Workshop in Mathematical Programming Model building in Mathematical Programming Oct. 10 – Nov. 14, 2006 Akiko Yoshise Materials are available at http://infoshako.sk.tsukuba.ac.jp/~yoshise/Course/MC/
Workshop in Mathematical Programming
Model building in Mathematical Programming Oct. 10 – Nov. 14, 2006 Akiko Yoshise
Materials are available athttp://infoshako.sk.tsukuba.ac.jp/~yoshise/Course/MC/
Schedule:
I. Oct. 10What is Mathematical ProgrammingHow to get XPRESS-MPCase study I
II. Oct. 17Some Special Types of Mathematical ProgrammingCase study I IAssignment #1
Due date Oct. 30
Schedule:III. Oct. 24:
Building Integer Programming ModelCase study III
Assignment #2Due date Nov. 20
IV. Nov. 1: Solving Linear Programming ModelSolving Integer Programming Model
V. Nov. 8:Discussions
VI. Nov. 15:Presentation of Assignment#2
Linear Programming Models
.0,0 ,12
,42 subject to32 Maximize
21
21
21
21
≥≥−≥+−
≤++
xxxx
xxxx
Minimax Objectives
sconstraintlinear alconvention subject to
Maximum Minimize ⎟⎟⎠
⎞⎜⎜⎝
⎛∑
jjiji
xa
sconstraintlinear alconvention
, allfor subject to Minimize
∑ ≤j
jij izxaz
Maxmini Objectives
sconstraintlinear alconvention subject to
Mimimum Maximize ⎟⎟⎠
⎞⎜⎜⎝
⎛∑
jjiji
xa
sconstraintlinear alconvention
, allfor subject to Maximize
∑ ≥j
jij izxaz
Ratio Objectives
∑∑∑
≤j jj
j jj
j jj
exd
xb
xa
subject to
Minimize)(or Maximize
∑
∑
∑
≤−
=
jjj
jjj
jjj
etwd
wb
wa
0
1, subject to
Minimize)(or Maximize
txwxb
t jjj jj
==∑
,1
Ratio Constraints easy
5.0
≤
∑∑
j jj
j jj
xb
xa
∑∑ ≤j jjj jj xbxa 0.5
00.5
≤− ∑∑ j jjj jj xbxa
Objectives including absolute values
idxb
xac
ij ijij
i j ijiji
allfor osubject t
Minimize
≤∑∑ ∑
idxb
izxa
izxa
zc
ij ijij
j iijij
j iijij
i ii
allfor
allfor 0
allfor 0 osubject t
Minimize
≤
≥+
≤−
∑∑∑∑
∑=j ijiji xaz
( )0≥ic
( )∞−⇒≤ togoes valueobjective 0ic
(applied) Minimax + absolute values
∑−j
jijiixab Maximum Minimize
∑
∑≥+−
≤−−
jjiji
jjiji
izxab
izxabz
allfor 0
allfor 0 subject to Minimize
zzxabz ij
jijii ≤−= ∑ ,
Hard and soft constraints
Hard constraints Soft constraints
∑
∑
∑
=
≥
≤
jjj
jjj
jjj
bxa
bxa
bxa
0 ,0 ,
0 ,
0 ,
≥≥≥+−
≥≥+
≥≤−
∑
∑
∑
vubvuxa
vbvxa
ubuxa
jjj
jjj
jjjvs
vs
vs
Assignment #2 (Due date: Oct. 31)
A company wishes to move some of its departments out of TokyoBenefits:
Cheaper housingGovernment incentives
Looses:Increasing the communication costs between departments
The company comprises five departmentsD1, D2, D3, D4, D5
The possible cities for location areTokyo, Tsukuba, Narita
Benefits to be derived from each relocation
D1 D2 D3 D4 D5
Tsukuba 20
10
30 10 5 25
Narita 30 15 5 15
Quantities of communication
D1 D2 D3 D4 D5D1 0.0 2.0 1.5 3.0D2 0.0 4.0 0.0D3 0.0 0.5D4 0.7
Cost per unit of communication
Tokyo Tsukuba Narita
Tokyo 10 25 50
Tsukuba 10 40
Narita 30
Where should each department be located so as to minimize the total cost per year?
Formulate the above problem into an optimization problem
Determine the variables, the objective function and the constraints
Describe your idea for solving your problem