Problem Sheet 2 - School of Maths and Stats Local Site - The ...

The University of Western AustraliaSchool of Mathematics & Statistics3OR: Operations ResearchProblem Sheet 2There are 3 Assignment 1 problems(starred).Please hand your solutions to allAssignment 1 problems to me by 5pm, Friday7/9/2010.1. A company produces dressed timber(smooth rectangular section) and moldings(fancy sections).Dressed timber isproduced from rough-sawn timber, whichhas been cut to approximately the correctsize, by planing.Moldings are producedfrom dressed timber by milling. The companyhas a contract to supply 200 tonneof rough-sawn timber, 500 tonne of dressedtimber and 80 tonne of molding. The companycan either produce the timber productsitself or purchase some or all the productsfrom others suppliers. The table belowstates the internal production costs($/tonne), that is, for example, the costof producing dressed timber from roughsawntimber, and external purchase costs($/tonne).Product Internal ExternalRough-sawn 35 60Dressed 10 75Molding 5 94Cutting Planing MillingRough-sawn 1 0 0Dressed 3 3 0Molding 2 2 3Capacity 1600 1500 1700The company has three sections for cutting,planing and milling.The table belowshows the capacity of the sections (totalproduct hours available) and processingtime required (hours/tonne) for eachproduct.(a) Write down a linear program to fulfillits contract at minimum cost.(b) Solve the linear program using software.(c) Describe the optimal productionstrategy for the company. What canyou say about the internal productioncosts and capacity constraints of thiscompany? What could the companydo to reduce its operation costs onsimilar contracts?2. Western Australian company needs towork out the production schedule for manufacturingsofas.Two resources, labourand material, are needed in the production.The company has designs of threedifferent models.The production engineeringdepartment has provided the followingdata:A B CLabour 7 3 6Material(kg per unit) 4 4 5Profits($ per unit) 4 2 3A supplier has been contracted to supplythe raw material, but with a maximum of250kg per day. The company has a labourforce of 120 hours per day. The problemfacing the company is to determine thedaily production rate of the various modelsin order to maximize the total profit.Formulate this decision-making problemas a linear programming problem and solvethe LP problem by software.3. Formulate each of the following problemsas an integer programming problem.min z = f 1 (x 1 ) + f 2 (x 2 ) subject to the followingconstraints:(i) x 1 ≥ 10 or x 2 ≥ 10.(ii) At least one of the inequalities, 2x 1 +1

x 2 ≥ 15, x 1 + x 2 ≥ 15 and x 1 + 2x 2 ≥ 15,holds.(iii) |x 1 − x 2 | = 0, 5 or 10.(iv) x 1 , x 2 ≥ 0.4. A company manufactures a product atm different factories with output capacitiesa i , i = 1, 2, ..., m. There are ndifferent retailers with demands b j , j =1, 2, ..., n for the product. The companyalso has p warehouses with capacitiesq k , k = 1, 2, ..., p and all the units ofthe product from the factories need to beshipped to the warehouses first, and thenshipped from the warehouses to the retailers.It is know that ∑ mi=1 a i ≥ ∑ nj=1 b jand that, for each k = 1, 2, ..., p, if the kthwarehouse is used, there is a fixed cost f kfor the company, irrespective of the quantityshipped to warehouse k. Let c ik andd kj denote the shipping costs from, respectively,factory i to warehouse k and warehousek to retailer j. Set up an LP for thecompany to minimise the total cost.5. A real estate development firm, Peter andJohnson, is considering five possible developmentprojects.The following tableshows the estimated long-run profit (netpresent value) that each project wouldgenerate, as well as the amount of investmentrequired to undertake the project, inunits of millions of dollars.Devel. project 1 2 3 4 5Estimated profit 1 1.8 1.6 0.8 1.4Capital required 6 12 10 4 8The owners of the firm, Dave Peterson andRon Johnson, have raised $20 million ofinvestment capital for the projects. Daveand Ron now want to select the combinationof projects that will maximise theirtotal estimated long-run profit without investingmore than $20 million. Formulatea BIP model for this problem and solve itby a software package.6. Consider the following integer nonlinearprogramming problem.maximize z = 4x 2 1 − x 3 1 + 10x 2 2 − x 4 2,subject to x 1 + x 2 ≤ 3,x 1 ≥ 0, x 2 ≥ 0,x 1 and x 2 are integers.The problem can be reformulated in twodifferent ways as an equivalent pure BIPproblem with six binary variables.(a) Formulate a BIP model for theproblem where the binary vari-{ables have the interpretation y ij =1 if xi = jSolve the resultingproblem by a computer0 otherwise.softwarepackage and thus find the solution tothe original problem.(b) Formulate a BIP model for theproblem where the binary vari-{ables have the interpretation y ij =1 if xi ≥ jSolve the resultingproblem by a computer0 otherwise.softwarepackage and thus find the solution tothe original problem.7. Consider the following IP:max z = 20x 1 + 10x 2 + 10x 2s.t. 2x 1 + 20x 2 + 4x 3 ≤ 15,6x 1 + 20x 2 + 4x 3 = 0,x 1 , x 2 , x 2 non-negative integers.Find out whether the optimal solutionsto this IP can be obtained by solving theLP relaxation problem and then rounding2

down or up the non-integer components ofthe solutions.8. Solve the following BIP problems using theBranch-and-Bound algorithm.(a)* max z = 2x 1 − x 2 + 5x 3 − 3x 4 + 4x 5(b) min z = 5x 1 + 6x 2 + 7x 3 + 8x 4 + 9x 5s.t. 3x 1 − x 2 + x 3 + x 4 − 2x 5 ≥ 2,(c) max z = x 1 + x 2x 1 + 3x 2 − x 3 − 2x 4 + x 5 ≥ 0,−x 1 − x 2 + 3x 3 + x 4 + x 5 ≥ 1,x i = 0 or 1, i = 1, 2, 3, 4, 5.s.t. 2x 1 + 5x 2 ≤ 16,6x 1 + 5x 2 ≤ 30,x 1 , x 2 non-negative integers.9. Solve the following MIP by the branchand-boundmethod.max z = 5x 1 + 4x 2 + 4x 3 + 2x 4s.t. x 1 + 3x 2 + 2x 3 + x4 ≤ 10,5x 1 + x 2 + 3x 3 + 2x 4 ≤ 15,x 1 + x 2 + x 3 + x 4 ≤ 6,x j ≥ 0, for j = 1, 2, 3, 4,x j is integer for j = 1, 2, 3.10. Solve the following IP problems using theCutting Plane approach.(a)* max z = x 1 + x 2 ,s.t. 2x 1 + x 2 ≤ 6,4x 1 + 5x 2 ≤ 20,x 1 , x 2 ≥ 0 and are integers .3(b) min z = 5x 1 + x 2 ,s.t. 3x 1 + x 2 ≥ 9,x 1 + x 2 ≥ 5,x 1 + 8x 2 ≥ 8,x 1 , x 2 ≥ 0 and are integers .s.t. 3x 1 − 2x 2 + 7x 3 − 5x 4 + 4x 5 ≤ 6,11. The research & development division ofx 1 − x 2 + 2x 3 − 4x 4 + 2x 5 ≤ 0,x i = 0 or 1, i = 1, 2, ..., 5.the Progressive Company has been developing4 possible new product lines. Managementmust now make a decision as towhich of these 4 products actually will beproduced and at what levels.Therefore,an operations research study has been requestedto find the most profitable productmix.A substantial cost is associated beginningthe production of any product, as given inthe 1st row of the following table. Management’sobjective is to find the productmix that maximizes the total profit.Product 1 2 3 4Start-up cost $K 50 40 70 60Marginal revenue $ 70 60 90 80Let the continuous decision variablesx 1 , x 2 , x 3 and x 4 be the production levelsof products 1, 2, 3 and 4, respectively.Management has imposed the followingpolicy constraints on these variables:(a) No more than 2 of the products canbe produced.(b) Either product 3 or 4 can be producedonly if either product 1 or 2is produced.(c) Either 5x 1 + 3x 2 + 6x 3 + 4x 4 ≤ 6, 000or 4x 1 + 6x 2 + 3x 3 + 5x 4 ≤ 6, 000.Introduce auxiliary binary variables to formulatea mixed BIP model for this problemand then use a computer program (eglp-solve) to solve the problem.

12. ∗ A convenience store chain wants to opena number of new stores in a district containing12 residential areas: A, B, ..., L. Itis required that the residences in any areashould be able to find a store within a distanceof 800 meters from the area. Thetable below is a list of the areas which arein the circle of radius 800m of a given one.Use an IP to find the minimum number ofstores the company needs to set up in thedistrict. (Feel free to use lp-solve or othersoftware packages to solve the IP.)the total payment.Also, use a softwarepackage to find the optimal solution.Maximum hoursS i Rate $ M T W T F1 10 6 0 6 0 72 10 0 6 0 6 03 9.9 4 8 3 0 54 9.8 5 5 6 0 45 10.8 3 0 4 8 06 10.3 0 6 0 6 3area neighbouring areas (within a distance of 800m)AA,C,E,G,H,IBB,H,ICA,C,G,H,IDD,JEA,E,GFF,J,KGA,C,E,GHA,B,C,H,IIA,B,C,H,IJD,F,J,K,LKF,J,K,LLJ,K,L13. UWA’s School of Maths & Stats employs 4Honours students (S 1 , S 2 , S 3 , S 4 ) and twoMasters students (S 5 , S 6 ) as demonstratorsin MCL which opens from 8am to10pm, Monday to Friday. The availabletimes and the agreed hourly rates for forthe demonstrators are listed in the table.There are also two other conditions:• During the opening hours there is atleast one demonstrator in MCL and• the minimum numbers of workinghours per week for the Honours andMasters students are respectively 8and 7.Set up an LP for determining the students’working days and hours so as to minimize4