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