INTEGER PROGRAMMING 1.224J/ESD.204J TRANSPORTATION OPERATIONS, PLANNING AND CONTROL: CARRIER SYSTEMS

Transcription

1 INTEGE POGAMMING 1.224J/ESD.204J TANSPOTATION OPEATIONS, PLANNING AND CONTOL: CAIE SYSTEMS Professor Cyntha Barnhart Professor Ngel H.M. Wlson Fall 2003

2 IP OVEVIEW Sources: -Introducton to lnear optmzaton (Bertsmas, Tstskls)- Chap 1 - Sldes Fall 2000

3 Outlne When to use Integer Programmng (IP) Bnary Choces Example: Warehouse Locaton Example: Warehouse Locaton 2 estrcted range of values Gudelnes for strong formulaton Set Parttonng models Solvng the IP Lnear Programmng relaxaton Branch-and bound Example 12/31/2003 Barnhart 1.224J 3

4 When to use IP Formulaton? IP (Integer Programmng) vs. MIP (Mxed Integer Programmng) Bnary nteger program Greater modelng power than LP Allows to model: Bnary choces Forcng constrants estrcted range of values Pecewse lnear cost functons 12/31/2003 Barnhart 1.224J 4

5 Example: Warehouse Locaton A company s consderng openng warehouses n four ctes: New York, Los Angeles, Chcago, and Atlanta. Each warehouse can shp 100 unts per week. The weekly fxed cost of keepng each warehouse open s $400 for New York, $500 for LA, $300 for Chcago, and $150 for Atlanta. egon 1 requres 80 unts per week, regon 2 requres 70 unts per week, and egon 3 requres 40 unts per week. The shppng costs are shown below. Formulate the problem to meet weekly demand at mnmum cost. From/To egon 1 egon 2 egon 3 New York Los Angeles Chcago Atlanta /31/2003 Barnhart 1.224J 5

6 Warehouse Locaton- Approach What are the decson varables? Varables to represent whether or not to open a gven warehouse (y=0 or 1) Varables to track flows between warehouses and regons: x What s the obectve functon? Mnmze (fxed costs+shppng costs) What are the constrants? Constrant on flow out of each warehouse Constrant on demand at each regon Constrant ensurng that flow out of a closed warehouse s 0. 12/31/2003 Barnhart 1.224J 6

7 Warehouse Locaton- Formulaton Let y be the bnary varable representng whether we open a warehouse (y =1) or not (y =0). x represents the flow from warehouse to regon c = weekly cost of operatng warehouse t = unt transportaton cost from to W = the set of warehouses; = the set of regons MIN ( c. y t. x ) s.t. W W x 100. y, W x b, x Z, y 0,1 Forcng constrant 12/31/2003 Barnhart 1.224J 7

8 Warehouse Locaton- Addtonal Constrants If the New York warehouse s opened, the LA warehouse must be opened y NYC y LA y LA y y 2 ATL 1 elatonshp constrant At most 2 warehouses can be opened elatonshp constrant elatonshp constrant ynyc yla or 0 Ether Atlanta or LA warehouse must be opened, but not both 12/31/2003 Barnhart 1.224J 8

9 Bnary Choces Model choce between 2 alternatves (open or closed, chosen or not, etc) Set x=0 or x=1 dependng on the chosen alternatve Can model fxed or set-up costs for a warehouse Forcng flow constrants f warehouse s not open, no flow can come out of t Can model relatonshps 12/31/2003 Barnhart 1.224J 9

10 Example: Warehouse Locaton 2 A company s lookng at addng one or more warehouses somewhere n the regons whch they serve. Each warehouse costs $c w per month to operate and can delver a total of u w unts per month. It costs $c to transport a unt from the plant n regon to the warehouse n regon. Furthermore, the delvery costs from a warehouse n regon to consumers n regon s zero. Warehouses can servce other regons, but the company must pay addtonal transportaton costs of $t per unt per addtonal regon crossed. So to delver 1 unt from a warehouse n regon 2 to a customer n regon 4 would cost $(2 t). Note that the cost to transport a good from warehouse 0 to warehouse s $( t), not $t. All unts must travel through a warehouse on ther way to the customer. Fnally, there s a monthly demand for d unts of the product n regon. Formulate the problem to determne where to locate the new warehouses so as to mnmze the total cost each month f the plant s located n regon p. 12/31/2003 Barnhart 1.224J 10

11 Example 2: Network epresentaton 1, u1 Customer regon 1 d1 Plant p 2, u2 3, u3 Customer regon 2 Customer regon 3 d3 d2 4, u4 Customer regon 4 d4 5,u5 Customer regon 5 d5 12/31/2003 Barnhart 1.224J 11

12 Example 2: Approach Decson Varables? y = whether or not we open a warehouse n regon z =flow from warehouse to regon x p =flow from plant p to warehouse. Obectve Functon? MIN (fxed costs+transportaton costs from plant to warehouse+transportaton costs from warehouse to regon) Constrants? balance constrants at each warehouse demand constrants for each regon capacty constrants at each warehouse. Leta=cost of delverng a unt from warehouse to regon, a=t. - Letcp=cost of transportng one unt from the plant to warehouse 12/31/2003 Barnhart 1.224J 12

13 Example 2: Formulaton Mn c w. y c p. x p a. z s.t. x x p z d, z x 0,, z p u y, w Z, ; y 0, 1 12/31/2003 Barnhart 1.224J 13

14 Example 2: Addtonal Constrants At most 3 warehouses can be opened y 3 y y rw3 rw1 rw3 rw2 If you open a warehouse n some regon rw1 or rw2, you must also open a warehouse n regon rw3 y y yrw1 yrw2 yrw or 1 12/31/2003 Barnhart 1.224J 14

15 Example 2: Addtonal Constrants A plant costs $c p per month to operate and can output u p unts per month. In ths case, a plant can delver drectly to customers n ts regon at no addtonal cost, however t cannot delver drectly to customers n other regons; all unts travelng out of the plant s regon must pass through a warehouse before ther delvery to the customer. Formulate the problem to fnd the optmal dstrbuton of plants and warehouses. Addtonal decson varables: w = whether or not we open a plant n regon u = amount of flow drectly from plant to regon (no cost) Obectve Functon Addtonal term to account for the cost of the plants evsed constrants Constrants range over all regons, not only regon p Add drect flow from plant to customers n same regon Add constrant that total flow leavng a plant s less than u p 12/31/2003 Barnhart 1.224J 15

16 Example 2: Network epresentaton 2 Plant 1 1, u1 Customer regon 1 d1 Plant 2 2, u2 Customer regon 2 d2 Plant 3 3, u3 Customer regon 3 d3 Plant 4 4, u4 Customer regon 4 d4 Plant 5 5,u5 Customer regon 5 d5 12/31/2003 Barnhart 1.224J 16

17 12/31/2003 Barnhart 1.224J 17 Example 2: Formulaton 2 y w Z z x u w u u x y u x x z d u z s t z a x c w c y c MIN p w p w 0,1, ;,,,,.,. 0,,.....

18 estrcted range of values estrct a varable x to take values n a set {a 1,, a m } Introduce m bnary varables y, =1..m and the constrants X s. t. 1.. m y y 1.. m a 1 y 0,1, 12/31/2003 Barnhart 1.224J 18

19 Gudelnes for strong formulaton Good formulaton n LP: small number of varables (n) and constrants (m), because computatonal complexty of problem grows polynomally n n and m LP: choce of a formulaton s mportant but does not crtcally affect ablty to solve the problem IP: Choce of formulaton s crucal! Example: aggregaton of demand (Warehouse) 12/31/2003 Barnhart 1.224J 19

20 Set Parttonng models Very easy to wrte, often very hard to solve All rules, even non-lnear, mpractcal rules can be respected Every obect s n exactly one set Huge number of varables (all feasble combnatons) 12/31/2003 Barnhart 1.224J 20

21 Lnear Programmng relaxaton elax the ntegralty constrant Examples: X n Z + becomes X 0 X n {0,1} becomes 0 X 1 If an optmal soluton to the relaxaton s feasble for the MIP (.e., X take on nteger values n the optmal soluton of the relaxaton) => t s also the optmal soluton to the MIP The LP relaxaton provdes a lower bound on the soluton of the IP Good formulatons provde a tght bound on the IP 12/31/2003 Barnhart 1.224J 21

22 Branch-and-Bounds: A soluton approach for bnary Integer programs Branch-and-bound s a smart enumeraton strategy: Wth branchng, all possble solutons are enumerated (e.g. 2 number of bnary varables ) Wth boundng, only a (usually) small subset of possble solutons are evaluated before a provably optmal soluton s found 12/31/2003 Barnhart 1.224J 22

23 Branch-and-Bound Algorthm Begnnng wth root node (mnmzaton): Bound: Solve the current LP wth ths and all restrctons along the (back) path to the root node enforced Prune If optmal LP value s greater than or equal to the ncumbent soluton => Prune If LP s nfeasble => Prune If LP s ntegral => Prune Branch Set some varable to an nteger value epeat untl all nodes pruned 12/31/2003 Barnhart 1.224J 23

24 Example Company XYZ produces products A, B, C and D. In order to manufacture these products, Company XYZ needs: A B C D Proft Avalablty Nals Screws Glue Company XYZ wants to know whch products t should manufacture. Let X P = 1 f product P s manufactured, 0 otherwse 12/31/2003 Barnhart 1.224J 24

25 Solvng the LP 12/31/2003 Barnhart 1.224J 25

26 Branch-and-Bound ZLP=-6.07 X2=0 X2= ZLP= ZLP=-5.92 X1=0 X1= ZLP= ZLP=-5.7 ESULT: X 1 =1; X 2 =0; X 3 =1; X 4 =1 =>Ob. Value= /31/2003 Barnhart 1.224J 26