Document

University of Thessaly
Department of Mechanical Engineering
Graduate-level course: “Production Systems”
Instructor: Prof. George Liberopoulos
Problem set on EPL and ELSP models
Problem 1. Consider the EPL problem with production rate P parts per unit time, demand rate D parts per
unit time, setup time τs, setup cost A euro per production run (or lot or cycle), production cost c euro per part,
holding cost rate h euro per part (held in inventory) per unit time. Backorders are allowed at a cost of b euro
per part (backordered or “missing”) per unit time. The decision variables are the production lot size Q in
parts per production run (cycle), which determines the height of the “triangle” characterizing the finishedgoods inventory trajectory, and the so-called “fill rate”, i.e., the percentage of demands satisfied immediately
from finished-goods inventory, which determines the position of the triangle relative to zero (see figure
below). Find the unconstrained and constrained optimal values of Q and F.
Finished goods
inventory
FT
(1–F)T
Q(1–D/P)
time
T=Q/D
Problem 2. Consider the EPL problem, where the following production lot size policies may be used:
A. Produce different quantities Q1 and Q2 in an alternating sequence, i.e., Q1, Q2, Q1, Q2, …, where Q1 > Q2.
B. Produce the same quantity Q in each cycle, i.e., Q, Q, Q, …, where Q = (Q1 + Q2)/2.
Show that policy B always results is a smaller average setup, production and inventory holding cost per unit
time than policy A.
Problem 3. Consider a manufacturer producing chocolate chip cookies to meet a constant deterministic
demand rate of D cookies per unit time over an infinite horizon. The cookies require two production steps: in
the first step, a Giant Raw Cookie Production Machine (GRCPM) makes raw cookies, one at a time, at a rate
P. In the second step, a number of raw cookies less than or equal to C are baked simultaneously for B time
units. Once the baking process starts, additional raw cookies cannot be baked until the oven is empty, but
additional raw cookie production can occur. The manufacturer pays a holding cost of € hr per raw cookie per
unit time (until the baking is completed), and then pays a holding cost of hc per baked cookie per unit time
until the baked cookie is sold, with hr ≤ hc. There is no setup cost in either the GRCPM or the oven. All
demand must be met. Note that as is typical for infinite horizon deterministic inventory problems, you can
assume any feasible finite initial inventory in the raw, cooked, and baking stages.
(a) What is a necessary and sufficient condition for the existence of a feasible production policy for this
problem?
(b) Define production cycle time to be the time it takes to produce Q ≤ C units followed by the time B it
takes to bake those units. Note that it is technically possible for production cycles to overlap, since
production can be started while cookies are still baking. Derive an expression for the optimal
production lot size Q, and the optimal time between the start of consecutive cycles.
(c) Derive an expression for the optimal unconstrained production lot size Q (the only constraint on Q is
Q ≤ C), and the optimal time between the start of consecutive cycles, assuming that there is a setup
cost of € A per production run in CRCPM.
1
Problem 4. A plastic foil manufacturer has a processing line that can produce three grades of plastic foils, A,
B, and C. The line can produce one grade at a time. The production rate for any grade is 200 tons per day.
The production line produces continuously 24 hours a day, 365 days a year. The market shares for the three
types are 50%, 40%, and 10%, respectively and the total demand rate for the three grades is denoted by D
tons per day. The setup cost from any grade to any other grade is € 100 and the setup time is 1 hour. The
production costs for grades A, B, and C are 10.000, 12.000, and 15.000 euro per ton, respectively. The
inventory holding cost rate is based on a 20% annual interest rate. Assume that the manufacturer uses the
method of powers of two to determine the optimal number of runs that each grade is produced in each cycle.
Use EXCEL to find and draw on a graph the optimal cycle length and the respective optimal average setup
and inventory holding cost (not the production cost) as a function of the total demand rate D, for values of D
= (140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195). How does the graph look like?
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Problem 4.
A
B
C
D =
195
r =
0,2
market share Di
Pi
ρi
ci
0,5 97,5
200 0,4875 10000
0,4
78
200
0,39 12000
0,1 19,5
200 0,0975 15000
0,975
D
140
145
150
155
160
165
170
175
180
185
190
195
T*
C*
1,657 1048,3
1,638 1046,6
1,621 1045,3
1,605 1044,4
1,59 1043,8
1,576 1043,5
1 563 1043,4
1,563
1043 4
1,667 1082,7
2,083 1244,1
2,778 1542,8
4,167 2187,2
8,333 4219,2
Tunc
hi
5,479
6,575
8,219
C*/1000
1,048
1,047
1,045
1,044
1,044
1,043
1 043
1,043
1,083
1,244
1,543
2,187
4,219
Ai
100
100
100
τi
Tiunc
0,04 0,855
0,04
0,8
0,04 1,176
Tmin= 0,8
Tmin
nj
njround mjround numer denom num
1,07
1
2 136,901
200 0,083
1
1
2 156,427
200 0,083
1,47
2
1 144,647
100 0,042
nmax
2
437,975
1000 0,208
Tunc = 1,511 Tmin= 8,3333 T* = ci*Di
975000
936000
292500
2203500
8,333
C* = 4219
9
8
7
6
5
T* vs D
4
C*/1000 vs D
3
2
1
0
135
145
155
165
175
185
195

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