ISyE 3104: Introduction to Supply Chain Modeling: Manufacturing and Warehousing
Instructor: 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
Xt
It
St
=
=
=
=
Ht
Ft
Ot
SCt
=
=
=
=
Total cost
=
demand in period t (D1 = 1,400, D2 = 1,600, …, D8 = 1,400)
amount of production in period t (X0 = 1,600)
inventory level at the end of period t (I0 = 200)
amount of stockout at the end of period t (these quantities are lost sales, i.e., they are not
carried over as backorders)
increase of production capacity in period t (in number of units)
decrease in production capacity in period t (in number of units)
overtime production in period t (in number of units)
amount of subcontracted unit in period t
Regular Production cost + Holding cost + Stockout cost + Hiring cost + Firing cost
+ Overtime cost + Subcontracting cost
Objective function:
8
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]
t 1
Subject to
I t-1 + X t + O t + SC t =
D t - St + I t
Xt =
X t-1 + H t - F t
Ot 
0.2 X t
It 
400
All variables are positive reals
Material Balance
Workforce Balance
Overtime limit
Inventory Limit
(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 = $50.
13.3
Period
Expected Demand
1
2
3
4
5
6
7
8
1,400
1,600
1,800
1,800
2,200
2,200
1,800
1,400
14,200
Plan A
Period Demand
1
2
3
4
5
6
7
8
1,200*
1,600
1,800
1,800
2,200
2,200
1,800
1,400
Production
(Result of
Previous
Month)
1,600
1,200
1,600
1,800
1,800
2,200
2,200
1,800
Hire
(Units)
Fire
(Units)
Extra
Cost ($)
400
30,000 <--(cost to go from 1,600 in Jan. to 1,200 in Feb.)
20,000 <--(cost to go from 1,200 in Feb. to 1,600 in Mar.)
10,000
20,000
30,000
400
200
400
400
14,200
$ 110,000 Total cost = $50*14,200 + $110,000 = $820,000
* Note: Period 1 demand was given as 1,400 units. Because we have 200 units in beginning inventory, the demand to be
met by production is only 1,200 units.
13.4
Period
0
1
2
3
4
5
6
7
8
Total (Units)
Unit Cost
Total Cost
13.5
Demand
1,400
1,600
1,800
1,800
2,200
2,200
1,800
1,400
Plan B
Production
Ending Inv.
Subcontract
1,400
1,400
1,400
1,400
1,400
1,400
1,400
1,400
11,200
@$50/unit
$560,000
200
200
0
0
0
0
0
0
0
200
@$20/unit
$4,000
400
400
800
800
400
2,800
@$125/unit
$350,000
Plan C
Production
Ending Inv.
Stockout
1,775
1,775
1,775
1,775
1,775
1,775
1,775
1,775
14,200
@$50/unit
$710,000
200
575
750
725
700
275
0
0
375
3,400
@$20/unit
$68,000
$914,000
(a)
Period
0
1
2
3
4
5
6
7
8
Total (Units)
Unit Cost
Total Cost
Demand
1,400
1,600
1,800
1,800
2,200
2,200
1,800
1,400
150
25
175
@$125/unit
$21,875
$799,875
All other things being equal, it would appear that Plan C, with a cost of $799,875, should be recommended
over Plan A (cost = $850,000) or Plan B (cost = $914,000)
(b)
Graph of Plan C
2,200
2,100
2,000
1,900
1,800
1,775
1,700
1,600
1,500
1,400
0
Jan
13.6
Feb
Mar
Apr
May
Jun
Jul
Aug
(a) Plan D: Maximum units in overtime = 0.20 x 1,600 = 320
Period
0
1
2
3
4
5
6
7
8
Total (Units)
Unit Cost
Total Cost
Demand
1,400
1,600
1,800
1,800
2,200
2,200
1,800
1,400
Plan D
Reg.
O.T.
(Units)
(Units)
1,600
1,600
1,600
1,600
1,600
1,600
1,600
1,600
12,800
@$50/unit
$640,000
320
320
200
840
@$100/unit
$84,000
End Inv.
(Units)
Stockouts
(Units)
200
400
400
200
200
1,200
@$20/unit
$24,000
280
280
560
@$100/unit
$56,000
$804,000
(b)
Period
0
1
2
3
4
5
6
7
8
Total (Units)
Unit Cost
Total Cost
Demand
1,400
1,600
1,800
1,800
2,200
2,200
1,800
1,400
Plan E
Production
Subcont (Units)
Ending Inv.
200
400
400
200
1,600
1,600
1,600
1,600
1,600
1,600
1,600
1,600
12,800
@$50/unit
1400
@$125/unit
200
1200
@$20/unit
$640,000
$175,000
$24,000
600
600
200
All other things being equal, it would appear that Plan D, with a cost of $802,000, should be recommended
over Plan E (cost = $839,000).
Note that of all the plans discussed, it would appear that Plan C, with a cost of $799,875, should
be recommended over all others.
$839,000
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
800
Stout
400
Winter Ale
750
Summer Brew
0
Octoberfest
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 W1 W11 W1 W1
0
2
3
Pale Ale
350 340 300 300 300 300 300 300 300 300 300 300 300
Stout
170 170 160 160 165 165 165 165 175 175 175 175 190
Winter Ale 320 330 310 310 265 265 265 265 180 180 180 180 150
Summer
40
Brew
Octoberfest
Product W1 W15 W1 W1 W1 W1 W2 W2 W2 W2 W24 W2 W2
4
6
7
8
9
0
1
2
3
5
6
Pale Ale
300 300 300 300 300 300 300 300 300 300 300 300 300
Stout
190 190 190 200 200 200 200 215 215 215 215 225 225
Winter Ale 150 150 150 50
Summer
40 40 40 100 100 100 100 170 170 170 170 225 225
Brew
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 lifecycles, 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 products
Step 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
5
T
 cX
Production cost =
i 1 t 1
where Xit is the number of unit of product i
it
produced at time t
5
Holding cost =
T
 h I
i 1 t 1

i it
5
Backordering cost =
T
 b I
i 1 t 1
5
Setup cost =
where Iit+ is the on-hand inventory of product i at time t
T

i it
5
where Iit- is the backorders of product i at time t
T
 s F P   s Max( F , F
i 1 t 1
1 it it
i 1 t 1
2
it
i ,t 1
)(1  Pit )
Where Fit is the number of fermentors occupied by product i at time t
Pit 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 threeweek maintenance, you will put schedule release on all three weeks.