ISyE 3104: Introduction to Supply Chain Modeling: Manufacturing and Warehousing
Instructor: Spyros Reveliotis
Spring 2002
Homework #5
Due Date: Friday, 4/5/02
Problem set:
1.
From your textbook: Problems 5.1, 5.2, 5.3, 5.4, 5.10 and 5.11 (for 5.11, just provide the formulation modeling the new situation).
5.1
Advantages of Aggregating Items
Less data is required for building and maintaining models.
Less computation required to solve model (larger systems can be modeled).
Forecasts of aggregate demand are usually more accurate than those of individual items.
Allows short term scheduling to react to current inventory levels when planning batch sizes and order release times.
5.2
Standard outputs of an aggregate production plan
Production quantity for each product family in each period.
Planned amount of each key resource to be used over the planned horizon.
Inventories and backlogs to be carried during the different periods.
Differences from the outputs of the disaggregate production plan:
Aggregate plan avoid solving unnecessary detailed data
5.3
Major tradeoffs examined in aggregate production planning models are:
The major tradeoffs are among inventory, workforce/equipment changes, workload changes/overtime, and product shortage. An example would be:

Inventory holding cost VS Shortage cost

Cost to modify capacity VS holding cost and shortage cost
5.4
Assume that the machine hours required for each product type are known.
Decision Variables:
X it
= amount of product i made in period t;
I it
= inventory level of product i at the end of period t;
W jt
= workforce level of worker type j in period t;
H jt
(F jt
) = Number of workers of type j Hired (Fired) effective period t.
Note that inventory variables can be eliminated from the model by noting I it

I i 0
 r t  

1
X ir
 t

1 r
D ir
and then replacing all I it
variables. a. X it
variables: (1500 products) x (18 months) x (4 weeks/month) = 108,000.
W jt
, H jt
, F jt
variables: (12 labor classes) x (18 months) x (4 wks/mth) x (3 vars) = 2,592.
Thus, approximately 110,000 variables. b. For the aggregated model we have 4 products, 1 labor class, 36 periods. This requires 144 X it
variables and 36 variables each for workforce level, hiring, and firing.

5.10
Let X it
= production of product family i in period t ;
S ijt
= shipments of product family i from the plant to warehouse j in period t ;
I ijt
= inventory of i at location j in period t ( j = 0 for plant, j > 0 indicates warehouse);
W t
= number of workers employed in period t ;
H t
(F t
) = number of workers hired (fired) in period t;
O t
= overtime hours scheduled in period t .
Minimize Cost = t
3 

1


20(120) W t

30 O t

400 H t

500 F t

2  i

1



0.8
I

0.6
j
2 

1
I ijt




 subject to:
5 X
1 t

6 X
2 t

120 W O t
 t

60 H t
,
 t ,
I

I
,

1

X it

S

S ,
 i, t ,
 i, j>0, t , I ijt

I


S ijt

D ijt
W t

W t

1

H t

F t
,
 t
I
1 jt

I
2 jt

3000 ,
 j t
O t

20 W t

20 H t
I

0 , I ijt

500 , X it

0
Labor Availability
Plant Inventory Balance
Warehouse Inventory Balance
Workforce Balance, W
0
= 275
Warehouse Storage Capacity
 t Overtime Limit (Assumes no new workers)
5.11
Assumption: Workers may not be hired or fired. (There will be no H t
and F t
.)
Minimize Cost = t
3 

1


20(120) W t

30 O t
 i
2 

1



0.8
I

0.6
j
2 

1
I ijt




 subject to:
5 X
1 t

6 X
2 t

120 W t

O t
,
 t ,
I

I
,

1

X it

S

S ,
 i, t ,
I ijt

I


S ijt

D ijt
 i, j>0, t ,
W t

0.99
W t

1
,
 t
I
1 jt

I
2 jt

3000 ,
 j t
O t

20 W t
I
 t

0 , I ijt

500 , X it

0
Labor Availability
Plant Inventory Balance
Warehouse Inventory Balance
Workforce Balance, W
0
= 275
Warehouse Storage Capacity
Overtime Limit
2.
A local firm manufactures children's toys. The projected demand over the next four months for one particular family of toy robots is as follows:
Month
July
August
September
October
Workdays Forecasted Demand
23
16
20
22
3,825
7,245
2,770
4,440

Assume that a normal workday is 8 hours. Hiring costs are $350 per worker and firing costs (including severance pay) are $850 per worker. Holding costs are $4.00 per aggregate unit held per month.
Assume that it requires an average of 1 hour and 40 minutes for one worker to assemble one toy.
Shortages are not permitted. Assume that the ending inventory for June was 600 of these toys and the manager wishes to have at least 800 units on hand at the end of October. Assume that the current workforce level is 35 workers. a.
Find the minimum constant workforce plan for the four months and the total hiring, firing and holding costs of that plan.
Solution:
Since there is a 600 units ending inventory in June, the demand for July will be reduced to be
3825– 600 = 3225 units. By the same way, the demand for October will be increased to be 4440 + 800
= 5240 units. This is based on First In First Out policy.
Month Demand Days Units/workr/mo.
July
August
Sept
Oct
3225
7245
2770
5240
23
16
20
22
110.4
76.8
96
105.6
Cumulative
Units/workr
110.4
187.2
283.2
388.8
Cumulative
Demand
3225
10470
Worker Req.
30
13240
18480
56
47
48
Note:
1. To calculate the number of units produced per worker for a particular month (column 4), we first calculate the number of units produced per day, which is equal to (8 hr)/(1 hr 40 min/ unit) =
4.8 units/day. Thus, the number of units produced per month is Days available * 4.8. For example, in July, unit/worker/mo. = 23 days * 4.8 units/day – 110.4 units/worker/mo.
2. To calculate the number of workers required for meeting the demand without any stockouts
(column 7), for every month, we divide the cumulative demand up to that month (column 6) by the cumulative number of units produced by a worker (column 5) over the corresponding time interval. Then, we know that for any given month, the cumulative demand up to that month will be met if we employ the number of workers indicated in the corresponding cell, and have them work full-time until that month, possibly building anticipatory inventories in certain months, to be consumed in the subsequent months. Obviously, if we want to experience no stockouts over the entire planning horizon , we must employ the maximum number of workers appearing in the aforementioned column (i.e., 56). These workers will need to work full-time for the months of
July and August, but they will be under-utilized in the sub-sequent months of September and
October.
Table below calculates the production based on 56 workers:
Ending Inv. Month Cumulative
Demand
July
August
3225
10470
Production Cumulative
Production
6182
4288
6182
10470
2957
0
Sept
Oct
13240
18480
2770
5240
13240
18480
0
800
The total inventory held is 2,957 + 0 + 0 + 800 = 3757 units, which results in an inventory holding cost of (3757)(4) = $15,028. The cost of hiring is (56-35)(350) = $7,350, giving a total cost of $15,028 +
$7,350 = $22,378. b.
Determine the plan that corresponds to demand chasing , i.e., the plan that changes the workforce level each month to most closely match the demand, and compute the cost(s) of this plan.

Month Demand Days Unit/worker/mo. Worker Req./mo.
July 3225 23 110.4 30
August
Sept
Oct
7245
2770
5240
16
20
22
76.8
96
105.6
95
29
50
Month Cumulative
Demand
July
August
Sept
Oct
3225
10470
13240
18480
Production Cumulative
3225
7245
2770
5240
Production
3312
10608
13392
18672
Ending Inv.
0
0
0
800
The total number of workers fired is 5+66 = 71 at a cost of (5+66)(850) = $60,350 and the total hired is 65+21 = 86 at a cost of (86)(350) = $30,100. inventory holding cost is (800)(4) = $3,200. The total cost of this plan is $60,350 + $30,100 + $3,200 = $93,650.
Note: Since the problem does not give us any information about the labor rates, we cannot compute any
(regular) labor costs (which is also part of the cost structure in the considered problem context).
Furthermore, these labor costs include also an "undertime" cost, since in certain months there is a certain amount of time paid to the workers for doing nothing - this is the results of rounding to the next intege / employing the "ceiling" function , in the computation of the worker requirements.
3.
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
Pale Ale
Avail. Inventory (in cases)
800
Stout
Winter Ale
Summer Brew
400
750
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 W10 W11 W12 W13
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
Summer Brew
Octoberfest
320 330 310 310 265 265 265 265 180 180 180 180 150
40
Product W14 W15 W16 W17 W18 W19 W20 W21 W22 W23 W24 W25 W26
Pale Ale
Stout
300 300 300 300 300 300 300 300 300 300
190 190 190 200 200 200 200 215 215 215
300 300 300
215 225 225
Winter Ale 150 150 150 50
Summer 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 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 $ s
1
if the utilized fermentor had been used previously for the production of the same type of beer, and $ s
2 dollars otherwise. ii.
Carrying one case of inventory from on week to the next costs $ h i
, i=1,…5
, depending on the type of beer. iii.
Backordering a case of beer for one week costs $ b i
, 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 = i
5 T 

1 t

1 cX it
where X it
is the number of unit of product i produced at time t
Holding cost =
5 T 
where I it
+ is the on-hand inventory of product i at time t i

1 t

1 h I i it

Backordering cost = i
5 T 

1 t

1 b I i it

where I it
is the backorders of product i at time t
Setup cost =
5 T  i

1 t

1 s F P
1 it it
 i
5 T 

1 t

1 s Max F F
2
( it
,
,

1
)(1

P it
)
Where F it
is the number of fermentors occupied by product i at time t
P it
is equal to 1 if F it
= F i, 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.

Extra credit (50%)
Provide a mathematical (mixed integer) programming formulation for production scheduling problem as it arises in the McGuinness case study presented in class. In your formulation, consider that backorders are not allowed , and the optimized objective is the minimization of the inventory holding cost (i.e., set up cost is not an issue - the main value of this formulation is that it would provide a definite answer on whether there exists a feasible production schedule or not).
Please, refer to the following (pdf) document: http://www.isye.gatech.edu/~spyros/courses/IE3104/mcguinness-prod-plan.pdf