Reference no: EM132411787

Supply Chain Inventory and Production Planning Assignment - University of Southern Denmark, Denmark

Problem 1 -

In the settings of the EOQ model it was assumed that no backorder is allowed and hence, if there is no lead time, replenishment order is placed when the inventory level reaches 0. However, it may be possible that customers are ready to accept some waiting time before they receive their orders. Of course, there is a penalty for having units of product in backorder; assume that this penalty is equal to q for every unit of product value in backorder for one unit of time.

(a) Discuss how this backordering possibility can affect the optimal replenishment policy, under the settings of the EOQ model. Under what condition it may be beneficial to have a backorder? Provide an illustration of the inventory level over time.

(b) Determine decision variables of the EOQ model with backorders and derive the expression of the total relevant cost as function of decision variables. Find the optimal values of determined decision variables that minimize the total relevant cost.

Problem 2 -

A company producing personal computers supplies various components from its suppliers and operates 52 weeks per year. A particular component is needed at the rate of 100 items per week. The company's inventory holding rate for this component is 20%. The administrative rate to place an order of any size to the manufacturer costs $50. The manufacturer, however, tries to have orders of larger size and comes up with the following discount policy.

Discount category	Quantity purchased	Price per unit
1	1 to 99	$100
2	100 to 199	$95
3	200 to 499	$90

Note that the price in each category applies to every unit purchased (for example, the price of purchasing 130 units is equal to 130 x 95).

(a) Determine the optimal order quantity according to the EOQ model with discounts.

(b) With this order quantity, how many orders will be placed annually? What will be the time interval between orders?

Problem 3 -

Fred runs a newsstand in the airport, where daily Financial Journal newspaper is sold. He purchases copies of the newspaper for $1:50 per copy, sells it for $2:50 each, and receives the refund of $0:50 for every unsold paper. The number of requests for newspaper ranges from 15 to 18 every day. Fred estimates that there are 15 requests in about 40% of days, 16 requests in 20% of days, 17 requests in 30% of days.

(a) How many copies should he order every day to minimize the expected cost?

(b) Suppose Fred does not identify the correct quantity, but uses the quantity 15 for ordering newspapers. What is his average loss compared to optimal policy?

Problem 4 -

Cannister, Inc., specializes in the manufacture of plastic containers. The data on the monthly sales of 10-ounce shampoo bottles for the past 5 years are as follows in Table 1.

Year 1 Year 2 Year 3 Year 4 Year 5

Demand Demand Demand Demand

Month	Year 1	Year 2	Year 3	Year 4	Year 5
	Demand	Demand	Demand	Demand	Demand
Jan	742	741	896	951	1030
Feb	697	700	793	861	1032
Mar	776	774	885	938	1126
Apr	898	932	1055	1109	1285
May	1030	1099	1204	1274	1468
Jun	1107	1223	1326	1422	1637
Jul	1165	1290	1303	1486	1611
Aug	1216	1349	1436	1555	1608
Sep	1208	1341	1473	1604	1528
Oct	1131	1296	1453	1600	1420
Nov	971	1066	1170	1403	1119
Dec	783	901	1023	1209	1013
Table 1: Monthly demand for shampoo for the past 5 years

(a) In Winter's method, assume that β= 0.3 and γ = 0.25, try to figure out an appropriate α you should use in the smoothing formula.

(b) Initialized the model by using the first three years data, forecast the demands of year 4 and 5 based on Winter's method.

(c) Choose a tracking signal discussed in class, calculate the signals for the forecasted periods, discuss whether there is a problem based on the signal.

(d) Assume that your are using a (s, Q) policy to manage the inventory of shampoo, and the lead time L = 3 months, how would you calculate the standard deviation of demand during the lead time. Notice that the problem here is different from the one discussed in class, where we have more than one SKU. Here, we have only one, but the spirit of estimating the relation σ_L = L^cσ₁ is the same to that discussed in class. Discuss whether the data here can be used to estimate σ_L, if yes, please implement your idea in a spreadsheet model and report the answer; if no, please explain why and what data you need to do the job.

Problem 5 -

DJI Technology Co. Ltd is a Chinese technology company headquartered in Shenzhen, Guangdong with factories throughout the world, and is a world-leading manufacturer of commercial unmanned aerial vehicles (commonly known as \drones") for aerial photography and videography. DJI is about to launch a new model called MAVIC mini to European Union (EU) market, which will be sold at 400 euro per unit. Based on historical demand for similar models, the marketing department has forecasted the monthly demand for the model for the next year shown in Table 2.

Month	Demand	Month	Demand
January	3800	July	8400
February	3900	August	9500
March	4400	September	4200
April	5600	October	3200
May	6200	November	3600
June	6400	December	8400
Table 2: Monthly demand for MAVIC Mini.

The manufacturing of this model for EU market is made at DJI's plant in EU, called DJI(EU) hereafter. The capacity of DJI(EU) is governed by the number of people on the production line. The plant operates for 20 days a month, 8 hours each day. One worker can assemble a drone every 2 hours. Materials cost for each drone totals 130 Euros. Workers are paid 20 Euros per hour and a 60 Euros per hour for overtime. The plant currently employs 50 workers. Overtime is limited to a maximum of 20 hours per month per employee. Assume DJI has a zero starting inventory, but would like to keep 1000 ending inventory. The holding cost for each unit is 20 euro per month, which will be inured if a unit is held in the inventory at the end of the month. If DJI(EU) cannot produce enough units for a month, the DJI has the option to ship some units from its division in China, but this will cost 210 euro in total for each unit shipped from China. In each month, DJI(EU) also has the option to hire extra workers, which costs 1300 euro to train each worker. DJI(EU) may also layoff  workers, which costs 4000 euro for each worker. Backorder and lost sale are not allowed.

Given the information above, please answer the questions below by assuming you are the S&OP manager of the DJI(EU).

(a) If you use a level strategy, what will be the plan, what is the total profit of implementing the plan? Please make a plan in a spreadsheet and do the calculation.

(b) If you use a chase strategy, what will be the plan, what is the total profit of implementing the plan? Please make a plan in a spreadsheet and do the calculation.

(c) Set up a LP problem first by write out the LP in mathematical form, you need to define notations for all variables and parameters you need, write out objective function as well as all constraints of the aggregate planning problem being faced with by the S&OP manager. (Profit maximization problem)

(d) Solve the above LP problem in a spreadsheet model. Discuss the plan from this LP model. What are the options you have used in the plan, what are the options you have not used, discuss why these options are not used and under which condition you might want to use them.

(e) Assume that training new workers requires a one month period, in this case, how should you change the spreadsheet model, please do so and solve the LP problem again.

(f) Marketing department informs you that the forecasted demand is the mean of demand, which has a normal distribution with a 400 standard deviation. Given this information, do you still want to keep the plan you solved from the LP optimization above, if yes, why, if not, please explain why and how you want to change.

(g) Consider the LP problem in part (c), if now backorder is allowed. Suppose that for each unit of backorder you need to pay 80 euro, do you think you should use this option in the plan. Solve the LP again and discuss why you should use it. Also, is it true that backorder and inventory level cannot be both positive, explain. If the backorder cost increases, to which value you should stop using this option?

(h) Again, assume no backorders now. Marketing research shows that if price is reduced by α% in a month, the demand in that month will be increased by β% due to the increase in consumption in that month. On top of this, the demands in the subsequent months will be forwarded to this month. The research shows that there exists a parameter such that if price is reduced in a month t, then the demand in any subsequent month s will be forwarded in the following way: if s <= t + 3, then γ^s-t% of demand in month s will be moved to the month t; if s > t + 3, then no demand will be moved forward. Also since the forecasted demands range for a finite horizon, if there is no forecasted demand in the subsequent months, you can assume that the demand is zero in that month. Suppose the research shows that there are two scenarios of price cutting shown in Table 3.

	Scenario 1	Scenario 2
α	10	15
Β	15	20
γ	10	15
Table 3: Price Cutting Scenarios

Suppose you can only cut price in one month. The marketing department proposes to reduce the price in one month during the third quarter of the year. But, you believe that it may be better to do it in the second quarter of the year. Please analyze and decide in which month you should cut the price and how much you should cut (assume you can only choose one from the two scenarios above).