scm excel based models

Modelling & SimulationSupply Chain Management

Excel Based Simulation Problems

1. Grocery Store: Single Server Queue

A small grocery store has only one check out counter. Customers arrive at this checkout counter at random times that are from 1 to 8 minutes apart. Each possible value of inter arrival time has the same probability of occurrence. The service times vary from 1 to 6 minutes. The probability of occurrences of each possible service time is given below:

Service Time

Mins Probability

1 0.12 0.23 0.34 0.255 0.16 0.05

The problem is to simulate the system first on paper for only twenty arrivals with the help of the random numbers given in the following table:

Random numbers for Arrivals Service

0.674 0.9080.846 0.4340.809 0.9550.220 0.5210.351 0.0080.138 0.7880.824 0.7140.283 0.3040.399 0.7590.746 0.9100.477 0.9110.498 0.7830.629 0.9000.360 0.8730.721 0.0560.525 0.8290.610 0.0700.758 0.7960.681 0.425

Excel Based Simulation 1

0.127 0.014

Then create the model with Excel and find following characteristics for the queue for 100 customers and match answers:

1 Average waiting time for customer in Queue 1.74 mins2 Probability (wait) 0.463 Probability of idle server 0.244 Server busy 0.765 Average service time 3.17 mins6 Average time between arrival 4.19 mins7 Average waiting time for those who wait 3.22 mins8 Average time customer spends in system 4.91 mins

2. Able – Baker Call Centre Problem: Two server queue

Consider a computer technical support centre where personnel take calls and provide service. The time between calls ranges from 1 to 4 minutes, with distribution shown in the table below:

Time between arrival

Mins Probability

1 0.252 0.43 0.24 0.15

There are two technical support people – Able and Baker. Able is more experienced and can provide service faster than Baker. The distribution of their service times are shown below:

Service Time of Baker

Mins Probability


Service Time of Able

Mins Probability

2 0.303 0.284 0.255 0.17

3 0.354 0.255 0.206 0.20

Times are usually a continuous measure. But for this and other time based examples we make them discrete for ease of explanation.

Able and Baker have reached an understanding that Able gets the call if both technical support people are idle – because Able has seniority.

The problem is to find out how well the system is working. To estimate first find the following characteristics of the queue system by simulating for 20 customers (using the random numbers given in the table below) and then write an Excel based model for 100 customers and compare answers given in the last column:

1 Average wait time 0.0002 Probability that a customer has to wait in Q 0.0003 Probability of idle time of Able 0.2224 Probability of idle time of Baker -0.0315 Average service time of Able 3.8576 Average service time of Baker 5.4177 Average time between arrival 2.4408 Average wait time for those who wait9 Average time in the system 2.538

3. Higher Education Fair

At a higher education fair more than 2000 students are likely to turn up. A leading private management institute from USA has set up a stall and deputed two executives to attend to admission related queries raised by students. One of the executives Samantha is very senior and the other executive George is only with couple of years of experience. Between them they have decided that if both are free, the next query will be handled by Samantha because she is senior and possibly provide correct answers to almost all questions. She is also bit faster than George.

The inter arrival distribution of queries from admission seekers are given in the following table:

Time between arrivals (Mins)

Probability

1 0.252 0.403 0.204 0.15

The service distribution times for Samantha & George are given below:


Samantha GeorgeService Time

(Mins)Probability

Service Time

(Mins)Probability

1 0.20 4 0.402 0.40 5 0.303 0.25 6 0.204 0.15 7 0.10

The problem is to find how well the arrangement will work? To estimate the system measures of performance, a simulation of first ten admission seekers is made by using following sets of random numbers for arrival of students with queries and service times for Samantha and George.

Random Numbers forArrival

of students

Service Times for Samantha

Service Times for

George21 7 9552 65 5342 69 4934 21 9483 40 5262 47 3154 18 954 20 1190 72 9215 74 8

It was decided that the system will be implemented only if both criteria are satisfied:

1. Average time for the students in the system is less than 6 minutes.

2. Average waiting time for those students who wait is less than 5 minutes.

What is your recommendation?

4. News Vendor Model

The problem of purchase and sale of newspaper is that the number of papers to be bought has to be decided in the morning. There is no opportunity of buying a second lot after assessing the day’s demand. Such a newsstand owner Rampukar buys news paper at Rs. 1.50 and sells for Rs. 2.50. News Papers not sold at the end of the day are disposed off at the rate of Rs. 4.00 for 20 news paper. News paper can be purchased in bundles of 10. Thus Rampukar can buy 50, 60, 70 …papers and so on. There are three types of news days: Good, Fair and Poor; they have the probability of occurrences as 0.30, 0.50 and 0.20. The distribution of demand is given in the following table

Demand Good Fair Poor40 0.03 0.10 0.4450 0.05 0.18 0.22


60 0.15 0.40 0.1670 0.20 0.20 0.1280 0.35 0.08 0.0090 0.15 0.04 0.00100 0.07 0.00 0.00

The news stand boy Rampukar had decided to buy 60 news papers daily. By using the allocated random numbers as given below, calculate the daily average profit for twenty days.

Random Number for

Type of News days Demand

98 9970 6391 1645 2991 357 3857 9049 7686 6137 1660 252 679 2750 734 4192 229 7459 1915 562 95

5. Reliability Problem

A milling machine has three different bearings that fail in service. The distribution of the life of each bearing is identical, as shown in the table. When a bearing fails, the mill stops, a repairperson is called and a new bearing is installed. The delay time of the repair person’s arriving at the milling machine is also a random variable having the distribution given in second table. Down time of the mill is estimated at $ 10 per minute. The direct on-site cost of the repair person is $ 30 per hour. It takes 20 minutes to change one bearing, 30 minutes to change two bearings, and 40 minutes to change three bearings. A


proposal has been made to replace all three bearings whenever a bearing fails. Management needs an evaluation of the proposal. The total cost per 10,000 bearing-hours will be used as the measure of performance.

Bearing Life DistributionBearing Life

(Hours)Probability of

faliure1000 0.101100 0.131200 0.251300 0.131400 0.091500 0.121600 0.021700 0.061800 0.051900 0.05

Delay Time DistributionDelay Time (minutes)

Probability

5 0.610 0.315 0.1

Inventory Replenishment

Continuous Review – Q Type

6. Uncertain Demand but Fixed Lead Time

Using following data build an Excel based simulation model to compute minimum total cost for fifty periods:

Average weekly demand = 400 units Standard Deviation of the weekly demand = 15 units Delivery lead time = 4 weeks Cycle Service Level 95% (Z value 1.65) Product cost = Rs. 500 Annual Carrying cost = 20% Cost of placing order = Rs. 490 Out of stock cost = Rs. 20

Assume opening stock to be 1390 units. The manager has assumed that the opening stock would be between 0 and 1600 and the ordering cost would be between 0 and


Rs.700. Calculate total cost for fifty periods. Optimise it by varying opening stock and ordering cost.(Answer: Opening stock 1556, Ordering cost Rs. 684 and the Optimised total cost is Rs.94326)

7. Uncertain Demand and Variable Lead Time

Using following data build an Excel based simulation model to compute minimum total cost for fifty periods:

Average weekly demand = 400 units Standard Deviation of the weekly demand = 15 units Delivery lead time = 4 weeks Standard deviation of the lead time = 2 weeks Cycle Service Level 95% (Z value 1.65) Product cost = Rs. 500 Annual Carrying cost = 20% Cost of placing order = Rs. 500 Out of stock cost = Rs. 20

Assume opening stock to be 1800 units. Management expects that the opening inventory will lie between 1400 and 0. Similarly the ordering cost will be between 0 and Rs. 500) Calculate total cost for fifty periods. Optimise it by varying opening stock and ordering cost.

(Answer: Opening stock 1398, Ordering cost Rs. 468 and the Optimised total cost is Rs.47102)

8. Fixed Period Review – P Type

Using following data build an Excel based simulation model to compute minimum total cost for sixty one day in March and April:

Average daily demand = 20 units Standard Deviation of the daily demand = 6 units Delivery lead time = 3 days Review period = 7 days (Every Saturday) Cycle Service Level 95% (Z value 1.65) Product cost = Rs. 500 Daily Carrying cost = Rs. 1 Cost of placing order = Rs. 150

Assume opening stock to be 100 units. Calculate total cost for sixty one days. Management feels that the cost of placing each order should be between Rs. 150 and Rs. 700. The management is also willing to start with an opening inventory between 0 and 150 units. Optimise it by varying opening stock and ordering cost.


(Answer: Opening stock 72, Ordering cost Rs. 150 and the total cost is Rs. 7124)

9. Forecasting – High, Medium & Low demand

A firm makes two types of gents shirts – Designer and Standard. The selling price for designer is Rs. 500 and that of standard is Rs. 200. The per unit costs are Rs. 90 and Rs. 85 respectively. The firm does not carry stock for more than a year. It sells off the old stock and recovers Rs. 30 for each designer and Rs. 20 for each standard shirt. The demand for the shirts is equally distributed between high, medium and low. Mean and the standard deviation of the three levels of demands are given below:

Type of Shirt

High Demand

Mean

Medium Demand

Mean

Low Demand

Mean

Demand Std Dev

Designer 2000 900 300 100Standard 5000 1500 800 250

Marketing department predicts that not more 1200 designer shirts should be produced but at least 2000 units of standard shirts should be made available. Develop a forecasting model for 50 demand states and estimate maximum average profit. Assume that there is no penalty for lost sales.(Answer: Order 1089 designer shirts, 2000 standard shirts to make maximum profit of Rs. 959848 for 50 states of demand)

10. Forecasting – Random demand

A firm makes two types of gents’ shirts – Designer and Standard. The selling price for designer is Rs. 2500 and that of standard is Rs. 800. The per unit costs are Rs. 1900 and Rs. 285 respectively. The firm does not carry stock for more than a year. It sells off the old stock and recovers Rs. 830 for each designer and Rs. 140 for each standard shirt. The demand for the shirts is random Mean and the standard deviation of the demand are given below:

Type of Shirt Mean Demand SD of Demand

Designer 600 160Standard 1000 170

Marketing department predicts that not more 800 designer shirts should be produced but at least 2000 units of standard shirts should be made available. Develop a forecasting model for 500 demand states and estimate maximum average profit. Assume that there is no penalty for lost sales.(Answer: Order 800 designer shirts, 2926 standard shirts to make maximum profit of Rs. 351777 for 500 states of demand)


11. Aggregate Planning – Chase

A firm produces a product that has six months demand cycle as shown in the table below:

January February March April May June

Demand Forecast

500 500 600 300 200 100

Workdays 22 19 21 21 22 20

Work Hours @ 8 per day

176 152 168 168 176 160

Each unit requires 10 worker hours to produce, at a labour cost of Rs. 6 per hour regular rate (or Rs. 9 per hour over time rate). The idle time of a worker costs the company a sum of Rs.2 per hour. There are currently 22 workers employed in the subject department. The hiring and training cost per worker is Rs. 400 per person. The layoff cost is Rs. 600 per person. The company policy is to have 10% of the demand forecast as the safety stock that becomes the beginning inventory for the next month. There are currently 100 units in the stock. The inventory carrying cost per unit per month is Rs. 2 only. Unit shortage or stock outs have been assigned a cost of Rs. 6 per unit per month.

Calculate the cost of the aggregate plan if the firm decides to vary workforce size to accommodate demand.

12. Aggregate Planning – Level

Prakash Industries produces a product that has six months demand cycle as shown in the table below:

January February March April May June

Demand Forecast

300 500 400 100 200 300

Workdays 22 19 21 21 22 20

Work Hours @ 8 per day

176 152 168 168 176 160

Each unit requires 10 worker hours to produce, at a labour cost of Rs. 6 per hour regular rate (or Rs. 9 per hour over time rate). The total cost per unit is estimated to be Rs. 200 but the units can be subcontracted at a cost of Rs. 208 per unit. There are currently 20 workers employed in the subject department. The hiring and training cost per worker is Rs. 300 per person. The layoff cost is Rs. 400 per person. The company policy is to have 20% of the demand forecast as the safety stock that becomes the beginning inventory for the next month. There are currently 50 units in the stock. The inventory carrying cost per


unit per month is Rs. 2 only. Unit shortage or stock outs have been assigned a cost of Rs. 20 per unit per month.

The firm must begin January with the 50 units on hand.

Calculate the cost of the aggregate plan if the firm decides to maintain constant work force of 20, and build inventory or incur stock out cost.

13. Seasonal Demand and discounted price – End of the season sale of sweater

A large departmental store plans to sell sweaters in its winter catalogue for $150 each. The manager expects demand to be normally distributed with mean 3000 units and standard deviation of 1000 units. Toward the end of the winter season the store sends out a sales catalogue with discounted prices on unsold items. The discounted price determines the demand in response to the sales catalogue. The manager anticipates that the sales catalogue will generate demand for sweaters with a mean of (1000 – 5p) and a standard deviation of (1000 – 5p)/3, where p is the discounted price charged. Any left over sweater after the sales catalogue are donated to charity. Each sweater costs the store $50. Thus the donation to the charity fetches $25 in tax benefits. The store incurs a cost of $5 per unsold sweater to store and transport them to the charity, resulting in a salvage value of $20 per sweater sent to charity. The manager has decided to charge a discount price of max [$25, ($150 – n/20)], where n is the number of sweaters left over after the winter catalogue sale. The manager would like to identify the number of sweaters that should be purchased at the start of the winter season.(Answer: 3000 sweater ordered, For 100 demand states Average number os sweaters discounted is 497, Average number donated to charity is 254, Average profit is 253007 and standard deviation of profit is 75435)

14. Seasonal Demand and discounted price – Hotel Rooms

Problem 1Consider an exclusive resort with 400 identical rooms. Full price of these rooms per night is Rs. 2000. Management is forecasting on pricing rooms for the next five-day holiday. The goal is to price rooms so as to maximise revenue. Based on past experience, management estimates the relationship between demand, D, and the price, p, by the linear functionD = 1000 – 0.5pThe standard deviation of this demand is estimated to be SD = (1000 – 0.5p)/4This implies that when the price is 1600, there is demand for 200 rooms, while if the price is 1200, there is demand for 400 rooms.

It is also assumed that the demand at the higher rate is randomly distributed with mean of 200 rooms and standard deviation of 60 rooms. To maximise revenue management would like to sell maximum rooms at the higher rate and then book at the maximum discounted rate of 1200 at which all the 400 rooms can be booked.(Answer: Book 206 rooms at 1588 to maximise revenue of Rs. 559,933)


Problem 2Consider a Hotel with 400 identical rooms. Full price of these rooms per night is Rs. 2000. Management is forecasting on pricing rooms for the next holiday season. The goal is to book rooms so as to maximise revenue. Based on past experience, management estimates that there is demand for all the 400 rooms at the discounted rate of Rs. 1200 per room. Management is considering two higher rates of Rs. 1600 and Rs. 1800. It is estimated that the mean demand at Rs. 1600 is of 200 rooms with standard deviation of 10 rooms. Similarly, the mean demand at Rs. 1800 per night is 50 rooms with standard deviation of 5 rooms.

To maximise revenue management would like to sell maximum rooms at the higher rates and then book at the maximum discounted rate of 1200 at which all the 400 rooms can be booked.(Answer: Book 50 rooms at Rs. 1800; 198 rooms at Rs. 1600; 151 rooms at Rs. 1200 yielding maximum revenue of Rs. 311,909)

15. Transportation and Trans-shipment Models

Capacitated Plant Location Model: Sun Oil is a manufacturer of petrochemical products with worldwide sales. The Vice President of Supply Chain is considering several alternatives to meet demand. One alternative is to set up plants in each region – advantages are lower transportation costs and avoidance of import duties but the disadvantages are not able to fully exploit economies of scale (plant will be sized to meet the local demand). Other alternative is to consolidate plants in few regions – this will improve economies of scale but would increase transportation costs and import duties.

Annual demands for each of the five regions are displayed below:

Cost and Demand Data for Sun Oil

Inputs - Costs, Capacities, Demands

Demand Region - Production & transportation costs per 1,000,000 unitsFixed Cost

Low Capacity

Fixed Cost

High Capacity

Supply Region N. America S. America Europe Asia Africa

N. America 81 92 101 130 115 6000 10 9000 20

S. America 117 77 108 98 100 4500 10 6750 20

Europe 102 105 95 119 111 6500 10 9750 20

Asia 115 125 90 59 74 4100 10 6150 20

Africa 142 100 103 105 71 4000 10 6000 20


Demand 12 8 14 16 7

Variable production, inventory and transportation costs are shown above. For example it costs $ 92,000 to produce one million units in North America and sell them in South America. Sun Oil is considering two different plant sizes in each location. Low capacity plants can produce 10 million units per year where as high capacity plants can produce 20 million units. Fixed costs for setting up low and high capacity plants at each of the locations are also displayed.

The Vice President would like to know what the low cost network would look like.

Network Optimization Models: Both TelecomOne and HighOptic are manufacturers of fibre optic telecommunication equipment. TelecomOne has focused on the eastern half of the United States. It has manufacturing plants located in Baltimore, Memphis, and Wichita and serves markets in Atlanta, Boston, and Chicago. HighOptic has targeted the western half of the United States and serves markets in Denver, Omaha and Portland. HighOptic has plants located in Cheyenne and Salt Lake City.

Plant capacities, market demand, variable production and transportation cost per thousand units shipped, and fixed costs per month at each plant are shown below:

Decision Variables for TelecomOptic

Inputs - Costs, Capacities, Demands

Demand Region - Production & transportation costs per 1,000 units (Thousand $) Monthly Capacity Thousand

Units

Monthly fixed cost Thousand

$Supply Region Atlanta Boston Chicago Denver Omaha Portland

Baltimore 1675 400 685 1630 1160 2800 18 7650

Memphis 380 1355 543 1045 665 2321 22 4100

Witchia 922 1646 700 508 311 1797 31 2200

Hi -Salt Lake City 1925 2400 1425 500 950 800 27 5000

Hi -Cheyenne 1460 1940 970 100 495 1200 24 3500

Demand 10 8 14 6 7 11

Management at both TelecomOne and HighOptic has decided to merge the two companies into a single entity to called TelecomOptic. Management feels that significant benefits will result if the two networks are merged appropriately.


TelecomOptic will have five factories from which to serve six markets. Management is debating whether all five factories are needed. They have assigned a supply chain team to study the network of combined company and identify the plants that should be shut down. Help them in making this decision.

Network Optimization Models: Deciding Plants and Warehouses together: Consider the following distribution system:

Single product

Two plants, referred to as P1 and P2

Plant P2 has an annual capacity of 60,000 units

The two plants have the same production costs

Two existing warehouses, referred to as W1 and W2, have identical warehouse handling costs

Three market areas: C1, C2 and C3, with demands of 50,000, 100,000 and 50,000 units respectively

Following table provides distribution cost per unit. For instance, distributing one unit from plant P1 to warehouse W2 costs $5.

Distribution costs per unit

Facility

WarehouseP1 P2 C1 C2 C3

W1 0 4 3 4 5

W2 5 2 2 1 2

Company’s objective is to find a distribution strategy that specifies the flow of products from the suppliers through the warehouses to the market areas without violating the plant P2 production capacity constraint that satisfies market demands, and minimizes the total distribution costs.

16. Supplier Relationship

(a) Retailer of woollen Garments

A retailer is contemplating to place order for the woollen garments for children for the next winter season. To avail bulk discount the retailer plans to order once for the entire season which lasts for barely three months. The average selling price of a particular type of garment sold is Rs. 150 per unit. The wholesale price paid by the retailer to the concerned manufacturer is Rs. 100 per unit. Instead of undergoing the hassles of storing woolen garments during off season and also to avoid the risk of fashion change and damage, the retailer prefers to dispose off unsold stock at the end


of the season at a discount store at Rs. 20 per unit. The manufacturer incurs fixed production cost of Rs. 50000 and the variable production cost per unit is Rs. 70. All other costs are not considered.

From the past experience the retailer estimates that the demand to be

Sr. No.

Demand in units

Probability

1. 1000 0.20

2. 2000 0.20

3. 3000 0.30

4. 5000 0.30

If the retailer has decided to order 4000 units of garments, determine which supply contract out of the following two is most beneficial to both retailer and manufacturer:

(1) Buyback Contract: Manufacturer offers to buy back the unsold garments from retailer for Rs. 35 per unit

(2) Revenue Sharing Contract: Manufacturer and retail have a revenue sharing arrangement in which the manufacturer agrees to decrease the wholesale price from Rs. 100 to Rs. 80 per unit and in return, the retailer provides ten percent of the product revenue to the manufacturer.

(b) Swimsuit: Sporting Goods ManufacturerConsider a company that designs, produces and sells summer fashion items such as swimsuits. About six months before summer, the company must commit itself to specific production quantities for all its products. Since there is no clear indication of how the market will respond to new designs, the company needs to use various tools to predict demand for each design and plan production and supply accordingly. In such settings the trade off are clear: overestimating customer demand will result in unsold inventory while underestimating customer demand will lead to inventory stock outs and loss of potential customers.

Based on past sales, knowledge of the industry, and economic conditions, the marketing department has developed a probabilistic forecast as follows:

Demand Probability

8000 0.110

10000 0.110

12000 0.275


14000 0.225

16000 0.185

18000 0.095

Average 13100

The forecast averages about 13,000, but there is a chance that demand will be greater or less than this.

Additional information available: Production cost per unit (C): $80 Selling price per unit (S): $125 Salvage value per unit (V): $20 Fixed production cost (F): $100,000

(1) If there is no opening inventory, how much is the estimated profit if 9000 units are produced? If 16000 units are produced?

(2) Effect of opening inventory: Calculate the production lot that will maximize profit if there are 5000 units as beginning inventory? If 10000 units are there as beginning inventory? Determine the level of opening inventory at which the company should produce nothing more?

Consider the same swimsuit example. This time we assume that there are two companies involved in the supply chain: a retailer who faces customer demand and a manufacturer who produces and sells swimsuits to the retailer.

Sequential Optimization: Extra information available: The variable cost of production per unit equals $ 35. How much should the retailer order from the manufacturer?

Buyback Contract: Suppose the manufacturer offers to buy unsold swimsuits from the retailer for $55. Estimate the total supply chain profit (sum of retailer and manufacturer profits) if the retailer places orders for 12000, 14000 or 16000 units. How much will be the increase in total profit with respect to the earlier case of sequential optimization?

Revenue Sharing Contract: Suppose the manufacturer and retailer have revenue sharing contract. The manufacturer agrees to decrease the wholesale price from $80 to $60, and, in return, the retailer provides 15 percent of the product revenue to the manufacturer. Estimate the share of total supply chain profit if the retailer orders for 14000 units in this case.

To this list add problems listed in the book Supply Chain Modeling by J. Shapiro.


scm excel based models

Documents

technical support people

salt lake city

excel based simulation

marketing department predicts

revenue sharing contract

maximum discounted rate

network optimization models

optimised total cost