Chapter 8 Inventory Models

8.1 Overview of Inventory Issues Proper control of inventory is crucial to the success of an enterprise. Typical inventory problems include: Basic inventory – Planned shortage Quantity discount – Periodic review Production lot size – Single period Inventory models are often used to develop an optimal inventory policy, consisting of: An order quantity, denoted Q. A reorder point, denoted R.

Type of Costs in Inventory Models Inventory analyses can be thought of as cost-control techniques. Categories of costs in inventory models: Holding (carrying costs) Order/ Setup costs Customer satisfaction costs Procurement/Manufacturing costs

Type of Costs in Inventory Models Holding Costs (Carrying costs): These costs depend on the order size Cost of capital Storage space rental cost Costs of utilities Labor Insurance Security Theft and breakage Deterioration or Obsolescence Ch = Annual holding cost per unit in inventory H = Annual holding cost rate C = Unit cost of an item Ch = H * C

Type of Costs in Inventory Models Order/Setup Costs These costs are independent of the order size. Order costs are incurred when purchasing a good from a supplier. They include costs such as Telephone Order checking Labor Transportation Setup costs are incurred when producing goods for sale to others. They can include costs of Cleaning machines Calibrating equipment Training staff Co = Order cost or setup cost

Type of Costs in Inventory Models Customer Satisfaction Costs Measure the degree to which a customer is satisfied. Unsatisfied customers may: Switch to the competition (lost sales). Wait until an order is supplied. When customers are willing to wait there are two types of costs incurred: Cb = Fixed administrative costs of an out of stock item ($/stockout unit). Cs = Annualized cost of a customer awaiting an out of stock item ($/stockout unit per year).

Type of Costs in Inventory Models Procurement/Manufacturing Cost Represents the unit purchase cost (including transportation) in case of a purchase. Unit production cost in case of in-house manufacturing. C = Unit purchase or manufacturing cost.

Demand in Inventory Models Demand is a key component affecting an inventory policy. Projected demand patterns determine how an inventory problem is modeled. Typical demand patterns are: Constant over time (deterministic inventory models) Changing but known over time (dynamic models) Variable (randomly) over time (probabilistic models) D = Demand rate (usually per year)

Inventory Classifications Inventory can be classified in various ways: Used typically by accountants at manufacturing firms. Enables management to track the production process. Items are classified by their relative importance in terms of the firm’s capital needs. Management of items with short shelf life and long shelf life is very different

Review Systems Two types of review systems are used: Continuous review systems. The system is continuously monitored. A new order is placed when the inventory reaches a critical point. Periodic review systems. The inventory position is investigated on a regular basis. An order is placed only at these times.

8.2 Economic Order Quantity Model - Assumptions Demand occurs at a known and reasonably constant rate. The item has a sufficiently long shelf life. The item is monitored using a continuous review system. All the cost parameters remain constant forever (over an infinite time horizon). A complete order is received in one batch.

The EOQ Model – Inventory profile The constant environment described by the EOQ assumptions leads to the following observation: The optimal EOQ policy consists of same-size orders. Q This observation results in the following inventory profile :

Cost Equation for the EOQ Model Total Annual Inventory Costs Total Annual Holding Costs Total Annual ordering Costs Total Annual procurement Costs = + + TC(Q) = (Q/2)Ch + (D/Q)Co + DC Q Ch The optimal order Size Q* =

Note: at the optimal order size total holding costs and ordering costs TV(Q) and Q* TV(Q) Constructing the total annual variable cost curve Add the two curves to one another Total Holding Costs Total annual holding and ordering costs * * Note: at the optimal order size total holding costs and ordering costs are equal * * o * Total ordering costs Q Q* The optimal order size

Sensitivity Analysis in EOQ models The curve is reasonably flat around Q*. Deviations from the optimal order size cause only small increase in the total cost. Q*

Cycle Time The cycle time, T, represents the time that elapses between the placement of orders. Note, if the cycle time is greater than the shelf life, items will go bad, and the model must be modified. T = Q/D

Number of Orders per Year To find the number of orders per years take the reciprocal of the cycle time N = D/Q Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy. How many orders are placed per year? N = 1000/250 = 4 orders. How often orders need to be placed (what is the cycle time)? T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.

Lead Time and the Reorder Point In reality lead time always exists, and must be accounted for when deciding when to place an order. The reorder point, R, is the inventory position when an order is placed. R is calculated by L and D must be expressed in the same time unit. R = L D

Lead Time and the Reorder Point – Graphical demonstration: Short Lead Time Inventory position Place the order now L R = Inventory at hand at the beginning of lead time

Lead Time and the Reorder Point – Graphical demonstration: Long Lead Time Outstanding order R = inventory at hand at the beginning of lead time + one outstanding order = demand during lead time = LD Inventory at hand L Place the order now

Safety stock Safety stocks act as buffers to handle: Higher than average lead time demand. Longer than expected lead time. With the inclusion of safety stock (SS), R is calculated by The size of the safety stock is based on having a desired service level. R = LD + SS

Safety stock Reorder Point L Place the order now R = LD Planned situation Actual situation Reorder Point Place the order now L R = LD

Safety stock Reorder Point ? L Place the order now R = LD + SS Actual situation ? Reorder Point LD SS=Safety stock Place the order now L The safety stock prevents excessive shortages. R = LD + SS

Inventory Costs Including safety stock Total Annual Inventory Costs Total Annual Holding Costs Total Annual ordering Costs Total Annual procurement Costs = + + TC(Q) = (Q/2)Ch + (D/Q)Co + DC + ChSS Safety stock holding cost

ALLEN APPLIANCE COMPANY (AAC) AAC wholesales small appliances. AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units. Management wishes to determine an optimal ordering policy for the Citron brand juicer

ALLEN APPLIANCE COMPANY (AAC) Data Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) Ch = $1.40 [HC = (14%)($10)] C = $10. H = 14% (10% ann. interest rate) + (4% miscellaneous) D = demand information of the last 10 weeks was collected:

ALLEN APPLIANCE COMPANY (AAC) Data The constant demand rate seems to be a good assumption. Annual demand = (120/week)(52weeks) = 6240 juicers.

AAC – Solution: EOQ and Total Variable Cost Current ordering policy calls for Q = 600 juicers. TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80 The EOQ policy calls for orders of size Savings of 16% Ö 2(6240)(12) 1.40 = 327.065 327 = Q* TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

AAC – Solution: Reorder Point and Total Cost Under the current ordering policy AAC holds 13 units safety stock (how come? Observe): AAC is open 5 day a week. The average daily demand = 120/week)/5 = 24 juicers. Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers. Reorder point without Safety stock = LD = 192. Current policy: R = 205. Safety stock = 205 – 192 = 13. For safety stock of 13 juicers the total cost is TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09 TV(327) + Procurement + Safety stock cost holding cost

AAC – Solution: Sensitivity of the EOQ Results Changing the order size Suppose juicers must be ordered in increments of 100 (order 300 or 400) AAC will order Q = 300 juicers in each order. There will be a total variable cost increase of $1.71. This is less than 0.5% increase in variable costs. Changes in input parameters Suppose there is a 20% increase in demand. D=7500 juicers. The new optimal order quantity is Q* = 359. The new variable total cost = TV(359) = $502 If AAC still orders Q = 327, its total variable costs becomes Only 0.4% increase TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13

AAC – Solution: Cycle Time For an order size of 327 juicers we have: T = (327/ 6240) = 0.0524 year. = 0.0524(52)(5) = 14 days. This is useful information because: Shelf life may be a problem. Coordinating orders with other items might be desirable. working days per week

AAC – Excel Spreadsheet =SQRT(2*$B$10*$B$14/$B$13) =1/E11 Copy to cell H12 =$B$15*$B$10+$B$16-INT(($B$15*$B$10+$B$16)/E10)*E10 Copy to cell H13 =E10/B10 Copy to cell H11 =(E10/2)*$B$13+($B$10/E10)*$B$14 Copy to cell H14 =$B$10*$B$11+E14+$B$13*B16 Copy to Cell H15

Service Levels and Safety Stocks

8.3 Determining Safety Stock Levels Businesses incorporate safety stock requirements when determining reorder points. A possible approach to determining safety stock levels is by specifying desired service level .

Two Types of Service Level Service levels can be viewed in two ways. The cycle service level The probability of not incurring a stockout during an inventory cycle. Applied when the likelihood of a stockout, and not its magnitude, is important for the firm. The unit service level The percentage of demands that are filled without incurring any delay. Applied when the percentage of unsatisfied demand should be under control.

The Cycle Service Level Approach In many cases short run demand is variable even though long run demand is assumed constant. Therefore, stockout events during lead time may occur unexpectedly in each cycle. Stockouts occur only if demand during lead time is greater than the reorder point.

The Cycle Service Level Approach To determine the reorder point we need to know: The lead time demand distribution. The required service level. In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by R = mL + zasL 1 –a = service level

The Cycle Service Level Approach P(DL<R) = 1 – a m=192 P(DL>R) = a R P(DL> R) = P(Z > (R – mL)/sL) = a. Since P(Z > Za) = a, we have Za = (R – mL)/sL, which gives… R = mL + zasL

AAC - Cycle Service Level Approach Assume that lead time demand is normally distributed. Estimation of the normal distribution parameters: Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week. Estimation of the variance of the weekly demand = Sample variance = 83.33 juicers2.

AAC - Cycle Service Level Approach To find mLand sL the parameters m (per week) and s (per week) must be adjusted since the lead time is longer than one week. Lead time is 8 days =(8/5) weeks = 1.6 weeks. Estimates for the lead time mean demand and variance of demand mL » (1.6)(120) = 192; s2L » (1.6)(83.33) = 133.33

AAC - Service Level for a given Reorder Point Let us use the current reorder point of 205 juicers. 205 = 192 + z (11.55) z = 1.13 From the normal distribution table we have that a reorder point of 205 juicers results in an 87% cycle service level.

AAC – Reorder Point for a given Service Level Management wants to improve the cycle service level to 99%. The z value corresponding to 1% right hand tail is 2.33. R = 192 + 2.33(11.55) = 219 juicers.

AAC – Acceptable Number of Stockouts per Year AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers. What is the equivalent service level for this strategy?

AAC – Acceptable Number of Stockouts per Year There will be an average of 6240/327 = 19.08 lead times per year. The likelihood of stockouts = 1/19 = 0.0524. This translates into a service level of 94.76%

The Unit Service Level Approach When lead time demand follows a normal distribution service level can be calculated as follows: Determine the value of z that satisfy the equation L(z) = aQ* / sL Solve for R using the equation R = mL + zsL

AAC – Cycle Service Level (Excel spreadsheet) =NORMINV(B7,B5,B6) =NORMDIST(B8,B5,B6,TRUE)

8.4 EOQ Models with Quantity Discounts Quantity Discounts are Common Practice in Business By offering discounts buyers are encouraged to increase their order sizes, thus reducing the seller’s holding costs. Quantity discounts reflect the savings inherent in large orders. With quantity discounts sellers can reward their biggest customers without violating the Robinson - Patman Act.

8.4 EOQ Models with Quantity Discounts Quantity Discount Schedule This is a list of per unit discounts and their corresponding purchase volumes. Normally, the price per unit declines as the order quantity increases. The order quantity at which the unit price changes is called a break point. There are two main discount plans: All unit schedules - the price paid for all the units purchased is based on the total purchase. Incremental schedules - The price discount is based only on the additional units ordered beyond each break point.

All Units Discount Schedule To determine the optimal order quantity, the total purchase cost must be included TC(Q) = (Q/2)Ch + (D/Q)Co + DCi + ChSS Ci represents the unit cost at the ith pricing level.

AAC - All Units Quantity Discounts AAC is offering all units quantity discounts to its customers. Data

Should AAC increase its regular order of 327 juicers, to take advantage of the discount?

AAC – All units discount procedure Step 1: Find the optimal order Qi* for each discount level “i”. Use the formula Step 2: For each discount level “i” modify Q i* as follows If Qi* is lower than the smallest quantity that qualifies for the i th discount, increase Qi* to that level. If Qi* is greater than the largest quantity that qualifies for the ith discount, eliminate this level from further consideration. Step 3: Substitute the modified Q*i value in the total cost formula TC(Q*i ). Step 4: Select the Q i * that minimizes TC(Q i*)

AAC – All units discount procedure Step 1: Find the optimal order quantity Qi* for each discount level “i” based on the EOQ formula

AAC – All Units Discount Procedure Step 2 : Modify Q i * $10/unit $9.75/unit $9.50 Q1* Q2* Q3* 599 336 1 299 300 331 999 600

AAC – All Units Discount Procedure Step 2 : Modify Q i * $10/unit Q3* Q3* Q3* Q3* Q3* $9.50 Q3* Q1* Q2* Q3* Q3* 336 1 299 300 331 999 600

AAC – All Units Discount Procedure Step 3: Substitute Q I * in the total cost function Step 4 AAC should order 5000 juicers

AAC – All Units Discount Excel Worksheet

8.4 Production Lot Size Model - Assumptions Demand rate is constant. Production rate is larger than demand rate. The production lot is not received instantaneously (at an infinite rate), because production rate is finite. There is only one product to be scheduled. The rest of the EOQ assumptions stay in place.

Production Lot Size Model – Inventory profile The optimal production lot size policy orders the same amount each time. This observation results in the inventory profile below:

Production Lot Size Model – Understanding the inventory profile The production increases the inventory at a rate of P. Demand accumulation during production run = DT1 Production time T1 Maximum inventory Maximum inventory = (P – D)T1 = (P – D)(Q/P) = Q(1 – D/P) The inventory increases at a net rate of P - D Production Lot Size = Q = PT1 The demand decreases the inventory at a rate of D. Demand accumulation during production run

Production Lot Size Model – Total Variable Cost The parameters of the total variable costs function are similar to those used in the EOQ model. Instead of ordering cost, we have here a fixed setup cost per production run (Co). In addition, we need to incorporate the annual production rate (P) in the model.

Production Lot Size Model – Total Variable Cost TV(Q) = (Q/2)(1 - D/P)Ch + (D/Q)Co P is the annual production rate The average inventory Ch(1-D/P) The Optimal Order Size Q* = 2DCo

Production Lot Size Model – Useful relationships Cycle time T = Q / D. Length of a production run T1 = Q / P. Time when machines are not busy producing the product T2 = T - T1 = Q(1/D - 1/P). Average inventory = (Q/2)(1-D/P).

FARAH COSMETICS COMPANY Farah needs to determine optimal production lot size for its most popular shade of lipstick. Data The factory operates 7 days a week, 24 hours a day. Production rate is 1000 tubes per hour. It takes 30 minutes to prepare the machinery for production. It costs $150 to setup the line. Demand is 980 dozen tubes per week. Unit production cost is $.50 Annual holding cost rate is 40%.

FARAH COSMETICS COMPANY – Solution Dozens Input for the total variable cost function D = 613,200 per year [(980 dozen/week (12)/ 7](365) Ch = 0.4(0.5) = $0.20 per tube per year. Co = $150 P = (1000)(24)(365) = 8,760,000 per year.

FARAH COSMETICS COMPANY – Solution Current Policy Currently, Farah produces in lots of 84,000 tubes. T = (84,000 tubes per run)/(613,200 tubes per year)= 0.137 years (about 50 days). T1 = (84,000 tubes per lot)/(8,760,000 tubes per year)= 0.0096 years (about 3.5 days). T2 = 0.137 - 0.0096 = 0.1274 years (about 46.5 days). TV(Q = 84,000) = (84,000/2) {1-(613,200/8,760,000)}(0.2) + 613,200/84,000)(150) = $8907.

FARAH COSMETICS COMPANY – Solution The Optimal Policy Using the input data we find TV(Q* = 31,499) = (31,499/2) [1-(613,200/8,760,000)](0.2) + (613,200/31,499)(150) = $5,850. The optimal order size (0.2)(1-613,200/8760,000) Q* = 2(613,200)(150) = 31,499

FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)

8.5 Planned Shortage Model When an item is out of stock, customers may: Go somewhere else (lost sales). Place their order and wait (backordering). In this model we consider the backordering case. All the other EOQ assumptions are in place.

Planned Shortage Model – the Total Variable Cost Equation The parameters of the total variable costs function are similar to those used in the EOQ model. In addition, we need to incorporate the shortage costs in the model. Backorder cost per unit per year (loss of goodwill cost) - Cs. Reflects future reduction in profitability. Can be estimated from market surveys and focus groups. Backorder administrative cost per unit - Cb. Reflects additional work needed to take care of the backorder.

Planned Shortage Model – the Total Variable Cost Equation The Annual holding cost = Ch[T1/T](Average inventory) = Ch[T1/T] (Q-S)/2 The Annual shortage cost = Cb(number of backorders per year) + Cs(T2/T)(Average number of backorders). To calculate the annual holding cost and shortage cost we need to find The proportion of time inventory is carried, (T1/T) The proportion of time demand is backordered, (T2/T). T1 T2 T

Finding T1/ T and T2/ T Q Average inventory = (Q - S) / 2 Proportion of time inventory exists = T1/T Q - S Q Q - S = (Q - S) / Q Q T2 T1 T1 Proportion of time shortage exists = T2/T T S T S = S / Q Average shortage = S / 2

Planned Shortage Model – The Total Variable Cost Equation Annual holding cost: Ch[T1/T](Q-S)/2 = Ch[(Q-S) /Q](Q-S)/2 = Ch(Q-S)2/2Q Annual shortage cost: Cb(Units in short per year) + Cs[T2/T](Average number of backorders) = Cb(S)(D/Q) + CsS2/2Q

Planned Shortage Model – The Total Variable Cost Equation The total annual variable cost equation The optimal solution to this problem is obtained under the following conditions Cs > 0 ; Cb < \/ 2CoCh / D (Q -S)2 D Q S2 2Q Ch + (Co + SCb) + TV(Q,S) = CS 2Q Holding costs Ordering costs Time independent backorder costs Time dependent backorder costs

Planned Shortage Model – The Optimal Inventory Policy The Optimal Order Size - Q* = Ch + Cs Cs x 2DCo (DCb)2 ChCs Ch The Optimal Backorder level S*= Q* Ch - DCb Ch + Cs Reorder Point R = L D - S*

SCANLON PLUMBING CORPORATION Scanlon distributes a portable sauna from Sweden. Data A sauna costs Scanlon $2400. Annual holding cost per unit $525. Fixed ordering cost $1250 (fairly high, due to costly transportation). Lead time is 4 weeks. Demand is 15 saunas per week on the average.

SCANLON PLUMBING CORPORATION Scanlon estimates a $20 goodwill cost for each week a customer who orders a sauna has to wait for delivery. Administrative backordrer cost is $10. Management wishes to know: The optimal order quantity. The optimal number of backorders. Backorder costs

SCANLON PLUMBING – Solution Input for the total variable cost function D = 780 saunas [(15)(52)] Co = $1,250 Ch = $525 Cs = $1,040 Cb = $10

SCANLON PLUMBING – Solution The optimal policy x (780)(10)2 (525)(1040) 525 2(780)(1250) 525+1040 1040 Q* = 74 » - _ S*= (74)(525) (780)(10) 525 + 1040 20 » R = (4 / 52)(780) - 20 = 40

SCANLON PLUMBING – Spreadsheet Solution

8.7 Review Systems – Continuous Review (R, Q) Policies The EOQ, production lot size, and planned shortage models assume that inventory levels are continuously monitored Items are sold one at a time.

8.7 Review Systems – Continuous Review (R, Q) Policies The above models call for order point (R) order quantity (Q) inventory policies. Such policies can be implemented by A point-of-sale computerized system. The two-bin system.

Continuous Review Systems (R, M) policies When items are not necessarily sold one at a time, the reorder point might be missed, and out of stock situations might occur more frequently. The order to level (R, M) policy may be implemented in this situation.

Continuous Review Systems (R,M) policies The R, M policy replenishes inventory up to a pre-determined level M. Order Q = Q* + (R – I) = (M – SS) + (R – I) each time the inventory falls to the reorder point R or below. (Order size may vary from one cycle to another).

Periodic Review Systems It may be difficult or impossible to adopt a continuous review system, because of: The high price of a computerized system. Lack of space to adopt the two-bin system. Operations inefficiency when ordering different items from the same vendor separately. The periodic review system may be found more suitable for these situations.

Periodic Review Systems Under this system the inventory position for each item is observed periodically. Orders for different items can be better coordinated periodically.

Periodic Review Systems (T,M) Policies In a replenishment cycle policy (T, M), the inventory position is reviewed every T time units. An order is placed to bring the inventory level back up to a maximum inventory level M. M is determined by Forecasting the number of units demanded during the review period T. Adding the desired safety stock to the forecasted demand.

Periodic Review Systems Calculation of the replenishment level and order size Q = M + LD – I M = TD + SS T =Review period L = Lead time SS= Safety stock Q = Inventory position D = Annual demand I = Inventory position

AAC operates a (T, M) policy Every three weeks AAC receives deliveries of different products from Citron. Lead time is eight days for ordering Citron’s juicers. AAC is now reviewing its juicer inventory and finds 210 in stock. How many juicers should AAC order for a safety stock of 30 juicers?

AAC operates a (T, M) policy – Solution Data Review period T = 3 weeks = 3/52 = .05769 years, Lead time = L = 8 days = 8/260 = .03077 years, Demand D = 6240 juicers per year, Safety stock SS = 30 juicers, Inventory position I = 210 juicers AAC operates 260 days a year. (5)(52) = 260.

AAC operates a (T, M) policy – Solution Review period demand = TD = ( 3/52)(6240) = 360 juicers, M = TD + SS = 360 + 30 = 390 juicers, Q = M + LD – I = 390 + .03077(6240) - 210 = 372 juicers.

AAC operates a (T, M) policy – Solution Replenishment level Order Order M = maximum inventory Inventory position Inventory position SS SS SS L L Review point Review point T Notice: I + Q is designed to satisfy the demand within an interval of T + L. To obtain the replenishment level add SS to I + Q.

8.8 Single Period Inventory Model - Assumptions Shelf life of the item is limited. Inventory is saleable only within a single time period. Inventory is delivered only once during a time period. Demand is stochastic with a known distribution. At the end of each period, unsold inventory is disposed of for some salvage. The salvage value is less than the cost per item. Unsatisfied demand may result in shortage costs.

The Expected Profit Function To find an optimal order quantity we need to balance the expected cost of over-ordering and under ordering. Expected Profit = S(Profit when Demand=X)Prob(Demand=X) x The expected profit is a function of the order size, the random demand, and the various costs.

The Expected Profit Function Developing an expression for EP(Q) Notation p = per unit selling price of the good. c = per unit cost of the good. s = per unit salvage value of unsold good. K = fixed purchasing costs Q = order quantity. EP(Q) = Expected Profit if Q units are ordered. Scenarios Demand X is less than the order quantity (X < Q). Demand X is greater than or equal to the order quantity (X ³Q.

The Expected Profit Function Scenario 1: Demand X is less than the units stocked, Q. Scenario 2: Demand X is greater than or equal to the units stocked. Profit = pX + s(Q - X) - cQ - K Profit = pQ - g(X - Q) - cQ - K EP(Q) = [pX+s(Q - X) - cQ - K]P(X) + [pQ - g(X - Q) - cQ - K]P(X)

The Optimal Solution To maximize the expected profit order Q* For the discrete demand case take the smallest value of Q* that satisfies the condition P(D £ Q*) ³ (p - c + g)/(p - s + g) For the continuous demand case find the Q* that solves F(Q*) = (p - c + g) /(p - s + g)

THE SENTINEL NEWSPAPER Management at Sentinel wishes to know how many newspapers to put in a new vending machine. Data Unit selling price is $0.30 Unit production cost is $0.38. Advertising revenue is $0.18 per newspaper. Unsold newspaper can be recycled and net $0.01. Unsatisfied demand costs $0.10 per newspaper. Filling a vending machine costs $1.20. Demand distribution is discrete uniform between 30 and 49 newspapers.

SENTINEL - Solution Input to the optimal order quantity formula c = 0.20 [0.38-0.18] s = 0.01 g = 0.10 K = 1.20 p+ g - c p+ g - s The probability of the optimal service level = 0.30 + 0.10 - 0.20 0.30 + 0.10 - 0.01 = 0.513 =

SENTINEL – Solution Finding the optimal order quantity Q* 1.0 P(D £ 39) = 0.50 P(D £ 40) = 0.55 0.513 0.55 0.50 Q* = 40 30 39 40 49

SENTINEL – Spreadsheet Solution =(B5+B8-B6)/(B5+B8-B7) =ROUNDUP(B10+E5*(B11-10),0) =(E6-B10+1)/(B11-B10+1)

WENDELL’S BAKERY Management in Wendell’s wishes to determine the number of donuts to prepare for sale, on weekday evenings Data Unit cost is $0.15. Unit selling price is $0.35. Unsold donuts are donated to charity for a tax credit of $0.05 per donut. Customer goodwill cost is $0.25. Operating costs are $15 per evening. Demand is normally distributed with a mean of 120, and a standard deviation of 20 donuts.

WENDELL’S BAKERY - Solution Input to the optimal order quantity formula p = $0.35 c = $0.15 s = $0.05 g = $0.25 K = $15.00 The optimal service level = p+ g - c p+ g - s 0.35+ 0.25 - 0.15 0.35+0.25 - 0.05 = 0.8182 =

WENDELL’S BAKERY - Solution Finding the optimal order quantity From the relationship F(Q*) = 0.8182 we find the corresponding z value. From the standard normal table we have z = 0.3186. The optimal order quantity is calculated by Q* = m + zs For Wendell’s Q* = 120 + (0.3186)(20) @ 138 .8182 m=120 Q*

WENDELL’S BAKERY - Solution Calculating the expected profit For the normal distribution L [(Q* - m ) /s] is obtained from the partial expected value table. For Wendell’s EP(138) = (0.35 - 0.05)(120) - (0.15 - 0.05)(138) - (0.35 + 0.25 - 0.05)x(20)L[(138 - 120) / 20] - 15 = $6.10 EP(Q*) = (p - s) m - (c - s)Q* - (p + g - s) (s)L[(Q* - m ) /s] - K L(0.9) = 0.1004

WENDELL’S BAKERY - Spreadsheet Solution =(B5+B8-B6)/(B5+B8-B7) =NORMINV(E5,B10,B11) =(B5-B7)*B10-(B6-B7)*E6-(B5+B8-B7)*B11*(EXP(-(((E6-B10)/B11)^2)/2)/((2*PI())^0.5)-((E6-B10)/B11)*(1-NORMSDIST((E6-B10)/B11)))-B9

WENDELL’S – The commission strategy When commission replaces fixed wages… Compare the maximum expected profit of two strategies: $0.13 commission paid per donut sold, $15 fixed wage per evening (calculated before). Calculate first the optimal quantity for the alternative policy. Check the expected difference in pay for the operator.

WENDELL’S – The commission strategy - Solution The unit selling price changes to c = 0.35 - 0.13 = $0.22 The optimal order: F(Q*) = (0.22 + 0.25 - 0.15) / (0.22 + 0.25 - 0.05)= 0.7616. Z = .71 Q* = m + zs = 120 + (0.71)(20) » 134 donuts.

WENDELL’S – The commission strategy - Solution Will the bakery’s expected profit increase? EP(134) = (0.22 - 0.05)(20) - (0.15 - 0.05)(134) - (0.22 + 0.25 - 0.05)x(20)L[(134 - 120) / 20] = $5.80 < 6.10 The bakery should not proceed with the alternative plan.

WENDELL’S – The commission strategy - Solution Comments The operator expected compensation will increase, but not as much as the bakery’s expected loss. An increase in the mean sales is probable when the commission compensation plan is implemented. This may change the analysis results.

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