MEM (Electrical) Spring 2012 Semester

EM-505: Operations Research (OR) Lecture #10Chapter 10: Deterministic Dynamic Programming Chapter 11: Deterministic Inventory Models

S. Zaffar Qasim Assistant Professor (CIS)1

Equipment Replacement Model The longer a machine stays in service, the higher is its maintenance cost, and the lower its productivity. When a machine reaches a certain age, it may be more economical to replace it. The problem thus reduces to determining the most economical age of a machine. Suppose that we are studying the machine replacement problem over a span of n years. At the start of each year, we decide whether to keep the machine in service an extra year or to replace it with a new one.

2

EM-505: Operations Research

1

Equipment Replacement Model Let r(t), c(t), and s(t) represent the yearly revenue, operating cost, and salvage value respectively of a t-year-old machine. The cost of acquiring a new machine in any year is I. The elements of the DP model are

3

Equipment Replacement Model

4

EM-505: Operations Research

2

Equipment Replacement Model (Example)

5

Equipment Replacement Model (Example) The determination of the feasible values for the age of the machine at each stage is somewhat tricky. Fig 1 summarizes the network representing the problem.

Fig 1

6

EM-505: Operations Research

3

Equipment Replacement Model (Example) At the start of year 1, we have a 3-year-old machine. We can either replace it (R) or keep it (K) for another year. At the start of year 2, if replacement occurs, the new machine will be 1 year old; otherwise, the old machine will be 4 years old. The same logic applies at the start of years 2 to 4.

7

Equipment Replacement Model (Example) If a 1-year-old machine is replaced at the start of year 2, 3, or 4, its replacement will be 1 year old at the start of the following year. Also, at the start of year 4, a 6-year-old machine must be replaced, and at the end of year 4 (end of planning horizon), we salvage (S) the machines.

8

EM-505: Operations Research

4

Equipment Replacement Model (Example) The network shows that at the start of year 2, the possible ages of the machine are 1 and 4 years. For the start of year 3, the possible ages are 1, 2 and 5 years, and For the start of year 4, the possible ages are 1, 2, 3, and 6 years. The solution of the network in fig 1 is equivalent to finding the longest route (i.e. maximum revenue) from the start of year 1 to the end of year 4.

9

Equipment Replacement Model (Example) Using the tabular form to solve the problem. All values are in thousands of dollars. Note that if a machine is replaced in year 4 (i.e. end of the planning horizon), its revenue will include the salvage value, s(t), of the replaced machine and the salvage value, s(1), of the replacement machine.

10

EM-505: Operations Research

5

Equipment Replacement Model (Example)

11

Equipment Replacement Model (Example)

12

EM-505: Operations Research

6

Equipment Replacement Model (Example)

13

Equipment Replacement Model (Example) Fig 2 summarizes the optimal solution. At the start of year 1, given t = 3, the optimal decision is to replace the machine. Thus, the new machine will be 1 year old at the start of year 2, and t = 1 at the start of year 2 calls for either keeping or replacing the machine. If it is replaced, the new machine will be 1 year old at the start of year 3; otherwise, the kept machine will be 2 years old. The process is continued in this manner until year 4 is reached.

The alternative optimal policies starting in year 1 are (R, K, K, R) and (R, R, K, K). The total cost is $55,300. 14

EM-505: Operations Research

7

Deterministic Inventory Models Inventory modeling deals with determining the level of a commodity that a business must maintain to ensure a smooth operation. The basis for the decision a model that balances the cost of capital resulting from holding too much inventory against the penalty cost from inventory shortage. The principal factor affecting the solution the nature of the demand: o deterministic or o probabilistic. In real life, demand is usually probabilistic, but in some cases the simpler deterministic approximation may be acceptable. 15

GENERAL INVENTORY MODEL The inventory problem involves placing and receiving orders of given sizes periodically. From this standpoint, an inventory policy answers two questions: How much to order? When to order? The basis for answering these questions is the minimization of the following inventory cost function:

16

EM-505: Operations Research

8

GENERAL INVENTORY MODEL Purchasing cost is the price per unit of an inventory item. At times, the item is offered at a discount if the order size exceeds a certain amount, which is a factor in deciding how much to order. Setup cost represents the fixed charge incurred when an order is placed regardless of its size. Increasing the order quantity reduces the setup cost associated with a given demand, but will increase the average inventory level and hence the cost of tied capital. On the other hand, reducing the order size increases the frequency of ordering and the associated setup cost. An inventory cost model balances the two costs.17

GENERAL INVENTORY MODEL Holding cost represents the cost of maintaining inventory in stock. It includes the interest on capital and the cost of storage, maintenance and handling. Shortage cost the penalty incurred when we run out of stock. It includes potential loss of income and the more subjective cost of loss in customers goodwill.

18

EM-505: Operations Research

9

GENERAL INVENTORY MODEL An inventory system may be based on periodic review (e.g., ordering every week or every month), in which new orders are placed at the start of each period. Alternatively, the system may be based on continuous review, where a new order is placed when the inventory level drops to a certain level, called the reorder point. An example of periodic review can occur in a gas station where new deliveries arrive at the start of each week. Continuous review occurs in retail stores where items (such as cosmetics) are replenished only when their level on the shelf drops to a certain level.19

ROLE OF DEMAND IN THE DEVELOPMENT OF INVENTORY MODELS In general, the analytic complexity of inventory models depends on whether the demand for an item is deterministic or probabilistic. Within either category, the demand may or may not vary with time. For example, the consumption of natural gas used in heating homes is a function of the time of the year, reaching its maximum in midwinter and tapering off during spring and summer months. Though this seasonal pattern repeats itself annually, the same month consumption may vary from year to year, depending, 20 for example, on the severity of weather.

EM-505: Operations Research

10

ROLE OF DEMAND

In practical situations the demand pattern in an inventory model may assume one of four types: i. Deterministic and constant (static) with time. ii. Deterministic and variable (dynamic) with time. iii. Probabilistic and stationary over time iv. Probabilistic and nonstationary over time This categorization assumes the availability of data that are representative of future demand.

21

ROLE OF DEMAND In terms of the development of inventory models, the first category is the simplest analytically and the fourth is the most complex. On the other hand, the first category is the least likely to occur in practice and the fourth is the most prevalent. In practice, we seek a balance between model simplicity and model accuracy, in the sense that we do not want to use a simplified model that does not reflect reality, or a complex that it is analytically intractable.

22

EM-505: Operations Research

11

i.

ROLE OF DEMAND How can we determine if a certain approximation of demand is acceptable? We can start by computing the mean and standard deviation of consumption for a specific period, say monthly. The coefficient of variation V = (Standard Deviation/Mean x 100) can then be used to determine the nature of demand using the following general guideline: If the average monthly demand is approximately constant for all months and V is reasonably small (20%) but approximately constant, o then the demand is probabilistic and stationary. iv. The only remaining case is the probabilistic nonstationary demand o which occurs when the means and coefficients of variation vary appreciably over time. In cases 3 and 4 additionally data usually are needed to determine the associated probability distributions.

24

EM-505: Operations Research

12

ROLE OF DEMAND Example 1 The data in table 1 provide the monthly (January through December) consumption of natural gas in a rural residential home over a span of 10 year (1990-1999). Whenever requested by a home owner, the natural gas supplier sends a truck to the site to fill a tank. The owner of the home decides the time and size of a delivery. From the standpoint of inventory modeling, it is reasonable to assume that each month represents a decision period in which the owner places an order. Our main concern here, however, is to analyze the nature of the demand.25

ROLE OF DEMAND

26

EM-505: Operations Research

13

ROLE OF DEMAND Example 1 An examination of the mean and the coefficient of variation, V, in table 1 reveals two results: Average consumption is dynamic (not constant) because it shows high average consumption during winter months relative to summer months. The coefficient of variation, V, is reasonably small (