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Solution to GPE homework #3
Designing a fruit juiceTask description
I Find concentrations for each ingredient.
I Respect upper and lower bounds for ingredients.
0.15 <= <= 1
0.08 <= <= 0.2
0.02 <= <= 0.05
0.05 <= <= 1
0.2 <= <= 1
I Stay within 10% of RDA values0.9RDA(n) <=
∑f∈F
vnfxf <= 1.1RDA(n)
I Minimize cost over year:∑
f∈F∑
m∈M cfmxf
Designing a fruit juiceIP model
min∑f∈F
∑m∈M
cfmxf
subject to
0.9RDA(n) <=∑f∈F
vnfxf <= 1.1RDA(n) ∀n ∈ N
lbf <= xf <= ubf ∀f ∈ F∑f∈F
xf = 1 ∀f ∈ F
Designing a fruit juiceResults for 1st task
5%
8%
2%
32%
53%
Total cost per year: 9355000For 1200000 bottles per year: 7 per bottle
Designing a fruit juice2nd task
I can take advantage of seasonal price variations: buy onemonth in advance
I cannot buy more than one month in advance because fruitgoes bad
I at end of year: storage empty
Introduce more variablesI bfm ∈ N: Buy bfm fruits of type f in month mI sfm ∈ N: Amount of fruits of type f bought in monthm that is
available in next month
Introduce more constraintsI bfm + sf ,m−1 − sfm − 100000xf = 0 form 6= JanuaryI bfm − sfm − 100000xf = 0 form = JanuaryI sfm <= 100000 · xf
Designing a fruit juiceIP model for 2nd task
min∑f∈F
∑m∈M
cfmbfm
subject to
0.9RDA(n) <=∑f∈F
vnfxf <= 1.1RDA(n) ∀n ∈ N
lbf <= xf <= ubf ∀f ∈ F∑f∈F
xf = 1 ∀f ∈ F
bfm + sf ,m−1 − sfm − 100000 · xf = 0 ∀f ∈ F, ∀m ∈ M,m 6= January
bfm − sfm − 100000 · xf = 0 ∀f ∈ F,m = January
sfm <= 100000 · xf ∀f ∈ F, ∀m ∈ M
Designing a fruit juiceResults for 2nd task
I concentrations stay the sameI sf ,December will be 0 because of minimization
%
1·104
2·104
3·104
4·104
Optimal apple buying policy
32000
January
32000
February
32000
March
32000
April
32000
May
32000
June
32000
July
32000
August
32000
September
64000
October
32000
November
0
December
price development
Designing a fruit juiceResults for 2nd task
I concentrations stay the sameI sf ,December will be 0 because of minimization
%
1·104
2·104
3·104
4·104
Optimal apple buying policy
32000
January
32000
February
32000
March
32000
April
32000
May
32000
June
32000
July
32000
August
32000
September
64000
October
32000
November
0
December
price development
Designing a fruit juiceResults for 2nd task
I concentrations stay the sameI sf ,December will be 0 because of minimization
%
1·104
2·104
3·104
4·104
Optimal apple buying policy
32000
January
32000
February
32000
March
32000
April
32000
May
32000
June
32000
July
32000
August
32000
September
64000
October
32000
November
0
December
price development
Designing a fruit juiceResults for 2nd task
%
1·104
2·104
3·104
4·104
Optimal banana buying policy
8000
January
8000
February
8000
March
8000
April
8000
May
8000
June
8000
July
8000
August
8000
September
8000
October
8000
November
8000
December
price development
%
1·104
2·104
3·104
4·104
Optimal lemon buying policy
2000
January
2000
February
2000
March
2000
April
2000
May
2000
June
2000
July
2000
August
2000
September
2000
October
2000
November
2000
December
price development
Designing a fruit juiceResults for 2nd task
%
1·104
2·104
3·104
4·104
Optimal orange buying policy
53000
January
53000
February
106000
March
53000
April
0
May
53000
June
106000
July
0
August
53000
September
53000
October
53000
November
53000
December
price development
Designing a fruit juiceResults for 2nd task
%
1·104
2·104
3·104
4·104
Optimal pineapple buying policy
5000
January
10000
February
0
March
5000
April
5000
May
10000
June
0
July
5000
August
10000
September
0
October
5000
November
5000
December
price development
Optimal objective value: 8953000Savings of about 4.49% compared to the first task
Designing a fruit juice3rd task
I two different ways to interpret the discount policyI apply discount for 10000 apples to every 10000−pack of apples
let ca be the original price for an apple.44100 apples cost 40000 ∗ 0.6 ∗ ca + 4000 ∗ 0.8 ∗ ca + 100 ∗ ca.
I or apply discount as soon as we buy more than 10000 applesto all of them44100 apples cost 44100 ∗ 0.6 ∗ ca.
Designing a fruit juice3 with first discount policy
Let S := {1, 100, 1000, 10000}.Let hs be the discount for buying s items at once,hs ∈ {0, 0.1, 0.2, 0.4}.
I Add variables dfms for all f ∈ F,m ∈ M, s ∈ S.I Add constraint bfm =
∑s∈S
s · dfms.
I Minimize ∑f∈F
∑m∈M
cfm∑s∈S
dfms ∗ (1− hs)
Designing a fruit juiceIP model for 3a
min∑f∈F
∑m∈M
cfm∑s∈S
dfms ∗ (1− hs)
subject to
0.9RDA(n) <=∑f∈F
vnfxf <= 1.1RDA(n) ∀n ∈ N
lbf <= xf <= ubf ∀f ∈ F∑f∈F
xf = 1 ∀f ∈ F
bfm + sf ,m−1 − sfm − 100000 · xf = 0 ∀f ∈ F, ∀m ∈ M,m 6= Januarybfm − sfm − 100000 · xf = 0 ∀f ∈ F,m = January
sfm <= 100000 · xf ∀f ∈ F, ∀m ∈ Mbfm =
∑s∈S
s · dfms ∀f ∈ F,m ∈ M
bfm ∈ N, sfm ∈ N, xf ∈ Q, dfms ∈ N
Designing a fruit juiceResults for 3a
I Optimal objective function value: 5485600.I Savings compared to first task: 70.54%.I Does not change optimal ingredients of fruit juice.I comparison of buying policies follows
Designing a fruit juice3a comparison of buying policies: apple
%
1·104
2·104
3·104
4·104
Optimal apple buying policy
32000
January
32000
February
32000
March
32000
April
32000
May
32000
June
32000
July
32000
August
32000
September
64000
October
32000
November
0
December
price development
%
1·104
2·104
3·104
4·104
Optimal apple buying policy with discounts a
50000
January
20000
February
50000
March
40000
April
0
May
32000
June
40000
July
50000
August
30000
September
40000
October
32000
November
0
December
price development
Designing a fruit juice3a comparison of buying policies: banana
%
1·104
2·104
3·104
4·104
Optimal banana buying policy
8000
January
8000
February
8000
March
8000
April
8000
May
8000
June
8000
July
8000
August
8000
September
8000
October
8000
November
8000
December
price development
%
1·104
2·104
3·104
4·104
Optimal banana buying policy with discounts a
10000
January
10000
February
10000
March
10000
April
0
May
10000
June
6000
July
10000
August
10000
September
10000
October
10000
November
0
December
price development
Designing a fruit juice3a comparison of buying policies: lemon
%
1·104
2·104
3·104
4·104
Optimal lemon buying policy
2000
January
2000
February
2000
March
2000
April
2000
May
2000
June
2000
July
2000
August
2000
September
2000
October
2000
November
2000
December
price development
%
1·104
2·104
3·104
4·104
Optimal lemon buying policy with discounts a
4000
January
0
February
4000
March
0
April
2000
May
4000
June
0
July
2000
August
2000
September
4000
October
0
November
2000
December
price development
Designing a fruit juice3a comparison of buying policies: pineapple
%
1·104
2·104
3·104
4·104
Optimal pineapple buying policy
5000
January
10000
February
0
March
5000
April
5000
May
10000
June
0
July
5000
August
10000
September
0
October
5000
November
5000
December
price development
%
1·104
2·104
3·104
4·104
Optimal pineapple buying policy with discounts a
10000
January
0
February
10000
March
0
April
10000
May
0
June
10000
July
0
August
10000
September
0
October
10000
November
0
December
price development
Designing a fruit juice3a comparison of buying policies: orange
%
1·104
2·104
3·104
4·104
Optimal orange buying policy
53000
January
53000
February
106000
March
53000
April
0
May
53000
June
106000
July
0
August
53000
September
53000
October
53000
November
53000
December
price development
%
1·104
2·104
3·104
4·104
Optimal orange buying policy with discounts a
60000
January
92000
February
60000
March
50000
April
10000
May
50000
June
100000
July
50000
August
10000
September
50000
October
51000
November
53000
December
price development
Designing a fruit juice3 with second discount policy
I Let di be binary.I Can use inequalities like bfm > 100dfm1, so dfm1 is 1 only if we
buy at least a 100 of fruit f in monthm.I Problem: Would have to put something like
cfmbfm − 0.1d1cfmbfm − 0.1d2cfmbfm − 0.2d3cfmbfm
into objective function, which is not linearI Can of course solve this by fixing bfm to a value computed in
earlier tasksI but this leads to a potentially non-optimal solutionI also, the model is then only used to evaluate the costsI leads to an objective value of 5573600
Designing a fruit juice3 with second discount policy
I have to linearize the nonlinear objectiveI do this similarly to other discount policyI Let S := {1, 100, 1000, 10000}, hs := {0, 0.1, 0, 2, 0.4}.I Introduce variables bfms integer for all f ∈ F,m ∈ M, s ∈ SI they are supposed to denote amount of fruit of type f in
monthm bought at discount hs:∑s∈S
bfms = bfm
I introduce variables dfms binary that regulate whether discounths applies
I only one discount applies:∑s∈S
dfms 6 1
I if discount applies, need to buy the minimum amount for it:bfms > s · dfms for all s ∈ S
I cannot buy at discount if discount doesn’t apply:bfms 6 200000 · dfms
3 with second discount policy,IP modelmin
∑f∈F
∑m∈M
cfm∑s∈S
hs · bfms
subject to
0.9RDA(n) <=∑f∈F
vnfxf <= 1.1RDA(n) ∀n ∈ N
lbf <= xf <= ubf ∀f ∈ F∑f∈F
xf = 1 ∀f ∈ F
bfm + sf ,m−1 − sfm − 100000 · xf = 0 ∀f ∈ F,∀m ∈ M,m 6= Januarybfm − sfm − 100000 · xf = 0 ∀f ∈ F,m = January
sfm <= 100000 · xf ∀f ∈ F,∀m ∈ M∑s∈S
bfms = bfm ∀f ∈ F,∀m ∈ M∑s∈S
dfms 6 1 ∀f ∈ F,m ∈ M
bfms > s · dfms ∀f ∈ F,m ∈ M,∀s ∈ Sbfms 6 200000 · dfms ∀f ∈ F,m ∈ M,∀s ∈ S
bfms ∈ N, bfm ∈ N, dfms ∈ {0, 1}, xf ∈ Q, sfm ∈ N
3 with second discount policy,solution
I leads to an optimal value of 5449200I 2.28% better than the solution fixing the values from the 2nd
taskI comparison of buying policies follows
Designing a fruit juice3b comparison of buying policies: apple
%
1·104
2·104
3·104
4·104
Optimal apple buying policy
32000
January
32000
February
32000
March
32000
April
32000
May
32000
June
32000
July
32000
August
32000
September
64000
October
32000
November
0
December
price development
%
1·104
2·104
3·104
4·104
Optimal apple buying policy with discounts b
32000
January
64000
February
0
March
64000
April
0
May
32000
June
64000
July
0
August
64000
September
32000
October
32000
November
0
December
price development
Designing a fruit juice3b comparison of buying policies: banana
%
1·104
2·104
3·104
4·104
Optimal banana buying policy
8000
January
8000
February
8000
March
8000
April
8000
May
8000
June
8000
July
8000
August
8000
September
8000
October
8000
November
8000
December
price development
%
1·104
2·104
3·104
4·104
Optimal banana buying policy with discounts b
10000
January
14000
February
0
March
14000
April
10000
May
0
June
10000
July
14000
August
0
September
14000
October
10000
November
0
December
price development
Designing a fruit juice3b comparison of buying policies: lemon
%
1·104
2·104
3·104
4·104
Optimal lemon buying policy
2000
January
2000
February
2000
March
2000
April
2000
May
2000
June
2000
July
2000
August
2000
September
2000
October
2000
November
2000
December
price development
%
1·104
2·104
3·104
4·104
Optimal lemon buying policy with discounts b
2000
January
2000
February
2000
March
2000
April
2000
May
2000
June
2000
July
2000
August
2000
September
2000
October
2000
November
2000
December
price development
Designing a fruit juice3b comparison of buying policies: pineapple
%
1·104
2·104
3·104
4·104
Optimal pineapple buying policy
5000
January
10000
February
0
March
5000
April
5000
May
10000
June
0
July
5000
August
10000
September
0
October
5000
November
5000
December
price development
%
1·104
2·104
3·104
4·104
Optimal pineapple buying policy with discounts b
10000
January
0
February
10000
March
0
April
10000
May
0
June
10000
July
0
August
10000
September
0
October
10000
November
0
December
price development
Designing a fruit juice3b comparison of buying policies: orange
%
1·104
2·104
3·104
4·104
Optimal orange buying policy
53000
January
53000
February
106000
March
53000
April
0
May
53000
June
106000
July
0
August
53000
September
53000
October
53000
November
53000
December
price development
%
1·104
2·104
3·104
4·104
Optimal orange buying policy with discounts b
53000
January
53000
February
106000
March
53000
April
0
May
53000
June
106000
July
0
August
53000
September
53000
October
53000
November
53000
December
price development
