Prodcution Planning Problems
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Example: Brute Production Process
Rylon Corporation manufactures Brute and Channelle perfumes from a raw material
costing $3/pound. Processing 1 pound of the material takes 1 hour and yields 3
ounces of Regular Brute and 4 ounces of Regular Chanelle. This perfume can be
sold directly or can be further processed to make luxury versions of the perfumes.
One ounce of Regular Brute can be sold for $7, but 3 hours of processing and $4
processing cost will convert it into one ounce of Luxury Brute, which sells for $18.
One ounce of Regular Chanelle can be sold for $6, but 2 hours of processing and $4
processing cost will convert it into one ounce of Luxury Chanelle, which sells for
$14. Rylon has available 4000 pounds of raw material, 6000 hours of processing
time, and will sell everything it produces.
Determine how Rylon can maximize its profit?
2
3
4
This formulation is incorrect. Observe that the variables x1 and x3 do not appear in
any of the constraints. This means that any point with x2 =x4= x5= 0 and x1 and x3
very large is in the feasible region. Points with x1 and x3 large can yield arbitrarily
large profits.
Thus, this LP is unbounded. Our mistake is that the current formulation does not
indicate that the amount of raw material purchased determines the amount of Brute
and Chanelle that is available for sale or further processing.
5
More specifically, from Figure (and the fact that 1 oz of processed Brute yields exactly 1
oz of Luxury Brute), it follows that
6
Similarly, from Figure it is clear that
7
Two constraints relate several decision variables. Students often omit constraints of
this type. As this problem shows, leaving out even one constraint may very well
lead to an unacceptable answer (such as an unbounded LP). If we combine two
constraint with the usual sign restrictions, we obtain the correct LP formulation.
8
Sailco Production Planning Problem
Sailco Corporation must determine how many sailboats should be produced
duruing each of the next four quarter (one quarter = three months). The demand
during each of the next four quarters is as follows; first quarter, 40 sailboats:
second quarter, 60 sailboats; third quarter, 75 sailboats; fourth quarter, 25
sailboats. Sailco must meet demands on time. At the beginning of the first
quarter, Sailco has an inventory of 10 sailboats. At the beginning of each
quarter, Sailco must decide how many sailboats should be produced during that
quarter. For simplicity, we assume that sailboats manufactured during a quarter
can be used to meet demand for that quarter. During each quarter, Sailco can
produce up to 40 sailboats with regular-time labor at a total cost of $400 per
sailboat. By having employees work overtime during a quarter, Sailco can
produce additional sailboats with overtime labor at total cost of $450 per
sailboat.
At the end of each quarter (after production has occurred and the current
quarter's demand has been satisfied), a carrying or holding cost of $20 per
sailboat is incurred. Use linear programming to determine a production schedule
to minimize the sum of production and inventory costs during the next four
quarters
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Sailco Production Planning Problem
✓ Sailco Co. must determine how many sailboats should be produced during
each of the next four quarters.
✓ Demand is known:
Q1: 40, Q2: 60, Q3: 75, Q4: 25.
✓ At the beginning of the first quarter, Sailco has an inventory of 10 boats.
✓ Sailco can produce 40 boats (each costs $400) per quarter with regular labor
hours. They can produce more if workers work overtime. These boats cost
$450 to produce.
✓ At the end of each quarter a carrying or holding cost of $20 per boat is
incurred.
✓ Sailco must meet demand on time. In other words, sailboats manufactured
during a quarter cannot be used to meet demand for previous quarters,
backordering is not allowed.
✓ Use linear programming to determine a production schedule to minimize the
sum of production and inventory costs during the next four quarters.
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Summary Table
Demand
Starting
Inventory
Production Limit
Per Quarter
Cost RegTime/Boat
(boat≤40)
Cost OverTime/Boat
(boat≥40)
Inventory
cost/boat
Q1
40
10
40
$400
$450
$20
Q2
60
40
$400
$450
$20
Q3
75
40
$400
$450
$20
Q4
25
40
$400
$450
$20
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Sailco Production Planning Problem
• For each quarter, Sailco must determine the number of sailboats that should be
produced by regular-time and by overtime labor.
• Decision variables :
– xt : number of sailboats produced by regular time labor during
quarter t
– yt : number of sailboats produced by overtime labor during quarter t
– It : number of sailboats on hand at the end of quarter t
• Parameters:
– dt : demand during period t
Not: The decision variables have been split into x and y because we need to think of
them as different types of products. The reason for this is because it is the same
product that can be manufactured in the same period at two different costs.
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Sailco Production Planning Problem
X1
X2
Y1
I0=10
X3
Y2
I1
Y3
Q3
Q2
d1
Y4
I3
I2
Q1
X4
d3
d2
Xt
I4
Q4
d4
Yt
It=It-1+Xt+Yt-dt
It-1
Qt
dt
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Sailco Production Planning Problem
• Inventory Balance Equation:
It = It-1+ xt + yt – dt (t = 1,2,3,4)
• Note that quarter t’s demand will be met on time if and only if
(iff ) It ≥ 0.
• Sailco’s constraints:
• Each period’s regular time production does not exceed 40.
• Inventory constraints for each time period.
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Complete LP Model
min z = 400x1 + 400x2 + 400x3 + 400x4 + 450y1 + 450y2 + 450y3 +
450y4 + 20I1 +20I2 +20I3+20I4
subject to
x1≤40; x2≤40; x3≤40; x4≤40;
x1 + y1 + 10 - 40 = I1
x2 + y2 + I1 - 60 = I2
x3 + y3 + I2 - 75 = I3
x4 + y4 + I3 - 25 = I4
It ,xt, yt ≥ 0, t=1,2,3,4
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Optimal Solution
40
40
0
10
10
Q1
40
40
10
35
60
0
0
0
Q2
25
Q3
75
0
Q4
25
16
Rolling Horizon
40
?
0
10
?10
Q1
?40
?
?
?
?60
?
?
?
Q2
?
Q3
?75
?
Q4
?25
17
Rolling Horizon
40
?
0
10
15
Q1
35
?
?
?60
?
?
?
Q2
?
?
Q3
?75
✓ Suppose quarter 1’s actual demand is 35 boats.
✓ Quarter 5 demand suppose the forecast is 36.
Rolling Horizon-dönen planlama ufku
?
Q5
Q4
?36
?25
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Rolling Horizon Model
min z = 400x2 + 400x3 + 400x4 + 400x5 + 450y2 + 450y3 + 450y4 +
450y5 + 20I2 +20I3 +20I4+20I5
s.t.
x2≤40; x3≤40; x4≤40; x5≤40;
x2 + y2 + 15 = I2 + 60
x3 + y3 + I2 = I3 + 75
x4 + y4 + I3 = I4 + 25
x5 + y5 + I4 = I5 + 36
It ,xt, yt ≥ 0, t=2,3,4,5
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