Chapter 5 Decision Theory

Mathematical Expectation or expected valueis the product of the probability for an event will occur and the amount to be received upon such an occurrence.

The Concept of Probability• It is more often that we are compelled to predict the outcome of

future events based on the repeated experiments of observation of the same events under the same condition. The probability of occurrence of the event(called success) and the probability of occurrence (called failure) is equal to 1 or 100%.

What is PROBABILITY?

Probability• is an event that ranges to 0 to 1, probability of success is 1 or 100%;

if the event cannot occur, is probability is 0.

Computation of mathematical expectation or expected valueLet P represent the probability value and X represent the amount of many mathematical expectation is computed as:

EV = P(X)

If the event has several possible outcome with probability……. and EV denotes a discrete variable.

Example 1: A fair coin is tossed. If the coin lands heads, Mr. A will receive

Php. 6.00, and pay Php. 4.00 if it lands tails. Find.

There is only 2 possible outcomes, heads or tails, the probability of heads is , and that of tails is also , so

Let and

Given:

Computation:

(This means the game is fair for Mr. A)

What is DECISION MAKING UNDER CERTAINTY?

Decision-making under certaintyThe possible future conditions will actually happen when the manager know which event will occur. The decision maker is to pick the alternative with the best pay off for the known event. The best alternative is the highest pay off if the pay offs are expressed in profits. If the pay offs are expressed as costs, the best alternative is the lowest pay off.

Example:Which is the best alternative if the probability of small demand

is estimated to be 0.4 and the probability of large demand is estimated to be 0.6?

Therefore: The large facility is the highest expected value which 612.

Possible Future

Demand

Alternative Low High Expected Value (EV)

Small Facility 300 370

Large Facility 301 900

What is DECISION MAKING UNDER UNCERTAINTY?

Decision making under uncertaintyWe assume the manager can list the possible events but cannot estimate their probabilities. Perhaps the lack of prior experience makes it difficult for the firm to estimate probabilities.

The following conditions will be used in decision rules:1. Maximin2. Maximax3. Laplace4. Minimax Regret

Maximin – choose the alternative that is the “best of the best.” This maximin approach is essentially a pessimistic one because it takes into account the worst possible outcome for each alternative.

Example: A. Maximin Alternative Worst PayoffSmall facilitys 300Large facility 180

Since P300.00 is the worst member, the pessimist would build a small facility.

Maximax – choose the alternative that is the “best of the worst.” The maximax approach is an optimistic, “go for it” strategy that has high expectations.

Example: B. MaximaxAlternative Best Payoff

Small facility 370Large facility 900

Since P900,000 is the best member, the optimist would build a large facility.

Laplace – choose the alternative with the best “weighted payoff.” The Laplace approach treats the states of nature as equally likely (equal probability) to each event.

Example: C. LaplaceAlternative Weighted Payoff

Small facility 0.5(300) + 0.5(370) = 335Large facility 0.5 (180) + 0.5(900) = 540

Since P540,000 is the best weighted payoff, the realist would build a large facility.

Minimax Regret – choose the alternative with the best “worst regret.” This approach seeks to minimize the difference between the given payoff and the best payoff for each state of nature.

Example: Minimax RegretTo solve this problem, find the difference between a given payoff and the best payoff. Let us consider, in the first column, the best payoff is 300, the remaining column must be subtracted from 300.

Alternative REGRET Maximum RegretLow Demand High Demand

Small Facility 300 – 300 = 0 900 – 370 = 530 530Large Facility 300 – 180 = 120 900 – 900 = 0 120

Since the column on the right shows the worst regret for each alternative, pick a large facility to minimize the maximum regret. The biggest regret is associated with having only a small facility and high demand.

What is DECISION MAKING UNDER RISK?

Decision making under riskThe manager has less information than with decision-making under certainty but more information than with decision-making under uncertainty. A widely used approach under such circumstances is the expected value. The expected value (EV) is the sum of the payoffs for an alternative where each payoff is weighted by the probability for the relevant state of mature.

Example:Which is the best alternative if the probability of small demand is estimated

to be 0.4 and the probability of large demand is estimated to be 0.60(60%)?

Possible Future DemandAlternative Low High Expected ValueSmall Facility 300 370 0.4(300)+0.6(370) = 342Large Facility 180 900 0.4(180)+0.6(900) = 612

Therefore: Large facility is the highest expected value which is 612.

What is a DECISION TREE?

Decision TreeA decision tree is a schematic model of alternative available to

the decision make along with their possible consequences. A decision tree is composed of a number of nodes that have branches emanating from it. Two types of nodes are used in such a tree: a square represent a decision point and a circle stands for a chance event.

Two types of nodes• Square represent a decision point• Circle stands for a chance event

After the tree has been drawn, analyzed from right to left, that is, start with the last decision that might be made. For each decision, choose the alternative that will yield the greatest return or the lowest cost if chance events follow a decision, choose the alternative cost. If chance events follow a decision, choose the alternative that has the greatest expected monetary value or connect cost.

ExampleThe manager of the company has to decide whether to prepare a bid or set. It cost P5,000.00 to prepare the bid. If the bid is submitted, the probability that the contract will be awarded is 60%. If the company is awarded the contract, it may earn an income of P60,000.00 if it succeeds or pay a fine of P15,000 if it fails. The probability of success is estimated to be 70%. Should the owner prepare the bid.

Solution:

At the end of the success branch, P60,000 is the return, but since there is a cost of P5,000 for bid preparation, then P5,000 should be subtracted. Since a fine of P15,000 is forced in case of failure, and there is also a cost of P5,000 for bid preparation, then -15,000-5,000 are indication at the end of the failure branch. The sign of -5,000 also be indicated at the end of the failure branch. If the contract is not awarded, there is a cost of P5,000 for bid preparation.

Continuation ----

Compute the E.V. Backward from position.

E.V. = .7(55,000)+.3(-20,000)= 38,500 – 6,000= 32,500

E.V. = .6(32,500) + .4(-5,000)= 19,500 – 2,000= 17,500

Expected Value of Perfect Information (EVPI)If a manager were able to determine which state of nature would occur, then one would know which decision is made. Once a manager knows which decision to make, the pay off increases and is now a certainty, not a probability. Because the payoff will increase with knowledge of which state of nature will occur, this knowledge has value. Therefore, we now look on how to determine the value information. The expected gain is the expected value of perfect information or EVPI.

Example:Compute for the expected payoff under certainty. The best payoff for the small capacity and large capacity are P370 and P900 respectively. Then combine by weighing each payoff by the probability of that state of nature and add the amounts. The expected payoff under certainty is,

.40(370) + .60(900) = 148 + 540 = P688The expected payoff under risk, as computer is P612EVPI = P688 – P612 = P76

Inventory ModelAn inventory is a stock or store of goods. It may be thought of as a

resources or as a list of some category of materials, machines, people, money, or information for some organization unit at some time. Inventories are added to and depleted from; if the additional and depletion processes are stopped, what remains is inventory. Alternatively, an inventory can be conceived as an idle usable resource.

Proper inventory managementA. The proper quantity of inventory to order at any given timeB. The proper time to order the quantity

Four basic costs are associated with inventories: purchase cost; holding or carrying cost; ordering cost, and shortage cost. Annual cost of inventory is the sum of the annual ordering cost and annual carrying cost.

Holding or Carrying CostHolding or carrying cost relates to physically holding items in

storage. This cost is proportional to the amount of inventory and the time over which it is held. They include interest, insurance, taxes, depreciation, obsolescence, deterioration, spoilage, pilferage, breakage, and warehousing cost (heat, light, rent, security). Holding cost also includes opportunity costs associated with having funds tied up in inventory that could be used elsewhere. It essentially represents the explicit and implicit cost of maintaining and owning the inventory. A significant component of carrying cost is the opportunity cost associated with owning the inventory. Holding or carrying cost is stated in either of two ways. One way is to specify cost as a percentage of unit price, and the other is to specify a peso amount per unit.

Ordering or Set-up costs Ordering costs are costs associated with ordering and receiving

inventory. This cost is incurred whenever an inventory is replenished. Generally, it is independent of the quantity replenished. Ordering cost is the term used for purchase or vendor models. These costs include determining how much is needed, typing up invoices, inspecting goods upon arrival for quality and quantity, moving the goods to temporary storage. Ordering costs are generally expressed as a peso amount per order, regardless of order size. In production models, the term “setup cost” is frequently used to include the cost of labor and materials used in setting up machinery for the production run.

Purchase of Direct Production Cost

The purchase cost is for vendor supply environments or direct production cost in case of items produced by user. In either situation the unit cost maybe constant for all replenishment quantities, or it may vary with the quantity purchased or produced. Businessmen frequently offer discounts or price breaks if the user purchases quantities that excess some specified quantity. Similarly, unit cost may decrease as larger production runs are made due to economics of scale.

Shortage CostsShortage costs result when demands exceed the supply of inventory

on hand. The cost can include the opportunity cost of not making a sale, loss of customer goodwill, lateness charges, and similar costs. Furthermore, if the shortage occurs in an item carried for internal use (e.g., to supply an assembly line), the cost of lost production or downtime is considered a shortage cost.

Shortage costs are computed differently depending on whether or not back ordering is possible. When back ordering is permitted, explicit costs are incurred for overtime, special clerical and administrative efforts, expediting, and special transportation. When the unavailable item is a finished good, there is often an implicit cost reflecting loss of goodwill. This is a difficult cost to measure, since it supposedly accounts for lost future sales.

How Much to Order: The economic order quantity Models

The question of how much to order is frequently determined by using an economic order quantity (EOQ) model. EOQ model identifies the optimal order, quantity in terms of minimizing the sum of certain annual costs which vary with order size.

The best known and most fundamental inventory model is the economic order quantity model. This model is applicable when the demand for the item has a constant or nearly constant rate and when the entire quantity ordered arrives in the inventory at one point in time. The constant demand rate condition means simply that the same number of units is taken from the inventory each period of time.

The how-much-to-order decision involves selecting an order quantity that draws a compromise between:

1. Keeping small inventories and ordering frequently,2. Keeping large inventories and ordering frequently.

The first alternative would result in undesirable high ordering costs, while the second alternative would result in undesirable high inventory holding costs.

Formula

The annual inventory holding or carrying costs can be calculated using the average inventory. That is, we can calculate the annual inventory carrying costs by multiplying the average inventory by cost of carrying one unit in the inventory for a year. Thus, the general equation for annual inventory carrying costs is as follows.

Formula:

Annual inventory Carrying Costs = (Average Inventory)(Annual holding cost per unit)Or

Annual carrying cost = Average inventory cost times percentage of carrying cost.

The annual ordering costs can be calculated by multiplying the number of order per year and the cost per order. Thus, the general equation for annual inventory cost is as follows:

Annual ordering cost = (Number of orders per year)(cost per order)

Thus the total annual cost = annual inventory carrying cost plus annual ordering cost.

Physical relation of the annual carrying cost, annual ordering cost, total annual cost, and economic order quantity.

Example: Suppose that R & C Beverage Company has a beverage product that has a

constant annual demand rate of 7,200 cases. A case of soft drink costs R & C P288. Ordering cost is P200 per order and inventory carrying cost is charged at 25% of the cost per unit. Identify the following aspects of the inventory policy:

a. Economic Order Quantityb. Annual Carrying Costc. Total Annual Cost

Solution:a. Given

Annual inventoy = 7200 casesCost per case of soft drinks = P288Cost per order = P200Percentage of carrying cost = 25%

Q

Q

Q

Q

B. Annual Carrying Cost = (Average inventory cost) (percentage of carrying cost)

Average inventory = ½ Economic order quantityAnnual Carrying Cost = ½ (200) x 0.25 (288)

= 100 x 72= P7,200

C. Annual Ordering cost = No. Of orders x Cost per order

No. Of Orders = Annual Inventory/EOQTotal Annual ordering cost = 36 x P200 = P7,200

D. Total Annual Cost = Annual Ordering cost + Annual Carrying CostTotal Annual Cost = P7,200 + P7,200

= P14,000