create or spend a certain

  amount of a resource.
• Deterministic DP


  Recursive solution to the problem
• Deterministic DP


  • The principle of optimality allows to solve the problem stage by stage recursively
  • The solution procedure first finds the optimal policy for the last stage
  • The solution for the last stage is normally trivial
  • A recursive relationship is established which identifies the optimal policy for stage t, given that stage t+1 has already been solved
  • The solution procedure starts at the end and moves backward stage by stage until it finds the optimal policy starting at the initial stage
• Optimal Policy


  An optimal policy decision at each stage for each of the possible states
• Probabilistic DP

  Solving Inventory Problems by DP Main characteristics:

  1. Time is broken up into periods. The demands for all periods are known in advance.

  2. At the beginning of each period, the firm must determine how many units should be

  produced.

  3. Production and storage capacities are limited.

  4. Each period’s demand must be met on time from inventory or current production.

  5. During any period in which production takes place, a fixed cost of production as well

  as a variable per- unit cost is incurred.

  6. The firm’s goal is to minimize the total cost of meeting on time the demands.
• Principle of Optimality


  Given the current state, the optimal decision for each of the remaining states does not depend on the previous states or decisions
• Geographical setting


  • The optimal route from a current city to the final destination does not depend on the way we got to the city
• Decision Theory, a

  discipline addressing

  important practices to

  assess a recommended

  course of action for the

  decision-maker.
• A system can be formulated as a dynamic programming problem only if the principle of optimality holds for it
• Concept and Decision-Making Scenarios

  − A process of selecting an act out of several available alternative

  courses of action judged to be the best action according to some pre-

  determined criteria.

  − Objective: The main aim of decision theory is to help the decision–

  maker in selecting best course of action from amongst the available

  course of action.
• Structure of Decision-Making:

  o Decision Maker – The decision maker refers to individual or a group of

  individuals responsible for making the choice of an appropriate courses of

  action amongst the available courses of action.

  o Courses of action – The alternatives courses of action or strategies are the acts

  that are available to decision maker.

  o Example – The number of units of a particular item to be ordered for stock.
• States of nature (outcomes)


  The events identify the occurrence which are outside of the decision maker's control and which determine the level of success for a given act
• States of nature (outcomes)


  • The level of market demand for a particular item
• Types of decision–making:

  o Decision making under certainty

  o Decision making under uncertainty

  o Decision making under risk
• Payoff (conditional profit values)


  Each combination of a course of action and a state of nature is associated with a payoff, which measures the net benefit to the decision maker that accrues from a given combination of decision alternatives and events
• Payoff table


  For a given problem, lists the states of nature (outcomes) and a set of given courses of action. For each combination of state of nature and courses of action, the payoff is calculated
• Regret or opportunity loss


  The opportunity loss has been defined to be the difference between the highest possible profit for a state of nature and the actual profit obtained for the particular action taken