Dai Park Textbook

Stochastic Manufacturing & Service Systems Jim Dai and Hyunwoo Park School of Industrial and Systems Engineering Georgia Institute of Technology October 19, 2011 2 Contents 1 Newsvendor Problem 1. 1 Pro? t Maximization 1. 2 Cost Minimization . 1. 3 Initial Inventory . . 1. 4 Simulation . . . . . . 1. 5 Exercise . . . . . . . 5 5 12 15 17 19 25 25 27 29 29 31 32 33 34 39 39 40 40 42 44 46 47 48 49 51 51 51 52 54 55 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Queueing Theory 2. 1 Introduction . . . . . . . 2. 2 Lindley Equation . . . . 2. 3 Tra? c Intensity . . . . . 2. 4 Kingman Approximation 2. 5 Little’s Law . . . . . . . 2. 6 Throughput . . . . . . . 2. 7 Simulation . . . . . . . . 2. 8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discrete Time Markov Chain 3. 1 Introduction . . . . . . . . . . . . . . . . . . . . 3. 1. 1 State Space . . . . . . . . . . . . . . . . 3. 1. 2 Transition Probability Matrix . . . . . . 3. 1. 3 Initial Distribution . . . . . . . . . . . . 3. 1. 4 Markov Property . . . . . . . . . . . . . 3. 1. 5 DTMC Models . . . . . . . . . . . . . . 3. 2 Stationary Distribution . . . . . . . . . . . . . 3. 2. 1 Interpretation of Stationary Distribution 3. 2. 2 Function of Stationary Distribution . . 3. 3 Irreducibility . . . . . . . . . . . . . . . . . . . 3. 3. 1 Transition Diagram . . . . . . . . . . 3. 3. 2 Accessibility of States . . . . . . . . . . 3. 4 Periodicity . . . . . . . . . . . . . . . . . . . . . 3. 5 Recurrence and Transience . . . . . . . . . . . 3. 5. 1 Geometric Random Variable . . . . . . 3. 6 Absorption Probability . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. 7 3. 8 3. 9 3. 0 Computing Stationary Distribution Using Cut Method Introduction to Binomial Stock Price Model . . . . . . Simulation . . . . . . . . . . . . . . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . 59 61 62 63 71 71 72 73 75 78 80 80 80 82 84 91 91 96 97 100 101 103 103 104 106 107 107 108 109 111 111 117 117 130 135 148 159 4 Poisson Process 4. 1 Exponential Distribution . . . . . . . 4. 1. 1 Memoryless Property . . . . 4. 1. 2 Comparing Two Exponentials 4. 2 Homogeneous Poisson Process . . . . 4. 3 Non-homogeneous Poisson Process . 4. Thinning and Merging . . . . . . . . 4. 4. 1 Merging Poisson Process . . . 4. 4. 2 Thinning Poisson Process . . 4. 5 Simulation . . . . . . . . . . . . . . . 4. 6 Exercise . . . . . . . . . . . . . . . . 5 Continuous Time Markov Chain 5. 1 Introduction . . . . . . . . . . . 5. 1. 1 Holding Times . . . . . 5. 1. 2 Generator Matrix . . . . 5. 2 Stationary Distribution . . . . 5. 3 M/M/1 Queue . . . . . . . . . 5. 4 Variations of M/M/1 Queue . . 5. 4. 1 M/M/1/b Queue . . . . 5. 4. 2 M/M/? Queue . . . . . 5. 4. 3 M/M/k Queue . . . . . 5. 5 Open Jackson Network . . . . . 5. 5. 1 M/M/1 Queue Review . 5. 5. 2 Tandem Queue . . . . . 5. 5. Failure Inspection . . . 5. 6 Simulation . . . . . . . . . . . . 5. 7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exercise Answers 6. 1 Newsvendor Problem . . . . . . . 6. 2 Queueing Theory . . . . . . . . . 6. 3 Discrete Time Markov Chain . . 6. 4 Poisson Process . . . . . . . . . . 6. 5 Continuous Time Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Newsvendor Problem In this course, we will learn how to design, analyze, and manage a manufacturing or service system with uncertainty. Our ? rst step is to understand how to solve a single period decision problem containing uncertainty or randomness. 1. 1 Pro? t Maximization We will start with the simplest case: selling perishable items. Suppose we are running a business retailing newspaper to Georgia Tech campus. We have to order a speci? c number of copies from the publisher every evening and sell those copies the next day.One day, if there is a big news, the number of GT people who want to buy and read a paper from you may be very high. Another day, people may just not be interested in reading a paper at all. Hence, you as a retailer, will encounter the demand variability and it is the primary un certainty you need to handle to keep your business sustainable. To do that, you want to know what is the optimal number of copies you need to order every day. By intuition, you know that there will be a few other factors than demand you need to consider. †¢ Selling price (p): How much will you charge per paper? Buying price (cv ): How much will the publisher charge per paper? This is a variable cost, meaning that this cost is proportional to how many you order. That is why it is denoted by cv . †¢ Fixed ordering price (cf ): How much should you pay just to place an order? Ordering cost is ? xed regardless of how many you order. †¢ Salvage value (s) or holding cost (h): There are two cases about the leftover items. They could carry some monetary value even if expired. Otherwise, you have to pay to get rid of them or to storing them. If they have some value, it is called salvage value. If you have to pay, it is called 5 6 CHAPTER 1.NEWSVENDOR PROBLEM holding cost. Hence , the following relationship holds: s = ? h. This is per-item value. †¢ Backorder cost (b): Whenever the actual demand is higher than how many you prepared, you lose sales. Loss-of-sales could cost you something. You may be bookkeeping those as backorders or your brand may be damaged. These costs will be represented by backorder cost. This is per-item cost. †¢ Your order quantity (y): You will decide how many papers to be ordered before you start a day. That quantity is represented by y. This is your decision variable. As a business, you are assumed to want to maximize your pro? t. Expressing our pro? t as a function of these variables is the ? rst step to obtain the optimal ordering policy. Pro? t can be interpreted in two ways: (1) revenue minus cost, or (2) money you earn minus money you lose. Let us adopt the ? rst interpretation ? rst. Revenue is represented by selling price (p) multiplied by how many you actually sell. The actual sales is bounded by the realized dema nd and how many you prepared for the period. When you order too many, you can sell at most as many as the number of people who want to buy. When you order too few, you can only sell what you prepared. Hence, your revenue is minimum of D and y, i. . min(D, y) or D ? y. Thinking about the cost, ? rst of all, you have to pay something to the publisher when buying papers, i. e. cf +ycv . Two types of additional cost will be incurred to you depending on whether your order is above or below the actual demand. When it turns out you prepared less than the demand for the period, the backorder cost b per every missed sale will occur. The amount of missed sales cannot be negative, so it can be represented by max(D ? y, 0) or (D ? y)+ . When it turns out you prepared more, the quantity of left-over items also cannot go negative, so it can be expressed as max(y ? D, 0) or (y ? D)+ .In this way of thinking, we have the following formula. Pro? t =Revenue ? Cost =Revenue ? Ordering cost ? Holding c ost ? Backorder cost =p(D ? y) ? (cf + ycv ) ? h(y ? D)+ ? b(D ? y)+ (1. 1) How about the second interpretation of pro? t? You earn p ? cv dollars every time you sell a paper. For left-over items, you lose the price you bought in addition to the holding cost per paper, i. e. cv + h. When the demand is higher than what you prepared, you lose b backorder cost. Of course, you also have to pay the ? xed ordering cost cf as well when you place an order. With this logic, we have the following pro? t function. Pro? t =Earning ?Loss =(p ? cv )(D ? y) ? (cv + h)(y ? D)+ ? b(D ? y)+ ? cf (1. 2) 1. 1. PROFIT MAXIMIZATION 7 Since we used two di? erent approaches to model the same pro? t function, (1. 1) and (1. 2) should be equivalent. Comparing the two equations, you will also notice that (D ? y) + (y ? D)+ = y. Now our quest boils down to maximizing the pro? t function. However, (1. 1) and (1. 2) contain a random element, the demand D. We cannot maximize a function of random element if we all ow the randomness to remain in our objective function. One day demand can be very high. Another day it is also possible nobody wants to buy a single paper. We have to ? ure out how to get rid of this randomness from our objective function. Let us denote pro? t for the nth period by gn for further discussion. Theorem 1. 1 (Strong Law of Large Numbers). Pr g1 + g2 + g3 +  ·  ·  · + gn = E[g1 ] n>? n lim =1 The long-run average pro? t converges to the expected pro? t for a single period with probability 1. Based on Theorem 1. 1, we can change our objective function from just pro? t to expected pro? t. In other words, by maximizing the expected pro? t, it is guaranteed that the long-run average pro? t is maximized because of Theorem 1. 1. Theorem 1. 1 is the foundational assumption for the entire course.When we will talk about the long-run average something, it involves Theorem 1. 1 in most cases. Taking expectations, we obtain the following equations corresponding to (1. 1) and ( 1. 2). E[g(D, y)] =pE[D ? y] ? (cf + ycv ) ? hE[(y ? D)+ ] ? bE[(D ? y)+ ] =(p ? cv )E[D ? y] ? (cv + h)E[(y ? D)+ ] ? bE[(D ? y)+ ] ? cf (1. 4) (1. 3) Since (1. 3) and (1. 4) are equivalent, we can choose either one of them for further discussion and (1. 4) will be used. Before moving on, it is important for you to understand what E[D? y], E[(y? D)+ ], E[(D ? y)+ ] are and how to compute them. Example 1. 1. Compute E[D ? 18], E[(18 ? D)+ ], E[(D ? 8)+ ] for the demand having the following distributions. 1. D is a discrete random variable. Probability mass function (pmf) is as follows. d Pr{D = d} 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 Answer: For a discrete random variable, you ? rst compute D ? 18, (18 ? D)+ , (D ? 18)+ for each of possible D values. 8 d CHAPTER 1. NEWSVENDOR PROBLEM 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 Pr{D = d} D ? 18 (18 ? D)+ (D ? 18)+ 10 8 0 15 3 0 18 0 2 18 0 7 18 0 12 Then, you take the weighted average using corresponding Pr{D = d} for each possible D. 1 1 1 1 1 125 (10) + (15) + (18) + (18) + (18) = 4 8 8 4 4 8 1 1 1 1 1 19 + E[(18 ?D) ] = (8) + (3) + (0) + (0) + (0) = 4 8 8 4 4 8 1 1 1 1 1 + E[(D ? 18) ] = (0) + (0) + (2) + (7) + (12) = 5 4 8 8 4 4 E[D ? 18] = 2. D is a continuous random variable following uniform distribution between 10 and 30, i. e. D ? Uniform(10, 30). Answer: Computing expectation of continuous random variable involves integration. A continuous random variable has probability density function usually denoted by f . This will be also needed to compute the expectation. In this case, fD (x) = 1 20 , 0, if x ? [10, 30] otherwise Using this information, compute the expectations directly by integration. ? E[D ? 18] = ? 30 (x ? 18)fD (x)dx (x ? 18) 10 18 = = 10 18 1 dx 20 1 20 dx + 30 (x ? 18) x 10 dx + 18 30 (x ? 18) 1 20 dx 1 20 dx = = x2 40 1 20 + 18 x=18 x=10 18x 20 18 x=30 x=18 The key idea is to remove the ? operator that we cannot handle by separating the integration interval into two. The other two expectations can 1. 1. PROFIT MAXIMIZATION be computed in a similar way. 9 ? E[(18 ? D)+ ] = 30 (18 ? x)+ fD (x)dx (18 ? x)+ 10 18 = = 10 18 1 dx 20 1 20 1 20 +0 30 (18 ? x)+ (18 ? x) 10 x2 2 x=18 dx + 18 30 (18 ? x)+ 0 18 1 20 dx = dx + 1 20 dx 18x ? = 20 x=10 ? E[(D ? 18)+ ] = 30 (18 ? x)+ fD (x)dx (x ? 8)+ 10 18 = = 10 18 1 dx 20 1 20 30 (x ? 18)+ 0 10 x2 2 dx + 18 30 (x ? 18)+ 1 20 dx 1 20 dx = =0 + 1 20 dx + 18 x=30 (x ? 18) ? 18x 20 x=18 Now that we have learned how to compute E[D? y], E[(y? D)+ ], E[(D? y)+ ], we have acquired the basic toolkit to obtain the order quantity that maximizes the expected pro? t. First of all, we need to turn these expectations of the pro? t function formula (1. 4) into integration forms. For now, assume that the demand is a nonnegative continuous random variable. 10 CHAPTER 1. NEWSVENDOR PROBLEM E[g(D, y)] =(p ? cv )E[D ? y] ? (cv + h)E[(y ? D)+ ] ? bE[(D ? y)+ ] ? f ? =(p ? cv ) 0 (x ? y)fD (x)dx ? ? (cv + h) 0 ? (y ? x)+ fD (x)dx ?b 0 (x ? y)+ fD (x)dx ? cf y ? =(p ? cv ) 0 xfD (x)dx + y y yfD (x)dx ? (cv + h) 0 ? (y ? x)fD (x)dx ?b y (x ? y)fD (x)dx ? cf y y =(p ? cv ) 0 xfD (x)dx + y 1 ? 0 y y fD (x)dx xfD (x)dx ? (cv + h) y 0 y fD (x)dx ? 0 y ? b E[D] ? 0 xfD (x)dx ? y 1 ? 0 fD (x)dx ? cf (1. 5) There can be many ways to obtain the maximum point of a function. Here we will take the derivative of (1. 5) and set it to zero. y that makes the derivative equal to zero will make E[g(D, y)] either maximized or minimized depending on the second derivative.For now, assume that such y will maximize E[g(D, y)]. We will check this later. Taking the derivative of (1. 5) will involve di? erentiating an integral. Let us review an important result from Calculus. Theorem 1. 2 (Fundamental Theorem of Calculus). For a function y H(y) = c h(x)dx, we have H (y) = h(y), where c is a constant. Theorem 1. 2 can be translated as follows for our case. y d xfD (x)dx =yfD (y) dy 0 y d fD (x)dx =fD (y) dy 0 (1. 6) (1. 7) Also remember the relationship between cd f and pdf of a continuous random variable. y FD (y) = fD (x)dx (1. 8) 1. 1. PROFIT MAXIMIZATION Use (1. 6), (1. 7), (1. ) to take the derivative of (1. 5). d E[g(D, y)] =(p ? cv ) (yfD (y) + 1 ? FD (y) ? yfD (y)) dy ? (cv + h) (FD (y) + yfD (y) ? yfD (y)) ? b (? yfD (y) ? 1 + FD (y) + yfD (y)) =(p + b ? cv )(1 ? FD (y)) ? (cv + h)FD (y) =(p + b ? cv ) ? (p + b + h)FD (y) = 0 If we di? erentiate (1. 9) one more time to obtain the second derivative, d2 E[g(D, y)] = ? (p + b + h)fD (y) dy 2 11 (1. 9) which is always nonpositive because p, b, h, fD (y) ? 0. Hence, taking the derivative and setting it to zero will give us the maximum point not the minimum point. Therefore, we obtain the following result. Theorem 1. 3 (Optimal Order Quantity).The optimal order quantity y ? is the smallest y such that FD (y) = p + b ? cv ? 1 or y = FD p+b+h p + b ? cv p+b+h . for continuous demand D. Looking at Theorem 1. 3, it provides the following intuitions. †¢ Fixed cost cf does not a? ect the o ptimal quantity you need to order. †¢ If you can procure items for free and there is no holding cost, you will prepare as many as you can. †¢ If b h, b cv , you will also prepare as many as you can. †¢ If the buying cost is almost as same as the selling price plus backorder cost, i. e. cv ? p + b, you will prepare nothing. You will prepare only upon you receive an order.Example 1. 2. Suppose p = 10, cf = 100, cv = 5, h = 2, b = 3, D ? Uniform(10, 30). How many should you order for every period to maximize your long-run average pro? t? Answer: First of all, we need to compute the criterion value. p + b ? cv 10 + 3 ? 5 8 = = p+b+h 10 + 3 + 2 15 Then, we will look up the smallest y value that makes FD (y) = 8/15. 12 1 CHAPTER 1. NEWSVENDOR PROBLEM CDF 0. 5 0 0 5 10 15 20 25 30 35 40 D Therefore, we can conclude that the optimal order quantity 8 62 = units. 15 3 Although we derived the optimal order quantity solution for the continuous demand case, Theorem 1. applies to t he discrete demand case as well. I will ? ll in the derivation for discrete case later. y ? = 10 + 20 Example 1. 3. Suppose p = 10, cf = 100, cv = 5, h = 2, b = 3. Now, D is a discrete random variable having the following pmf. d Pr{D = d} 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 What is the optimal order quantity for every period? Answer: We will use the same value 8/15 from the previous example and look up the smallest y that makes FD (y) = 8/15. We start with y = 10. 1 4 1 1 3 FD (15) = + = 4 8 8 1 1 1 1 FD (20) = + + = 4 8 8 2 1 1 1 1 3 FD (25) = + + + = 4 8 8 4 4 ? Hence, the optimal order quantity y = 25 units.FD (10) = 8 15 8 < 15 8 < 15 8 ? 15 < 1. 2 Cost Minimization Suppose you are a production manager of a large company in charge of operating manufacturing lines. You are expected to run the factory to minimize the cost. Revenue is another person’s responsibility, so all you care is the cost. To model the cost of factory operation, let us set up variables in a slightly di? erent way. 1. 2. COST MINIMIZATION 13 †¢ Understock cost (cu ): It occurs when your production is not su? cient to meet the market demand. †¢ Overstock cost (co ): It occurs when you produce more than the market demand.In this case, you may have to rent a space to store the excess items. †¢ Unit production cost (cv ): It is the cost you should pay whenever you manufacture one unit of products. Material cost is one of this category. †¢ Fixed operating cost (cf ): It is the cost you should pay whenever you decide to start running the factory. As in the pro? t maximization case, the formula for cost expressed in terms of cu , co , cv , cf should be developed. Given random demand D, we have the following equation. Cost =Manufacturing Cost + Cost associated with Understock Risk + Cost associated with Overstock Risk =(cf + ycv ) + cu (D ? )+ + co (y ? D)+ (1. 10) (1. 10) obviously also contains randomness from D. We cannot minimize a random objective itself. Instead, based on Theorem 1. 1, we will minimize expected cost then the long-run average cost will be also guaranteed to be minimized. Hence, (1. 10) will be transformed into the following. E[Cost] =(cf + ycv ) + cu E[(D ? y)+ ] + co E[(y ? D)+ ] ? ? =(cf + ycv ) + cu 0 ? (x ? y)+ fD (x)dx + co 0 y (y ? x)+ fD (x)dx (y ? x)fD (x)dx (1. 11) 0 =(cf + ycv ) + cu y (x ? y)fD (x)dx + co Again, we will take the derivative of (1. 11) and set it to zero to obtain y that makes E[Cost] minimized.We will verify the second derivative is positive in this case. Let g here denote the cost function and use Theorem 1. 2 to take the derivative of integrals. d E[g(D, y)] =cv + cu (? yfD (y) ? 1 + FD (y) + yfD (y)) dy + co (FD (y) + yfD (y) ? yfD (y)) =cv + cu (FD (y) ? 1) + co FD (y) ? (1. 12) The optimal production quantity y is obtained by setting (1. 12) to be zero. Theorem 1. 4 (Optimal Production Quantity). The optimal production quantity that minimizes the long-run average cost is the smallest y such tha t FD (y) = cu ? cv or y = F ? 1 cu + co cu ? cv cu + co . 14 CHAPTER 1. NEWSVENDOR PROBLEM Theorem 1. can be also applied to discrete demand. Several intuitions can be obtained from Theorem 1. 4. †¢ Fixed cost (cf ) again does not a? ect the optimal production quantity. †¢ If understock cost (cu ) is equal to unit production cost (cv ), which makes cu ? cv = 0, then you will not produce anything. †¢ If unit production cost and overstock cost are negligible compared to understock cost, meaning cu cv , co , you will prepare as much as you can. To verify the second derivative of (1. 11) is indeed positive, take the derivative of (1. 12). d2 E[g(D, y)] = (cu + co )fD (y) dy 2 (1. 13) (1. 13) is always nonnegative because cu , co ? . Hence, y ? obtained from Theorem 1. 4 minimizes the cost instead of maximizing it. Before moving on, let us compare criteria from Theorem 1. 3 and Theorem 1. 4. p + b ? cv p+b+h and cu ? cv cu + co Since the pro? t maximization problem solved previously and the cost minimization problem solved now share the same logic, these two criteria should be somewhat equivalent. We can see the connection by matching cu = p + b, co = h. In the pro? t maximization problem, whenever you lose a sale due to underpreparation, it costs you the opportunity cost which is the selling price of an item and the backorder cost.Hence, cu = p + b makes sense. When you overprepare, you should pay the holding cost for each left-over item, so co = h also makes sense. In sum, Theorem 1. 3 and Theorem 1. 4 are indeed the same result in di? erent forms. Example 1. 4. Suppose demand follows Poisson distribution with parameter 3. The cost parameters are cu = 10, cv = 5, co = 15. Note that e? 3 ? 0. 0498. Answer: The criterion value is cu ? cv 10 ? 5 = = 0. 2, cu + co 10 + 15 so we need to ? nd the smallest y such that makes FD (y) ? 0. 2. Compute the probability of possible demands. 30 ? 3 e = 0. 0498 0! 31 Pr{D = 1} = e? 3 = 0. 1494 1! 32 ? Pr{D = 2} = e = 0. 2241 2! Pr{D = 0} = 1. 3. INITIAL INVENTORY Interpret these values into FD (y). FD (0) =Pr{D = 0} = 0. 0498 < 0. 2 FD (1) =Pr{D = 0} + Pr{D = 1} = 0. 1992 < 0. 2 FD (2) =Pr{D = 0} + Pr{D = 1} + Pr{D = 2} = 0. 4233 ? 0. 2 Hence, the optimal production quantity here is 2. 15 1. 3 Initial Inventory Now let us extend our model a bit further. As opposed to the assumption that we had no inventory at the beginning, suppose that we have m items when we decide how many we need to order. The solutions we have developed in previous sections assumed that we had no inventory when placing an order.If we had m items, we should order y ? ? m items instead of y ? items. In other words, the optimal order or production quantity is in fact the optimal order-up-to or production-up-to quantity. We had another implicit assumption that we should order, so the ? xed cost did not matter in the previous model. However, if cf is very large, meaning that starting o? a production line or placing an order i s very expensive, we may want to consider not to order. In such case, we have two scenarios: to order or not to order. We will compare the expected cost for the two scenarios and choose the option with lower expected cost.Example 1. 5. Suppose understock cost is $10, overstock cost is $2, unit purchasing cost is $4 and ? xed ordering cost is $30. In other words, cu = 10, co = 2, cv = 4, cf = 30. Assume that D ? Uniform(10, 20) and we already possess 10 items. Should we order or not? If we should, how many items should we order? Answer: First, we need to compute the optimal amount of items we need to prepare for each day. Since cu ? cv 1 10 ? 4 = , = cu + co 10 + 2 2 the optimal order-up-to quantity y ? = 15 units. Hence, if we need to order, we should order 5 = y ? ? m = 15 ? 10 items. Let us examine whether we should actually order or not. . Scenario 1: Not To Order If we decide not to order, we will not have to pay cf and cv since we order nothing actually. We just need to conside r understock and overstock risks. We will operate tomorrow with 10 items that we currently have if we decide not to order. E[Cost] =cu E[(D ? 10)+ ] + co E[(10 ? D)+ ] =10(E[D] ? 10) + 2(0) = $50 16 CHAPTER 1. NEWSVENDOR PROBLEM Note that in this case E[(10 ? D)+ ] = 0 because D is always greater than 10. 2. Scenario 2: To Order If we decide to order, we will order 5 items. We should pay cf and cv accordingly. Understock and overstock risks also exist in this case.Since we will order 5 items to lift up the inventory level to 15, we will run tomorrow with 15 items instead of 10 items if we decide to order. E[Cost] =cf + (15 ? 10)cv + cu E[(D ? 15)+ ] + co E[(15 ? D)+ ] =30 + 20 + 10(1. 25) + 2(1. 25) = $65 Since the expected cost of not ordering is lower than that of ordering, we should not order if we already have 10 items. It is obvious that if we have y ? items at hands right now, we should order nothing since we already possess the optimal amount of items for tomorrow’s op eration. It is also obvious that if we have nothing currently, we should order y ? items to prepare y ? tems for tomorrow. There should be a point between 0 and y ? where you are indi? erent between order and not ordering. Suppose you as a manager should give instruction to your assistant on when he/she should place an order and when should not. Instead of providing instructions for every possible current inventory level, it is easier to give your assistant just one number that separates the decision. Let us call that number the critical level of current inventory m? . If we have more than m? items at hands, the expected cost of not ordering will be lower than the expected cost of ordering, so we should not order.Conversely, if we have less than m? items currently, we should order. Therefore, when we have exactly m? items at hands right now, the expected cost of ordering should be equal to that of not ordering. We will use this intuition to obtain m? value. The decision process is s ummarized in the following ? gure. m* Critical level for placing an order y* Optimal order-up-to quantity Inventory If your current inventory lies here, you should order. Order up to y*. If your current inventory lies here, you should NOT order because your inventory is over m*. 1. 4. SIMULATION 17 Example 1. 6.Given the same settings with the previous example (cu = 10, cv = 4, co = 2, cf = 30), what is the critical level of current inventory m? that determines whether you should order or not? Answer: From the answer of the previous example, we can infer that the critical value should be less than 10, i. e. 0 < m? < 10. Suppose we currently own m? items. Now, evaluate the expected costs of the two scenarios: ordering and not ordering. 1. Scenario 1: Not Ordering E[Cost] =cu E[(D ? m? )+ ] + co E[(m? ? D)+ ] =10(E[D] ? m? ) + 2(0) = 150 ? 10m? 2. Scenario 2: Ordering In this case, we will order.Given that we will order, we will order y ? ?m? = 15 ? m? items. Therefore, we will start tomorrow with 15 items. E[Cost] =cf + (15 ? 10)cv + cu E[(D ? 15)+ ] + co E[(15 ? D)+ ] =30 + 4(15 ? m? ) + 10(1. 25) + 2(1. 25) = 105 ? 4m? At m? , (1. 14) and (1. 15) should be equal. 150 ? 10m? = 105 ? 4m? ? m? = 7. 5 units (1. 15) (1. 14) The critical value is 7. 5 units. If your current inventory is below 7. 5, you should order for tomorrow. If the current inventory is above 7. 5, you should not order. 1. 4 Simulation Generate 100 random demands from Uniform(10, 30). p = 10, cf = 30, cv = 4, h = 5, b = 3 1 p + b ? v 10 + 3 ? 4 = = p + b + h 10 + 3 + 5 2 The optimal order-up-to quantity from Theorem 1. 3 is 20. We will compare the performance between the policies of y = 15, 20, 25. Listing 1. 1: Continuous Uniform Demand Simulation # Set up parameters p=10;cf=30;cv=4;h=5;b=3 # How many random demands will be generated? n=100 # Generate n random demands from the uniform distribution 18 Dmd=runif(n,min=10,max=30) CHAPTER 1. NEWSVENDOR PROBLEM # Test the policy where we order 15 it ems for every period y=15 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 33. 4218 # Test the policy where we order 20 items for every period y=20 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 44. 37095 # Test the policy where we order 25 items for every period y=25 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 32. 62382 You can see the policy with y = 20 maximizes the 100-period average pro? t as promised by the theory. In fact, if n is relatively small, it is not guaranteed that we have maximized pro? t even if we run based on the optimal policy obtained from this section.The underlying assumption is that we should operate with this policy for a long time. Then, Theorem 1. 1 guarantees that the average pro? t will be maximized when we use the optimal ordering policy. Discrete demand case can also be simulated. Suppose the demand has the following distribution. All other parameters remain same. d Pr{D = d} 10 1 4 15 1 8 20 1 4 25 1 8 30 1 4 The theoretic optimal order-up-to quantity in this case is also 20. Let us test three policies: y = 15, 20, 25. Listing 1. 2: Discrete Demand Simulation # Set up parameters p=10;cf=30;cv=4;h=5;b=3 # How many random demands will be generated? =100 # Generate n random demands from the discrete demand distribution Dmd=sample(c(10,15,20,25,30),n,replace=TRUE,c(1/4,1/8,1/4,1/8,1/4)) # Test the policy where we order 15 items for every period y=15 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 19. 35 # Test the policy where we order 20 items for every period y=20 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 31. 05 # Test the policy where we order 25 items for every period 1. 5. EXERCISE y=25 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) > 26. 55 19There are other distributions such as triangular, normal, Poisson or binomial distributions available in R. When you do your senior project, for example, you will observe the demand for a departm ent or a factory. You ? rst approximate the demand using these theoretically established distributions. Then, you can simulate the performance of possible operation policies. 1. 5 Exercise 1. Show that (D ? y) + (y ? D)+ = y. 2. Let D be a discrete random variable with the following pmf. d Pr{D = d} Find (a) E[min(D, 7)] (b) E[(7 ? D)+ ] where x+ = max(x, 0). 3. Let D be a Poisson random variable with parameter 3.Find (a) E[min(D, 2)] (b) E[(3 ? D)+ ]. Note that pmf of a Poisson random variable with parameter ? is Pr{D = k} = ? k e . k! 5 1 10 6 3 10 7 4 10 8 1 10 9 1 10 4. Let D be a continuous random variable and uniformly distributed between 5 and 10. Find (a) E[max(D, 8)] (b) E[(D ? 8)? ] where x? = min(x, 0). 5. Let D be an exponential random variable with parameter 7. Find (a) E[max(D, 3)] 20 (b) E[(D ? 4)? ]. CHAPTER 1. NEWSVENDOR PROBLEM Note that pdf of an exponential random variable with parameter ? is fD (x) = ? e x for x ? 0. 6. David buys fruits and vegetables wholesal e and retails them at Davids Produce on La Vista Road.One of the more di? cult decisions is the amount of bananas to buy. Let us make some simplifying assumptions, and assume that David purchases bananas once a week at 10 cents per pound and retails them at 30 cents per pound during the week. Bananas that are more than a week old are too ripe and are sold for 5 cents per pound. (a) Suppose the demand for the good bananas follows the same distribution as D given in Problem 2. What is the expected pro? t of David in a week if he buys 7 pounds of banana? (b) Now assume that the demand for the good bananas is uniformly distributed between 5 and 10 like in Problem 4.What is the expected pro? t of David in a week if he buys 7 pounds of banana? (c) Find the expected pro? t if David’s demand for the good bananas follows an exponential distribution with mean 7 and if he buys 7 pounds of banana. 7. Suppose we are selling lemonade during a football game. The lemonade sells for $18 per g allon but only costs $3 per gallon to make. If we run out of lemonade during the game, it will be impossible to get more. On the other hand, leftover lemonade has a value of $1. Assume that we believe the fans would buy 10 gallons with probability 0. 1, 11 gallons with probability 0. , 12 gallons with probability 0. 4, 13 gallons with probability 0. 2, and 14 gallons with probability 0. 1. (a) What is the mean demand? (b) If 11 gallons are prepared, what is the expected pro? t? (c) What is the best amount of lemonade to order before the game? (d) Instead, suppose that the demand was normally distributed with mean 1000 gallons and variance 200 gallons2 . How much lemonade should be ordered? 8. Suppose that a bakery specializes in chocolate cakes. Assume the cakes retail at $20 per cake, but it takes $10 to prepare each cake. Cakes cannot be sold after one week, and they have a negligible salvage value.It is estimated that the weekly demand for cakes is: 15 cakes in 5% of the weeks, 1 6 cakes in 20% of the weeks, 17 cakes in 30% of the weeks, 18 cakes in 25% of the weeks, 19 cakes in 10% of the weeks, and 20 cakes in 10% of the weeks. How many cakes should the bakery prepare each week? What is the bakery’s expected optimal weekly pro? t? 1. 5. EXERCISE 21 9. A camera store specializes in a particular popular and fancy camera. Assume that these cameras become obsolete at the end of the month. They guarantee that if they are out of stock, they will special-order the camera and promise delivery the next day.In fact, what the store does is to purchase the camera from an out of state retailer and have it delivered through an express service. Thus, when the store is out of stock, they actually lose the sales price of the camera and the shipping charge, but they maintain their good reputation. The retail price of the camera is $600, and the special delivery charge adds another $50 to the cost. At the end of each month, there is an inventory holding cost of $25 fo r each camera in stock (for doing inventory etc). Wholesale cost for the store to purchase the cameras is $480 each. (Assume that the order can only be made at the beginning of the month. (a) Assume that the demand has a discrete uniform distribution from 10 to 15 cameras a month (inclusive). If 12 cameras are ordered at the beginning of a month, what are the expected overstock cost and the expected understock or shortage cost? What is the expected total cost? (b) What is optimal number of cameras to order to minimize the expected total cost? (c) Assume that the demand can be approximated by a normal distribution with mean 1000 and standard deviation 100 cameras a month. What is the optimal number of cameras to order to minimize the expected total cost? 10.Next month’s production at a manufacturing company will use a certain solvent for part of its production process. Assume that there is an ordering cost of $1,000 incurred whenever an order for the solvent is placed and the solvent costs $40 per liter. Due to short product life cycle, unused solvent cannot be used in following months. There will be a $10 disposal charge for each liter of solvent left over at the end of the month. If there is a shortage of solvent, the production process is seriously disrupted at a cost of $100 per liter short. Assume that the initial inventory level is m, where m = 0, 100, 300, 500 and 700 liters. a) What is the optimal ordering quantity for each case when the demand is discrete with Pr{D = 500} = Pr{D = 800} = 1/8, Pr{D = 600} = 1/2 and Pr{D = 700} = 1/4? (b) What is the optimal ordering policy for arbitrary initial inventory level m? (You need to specify the critical value m? in addition to the optimal order-up-to quantity y ? . When m ? m? , you make an order. Otherwise, do not order. ) (c) Assume optimal quantity will be ordered. What is the total expected cost when the initial inventory m = 0? What is the total expected cost when the initial inventory m = 700? 22 CHAPTER 1. NEWSVENDOR PROBLEM 11.Redo Problem 10 for the case where the demand is governed by the continuous uniform distribution varying between 400 and 800 liters. 12. An automotive company will make one last production run of parts for Part 947A and 947B, which are not interchangeable. These parts are no longer used in new cars, but will be needed as replacements for warranty work in existing cars. The demand during the warranty period for 947A is approximately normally distributed with mean 1,500,000 parts and standard deviation 500,000 parts, while the mean and standard deviation for 947B is 500,000 parts and 100,000 parts. (Assume that two demands are independent. Ignoring the cost of setting up for producing the part, each part costs only 10 cents to produce. However, if additional parts are needed beyond what has been produced, they will be purchased at 90 cents per part (the same price for which the automotive company sells its parts). Parts remaining at the end of the warr anty period have a salvage value of 8 cents per part. There has been a proposal to produce Part 947C, which can be used to replace either of the other two parts. The unit cost of 947C jumps from 10 to 14 cents, but all other costs remain the same. (a) Assuming 947C is not produced, how many 947A should be produced? b) Assuming 947C is not produced, how many 947B should be produced? (c) How many 947C should be produced in order to satisfy the same fraction of demand from parts produced in-house as in the ? rst two parts of this problem. (d) How much money would be saved or lost by producing 947C, but meeting the same fraction of demand in-house? (e) Is your answer to question (c), the optimal number of 947C to produce? If not, what would be the optimal number of 947C to produce? (f) Should the more expensive part 947C be produced instead of the two existing parts 947A and 947B. Why? Hint: compare the expected total costs.Also, suppose that D ? Normal( µ, ? 2 ). q xv 0 (x?  µ)2 1 e? 2? 2 dx = 2 q (x ?  µ) v 0 q (x?  µ)2 1 e? 2? 2 dx 2 + µ =  µ2 v 0 (q?  µ)2 (x?  µ)2 1 e? 2? 2 dx 2 t 1 v e? 2? 2 dt +  µPr{0 ? D ? q} 2 2 where, in the 2nd step, we changed variable by letting t = (x ?  µ)2 . 1. 5. EXERCISE 23 13. A warranty department manages the after-sale service for a critical part of a product. The department has an obligation to replace any damaged parts in the next 6 months. The number of damaged parts X in the next 6 months is assumed to be a random variable that follows the following distribution: x Pr{X = x} 100 . 1 200 . 2 300 . 5 400 . 2The department currently has 200 parts in stock. The department needs to decide if it should make one last production run for the part to be used for the next 6 months. To start the production run, the ? xed cost is $2000. The unit cost to produce a part is $50. During the warranty period of next 6 months, if a replacement request comes and the department does not have a part available in house, it has to buy a part from the spot-market at the cost of $100 per part. Any part left at the end of 6 month sells at $10. (There is no holding cost. ) Should the department make the production run? If so, how many items should it produce? 4. A store sells a particular brand of fresh juice. By the end of the day, any unsold juice is sold at a discounted price of $2 per gallon. The store gets the juice daily from a local producer at the cost of $5 per gallon, and it sells the juice at $10 per gallon. Assume that the daily demand for the juice is uniformly distributed between 50 gallons to 150 gallons. (a) What is the optimal number of gallons that the store should order from the distribution each day in order to maximize the expected pro? t each day? (b) If 100 gallons are ordered, what is the expected pro? t per day? 15. An auto company is to make one ? al purchase of a rare engine oil to ful? ll its warranty services for certain car models. The current price for the engine oil is $1 per g allon. If the company runs out the oil during the warranty period, it will purchase the oil from a supply at the market price of $4 per gallon. Any leftover engine oil after the warranty period is useless, and costs $1 per gallon to get rid of. Assume the engine oil demand during the warranty is uniformly distributed (continuous distribution) between 1 million gallons to 2 million gallons, and that the company currently has half million gallons of engine oil in stock (free of charge). a) What is the optimal amount of engine oil the company should purchase now in order to minimize the total expected cost? (b) If 1 million gallons are purchased now, what is the total expected cost? 24 CHAPTER 1. NEWSVENDOR PROBLEM 16. A company is obligated to provide warranty service for Product A to its customers next year. The warranty demand for the product follows the following distribution. d Pr{D = d} 100 . 2 200 . 4 300 . 3 400 . 1 The company needs to make one production run to satisfy the wa rranty demand for entire next year. Each unit costs $100 to produce; the penalty cost of a unit is $500.By the end of the year, the savage value of each unit is $50. (a) Suppose that the company has currently 0 units. What is the optimal quantity to produce in order to minimize the expected total cost? Find the optimal expected total cost. (b) Suppose that the company has currently 100 units at no cost and there is $20000 ? xed cost to start the production run. What is the optimal quantity to produce in order to minimize the expected total cost? Find the optimal expected total cost. 17. Suppose you are running a restaurant having only one menu, lettuce salad, in the Tech Square.You should order lettuce every day 10pm after closing. Then, your supplier delivers the ordered amount of lettuce 5am next morning. Store hours is from 11am to 9pm every day. The demand for the lettuce salad for a day (11am-9pm) has the following distribution. d Pr{D = d} 20 1/6 25 1/3 30 1/3 35 1/6 One lettu ce salad requires two units of lettuce. The selling price of lettuce salad is $6, the buying price of one unit of lettuce is $1. Of course, leftover lettuce of a day cannot be used for future salad and you have to pay 50 cents per unit of lettuce for disposal. (a) What is the optimal order-up-to quantity of lettuce for a day? b) If you ordered 50 units of lettuce today, what is the expected pro? t of tomorrow? Include the purchasing cost of 50 units of lettuce in your calculation. Chapter 2 Queueing Theory Before getting into Discrete-time Markov Chains, we will learn about general issues in the queueing theory. Queueing theory deals with a set of systems having waiting space. It is a very powerful tool that can model a broad range of issues. Starting from analyzing a simple queue, a set of queues connected with each other will be covered as well in the end. This chapter will give you the background knowledge when you read the required book, The Goal.We will revisit the queueing the ory once we have more advanced modeling techniques and knowledge. 2. 1 Introduction Think about a service system. All of you must have experienced waiting in a service system. One example would be the Student Center or some restaurants. This is a human system. A bit more automated service system that has a queue would be a call center and automated answering machines. We can imagine a manufacturing system instead of a service system. These waiting systems can be generalized as a set of bu? ers and servers. There are key factors when you try to model such a system.What would you need to analyze your system? †¢ How frequently customers come to your system? > Inter-arrival Times †¢ How fast your servers can serve the customers? > Service Times †¢ How many servers do you have? > Number of Servers †¢ How large is your waiting space? > Queue Size If you can collect data about these metrics, you can characterize your queueing system. In general, a queueing system can be denoted as follows. G/G/s/k 25 26 CHAPTER 2. QUEUEING THEORY The ? rst letter characterizes the distribution of inter-arrival times. The second letter characterizes the distribution of service times.The third number denotes the number of servers of your queueing system. The fourth number denotes the total capacity of your system. The fourth number can be omitted and in such case it means that your capacity is in? nite, i. e. your system can contain any number of people in it up to in? nity. The letter â€Å"G† represents a general distribution. Other candidate characters for this position is â€Å"M† and â€Å"D† and the meanings are as follows. †¢ G: General Distribution †¢ M: Exponential Distribution †¢ D: Deterministic Distribution (or constant) The number of servers can vary from one to many to in? nity.The size of bu? er can also be either ? nite or in? nite. To simplify the model, assume that there is only a single server and we have in? ni te bu? er. By in? nite bu? er, it means that space is so spacious that it is as if the limit does not exist. Now we set up the model for our queueing system. In terms of analysis, what are we interested in? What would be the performance measures of such systems that you as a manager should know? †¢ How long should your customer wait in line on average? †¢ How long is the waiting line on average? There are two concepts of average. One is average over time.This applies to the average number of customers in the system or in the queue. The other is average over people. This applies to the average waiting time per customer. You should be able to distinguish these two. Example 2. 1. Assume that the system is empty at t = 0. Assume that u1 = 1, u2 = 3, u3 = 2, u4 = 3, v1 = 4, v2 = 2, v3 = 1, v4 = 2. (ui is ith customer’s inter-arrival time and vi is ith customer’s service time. ) 1. What is the average number of customers in the system during the ? rst 10 minutes? 2 . What is the average queue size during the ? rst 10 minutes? 3.What is the average waiting time per customer for the ? rst 4 customers? Answer: 1. If we draw the number of people in the system at time t with respect to t, it will be as follows. 2. 2. LINDLEY EQUATION 3 2 1 0 27 Z(t) 0 1 2 3 4 5 6 7 8 9 10 t E[Z(t)]t? [0,10] = 1 10 10 Z(t)dt = 0 1 (10) = 1 10 2. If we draw the number of people in the queue at time t with respect to t, it will be as follows. 3 2 1 0 Q(t) 0 1 2 3 4 5 6 7 8 9 10 t E[Q(t)]t? [0,10] = 1 10 10 Q(t)dt = 0 1 (2) = 0. 2 10 3. We ? rst need to compute waiting times for each of 4 customers. Since the ? rst customer does not wait, w1 = 0.Since the second customer arrives at time 4, while the ? rst customer’s service ends at time 5. So, the second customer has to wait 1 minute, w2 = 1. Using the similar logic, w3 = 1, w4 = 0. E[W ] = 0+1+1+0 = 0. 5 min 4 2. 2 Lindley Equation From the previous example, we now should be able to compute each customerâ€℠¢s waiting time given ui , vi . It requires too much e? ort if we have to draw graphs every time we need to compute wi . Let us generalize the logic behind calculating waiting times for each customer. Let us determine (i + 1)th customer’s waiting 28 CHAPTER 2. QUEUEING THEORY time.If (i + 1)th customer arrives after all the time ith customer waited and got served, (i + 1)th customer does not have to wait. Its waiting time is 0. Otherwise, it has to wait wi + vi ? ui+1 . Figure 2. 1, and Figure 2. 2 explain the two cases. ui+1 wi vi wi+1 Time i th arrival i th service start (i+1)th arrival i th service end Figure 2. 1: (i + 1)th arrival before ith service completion. (i + 1)th waiting time is wi + vi ? ui+1 . ui+1 wi vi Time i th arrival i th service start i th service end (i+1)th arrival Figure 2. 2: (i + 1)th arrival after ith service completion. (i + 1)th waiting time is 0.Simply put, wi+1 = (wi + vi ? ui+1 )+ . This is called the Lindley Equation. Example 2. 2. Given the f ollowing inter-arrival times and service times of ? rst 10 customers, compute waiting times and system times (time spent in the system including waiting time and service time) for each customer. ui = 3, 2, 5, 1, 2, 4, 1, 5, 3, 2 vi = 4, 3, 2, 5, 2, 2, 1, 4, 2, 3 Answer: Note that system time can be obtained by adding waiting time and service time. Denote the system time of ith customer by zi . ui vi wi zi 3 4 0 4 2 3 2 5 5 2 0 2 1 5 1 6 2 2 4 6 4 2 2 4 1 1 3 4 5 4 0 4 3 2 1 3 2 3 1 4 2. 3. TRAFFIC INTENSITY 9 2. 3 Suppose Tra? c Intensity E[ui ] =mean inter-arrival time = 2 min E[vi ] =mean service time = 4 min. Is this queueing system stable? By stable, it means that the queue size should not go to the in? nity. Intuitively, this queueing system will not last because average service time is greater than average inter-arrival time so your system will soon explode. What was the logic behind this judgement? It was basically comparing the average inter-arrival time and the average serv ice time. To simplify the judgement, we come up with a new quantity called the tra? c intensity. De? nition 2. 1 (Tra? Intensity). Tra? c intensity ? is de? ned to be ? = 1/E[ui ] ? =  µ 1/E[vi ] where ? is the arrival rate and  µ is the service rate. Given a tra? c intensity, it will fall into one of the following three categories. †¢ If ? < 1, the system is stable. †¢ If ? = 1, the system is unstable unless both inter-arrival times and service times are deterministic (constant). †¢ If ? > 1, the system is unstable. Then, why don’t we call ? utilization instead of tra? c intensity? Utilization seems to be more intuitive and user-friendly name. In fact, utilization just happens to be same as ? if ? < 1.However, the problem arises if ? > 1 because utilization cannot go over 100%. Utilization is bounded above by 1 and that is why tra? c intensity is regarded more general notation to compare arrival and service rates. De? nition 2. 2 (Utilization). Utilization is de? ned as follows. Utilization = ? , 1, if ? < 1 if ? ? 1 Utilization can also be interpreted as the long-run fraction of time the server is utilized. 2. 4 Kingman Approximation Formula Theorem 2. 1 (Kingman’s High-tra? c Approximation Formula). Assume the tra? c intensity ? < 1 and ? is close to 1. The long-run average waiting time in 0 a queue E[W ] ? E[vi ] CHAPTER 2. QUEUEING THEORY ? 1 c2 + c2 a s 2 where c2 , c2 are squared coe? cient of variation of inter-arrival times and service a s times de? ned as follows. c2 = a Var[u1 ] (E[u1 ]) 2, c2 = s Var[v1 ] (E[v1 ]) 2 Example 2. 3. 1. Suppose inter-arrival time follows an exponential distribution with mean time 3 minutes and service time follows an exponential distribution with mean time 2 minutes. What is the expected waiting time per customer? 2. Suppose inter-arrival time is constant 3 minutes and service time is also constant 2 minutes. What is the expected waiting time per customer?Answer: 1. Tra? c intensity is ? = 1/E[ui ] 1/3 2 ? = = = .  µ 1/E[vi ] 1/2 3 Since both inter-arrival times and service times are exponentially distributed, E[ui ] = 3, Var[ui ] = 32 = 9, E[vi ] = 2, Var[vi ] = 22 = 4. Therefore, c2 = Var[ui ]/(E[ui ])2 = 1, c2 = 1. Hence, s a E[W ] =E[vi ] =2 ? c2 + c2 s a 1 2 2/3 1+1 = 4 minutes. 1/3 2 2. Tra? c intensity remains same, 2/3. However, since both inter-arrival times and service times are constant, their variances are 0. Thus, c2 = a c2 = 0. s E[W ] = 2 2/3 1/3 0+0 2 = 0 minutes It means that none of the customers will wait upon their arrival.As shown in the previous example, when the distributions for both interarrival times and service times are exponential, the squared coe? cient of variation term becomes 1 from the Kingman’s approximation formula and the formula 2. 5. LITTLE’S LAW 31 becomes exact to compute the average waiting time per customer for M/M/1 queue. E[W ] =E[vi ] ? 1 Also note that if inter-arrival time or service time distribution is deterministic, c2 or c2 becomes 0. a s Example 2. 4. You are running a highway collecting money at the entering toll gate. You reduced the utilization level of the highway from 90% to 80% by adopting car pool lane.How much does the average waiting time in front of the toll gate decrease? Answer: 0. 8 0. 9 = 9, =4 1 ? 0. 9 1 ? 0. 8 The average waiting time in in front of the toll gate is reduced by more than a half. The Goal is about identifying bottlenecks in a plant. When you become a manager of a company and are running a expensive machine, you usually want to run it all the time with full utilization. However, the implication of Kingman formula tells you that as your utilization approaches to 100%, the waiting time will be skyrocketing. It means that if there is any uncertainty or random ? ctuation input to your system, your system will greatly su? er. In lower ? region, increasing ? is not that bad. If ? near 1, increasing utilization a little bit can lead to a disaster. Atl anta, 10 years ago, did not su? er that much of tra? c problem. As its tra? c infrastructure capacity is getting closer to the demand, it is getting more and more fragile to uncertainty. A lot of strategies presented in The Goal is in fact to decrease ?. You can do various things to reduce ? of your system by outsourcing some process, etc. You can also strategically manage or balance the load on di? erent parts of your system.You may want to utilize customer service organization 95% of time, while utilization of sales people is 10%. 2. 5 Little’s Law L = ? W The Little’s Law is much more general than G/G/1 queue. It can be applied to any black box with de? nite boundary. The Georgia Tech campus can be one black box. ISyE building itself can be another. In G/G/1 queue, we can easily get average size of queue or service time or time in system as we di? erently draw box onto the queueing system. The following example shows that Little’s law can be applied in broade r context than the queueing theory. 32 CHAPTER 2. QUEUEING THEORY Example 2. 5 (Merge of I-75 and I-85).Atlanta is the place where two interstate highways, I-75 and I-85, merge and cross each other. As a tra? c manager of Atlanta, you would like to estimate the average time it takes to drive from the north con? uence point to the south con? uence point. On average, 100 cars per minute enter the merged area from I-75 and 200 cars per minute enter the same area from I-85. You also dispatched a chopper to take a aerial snapshot of the merged area and counted how many cars are in the area. It turned out that on average 3000 cars are within the merged area. What is the average time between entering and exiting the area per vehicle?Answer: L =3000 cars ? =100 + 200 = 300 cars/min 3000 L = 10 minutes ? W = = ? 300 2. 6 Throughput Another focus of The Goal is set on the throughput of a system. Throughput is de? ned as follows. De? nition 2. 3 (Throughput). Throughput is the rate of output ? ow from a system. If ? ? 1, throughput= ?. If ? > 1, throughput=  µ. The bounding constraint of throughput is either arrival rate or service rate depending on the tra? c intensity. Example 2. 6 (Tandem queue with two stations). Suppose your factory production line has two stations linked in series. Every raw material coming into your line should be processed by Station A ? rst.Once it is processed by Station A, it goes to Station B for ? nishing. Suppose raw material is coming into your line at 15 units per minute. Station A can process 20 units per minute and Station B can process 25 units per minute. 1. What is the throughput of the entire system? 2. If we double the arrival rate of raw material from 15 to 30 units per minute, what is the throughput of the whole system? Answer: 1. First, obtain the tra? c intensity for Station A. ?A = ? 15 = = 0. 75  µA 20 Since ? A < 1, the throughput of Station A is ? = 15 units per minute. Since Station A and Station B is linked in series, the throughput of Station . 7. SIMULATION A becomes the arrival rate for Station B. ?B = ? 15 = = 0. 6  µB 25 33 Also, ? B < 1, the throughput of Station B is ? = 15 units per minute. Since Station B is the ? nal stage of the entire system, the throughput of the entire system is also ? = 15 units per minute. 2. Repeat the same steps. ?A = 30 ? = = 1. 5  µA 20 Since ? A > 1, the throughput of Station A is  µA = 20 units per minute, which in turn becomes the arrival rate for Station B. ?B =  µA 20 = 0. 8 =  µB 25 ?B < 1, so the throughput of Station B is  µA = 20 units per minute, which in turn is the throughput of the whole system. 2. 7 SimulationListing 2. 1: Simulation of a Simple Queue and Lindley Equation N = 100 # Function for Lindley Equation lindley = function(u,v){ for (i in 1:length(u)) { if(i==1) w = 0 else { w = append(w, max(w[i-1]+v[i-1]-u[i], 0)) } } return(w) } # # u v CASE 1: Discrete Distribution Generate N inter-arrival times and service times = sample( c(2,3,4),N,replace=TRUE,c(1/3,1/3,1/3)) = sample(c(1,2,3),N,replace=TRUE,c(1/3,1/3,1/3)) # Compute waiting time for each customer w = lindley(u,v) w # CASE 2: Deterministic Distribution # All inter-arrival times are 3 minutes and all service times are 2 minutes # Observe that nobody waits in this case. 4 u = rep(3, 100) v = rep(2, 100) w = lindley(u,v) w CHAPTER 2. QUEUEING THEORY The Kingman’s approximation formula is exact when inter-arrival times and service times follow iid exponential distribution. E[W ] = 1  µ ? 1 We can con? rm this equation by simulating an M/M/1 queue. Listing 2. 2: Kingman Approximation # lambda = arrival rate, mu = service rate N = 10000; lambda = 1/10; mu = 1/7 # Generate N inter-arrival times and service times from exponential distribution u = rexp(N,rate=lambda) v = rexp(N,rate=mu) # Compute the average waiting time of each customer w = lindley(u,v) mean(w) > 16. 0720 # Compare with Kingman approximation rho = lambda/mu (1/mu)*(rho/(1-rho)) > 16. 33333 The Kingman’s approximation formula becomes more and more accurate as N grows. 2. 8 Exercise 1. Let Y be a random variable with p. d. f. ce? 3s for s ? 0, where c is a constant. (a) Determine c. (b) What is the mean, variance, and squared coe? cient of variation of Y where the squared coe? cient of variation of Y is de? ned to be Var[Y ]/(E[Y ]2 )? 2. Consider a single server queue. Initially, there is no customer in the system.Suppose that the inter-arrival times of the ? rst 15 customers are: 2, 5, 7, 3, 1, 4, 9, 3, 10, 8, 3, 2, 16, 1, 8 2. 8. EXERCISE 35 In other words, the ? rst customer will arrive at t = 2 minutes, and the second will arrive at t = 2 + 5 minutes, and so on. Also, suppose that the service time of the ? rst 15 customers are 1, 4, 2, 8, 3, 7, 5, 2, 6, 11, 9, 2, 1, 7, 6 (a) Compute the average waiting time (the time customer spend in bu? er) of the ? rst 10 departed customers. (b) Compute the average system time (waiting time plus service time) of the ? st 10 departed customers. (c) Compute the average queue size during the ? rst 20 minutes. (d) Compute the average server utilization during the ? rst 20 minutes. (e) Does the Little’s law of hold for the average queue size in the ? rst 20 minutes? 3. We want to decide whether to employ a human operator or buy a machine to paint steel beams with a rust inhibitor. Steel beams are produced at a constant rate of one every 14 minutes. A skilled human operator takes an average time of 700 seconds to paint a steel beam, with a standard deviation of 300 seconds.An automatic painter takes on average 40 seconds more than the human painter to paint a beam, but with a standard deviation of only 150 seconds. Estimate the expected waiting time in queue of a steel beam for each of the operators, as well as the expected number of steel beams waiting in queue in each of the two cases. Comment on the e? ect of variability in service time. 4. The arrival rate of customers to an ATM machi ne is 30 per hour with exponentially distirbuted in- terarrival times. The transaction times of two customers are independent and identically distributed.Each transaction time (in minutes) is distributed according to the following pdf: f (s) = where ? = 2/3. (a) What is the average waiting for each customer? (b) What is the average number of customers waiting in line? (c) What is the average number of customers at the site? 5. A production line has two machines, Machine A and Machine B, that are arranged in series. Each job needs to processed by Machine A ? rst. Once it ? nishes the processing by Machine A, it moves to the next station, to be processed by Machine B. Once it ? nishes the processing by Machine B, it leaves the production line.Each machine can process one job at a time. An arriving job that ? nds the machine busy waits in a bu? er. 4? 2 se? 2? s , 0, if s ? 0 otherwise 36 CHAPTER 2. QUEUEING THEORY (The bu? er sizes are assumed to be in? nite. ) The processing times fo r Machine A are iid having exponential distribution with mean 4 minutes. The processing times for Machine B are iid with mean 2 minutes. Assume that the inter-arrival times of jobs arriving at the production line are iid, having exponential distribution with mean of 5 minutes. (a) What is the utilization of Machine A?What is the utilization of Machine B? (b) What is the throughput of the production system? (Throughput is de? ned to be the rate of ? nal output ? ow, i. e. how many items will exit the system in a unit time. ) (c) What is the average waiting time at Machine A, excluding the service time? (d) It is known the average time in the entire production line is 30 minutes per job. What is the long-run average number of jobs in the entire production line? (e) Suppose that the mean inter-arrival time is changed to 1 minute. What are the utilizations for Machine A and Machine B, respectively?What is the throughput of the production system? 6. An auto collision shop has roughly 10 cars arriving per week for repairs. A car waits outside until it is brought inside for bumping. After bumping, the car is painted. On the average, there are 15 cars waiting outside in the yard to be repaired, 10 cars inside in the bump area, and 5 cars inside in the painting area. What is the average length of time a car is in the yard, in the bump area, and in the painting area? What is the average length of time from when a car arrives until it leaves? 7. A small bank is sta? d by a single server. It has been observed that, during a normal business day, the inter-arrival times of customers to the bank are iid having exponential distribution with mean 3 minutes. Also, the the processing times of customers are iid having the following distribution (in minutes): x Pr{X = x} 1 1/4 2 1/2 3 1/4 An arrival ? nding the server busy joins the queue. The waiting space is in? nite. (a) What is the long-run fraction of time that the server is busy? (b) What the the long-run average waiting tim e of each customer in the queue, excluding the processing time? c) What is average number of customers in the bank, those in queue plus those in service? 2. 8. EXERCISE (d) What is the throughput of the bank? 37 (e) If the inter-arrival times have mean 1 minute. What is the throughput of the bank? 8. You are the manager at the Student Center in charge of running the food court. The food court is composed of two parts: cooking station and cashier’s desk. Every person should go to the cooking station, place an order, wait there and pick up ? rst. Then, the person goes to the cashier’s desk to check out. After checking out, the person leaves the food court.The coo

External causes for Enron to collapse Essay

1) Deregulation Deregulation of the U.S. energy industry made possible Enron’s emergence as a major corporation, but also ultimately may have contributed to its collapse. The company successfully seized the opportunity created by deregulation to create a new business as a market maker in natural gas and other commodities. Enron successfully influenced policymakers to exempt the company from various regulatory rules, for example in the field of energy derivatives. This allowed Enron to enter various trading markets with virtually no government oversight. Arguably, regulation might have prevented Enron from taking some of the risks and making some of the mistakes which it did. While deregulation may initially have helped Enron, by allowing it to create and enter new markets, it later hurt the company by removing the very restraints that might have kept it from becoming fatally overextended. 2) Lax regulatory enforcement Arguably, government regulatory agencies failed to exercise sufficient oversight or to enforce the rules that were on the books. Regulatory bodies that failed to enforce the rules governing Enron’s actions included the Securities and Exchange Commission (SEC), the Federal Energy Regulatory Commission (FERC), and the Commodities Futures Trading Commission (CFEC). 3) Weak and ambiguous accounting standards Hindsight makes it fairly clear that the accounting standards promulgated by the Financial Accounting Standards Board (FASB) were too weak and too ambiguous with respect to the complex trading transactions and financial structures that Enron established and operated. Two areas stand out as ones of particular concern. First, the rules apparently permitted the widespread use of market-to-market (MTM) accounting in areas for which it was not originally intended. Second, the 3 percent rule for outside ownership of SPEs was arguably too low to maintain genuine independence. An underlying issue was that corporate practice (e.g., sophisticated online trading of complex financial derivatives) had outpaced the work of the rules makers,  leading to the application of rules in situations for which they were not originally designed. 4) A lack of independence on the part of the company’s auditors and law firms working for the company A key external issue was conflict of interest on the part of accounting and law firms working for Enron. Arthur Andersen, the company’s accounting firm, arguably had a conflict of interest in that Arthur Andersen provided both external audit services and internal consulting for Enron. If Arthur Andersen were to challenge the propriety of Enron’s financial statements in its annual audit, it ran the risk of jeopardizing its lucrative consulting and â€Å"inside† accounting work for its client. Moreover, relations between the two firms were unusually close, possibly undermining Arthur Andersen’s objectivity and independence. Similarly, Vinson & Elkins, Enron’s outside law firm, was seemingly under pressure not to question the legality of the Special Purpose Entities (SPEs) too closely, since Enron was a major client of the firm. 5) Inadequate campaign finance and lobbyist rules. Enron made extensive legal use of various techniques of political influence, including engaging the services of lobbyists, making extensive contributions to political campaigns, particularly using soft money, and hiring former government officials. One of the external causes, then, may have been campaign finance and other rules that permitted such legal exercise of corporate influence in policymaking. 6) Weak stakeholder oversight. A case can be made that external stakeholders–especially large institutional investors such as pension and mutual funds–failed to exercise due diligence. These institutional investors were happy to make handsome returns on their extensive investments in Enron in the late 1990s, but failed to become actively involved in corporate governance at the company until it was  too late.

Battle Analysis for Bull Run

The battle itself was fought on July 21st, 1861, though the Union Army began executing its movements to Virginia almost a week prior. The Civil War divided the states in simple terms of a Union north and a Confederate south, with a couple undecided states in the middle. The President of the Union was Abraham Lincoln and the Confederate President was Jefferson Davis. Months prior to Bull Run President Lincoln had appointed Brigadier General Irwin McDowell to command the Army of Northeastern Virginia. McDowell was a Mexican-American War veteran and West Point graduate. The commander of the Confederate Army of the Potomac was Brigadier General P. G. T. Beauregard, who was dubbed â€Å"The Hero of Sumter. † He was also commended for valor in the Mexican-American war and like McDowell, a graduate of West Point. The two were classmates at one point. Only months after the start of the war at Fort Sumter, the Northern public pressed to march and capture the Confederate capital of Richmond, Virginia, which could bring an early end to the war. Against his better judgment, BG McDowell yielded to the political pressure and on July 16, 1861, the general departed Washington with the largest field army yet gathered on the North American continent. The Confederates found themselves at a disadvantage in mass initially, and BG McDowell wanted to keep that advantage. He ordered Union MG Robert Pattersons Army to engage BG Joseph Johnstons Army in the Shenandoah Valley, about 50 miles northwest of Manassas. The Union objective was to overwhelm the Confederate forces with a distraction flank attack to the right and a swift surprise flank to the left. With the reinforcements choked off, BG McDowell’s ambitious plan would put his Army in the Confederate capital by the end of the day. The Confederates, however, had been planning to attack the Union left, and if the attack had gone as planned it might have led to a clockwise rotation of the forces. Hundreds of excited spectators in horse-drawn carriages flocked from Washington D. C. to Manassas to watch what they thought to be a speedy Union Army defeat the Confederacy. Both the spectators and the Union Army would leave Bull Run in a hectic retreat back to Washington D. C. Each force had two Armies, one to the east and one to the west. For the Union, BG McDowell commanded the 36,000 Army of Northeastern Virginia Union troops in the east. MG Patterson commanded the 18,000 troops in the west. Within BG McDowell’s Army of five divisions there were several elements that consisted of: The 11th, 13th, 14th, 38th, and 69th New York, the 3rd, 4th, and 5th Maine, the 1st Minnesota, the 5th and 11th Massachusetts, the 1st Michigan, the 1st Vermont, the 2nd Wisconsin, with Griffin and Ricketts Artillery Brigades. BG Beauregard’s Confederate Army of the Potomac consisted of 21,000 troops in the east. BG Johnston’s four Brigades of 12,800 troops were in the Shenandoah Valley to the west and were critical reinforcements. BG Beauregard’s force of six Brigades consisted of: The 2nd, 4th, 5th, 8th, 18th, 27th, 33rd, and 49th Virginia, the Hampton Legion, the 6th North Carolina, the 7th Georgia, the 4th Alabama, Stuart’s Calvary, Elzey Regiment, Early Regiment, and the 7th and 8th South Carolina. The weapon technology used was fairly similar for both sides. Both the Union and Confederate Army relied on simple single-shot Pattern 1853 Enfield Muskets for their infantrymen. The revolvers used by the Union were mainly the new Colt Army Model 1860, and the Confederates preferred the older Colt 1851 Navy Revolver. A variety of bayonets were also an integral part of the infantrymans gear. Typically, these were socket or ring bayonets, intended to be attached to the end of the musket or rifle, and not wielded separately like a knife. The Confederate Calvary would also employ a Sabre, which was a long, lightweight single-edged slashing sword. Field Artillery also played an important role for both sides. The Union used 10-30 pound Parrott Rifles, 12 pound Napoleon smoothbores, 12 pound Howitzers, and 13 pound James Rifles. The Confederates had 6 pound guns, 6 pound rifles, 12 pound Howitzer, 10 pound Parrott Rifles, and 6 pound Cadet Guns. Both Generals had planned offensives. Much of the intelligence was concentrated on reporting the mass of the opposing forces rather than each other’s strategy. BG McDowell wanted a concentrated attack on the Confederate left flank, while BG Beauregard had planned to strike the Union left flank. From Washington D. C. the Union troops had marched southwest into Virginia, and it was at Centreville on July 20th, that BG McDowell decided to rest his weary, overheated troops and concentrate his forces. The same day, BG Johnston’s troops to the west in the Shenandoah Valley received word of the Union advances and they immediate slipped away to reinforce BG Beauregard. He never met MG Patterson’s forces. An hour after BG Johnston’s departure, MG Patterson wired BG McDowell saying he had managed to keep BG Johnston’s Army in the Shenandoah. Shortly after entering Centreville on the 20th, BG Tyler would disobey his orders and send his troops to attack the Confederate front along Bull Run. The attack was easily repulsed. With the Confederate troops dug in across the bank of Bull Run, and the majority of BG Beauregard’s force were behind them. The Union troops marched from Centreville at 0230 on July 21st. BG Tyler was ordered to initiate a diversion to the northwest at Stony Bridge at 0600. The diversion was quickly crushed by COL Evan’s Confederate forces and the feign fails. At 0830 the bayonets of McDowell’s flanking troops were spotted by one of COL Evan’s soldiers and he was warned of the Union plan to flank him. BG McDowell’s troops continued on to the left down bad roads, which would destroy his timeframe to ford Bull Run at Sudley Springs. COL Heinzelman’s Union division also missed the trail at Poplar Ford, and they were forced to stack up behind COL Hunter’s division also fording at Sudley Springs, further downstream. They arrived there at 0930, hours behind schedule.

A report on a piece of qualitative research - The Impact of Price Essay - 1

A report on a piece of qualitative research - The Impact of Price Changes on the Brand Equity of Toyota in Saudi Arabia - Essay Example This chapter will explain the research methodology used for achieving the goals of this research study and the justifications for the methodology chosen. A qualitative case study is chosen for accomplishing the objectives of this study since it emphasizes the experiences and perspectives of the consumers who have purchased Toyota vehicles or other similar vehicles in Saudi Arabia. In this regard a qualitative study involves the exploration of a specific social setting or phenomenon involving the collection of â€Å"detailed, in-depth data† from multiple sources including interviews, observations, open-ended questionnaires and secondary data such as reports and records (Creswell, 2009, p. 43). A qualitative case study allows for an â€Å"analysis of a† specific â€Å"phenomenon† which may be programs, institutions, individuals, or social groups (Merriam, 2009, p. x). Since this study involves the collection of data relative to the experiences and perspectives of Toyota consumers in Saudi Arabia relative to price changes and its impact on brand and value, this study is a qualitative case study. Specifically, this s tudy investigates a bonded system. Choi and Hong (2002) conducted a qualitative case study using in-depth, semi-structured interviews among managers in Acura, Honda and DaimlerChrysler to determine the impact of operations cost on the structure of the supply network. In this regard, a qualitative case study was useful for gaining an understanding of how cost influenced the behavior of those directly impacted by costs. Likewise, my case study seeks to determine the impact of price changes on the behavior of consumers who are directly impacted by price changes. Beach, Muhlemann, Price, Paterson and Sharp (2001) argue that any research that can impact production channels and management decisions in production management is best suited to qualitative studies. This is because

Racism Essay Example | Topics and Well Written Essays - 500 words

Racism - Essay Example Even with the thought the foreign workers are crucial in his country's economic well-being, the student still felt bothered and frustrated. What is more significant is that many students agreed with the fellow and his statement. The Singaporean experience is an excellent example of race relations. Its population is small and the dynamics of the relationship within its society is easily recognized because of it. Any conflict or significant development immediately comes to the surface. Today, more than a quarter of its population is composed of foreign residents, who, for their part, come from various countries and cultures. (Chong 2010, p. 145) By inviting all these peoples into the country, the Singaporean government is forcing them to live side by side each other in addition to living within the Singaporean community. The student's perspective at NTU told much about racial prejudice. As a citizen, he expects to be put above the rest, particularly in the governmental agenda. With the sizable number of foreigners, however, his economic and political influence in policy networks is threatened. This aggravates his personal racial biases. The result is unfortunate if we are to imagine how this student and similar Singaporeans would interact with other nationalities.

Autobiographical essay------Describe your past experiences and future Essay

Autobiographical ------Describe your past experiences and future plans, showing how the degree ( computer information syste - Essay Example I started as a database operator and gradually rose to be senior database administrator. I have worked for IBM, GM, Star Alliance (Sheraton Group). I also have the appropriate experience in sql server, Oracle, db2, sysbase, people’s soft and SAP. While pursing my bachelor’s degree, I gained expertise in almost all the important programming languages like C, C++, Java and Assembly Language. I possess over 9 years of Oracle Production DBA and MS SQL Server 2000, 2005 experience with Oracle 10g, 9i, 8i and UNIX/Red Hat Linux/Fedore/ Sun Solaris/AIX and Microsoft 2000 and 2003. I acquired over 2 years experience as an Oracle Application Server, Forms, Reports, and Data Warehouse. I hold an excellent DBA expertise with Production Database and Data Warehouse Administration. I am also proficient in Administration, Productivity and Modeling Tools. I am quite proficient at programming and Production Database Daily Administration. I do have the expertise in High Availability Tech nologies like Oracle 9i RAC, Oracle 10g RAC, Data Guard/Standby Database, Database Replication/Clone including the expertise in Database Health Maintenance. . Not to say, my panache for gaining technical proficiency was always equally matched by my drive to sharpen my soft skills.

Contract law assessed coursework Essay Example | Topics and Well Written Essays - 1500 words

Contract law assessed coursework - Essay Example These apartments prove to be in great demand, and Eileen and Paul have a change of mind and ask Anne to either pay rent or vacate the flat. They also ask Mike to enhance the rent with retrospective effect, from the time of completion of the flats. For advising Eileen and Paul in respect of their problems with Mike and Anne, the following issues have to be considered. Whether, there is any legally binding contract between Anne and Eileen and Paul. Whether Eileen and Paul can demand the arrears of rent from Mike, with retrospective effect. The principle of Promissory Estoppel has to be examined for answering these issues. In general, consideration is a very important factor in contracts, and renders a promise enforceable. The promisee has to provide something in exchange for the promise, which is termed as consideration. In the absence of consideration, a promise is in general, rendered unenforceable. In essence, the promisee has to provide something to the promisor, in exchange for th e promise (Capper, 2008, p. 105). In our problem, Mike was paying rent at a lower rate, since the construction work was in progress in the building. However, he had been paying a much higher rent, initially. The following case law indicates the attitude of the courts in deciding issues related to the principle of promissory estoppel. In Williams v Roffey, the court held that the performance of a previous contractual duty was consideration for a subsequent contract. The reasoning behind this ruling is the proper performance of the original contractual obligation would give rise to a practical benefit for the parties to the contract (Williams v Roffey Brothers & Nicholls (Contractors) Ltd, 1991).If the promisee performs something with regard to an earlier contract, which benefits the promisor, then the performance is considered as good consideration. The only requirement is that the performance must have resulted in some practical benefit to the promisor. However, in Re Selectmove Ltd , the appellate court held that the promise of the plaintiff to the Inland Revenue required the payment of arrears. As there were no immediate payments by the plaintiff, there was no good consideration (Re Selectmove Ltd, 1995). In Central London Property Trust Ltd V. High Trees House Ltd, sparse occupancy, occasioned by the World War, had caused the landlord to charge reduced rent. Subsequently, occupancy increased to the extent that there were no vacant flats. The tenants opposed the landlord’s attempt to charge the higher rent, and the court ruled that the tenants had to pay higher rent from the time of full occupancy (Central London Property Trust Ltd v High Trees House Ltd, 1947). This case constitutes the best decision in the Commonwealth and England. It deems reliance to be the basis for altering a contract (Teeven, 2002, p. 350). However, higher rent was not permitted from the very beginning; this constitutes the principle of promissory estoppels. Moreover, in Tool Me tal Manufacturing Co Ltd v. Tungsten Electric Co Ltd, the patent owners had promised to defer periodic payments due to them, from the outbreak of war (Tool Metal Manufacturing Co Ltd v. Tungsten Electric Co Ltd , 1955). The House of Lords held this promise to be binding during the period of suspension. This decision suggests that the principle of estoppel is in general, suspensory. As per the decision in Central London Property Trust Ltd, Mike has to pay the enhanced rent only after the completion of the construction of flats. However, Eileen and Paul demanded him to

Human Resource Management & Organisational analysis MSc Personal Statement

Human Resource Management & Organisational analysis MSc - Personal Statement Example My goal is to practice the best Human Resource in the organizations I will work for. I am a graduate from Hult International Business School in London where I studied Bachelor of International Business Administration and specialized in Management. I have worked in the bank of Respublika as an accounting specialist and I enjoyed my role as I got to interact with customers and making transactions for them. I have also worked for Look magazine in the UK and I was glad to share ideas on how to redo their Website and journal to make accessibility for customers easy as well as sharing ideas on how to increase profits. I trust that my decision to purse the course in King’s College is one of best decisions I have made in my life. This is because of professional lecturers, state of art equipments, the conducive learning environment, and the good reputation of the institution. I am confident that as I undertake the course in King’s College, I will accomplish my goals and I will be in a better position to face the Human Resource practices in any organization. I intend to work hard and in the end achieve the best results and be marketable

Strategic Plan Developement Paper Essay Example | Topics and Well Written Essays - 500 words

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Inter cultural communication Assignment Example | Topics and Well Written Essays - 500 words

Inter cultural communication - Assignment Example Furthermore, as has been discussed during the course of the semester, before being level of effective intercultural communication to take place, or the level of the nation all that, a relationship must at first exists. The depth and complexity of this relationship varies between individuals that issue(s) that are being discussed. However, in the event that the stakeholder wishes to have an effective level of communication with another partner, it is absolutely required that a relationship built on respect must exist first. Although this determinant alone does not guarantee that a level of agreement will be reached, the quality and depth of the communication, and the subsequent respect and trust that is fostered, will maximize the likelihood for this to take place. Accordingly, the need to focus upon this metric has encouraged many multinational firms and different governments to seek to build cultural appreciation and respect; prior to expecting a level of success with respect to com munication to be exhibited. Ultimately, human beings are extraordinarily impressionable creatures. As has been proven time and again, the culture and environment within which an individual is raised by profound and definitive impact with respect to the way in which they integrate with the world around them. With that being understood, it must also be understood that human beings are not programmable robots. Ultimately, for the thousands of individuals that might be influenced by particular culture and shoes to integrate with it, there may be a handful that rejected entirely; choosing to follow their own in life. However, instead of making the fundamental to stay at this juncture and saying that not all individuals are impacted by culture or are able to resist it, it must be noted that even those individuals who rejected entirely are impacted by. As such, even though culture can be resisted, the impact upon the individual is nonetheless profound as

Activity Based Costs Implementation for a Not-for-Profit Essay Example for Free

Activity Based Costs Implementation for a Not-for-Profit Essay The author was tasked with setting up an activity based costing (ABC) system for a not-for-profit organization. The first thing done by the author was to use the internet to research the use of ABC systems for non-profits. The result was the conclusion by the author that QuickBooks’ class feature could be used to track expenses, revenues and balance sheet costs for the implementing ABC. Income, Revenue and Balance Sheet reports are then prepared by class to see the result. Once the decision was made to use the class feature of QuickBooks, the author determined that the ABC system should accumulate costs into activity cost pools designed to correspond to the non-profit organizations major activities or business processes. The author determined that the costs in each pool would be largely caused by a single factor – the cost driver. In activity based costing (ABC), an activity cost driver is something that drives the cost of a particular activity. A factory, for example, may have running machinery as an activity. The activity cost driver associated with running the machinery could be machine operating hours, which would drive the costs of labor, maintenance and power consumption of running the machinery activity. From his research, the author found there are currently no comprehensive manuals to provide off-the-shelf instructions on how to install an ABC system in an organization. Each set of programs and activities, as well as each type of cost, presents different issues and problems. The author anticipated that many of the allocation issues faced by a not-for-profit would be similar to those faced by industry implementing an ABC system. On the other hand, the author determined that flexibility is the essence when implementing an ABC system in a not-for-profit organization. The purpose of ABC should be to provide decision-useful information, not to develop a pure measure of costs. ABC can provide interesting insights into the costs of programs and activities. ABC may highlight changes that have taken place gradually over time of which the manager may not be aware. The rational for using ABC is to allocate indirect costs to goods or services based, not simply on what is convenient, such as direct labor, but on the factors by which they are most influenced. Costs of support services should be allocated on the basis of the factors that most directly affect their magnitude. As demand for increased accountability becomes more intense for an organization, such organization must demonstrate that the benefits of the programs and activities in which they engage are commensurate with their costs. Accordingly, not-for-profit organizations need accounting systems that properly measure and report these costs.

Kant on Intuition Essay Example for Free

Kant on Intuition Essay Introduction Kant seems to have adapted the Spinozan trichotomy of spiritual activity. (Rocca, 77) In addition to sensible (empirical) intuition and understanding, Kant introduces pure intuition. The principles of this a priori, supra-empirical sensibility are dealt with by the transcendental aesthetic, a discipline which establishes that there are two pure forms of sensible intuition, serving as principles of a priori knowledge, namely, space and time. (Hayward, 1) Space is a necessary a priori representation, which underlies all outer intuitions (Hayward, 1); in particular, in order to perceive a thing, we must be in the possession of the a priori notion of space. Nor is time an empirical concept: it is the form of the inner sense, and is a necessary representation that underlies all intuitions. (Ewing 24) Pure intuition, unaided by the senses and, moreover, constituting the very possibility of sense experience, is for Kant the source of all synthetic a priori judgments. These include the synthetic judgments of geometry, which is for Kant the a priori science of physical space, and arithmetic, which he regards as based on counting, a process that takes time. Moreover, if for Aristotle, Descartes and Spinoza intuition was a mode of knowing first truths, it is for Kant no less than the possibility of outer experience. The faculty by means of which man creates geometries and theories is reason certainly sustained in some cases by sensible intuition, though not by any mysterious pure intuition. However, the products of reason are not all of them self-evident and definitive. Kantian time had a similar fate. We now consider that the characterization of time as the a priori form of the inner sense is psychologistic, and we reject the radical separation between time and physical space. The theories of relativity have taught us that the concepts of physical space and time are neither a priori nor independent from one another and from the concepts of matter and field. Infallibilism is, of course, one of the sources of Kantian intuitionism. Further sources are psychologism and the correct acknowledgment that sensible experience is insufficient for building categories (e. g. , the category of space). Instead of supposing that man builds concepts which enable him to understand the raw experience he like other animals has, Kant holds dogmatically and, as we now know, in opposition to contemporary animal and child psychology, that outer experience is possible only by the representation that has been thought. (Hahn, 89) Of all the influential contributions of Kant, his idea of pure intuition has proved to be the least valuable, but not, unfortunately, the least influential. Contemporary Intuitionism If Cartesian and Spinozan intuitions are forms of reason, Kantian intuition transcends reason, and this is why it constitutes the germ of contemporary intuitionism, in turn a gateway to irrationalism. There are, to be sure, important differences. While Kant admitted the value of sensible experience and of reason, which he regarded as insufficient but not as impotent, contemporary intuitionists tend to revile both. Whereas Kant fell into intuitionism because he realized the limitation of sensibility and the exaggerations of traditional rationalism, and because he misunderstood the nature of mathematics, intuitionists nowadays do not attempt to solve a single serious problem with the help of either intuition or its concepts; rather, they are anxious to eliminate intellectual problems, to cut down reason and planned experience, and to fight rationalism, empiricism, and materialism. This anti-intellectualist brand of intuitionism grew during the Romantic period (roughly, the first half of the nineteenth century) directly from the Kantian seed, but it did not exert a substantial influence until the end of the century, when it ceased being a sickness of isolated professors and became a disease of culture. Sensible intuition and geometrical intuition, or the capacity for spatial representation or visual imagination, have very few defenders in mathematics nowadays, because it has been shown once and for all that they are as deceptive logically as they are fertile heuristically and didactically. Therefore, what is usually called mathematical intuitionism does not rely on sensible intuition. It is now well understood that mathematical entities, relations, and operations, do not all originate in sensible intuition; it is realized that they are conceptual constructions that may altogether lack empirical correlates, even though some of them may serve as auxiliaries in theories about the world, such as physics. It is also recognized that self-evidence does not work as a criterion of truth, and that proofs cannot be shown by figures alone, because arguments are invisible. In particular, it is no longer required that axioms be self-evident; on the contrary, because they are almost always richer than the theorems they are designed to explain, axioms are often less evident than the theorems they give rise to, and are therefore apt to appear later than the theorems in the historical development of theories. Thus it is easier to obtain theorems on equilateral triangles than to establish general propositions about triangles. Mathematical intuitionism is best understood if it is regarded as a current that originated among mathematicians (a) as a reaction against the exaggerations of logicism and formalism; (b) as an attempt to rescue mathematics from the shipwreck that, at the beginning of our century, the discovery of the paradoxes in set theory seemed to forecast; (c) as a minor product of Kantian philosophy of pure intuition. It is only indebted to Kant, who was as much a rationalist and an empiricist as he was an intuitionist; and even what mathematical intuitionism owes to Kant may be left aside without fear of seriously misunderstanding the theory as has been recognized by Heyting, (Heyting 13) although Brouwer might not agree. The debt of mathematical intuitionism to Kant boils down to two ideas: (a) time though not space according to neointuitionists is an a priori form of intuition and is essentially involved in the number concept, which is generated by the operation of counting; (b) mathematical concepts are essentially constructible: they are neither mere marks (formalism) nor are they apprehensible by their being ready-made (Platonic realism of ideas); they are the work of human minds. The first assertion is unmistakably Kantian, but the second will be granted by many non-Kantian thinkers. Those mathematicians who are sympathetic with mathematical intuitionism tend to accept the second thesis while ignoring the first. Since a large part of mathematics may be built on the arithmetic of natural numbers, which would be generated by the intuition of time, it follows that the apriority of time does not only qualify the properties of arithmetic as synthetic a priori judgments, but it does the same for those of geometry, though certainly along an extended conceptual chain. The sole basal intuition would, then, suffice to engender step by step and in a constructive or recursive form not merely by means of creative definitions or by resorting to indirect proof the whole of mathematics or, rather, the mathematics allowed by mathematical intuitionism, which is only a portion of classical (pre-intuitionist) mathematics. It is true that Kant maintained that mathematics is the rational knowledge obtained from the construction of concepts. But what Kant meant by construction was not, for instance, the formation of an algorithm for the effective computation or construction of an expression like 100 100100, but rather the exhibition of the pure intuition corresponding to the concept in question. (Black 190) For Kant, to build a concept means to give its corresponding a priori intuition which, if possible, would be a psychological operation whereas, for mathematical intuitionism, the construction may be entirely logical, to the point that it may consist in the deduction of a contradiction. The ultimate foundation of all mathematical concepts, which for Kant and Brouwer alike must be intuitive, is quite another matter. Unlike Kant, the mathematical intuitionist will require that only the basic ideas be intuitive. With regard to the assertion that the basic intuition is prelinguistic, it seems definitely inconsistent with the findings of contemporary psychology, according to which every thought is symbolical, i. e. , accompanied by visual or verbal signs. Finally, the existence of Brouwers basic intuition (Stigt, 45) is at least as problematic as the existence of mathematical objects. (Curry 6) Mathematical intuitionism has both positive and negative elements. The former, the realistic elements, concern logic and the psychology of mathematics; the negative constituents are aprioristic and limiting, concern the foundations and methods of mathematics. Conclusion The debt of mathematical intuitionism to philosophical intuitionism is not large and, at any rate, what is involved is Kant’s intuitionism and not the anti-intellectualist intuitionism of many Romantics and post-Romantics. Besides, the contacts between mathematical and philosophical intuitionism are precisely those which the majority of mathematicians would not accept. The working mathematician, if he is concerned with the philosophy of mathematics at all, does not sympathize with intuitionism, because it looks for an a priori foundation or justification, or because it praises an obscure basic intuition as the source of mathematical creation, or because it claims that such an intuitive foundation is the sole warrant of certainty. Mathematical and logical intuitionisms are prized to some extent despite their peculiar dogmas, because they have contributed to the disintegration of alternative dogmas, particularly the formalist and the logicist ones. Works Cited Black Max. The Nature of Mathematics: London: Routledge Kegan Paul, 1933. 191 Curry Haskell B. Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North-Holland, 1951. Ewing A. C. Reason and Intuition, Proceedings of the British Academy, XXVII (1941) Hahn Hans. The Crisis of Intuition in The World of Mathematics. Edited by J R Newman New York: Simon Schuster, 1956 Hayward, Malcolm: The Geopolitics of Colonial Space: Kant and Mapmaking. Article accessed on 12/04/2007 from http://www. english. iup. edu/mhayward/Recent/Kant. htm Heyting A. â€Å"Heyting, Intuitionism in Mathematics, ( 1958), 13. Kant Immanuel. Kritik der reinen Vernunft (1781, 1787). Edited by R. Schmidt. Hamburg: Meiner, 1952. Translated by N. Kemp Smith . Immanuel Kants Critique of Pure Reason. London: Macmillan, 1929. Rocca, Della Michael. 1996. Representation and the Mind-Body Problem in Spinoza. Oxford University Press. Stigt, W. P. van 1990, Brouwers Intuitionism, Amsterdam: North-Holland, 1990.