Session 8a

Session 8a

Overview Operations Simulation Models • Reliability Analysis • RANK, VLOOKUP, MIN • Inventory Order Quantities • Single Product (MAX) • Multiple Products with Correlated Demand • Project management • PERT Analysis Decision Models -- Prof. Juran

Example 1: Reliability Consider a device that requires two batteries to function. If either of these batteries dies, the device will not work. Currently there are two brand new batteries in the device, and there are three extra brand new batteries. Each battery, once it is placed in the device, lasts a random amount of time that is lognormally distributed with mean 20 hours and standard deviation 5 hours. When any of the batteries in the device dies, it is immediately replaced by an extra (if an extra is still available). Simulate the time the device can last with the batteries currently available. What is the estimated probability that the device will last longer than 50 hours? Decision Models -- Prof. Juran

Decision Models -- Prof. Juran

B3:B4 contain the parameters for the random lifetimes of batteries. D1:I6 keep track of all five batteries, including when they start (and in what order) and when they fail (and in what order). Note that the random variables are G2:G6, which will be simulated by Crystal Ball. These are the “lifetimes” of the batteries. Decision Models -- Prof. Juran

We calculate the fail time of a battery by using the formula: Fail time = Start time + Lifetime Decision Models -- Prof. Juran

For example, in this realization, Batteries A and B both start at time 0. Battery B fails at time 26.41, and is replaced by Battery C (see row 10). Battery A fails at time 28.23 and is replaced by Battery D (row 11). Battery D fails at time 50.75 and is replaced by E (D started at time 28.23 and lasted 22.52; see row 12). Finally, Battery C fails at time 52.08 (it started at time 26.41 and lasted 25.67; see row 13). At this point the device fails, having lasted 52.08 hours. Decision Models -- Prof. Juran

Using the frequency chart’s “certainty” feature, we can drag the lower “grabber” to a point where the left window reads 50 (or Crystal Ball will allow you simply to type 50 into the left window). The certainty window indicates the estimated area under the curve in the blue shaded region. In this case, we estimate a 13.55% probability that the device will last longer than 50 hours. Decision Models -- Prof. Juran

Example 2: Blockbuster Publishers At present, 2000 copies of the book are in stock, and Blockbuster must determine how many copies of the book to print for the next year. Demand has a triangular distribution with parameters 4000, 6000, and 9000. Each copy sold during brings the publisher a revenue of $35. Any copies left at the end of the year can be sold for $5. The cost of printing the book is $50,000 plus $15 per book printed. Decision Models -- Prof. Juran

Somewhat arbitrarily, we have decided to consider possible order quantities of 0, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 (B1:I1). For each possible order quantity, we have a fairly simple income statement-like analysis in rows 2 through 9. All columns get their demand number from B3. (Note that the demand level in B3 will be a random variable later.) The lower part of the spreadsheet (rows 11 through 16) contains the basic parameters of the problem. Decision Models -- Prof. Juran

We run for 1000 trials, and then click on the extract data button. Decision Models -- Prof. Juran

Does your answer change if 4000 copies are currently in stock? Decision Models -- Prof. Juran

Example 3: Walton Bookstore Consider two competing products sold by a company. Sales of either product tend to take away sales from the other product. That is, demands for the two products are negatively correlated. The company first places an order for each product. Then during a period of time, there is demand D1 for product 1 and demand D2 for product 2. These demands are normally distributed with means 1000 and 1200 and standard deviations 250 and 350. The correlation between D1 and D2 is , where  is a negative number between –1 and 0. Decision Models -- Prof. Juran

The unit cost of each product is $7.50, the unit price for each product is $10, and the unit refund for any unit of either product not sold is $2.50. The company must decide how many units of each product to order. Use simulation to help the company by experimenting with different order quantities. Try this for  = -0.3,  = -0.5, and  = -0.7. What recommendations can you give about the “best” order quantities as the demands become more highly correlated (in a negative direction)? Decision Models -- Prof. Juran

We have laid out (somewhat arbitrarily) 16 different combinations of order quantities for the two products (B2:Q3). Each of the columns from B to Q represents a model of one order quantity strategy. Decision Models -- Prof. Juran

Parameter Description Characteristics Min Minimum Value Any number - ∞ to ∞ Max Maximum Value Any number - ∞ to ∞ Alpha ( α ) Shape Factor Must be > 0 Beta ( β ) Shape Factor Must be > 0 Beta Distribution The Beta distribution is a continuous probability distribution defined by four parameters: Decision Models -- Prof. Juran

The Beta distribution is popular among simulation modelers because it can take on a wide variety of shapes, as shown in the graphs above. The Beta can look similar to almost any of the important continuous distributions, including Triangular, Uniform, Exponential, Normal, Lognormal, and Gamma. For this reason, the Beta distribution is used extensively in PERT, CPM and other project planning/control systems to describe the time to completion of a task. Decision Models -- Prof. Juran

PERT Approximations The project management community has evolved approximations for the Beta distribution which allow it to be handled with three parameters, rather than four. The three parameters are the minimum, mode, and maximum activity times (usually referred to as the optimistic, most-likely, and pessimistic activity times). This doesn’t give exactly the same results as the mathematically-correct version, but has important practical advantages. Most real-life managers are not comfortable talking about things like probability functions and Greek-letter parameters, but they are comfortable talking in terms of optimistic, most-likely, and pessimistic. Decision Models -- Prof. Juran

3-step Procedure Decision Models -- Prof. Juran

Beta Distributions in Crystal Ball Decision Models -- Prof. Juran

Operations Example: Project Management (PERT) Sharon Katz is project manager in charge of laying the foundation for the new Brook Museum of Art in New Haven, Connecticut. Liya Brook, the benefactor and namesake of the museum, wants to have the work done within 41 weeks, but Sharon wants to quote a completion time that she is 90% confident of achieving. The contract specifies a penalty of $10,000 per week for each week the completion of the project extends beyond week 43. Decision Models -- Prof. Juran

Here’s an activity-on-arc diagram of the problem: Decision Models -- Prof. Juran

To use a beta distribution, we would start a spreadsheet model like this, calculating the mean and standard deviation using the PERT formulas: Decision Models -- Prof. Juran

Calculating shape and scale parameters: Decision Models -- Prof. Juran

Model Overview A section for simulating the times of the activities A section to keep track of each path through the network, to identify the critical path in each simulated project completion A section to keep track of each node and when it occurs Decision Models -- Prof. Juran

Here’s the section keeping track of the activity times. The numbers in column I will be CB assumption cells. Decision Models -- Prof. Juran

Session 8a

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