Principles Of Inventory Management When You Are Down To Four Order More 1st Edition John A. Muckstadt, Amar Sapra - Solutions

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Llenroc Electronics consumes a certain type of component at an average of 100 units per week. Usage varies from week to week. Data indicate that weekly demand is normally distributed with a standard deviation of 30 units. Usage is independent from week to week. Llenroc manages the component using
When determining whether or not to place an order in a periodic review management environment, a manager checks the inventory position at the beginning of each period. If it is a value of s or lower, an order is placed to increase its value to S. Thus, the inventory position random variable can
Suppose a firm manages its inventory using a periodic review system. The fixed cost of placing an order is $12 and the holding cost is $1 per unit per period. Furthermore, the shortage cost per unit per period is $5. The procurement lead time is zero.(a) The demand in each period is 2 units with
Solve Problems 10.2 and 10.3 by taking the demand to be normally distributed where the mean and standard deviation for periods 3 and 4 are the same as in Problem 10.2.That is, the mean for period 3 is µ3 = 125, and the standard deviation for period 3 is σ3 = 43.30.The mean for period 4 is µ4 =
In the previous problem, will your decision change if the order placement cost were $200? Once again, determine the order quantity if necessary.
Consider the inventory management at a retailer of an electronic product whose product-life is estimated to be two years. For simplicity we divide this horizon into four decision-making epochs. The lead time is zero period in length. We are now in period 2 and the demand in this period is going to
In Step 3 of the approximation algorithm in Section 10.2.1, show that an order is placed in period t only if the net inventory at the beginning of period t,xt , is less than the demand in period t,dt .
In Section 9.2 we developed an exact model for the (Q,r) policy when demand was a stationary Poisson process and when backorders are allowed. But suppose all demand occurring when the system is out of stock is lost. Further suppose τ = 0.In this case, modify the results obtained in Section 9.2.In
The Llenroc hospital orders a certain pharmaceutical. The supplier of this pharmaceutical sells them to the Llenroc hospital in cartons containing 12 doses. Demand for this pharmaceutical occurs according to a Poisson process at a rate of 4 doses per day. Assume the clinic in which it is used
Suppose demand over a lead time is Laplace distributed. Using the data in Problem 9.20, find Q∗ and r∗. Compare the results with those obtained from Problem 9.20.
Suppose demand over a lead time for a part is normally distributed with an expected demand of 20 units and a standard deviation of 5 units. The fixed ordering cost is $100, the annual demand rate is 500 units, the cost to hold a unit in stock for a year is $10, and the cost per backorder incident
In Problem 9.18, suppose there is no backorder cost, but there is a constraint on the fill rate. Suppose the goal is to minimize the average annual cost of ordering and holding inventory subject to the constraint that the fill rate must be at least .99, that is, Pout ≤ .01. Find Q∗ and r∗ in
Llenroc Electronics stocks parts used by airlines. One such part has a demand rate of 12 units per year, and demands arrive according to a Poisson process. For every order that is placed, Llenroc incurs a fixed order cost of $40. The cost to carry one unit of stock of this item for a year is $100,
Assume demand arrives according to a Poisson process. Further assume that lead times are gamma distributed and that orders do not cross. That is, procurement orders arrive in the sequence in which they are placed. Develop the cost model for this case.
Solve the first example problem in Section 9.1.5 assuming lead time demand is exponentially distributed. Let the mean of this distribution equal the mean of the normal distribution in that example. All other parameter values remain the same.
Suppose the demand over a lead time follows an exponential distribution. Develop equations to find Q∗ and r∗ for the model presented in Section 9.1.3.
An example of the heuristic approximation algorithm for finding Q and r was given in Section 9.1.5. In the first example in that section we assumed that demand is normally distributed. Suppose demand over a lead time is approximated by a Laplace distribution with the same parameter values. Find
Suppose demands on a system occur according to a Poisson process. Furthermore, suppose the procurement lead time is a known constant, τ. When demands arise and the system is out of stock they are lost. Let us assume that a (Q,r) policy is used to manage the system’s stock. As in the backorder
Llenroc Electronics (LE) stocks circuit boards found in radar sets used by commercial airlines. The inventory of these boards and many other electronic items used by airlines are held in LE’s national warehouse. One such board has a normal purchase price of $100. Since this board is widely used,
Suppose demands for an item are generated by a compound Poisson process. The customer arrival rate is λ units per unit of time. Furthermore, the number of units ordered per arriving customer has a geometric distribution. Construct the probability distribution for the number of units required to
Suppose p(x;λt) represents the probability that x units are demanded through a period of length t. Let λt be the expected demand over this period, where λ is the demand rate. Furthermore, let σ2 x ·t represent the variance of this demand process over this period of time. Suppose the
Suppose φ(·) and Φ¯(·) represent the standard normal density and complementary cumulative distribution functions, respectively. Show that *x(x) dx = 0 (r), x0(x) dx = $(r)+ro(r), (x) dx=0(r)-ro(r), and [ x(x) dx = r(r) + [(r)+ro(r)] [(1-2)(r)+ro(r)].
In Section 9.1 we developed an approximation model based on the assumption that demand in excess of supply is backordered. Suppose now that demand is lost when inventory is not available to meet demand. The goal now is to find the values of Q and r that minimize the average annual fixed ordering
Assume that n items are stocked at a particular warehouse. The warehouse manager is concerned about the number of shipments from suppliers received each year and wants the inventory policy to limit the expected number of such receipts. Suppose λi, hi, τi, π¯i, and σi represent, respectively,
Suppose the Laplace distribution is used to represent the demand over a lead time.Further, assume that π > 0 and π¯ = 0. Develop a method for finding Q∗ and r∗ in this case. Review the discussion in Section 9.2.9.2 as a guide for developing your procedure.
Prove that the Hessian matrix corresponding to the function B(Q,r) is positive semi-definite.
Show that P(j;t)=(At)P(r1;t)+(1 r)P(r;t). j=
Suppose customer demand is generated by a Poisson process with rate λ. Assume the procurement lead time is a constant τ. Then show that G(j)=((+1))P(u;) u=j+1 (27) -P(j-1;)-(at)jP(j;t) 2 + 2 (+1) jP(j+1;t),
Recall the approximation model developed in Section 9.1.6 for setting Q and r:subject to F¯(r) ≤a. Discuss the effect of using this model to find the optimal values for Q and r. min th
Assume there are three items whose stock levels need to be determined in a twoechelon system.Removals of item i at base j occur according to a Poisson process at rate λi j units per day. When a removal occurs at a base, three events take place simultaneously and instantaneously. First, a unit of
In Section 8.1.2.1 we derived expressions for the mean and variance for the number of units in resupply for an LRU at a base. The analysis was based on the assumption that the demand process is a Poisson process. Suppose that the demand process is negative binomially distributed rather than Poisson
Suppose we are managing the two-echelon system discussed in Section 8.1.1. Demands at each of 10 bases for an item occur according to a Poisson process with a rate of .5 units per day. The average depot-to-base resupply time, Ai j, for the item is 5 days for all 10 bases, and the average depot
Suppose we have a two-echelon base-depot system as discussed in Section 8.1.1.The demand process at each of 10 bases is a Poisson process, where each base has a demand rate of .5 units per day. Suppose the average depot-to-base order and ship time is either 1, 5, or 10 days. Further assume the
Suppose the demand process is a compound Poisson process where the order arrival rate is λ and where the order size is represented by a random variable, N, having a logarithmic distribution. That is, P(N = n) = −(1−p)n n (ln p)−1, where 0 < p < 1, n ≥ 1.Furthermore, suppose λ = −ln
Suppose a firm manages n item types at a single location. Assume an (s−1,s) policy is used for each of these items. Requests for these items are placed by customers on the firm. Each request corresponds to a failure of a single unit of a particular part type. The failed parts are repaired at a
Llenroc Industries owns a fleet of aircraft and wants to determine how many spares of each of N components to buy. Aircraft failures occur according to a Poisson process.Each failure is due to the failure of a single component and the probability that it is due to component i is λi/∑j λj, where
Suppose there are ten critical items on an aircraft. Compute the expected number of non-operational aircraft for several combinations of stock levels for these items assuming the demand process is Poisson. Expected demands over the lead time for these ten items are 10, 7, 2, 1, 0.7, 0.5, 0.3, 0.1,
Plot the backorder, ready rate, and fill rate functions (B(s),R(s),F(s)) for a singlestage inventory system managed using a continuous review (s−1,s) policy. Suppose the lead time demand is either Poisson distributed or negative binomially distributed with expected lead time demand being either
Prove an extension to Palm’s theorem when the arrival process is a compound Poisson process and where every customer is willing to wait τ time units for delivery of its order.
Plot the probabilities for a random variable that has a negative binomial distribution where its mean assumes values 1, 5, 25. For each mean, construct these plots when the variance-to-mean ratios are 1.01, 3, 10. How do these probabilities compare with the corresponding probabilities for a Poisson
Suppose that time is divided into periods of equal length. Demand in each period is Poisson distributed with a mean of λ units. An order-up-to policy is followed for managing inventories. Replenishment lead times are assumed to be independent and identically distributed with a mean of D periods.
Find the optimal stock level for the situation described in Problem 6.8, assuming a shortage cost of $1,000 and a holding cost of $100 per unit per period.
Suppose in the previous problem that capacity is one unit with probability .1, two units with probability .8, and three units with probability .1. Construct the shortfall probability distribution.
Suppose demands over an infinite planning horizon are independent and identically distributed from period to period. Assume the production and inventory system operates in the manner described in Section 6.3.1. Assume production capacity in each period is 2 units. The demand is either 0, 1, 2, or 3
In Section 6.2 we discussed the difference between the lost sales and the backorder cases. Suppose the lead time is two periods. Construct a dynamic programming model that can be used to find the optimal solution to the lost sales problem in this case. Use the notation found in Section 6.2 when
In Section 6.3 we developed an expression for the expected cost per period, F(s).Prove that F(s) is a convex function.
Let us consider the oil rig example discussed in Section 6.4.In that example, there are two stages, the onshore and oil rig facilities. Let us assume that an order placed by the material clerk on the oil rig at the beginning of a period nominally arrives at the beginning of the next period at the
Llenroc Electronics repairs components found in radar sets for the Air Force. One particular component in the radar set is costly and fails frequently. When Llenroc receives failed parts it is not capable of repairing all of them. Units awaiting repair are called reparables. By policy, Llenroc
The model developed in Section 6.2 was based on the assumption that the cost data were constant over time. Suppose τ = 1 and the costs are period-dependent. Derive results similar to those found in Section 6.2 for this case. Develop the dynamic programming model and corresponding recursions.
Assume the data are the same as in Exercise 6.1 except now the demand in each period has a negative binomial distribution with mean 5.(a) Suppose the variances in each of the 6 periods are 25, 25, 15, 10, 6 and 6. Find the optimal values for ¯si and s∗i , i = 1,...,6.(b) Suppose the variances
In Section 6.2 we developed a dynamic programming approach for finding the optimal order-up-to values over a finite planning horizon of N periods when there are no fixed ordering costs. Suppose that N = 6, and h = 1 and b = 10. Let α = 1 and assume the unit cost is constant over time. Suppose
In Section 5.3 we discussed an example in which a buyer had to determine how many handbags to purchase for a holiday season. Using the data in that example, find the optimal order quantity assuming demand is Poisson distributed with mean of 175 units rather uniformly distributed over the range 100
A buyer of clothing items must purchase these items well in advance of the selling season. The store’s manager recognizes that sales volume depends on the advertised selling price. If the manager sets the advertised selling price to be qi, then the probability distribution of the sales has
A fashion item must be purchased several months prior to the selling season.Purchases of the item can occur at only this time. Suppose the purchase price is c dollars per unit and the in-season selling price is q dollars per unit. Suppose s represents the procurement quantity and x the number of
In Exercise 5.12, suppose ai and bi are measured in cases, where a case contains 12 bottles. Suppose further that purchases are made in case quantities. Assuming demand for each wine type is negative binomially distributed, construct a marginal analysis type of procedure that can be employed to
A buyer for a wine shop visits Italy every year and places an order with producers for a wide range of wines. Some are quite expensive while others are moderately expensive. However, the buyer has only this one chance to buy these wines since other buyers will likely purchase the remaining
Construct an appropriate model and solution procedure that can be employed so that the expected number of customer repairs that can be completed using the allocation of parts to technicians is maximized considering part availability and volume constraints.
Each day technicians at Llenroc Electronics repair equipment located at customer sites. Since these repair actions are often accomplished by removing and replacing defective parts, the technicians are required to take parts with them to complete the repairs in a timely manner. In Exercise 5.10 we
Llenroc Electronics repairs equipment at customer locations. Customers call and report a failure of a system (e.g., copy machine) along with information about the nature of the failure. On the basis of this information, Llenroc dispatches a repair technician to the customer location along with a
A submarine is being stocked for a lengthy mission. Certain parts are carried on the vessel in case repairs are required while at sea. Three such critical parts are stored in a compartment that has limited space. The three part types all have very similar geometries. The capacity of the compartment
Each week Brian visits the fish market in Boston and procures quantities of a variety of fish that he will take to sell in Ithaca, NY. He has capacity to store only 1000 pounds of fish in his truck. Suppose he concentrates on four types of fish that appear to be best choices today. Suppose demand
Use the method developed in Problem 5.6 to solve Problem 5.1 when the unit price is $12.00 if 125 or fewer units are purchased but is $8.00 per unit if more than 125 units are purchased.
Suppose the unit procurement cost is a function of the quantity purchased. In particular, suppose that there is a set of values 0 = a0 < a1 < ··· < an such that the purchase price per unit is ci if the quantity ordered q satisfies ai−1 < q ≤ ai, with c1 > c2 > ··· > cn. Assuming the
In Exercises 5.3 and 5.4 we assumed that the subsequent purchase cost of the part is $25,000, independent of the year in which such an order might be placed. Suppose the per-unit subsequent costs are estimated to be $25,000, $26,000, $28,000, $30,000,$33,000, $35,000, and $40,000 for years 1
In Problem 5.3 we assumed that the demand rate per year for the part is a constant.Suppose that this is not the case. Let Λ(t) represent the expected remaining cumulative demand for the part given the remaining useful life of the part is t. Then the probability that n remaining units are demanded,
A manufacturer of a part, purchased by Llenroc Electronics (LE) for its airline customers, has just informed LE that it is going to terminate production of the part type. The manufacturer has informed LE that it has one last opportunity to obtain the part at the normal production rate price. The
Again, re-solve the problem.
Re-solve Problem 5.1 under the assumption that demand is Poisson distributed with a mean of 100 units. Now assume demand is uniformly distributed with the mean and standard deviation as stated in Problem
A distributor of seasonal items wants to establish the best stock level to purchase.The distributor estimates that the demand for an item during the season will be normally distributed with a mean of 100 units and a standard deviation of 10 units. The items will sell for $25.00 and will cost the
Suppose the variable and fixed procurement costs and the per-unit per-period holding costs are time-dependent. Denote them byCt, Kt, and ht , respectively. As usual, let dt represent the demand in period t, and assume dt > 0 for all t = 1,...,T. Generalize the Wagelmans–Hoesel–Kolen algorithm
Discuss in detail why the Wagner–Whitin algorithm requires calculations proportional to T2 in the worst case.
Consider a computer making firm that uses a Materials Requirement Planning(MRP) system for desktops. To illustrate how the two optimization methods or the heuristics could be incorporated into an MRP system, consider the following simple product structure for a desktop.Assume that we are in week 1
Mark the following statement as true or false and justify your answer.Suppose the setup cost in the dynamic lot sizing problem is stationary over time. If we increase the setup cost, the total number of orders placed over the horizon is likely to decrease in the Wagner–Whitin algorithm.
Consider a company facing a demand pattern and costs as follows:Month 1 2 3 4 5 6 Demand 50 20 40 20 25 60 Take the single-item cost as C = $5, the holding cost rate as I = .20, and the setup cost as K = $50. Determine a replenishment strategy using the Silver–Meal heuristic and the least unit
Consider the Wagner–Whitin algorithm. Recall that we used j∗(t) to denote the period in which an order is placed to satisfy period t’s demand. We know that j∗(t +1) ≥ j∗(t). That is, the order for period t + 1 in the t + 1-period problem is placed in period j∗(t) or later.In the
Prove the following statements in the context of a deterministic lot sizing problem.(a) In the optimal policy, if dt is satisfied by an order placed in period j, j < t, then dj+1,dj+2,...,dt−1 are also satisfied by the same order.(b) Suppose the optimal policy is unique (this is just to keep the
Consider the deterministic lot sizing problem.(a) Show that in any optimal policy xt(yt −xt) = 0 for all t.(b) Prove that either the order quantity in period t is zero or it is equal to dt + dt+1 +···+dk for some k ≥ t.
Repeat the above questions with the following data set.Week 1 2 3 4 5 6 7 8 9 10 11 12 Demand 69 29 36 61 61 26 34 67 45 67 79 56 Setup Cost 85 102 102 101 98 114 105 86 119 110 98 114 Take h = 1. Note that the setup cost varies over time.
A component used in a manufacturing facility is ordered from an outside supplier.The component is used in a variety of finished items and therefore the demand is high.Forecast demand (in thousands) over the next 8 weeks is:Week 1 2 3 4 5 6 7 8 Demand 21 33 29 15 9 41 50 45 Each component costs 70
A manufacturer produces a variety of integrated circuits. The cost of setting up to produce a lot of one particular type of integrated circuit is $4,000. The variable production cost is estimated to be $8. By collaborating with its customers, the manufacturer estimates that the demand during the
When developing the Wagner–Whitin model and establishing properties affecting the computational procedure for finding the optimal procurement policy, we assumed that costs were stationary through time. Suppose purchasing and holding costs are timedependent. Develop a forward dynamic programming
The cost model that we developed charged holding costs based on the period ending on-hand inventory level. Suppose we consider charging holding costs continually through time. Thus if the inventory level is x at time t in period j, we would charge holding costs at that instant proportional to the
Show that the joint replenishment problem can be represented as a serial system problem.
Let f : ℜ → ℜ be a convex function and let x1, x2, and x3 be three points in its domain such that x1 ≤ x2 ≤ x3. Show that f(x2) ≤ max{ f(x1), f(x3)}.
(Source: http://bmrc.berkeley.edu/courseware/ICMfg92/text/fab-6) There are four major manufacturing stages in the production of an Integrated Chip (IC)– Wafer Fabrication– Wafer Probe– Chip Assembly– Chip Test Consider an IC that is used in a mature product and whose demand for the next
Consider the system in Section 3.3.1. Suppose the demands in the regional warehouses are not identical, but are equal to 100, 125, 200, 80, 150, and 225, respectively.Assume the fixed cost and holding costs are as in the example. Determine the optimal PO2 reorder intervals assuming the base
Suppose the demands for each of the three products in the last problem were doubled. Compute the new optimal PO2 policy.
A deli orders three different types of patties (vegan, tuna, and beef) from a local manufacturer for use in sandwiches. The demand for vegan, tuna, and beef patties is 20, 30, and 125 per week, respectively. The order placement is rather easy; the store manager just goes to the manufacturer and
In a serial system, is it enough to constrain the reorder intervals such that Ti ≥ Ti−1 to ensure nestedness?
A regional retail chain receives shipments from manufacturers at its central warehouse and in turn sends the shipments to the retail stores. Consider a miniature version of this problem having three retail stores and a single item, ceiling lamps. The annual demand for ceiling lamps is 100 at store
For a four-stage serial system, compute the on-hand holding costs if the echelon holding costs are h′1 = 1, h′2 = 1.25, h′3 = 0.4, and h′4 = 0.6 (in $ per unit per year).
Consider a five-stage serial system. Suppose the on-hand holding costs are as follows: h1 = 4.5, h2 = 4.25, h3 = 3, h4 = 2, and h5 = 0.5 (in $ per unit per year). Compute the echelon holding cost for each stage.
The Campus Store of a university notes that the bi-weekly demand for folders is almost steady throughout the year and is equal to 30 per 2 weeks; the number of such bi-weekly periods is 26 per year. The demand slumps when the university is closed for short breaks but the surge in demand right after
Compute the optimal power-of-three reorder interval for a single-stage system.Show that the cost of this policy is at most 15.47% more than the cost of the optimal policy.
The procurement department of Llenroc Electronics purchases the component types from a particular supplier. The fixed cost of placing an order for any of three types of components is $100. The inventory holding cost factor, I, is .20. Other relevant data are presented in the following table.Item 1
Llenroc Electronics stocks n items. These items are ordered from a group of suppliers. The cost of placing an order for items of type i is Ki. The unit purchase cost for an item of type i is Ci. The holding cost factor I is the same for all item types. Suppose a maximum of b orders can be processed
Llenroc Automotive produces power steering units in production lots. These units go into Jaguar’s 4.2 liter engine models and have a demand rate estimated to be 20,000 units per year. Llenroc makes a variety of power steering units in addition to the one used in this model of Jaguars. Switching
The basic EOQ model permitted the value of Q∗ to be any positive real number.Suppose we restrict Q to be integer-valued. Develop an approach for finding Q∗ in this case.
Recall that in the algorithm for the all units discount model we stop as soon as we identify a discount for which the optimal EOQ, Q∗j, is feasible. An alternative stopping criterion is as follows. In Step 3, compute Z(Q∗j) and stop if Z(Q∗j) ≥ Zmin. While this stopping criterion involves
Consider the basic EOQ model. Suppose the order is delivered in boxes each of which holds A units. Thus, the order quantity can only be a multiple of A. Using the convexity of the cost function in Q, compute the optimal order quantity.

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Principles Of Inventory Management When You Are Down To Four Order More 1st Edition John A. Muckstadt, Amar Sapra - Solutions

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