ISyE 3104: Introduction to Supply Chain Modeling: Manufacturing and WarehousingInstructor: Spyros Reveliotis

Spring 2003

Solutions to Homework #7

DISCUSSION QUESTIONS

1. (a) Aggregate planning seeks to develop (aggregate) production and capacity plans over a planning horizon typically expanding from 3 to 18 months.(b) Demand options—attempt to “manipulate” demand so that it is better aligned to the

company’s production capacity Influence demand: employ advertising, promotion, personal selling, price cuts,

etc., to either change the magnitude of demand, or shift it in time Back order during high demand periods Counter-seasonal product mixing: produce lawn mowers and snow blowers

(c) Capacity options – attempt to “absorb” fluctuations in demand by producing in advance or adjusting production capacity

Change inventory levels Vary size of workforce by hiring and firing Vary production rate through use of overtime or idle time Subcontract Use part-time workers

3. Mathematical models are not more widely used because they tend to be relatively complex and are seldom understood by those persons performing the aggregate planning activities. The second issue is especially important given the fact that in order to establish computational tractability, the mathematical problems are based on simplifications of the original problem, and therefore, any obtained solution must be assessed and, if necessary, modified for feasibility, while maintaining near-optimality (and this requires some analytical maturity).

5. Aggregate planning is usually considered to encompass several production cycles. Some products, large ships or nuclear reactors, for example, have very long production cycles (years or tens of years). Other products, such as lawn mowers or jewelry have production cycles measured in days.

6. The Master Production Schedule (MPS) is produced by “disaggregating” the aggregate plan. In essence, the MPS must take into consideration the capacity and aggregate production plan developed in the aggregate planning stage as constraints to be observed by any feasible detailed production schedule. On the other hand, feasibility problems experienced in the MPS phase might trigger the reconsideration of the aggregate plan.

LP Formulation for Problem 13.3-13.6

Let Dt = demand in period t (D1 = 1,400, D2 = 1,600, …, D8 = 1,400)Xt = amount of production in period t (X0 = 1,600)I t = inventory level at the end of period t (I0 = 200)St = amount of stockout at the end of period t (these quantities are lost sales, i.e., they are not

carried over as backorders)Ht = increase of production capacity in period t (in number of units)Ft = decrease in production capacity in period t (in number of units)Ot = overtime production in period t (in number of units)SCt = amount of subcontracted unit in period t

Total cost = Regular Production cost + Holding cost + Stockout cost + Hiring cost + Firing cost+ Overtime cost + Subcontracting cost

Objective function:

Min total cost = [c*Xt + 20I t + 100S t + (50=5000/100)H t +(75=7500/100)F t+ (c+50)O t +(c+75)SC t]

Subject to I t-1 + X t + O t + SC t = D t - St + I t Material Balance

X t = X t-1 + H t - F t Workforce BalanceO t 0.2 X t Overtime limitI t 400 Inventory Limit

All variables are positive reals

(In fact, the variables should be integer, but this is one of the “approximations”/simplifications that we introduce in order to establish computational tractability; the obtained solution should subsequently be rounded off to a feasible integer solution. This type of rounding is expected to provide a near-optimal solution since the variable values in the optimal solution of the the above LP, will be large numbers, in general, and therefore, perturbing them by a fractional quantity will not take us very far away from the optimal point. Such an argument would not hold, though, if the variables had small values – in the order of unity.)Finally, in the above formulation, c denotes the unit production cost when production is carried out through regular labor; unfortunately, this quantity is not provided in the problem data. Taking into consideration the regular production cost (and therefore knowing c) is also important for obtaining a more accurate estimate for the total cost incurred by the various plans examined in problems 13.3-13.6, and for proceeding to comparisons regarding the cost-effectiveness of these plans. For the subsequent calculations, it is assumed that c = $50.

(b) Graph of Plan C

2,200

2,100

2,000

1,900

1,800

1,700

1,600

1,500

1,400

Jan Feb Mar Apr May Jun Jul Aug0

1,775

1. Computing a Master Production Schedule for the McGuinnes & Co. Microbrewery.

In this problem you are invited to use the tabular (spreadsheet-based) approach to develop a 6-month (26 weeks) MPS for the McGuinness microbrewery case study presented in class. A complete description of the case study and the spreadsheet discussed in class that can support your calculations, can be downloaded from the course Web-site (http://www.isye.gatech.edu/~spyros/courses/IE3104/course_materials.html). A detailed description of the faced situation is as follows:a. At the beginning of 2001, McGuinness & Co. had in its inventory the following

quantities from each of the five products:Product Avail. Inventory (in

cases)Pale Ale 800Stout 400Winter Ale 750Summer Brew 0Octoberfest 0

Furthermore, at that point, the company was brewing a full fermentor for each of the first two products, with these production lots requiring another week of fermentation for the pale ale (i.e., this lot would be available at the beginning of week 2), and two more weeks of fermentation for the stout (i.e., this lot would be available at the beginning of week 3).Finally, using the forecasting techniques discussed in class, and the information available in his order records, Mr. McGuinness was contemplating that the demand for each of the company five products over the next six months (26 weeks), would evolve as follows:

Product W1 W2 W3 W4 W5 W6 W7 W8 W9 W10

W11 W12

W13

Pale Ale 350 340 300 300 300 300 300 300 300 300 300 300 300Stout 170 170 160 160 165 165 165 165 175 175 175 175 190Winter Ale 320 330 310 310 265 265 265 265 180 180 180 180 150Summer Brew

40

OctoberfestProduct W1

4W15 W1

6W17

W18

W19

W20

W21

W22

W23

W24 W25

W26

Pale Ale 300 300 300 300 300 300 300 300 300 300 300 300 300Stout 190 190 190 200 200 200 200 215 215 215 215 225 225Winter Ale 150 150 150 50Summer Brew

40 40 40 100 100 100 100 170 170 170 170 225 225

Octoberfest

Using the above data, and the information in Word document describing the McGuinness & Co. Microbrewery case study regarding (i) the production (fermentation) lead times for each brew, (ii) the production capacity of the

microbrewery, and (iii) the company policy regarding the maximal allowed shelf-life, develop a production schedule for the next 26 weeks. In the preparation of this schedule you should also consider that, under the current operational conditions, the company produces the various products only in lots of a full or half fermentor (this was, indeed, the situation for the senior design project!)

In your work, you can utilize the spreadsheet presented in class (and available at the course Web-site). Remember, that the green cells in this spreadsheet correspond to the input data required for the problem definition, the orange/brown cells introduce your scheduling decisions (i.e., when to initiate production for each product, and how much to produce in each lot) in the overall computation, while the remaining white cells implement all the additional computations needed in order to evaluate the feasibility and quality / performance of your schedule. Also, as indicated in class, the overall spreadsheet computation is organized in a number of segments, with the top segment assessing the feasibility of any tentative schedule w.r.t. the available production capacity (i.e., fermentor capacity and availability), while the remaining segments compute the projected inventory position for each product, based on the current product availability (i.e., initial inventory position and scheduled receipts), its demand profile, and the contemplated scheduled releases (i.e., the (tentative) production plan w.r.t. this product).

Notice that there is a possibility that you will find out that the satisfaction of the entire (projected) demand with the current production resources and product availability is impossible. In that case, and considering the fact that the company experiences intense growth, the best strategy for Mr. McGuinness would be to expand its production capacity by buying and installing another (or possibly more) fermentor(s). However, in your work consider that, due to financial constraints, the company cannot afford the purchase of another fermentor in the immediate future, and discuss how Mr. McGuinness should deal with any arising schedule infeasibilities.At the end of this part of your work, please, turn in (i) the proposed production schedule, (ii) the resulting spreadsheet, and (iii) a supporting document explaining and justifying your proposition.

Solution:For the specified problem data, there is no feasible production schedule that can satisfy the entire product demand over the considered planning horizon. Hence, assuming that no additional production capacity (fermentors) can be installed during the considered time interval, we must consider how to accommodate the schedule infeasibility, by deciding which part of the demand should be left uncovered.

This decision should take into consideration the product phases w.r.t. (i) their entire life-cycles, as well as (ii) their seasonal cycles, and their implications for the company’s marketing and distribution policies. One way to reason about this problem is as follows:

1. The pale ale seems to be the major company product (i.e., the most well-established and with the most extensive circulation), having reached its mature phase. Hence, the company should maintain a stable and responsive production for this product. Furthermore, the current product expansion to a new market (Northeast) implies that the company must be careful with its new undertaken obligations (deals and/or

contracts with the new distributors and customers). Based on these considerations, it seems that the demand for pale ale should be met on its entirety.

2. The stout is a brew that is in its growth phase, and it is developing to the second major product for the company. Therefore, it is to the company’s advantage to promote aggressively this product. Allowing for planned shortages does not seem to support the company interests.

3. Similarly, the summer brew is a product in its growth phase. Furthermore, the product demand in the considered planning horizon corresponds to the “opening period” for the product. As a result, the company should try to meet the entire expected demand for this product, as well, since in this way it would promote and secure the product position in the market.

4. Finally, the winter ale is a product that is rather well-established; the product has been around for a while, and its annual demand presents some stability. At the same time, this is a seasonal product for the company, and the estimated product demand for the considered planning horizon – especially the last few weeks - corresponds to the closing phase of the product seasonal cycle. The previous two remarks imply that the company might be able to “take a hit” w.r.t. this product, by terminating the product distribution a little earlier than planned for this year, without this fact affecting significantly the product marketability and the company image.

To initialize the excel spreadsheet, we will follow the following steps:Step 1: Specify the number of fermentors, amount produced per fermentors, and product shelf life.

Step 2: Put in the fermentation time and the initial inventory level, i.e. 2 week fermentation time and initial inventory level of 800 for pale ale.

Step 3: Key in the given demand of each productsStep 4: Introduce the scheduled receipts, taking into consideration that the company is currently brewing one fermentor of pale ale and one fermentor of stout, to be ready by weeks 2 and 3 respectively.

Now we are ready to experiment with the spreadsheet in order to determine the scheduled releases that will "minimize" the implications of the experienced shortages.

A schedule developed according to the logic outlined above is as follows:

b. Suppose that Mr. McGuinness hired the services of an accountant who, after some thoughtful analysis of the company operations, informed him that his operational cost breakdown is as follows:

i. Every initiation of a new production lot costs $s1 if the utilized fermentor had been used previously for the production of the same type of beer, and $s2

dollars otherwise.ii. Carrying one case of inventory from on week to the next costs $hi, i=1,…5,

depending on the type of beer.iii. Backordering a case of beer for one week costs $bi, i=1,…,5, depending on

the type of product being backordered.iv. Finally, the variable production cost of producing one case of beer is equal to

$c.

Develop a formula that estimates the total cost of the production schedule developed in part (a). (As you can see, once this formula is implemented in the provided spreadsheet, it can allow the development of MPS's that are evaluated and "optimized" according to the more "standard" considerations of inventory control theory.)

Solution:

Total cost = Production cost + Holding cost + Backordering cost + Setup cost

Production cost = where Xit is the number of unit of product i

produced at time t

Holding cost = where Iit+ is the on-hand inventory of product i at time t

Backordering cost = where Iit- is the backorders of product i at time t

Setup cost =

Where Fit is the number of fermentors occupied by product i at time tPit is equal to 1 if Fit = Fi, t-1; 0 otherwise

c. An additional concern that arises in the development of feasible production schedules is the accommodation of any preventive maintenance of the production equipment that might reduce the nominal production capacity over certain periods of the planning horizon. Suppose the McGuinness & Co. Micro brewery runs a preventive maintenance program that necessitates the removal of each fermentor for certain predetermined time intervals over the considered planning horizon (measured in weeks - typically in the order of one to three weeks since these fermentors might have to be taken to the facility of the sub-contractor supporting the maintenance

program). Discuss how you would modify the provided Excel spreadsheet in order to incorporate the maintenance effects in your calculations.

Solution:

There are two possible ways to accommodate the preventive maintenance.

1. We can replace the “number of fermentors” cell on the very top of the spreadsheet, by an entire input row that will allow us to state the fermentor availability on a period-by-period basis.

2. We can add another imaginary (fictitious) product that would occupy the fermentors during the periods of scheduled maintenance. For this example, we can assume and use the Octoberfest Lager as a maintenance requiring product. You need to specify the fermentation time to be 1 week. So, if you have a three-week maintenance, you will put schedule release on all three weeks.