MIS 463: Decision Support Systems for Business Review of Linear Programming and Applications Aslı Sencer

BIS 517-Aslı Sencer2 Basic LP Models: Product Mix Production System for tables and chairs. Resource Unit Requirements Amount Available in a Period TableChair Wood (ft) Labor (hrs) Unit profit$6$8

BIS 517-Aslı Sencer3 Formulating a Linear Problem Define variables: : number of tables produced in a period : number of chairs produced in a period Define constraints: Define Objective Function

BIS 517-Aslı Sencer4 Basic LP Models: Feed Mix Two types of seeds are mixed to formulate the wheat of wild birdseed. Nutritional Item Proportional Content Total Requirement Buckwheat Sunflower wheat Fat.04.06≥480 lb Protein.12.10≥1200 lb Roughage.10.15≤1500 Cost per lb$.18$.10

BIS 517-Aslı Sencer5 LP Formulation

BIS 517-Aslı Sencer6 Applications of LP:Transportation Models Sporting goods company Capacity Plants Warehouses Demand Juarez Seoul Tel Aviv Yokohama Phoenx NY Frankfurt

BIS 517-Aslı Sencer7 LP:Transportation Models (cont’d.) From Plant Destination FrankfurtNYPhoenixYokohama Juarez $19$7$3$21 Seoul Tel Aviv Shipping Costs per pair of skis What are the optimal shipping quantities from the plants to the warehouses, if the demand has to be met by limited capacities while the shipping cost is minimized?

BIS 517-Aslı Sencer8 LP:Transportation Models (cont’d.) X ij : Number of units shipped from plant i to warehouse j. i=1,2,3 and j=1,2,3,4. Minimize shipping costs=19X 11 +7X 12 +3X X X 21 +7X 22 +3X X X X X X 34 From Plant Destination Capacity FrankfurtNYPhoenixYokohama JuarezX11X12X13X14100 SeoulX21X22X23X24300 Tel AvivX31X32X33X34200 Demand

BIS 517-Aslı Sencer9 LP:Transportation Models (cont’d.) subject to #shipped from a plant can not exceed the capacity: X 11 +X 12 +X 13 +X 14 ≤ 100 (Juarez Plant) X 21 +X 22 +X 23 +X 24 ≤ 300 (Seoul Plant) X 31 +X 32 +X 33 +X 34 ≤ 200 (Tel Aviv Plant) #shipped to a warehouse can not be less than the demand: X 11 +X 21 +X 31 +X 41 ≥ 150 (Frankfurt) X 12 +X 22 +X 32 +X 42 ≥ 100 (NY) X 13 +X 23 +X 33 +X 43 ≥ 200 (Phoenix) X 14 +X 24 +X 34 +X 44 ≥ 150 (Yokohama) Nonnegativity X ij ≥0 for all i,j.

BIS 517-Aslı Sencer10 Capacity Plants Warehouses Demand Juarez Seoul Tel Aviv Yokohama Phoenx NY Frankfurt LP:Transportation Models (cont’d.) Optimal Solution: Optimal cost=$6,

BIS 517-Aslı Sencer11 LP: Marketing Applications How to allocate advertising budget between mediums such as TV, radio, billboard or magazines? Ex: Real Reels Co. Allocated ad. Budget=$100,000 PlayboyTrueEsquire Readers10 million6 million4 million Significant Buyers 10%15%7% Cost per ad$10,000$5,000$6,000 Exposures per ad 1,000,000900,000280,000 No more than 5 ads in True and at least two ads in Playboy and Esquire

BIS 517-Aslı Sencer12 LP: Marketing Applications (cont’d.)

BIS 517-Aslı Sencer13 LP: Assignment Models Assignment of a set of workers to a set of jobs Individual Time required to complete one job DrillingGrindingLathe Ann5min10min Bud10515 Chuck15 10

BIS 517-Aslı Sencer14 LP: Assignment Models (cont’d.)

BIS 517-Aslı Sencer15 LP:Labor Planning Addresses staffing needs over a specific time period. Hong Kong Bank of Commerce: 12 Full time workers available, but may fire some. Use part time workers who has to work for 4 consequtive hours in a day. Luch time is one hour between 11a.m. and 1p.m. shared by full time workers. Total part time hours is less than 50% of the day’s total requirement. Part-timers earn $4/hr (=$16/day) and full timers earn $50/day.

BIS 517-Aslı Sencer16 LP:Labor Planning (Cont’d.) Time PeriodMinimum labor required 9a.m.-10a.m.10 10a.m.-11a.m.12 11a.m.-noon14 Noon-1p.m.16 1p.m.-2p.m.18 2p.m.-3p.m.17 3p.m.-4p.m.15 4p.m.-5p.m.10

BIS 517-Aslı Sencer17 LP:Labor Planning (cont’d.) Alternative Optimal Solution F=10, P 2 =2, P 3 =7, P 4 =5 F=10, P 1 =6, P 2 =1, P 3 =2, P 4 =5 at a cost of $724/day

BIS 517-Aslı Sencer18 Solving Linear Programs with a Spreadsheet Write out the formulation table Put the formulation table into a spreadsheet Use Excel’s Solver to obtain a solution

Step 1: The Formulation Table The formulation table arranges the problem in a tabular format, as shown below for the Microcircuit Production Plan.

Step 2: The Excel Spreadsheet The numbers in the Excel spreadsheet come from the formulation table.

BIS 517-Aslı Sencer21 Step 3: Expanded Spreadsheet The expanded spreadsheet contains the formulas necessary to use Solver. Put =SUMPRODUCT(B4:F4,$B$15:$F$15) in cell J4 and copy it down to cell J12. Cell J4 gives the value of the objective function. The solution is found here (the values of the decision variables).

BIS 517-Aslı Sencer22 Using Excel’s Solver to Solve Linear Programs Click on Tools on the menu bar, select the Solver option, and the Solver Parameters dialog box shown next appears.

BIS 517-Aslı Sencer23 Solver Parameters Dialog Box 1. Enter the value of the objective function, J4, in the Target Cell line, either with or without the $ sign. 2. The Target Cell is to be maximized so click on Max in the Equal To line. 3. Enter the decision variables in the By Changing Cells line, B15:F The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner. NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.

BIS 517-Aslı Sencer24 The Add Constraint Dialog Box To represent the constraints in rows 5 - 8: 1. Enter J5:J8 (or $J$5:$J$8) in the Cell Reference line. This is the total amount of these resources used. To represent the constraints in rows 5 - 8: 1. Enter J5:J8 (or $J$5:$J$8) in the Cell Reference line. This is the total amount of these resources used. 3. Enter the amounts of the resources available H5:H8 in the Constraint line (or =$H$5:$H$8). 4. Click Add and repeat Steps if another constraint is to be added. If this is the last constraint, click OK. Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint. If you need to change a constraint, the Change Constraint dialog box functions just like this one. 2. Enter <= as the sign because the resources used must be equal to or less than the amounts available, given next in Step 3. If another sign is needed, see the next slide.

BIS 517-Aslı Sencer25 The Solver Options Dialog Box Click on the Options button in the Solver Parameters dialog box to check the Solver Options dialog box to ensure that the Assume Linear Model and Assume Non-Negative boxes are checked.

BIS 517-Aslı Sencer26 Solver Results Dialog Box (Figure 9-9) Be sure to check the message in the Solver Results dialog box. In this case it indicates that a solution has been found. What happens when Solver does not find a solution will be discussed latter. Click OK and the spreadsheet with the solution, shown next, is obtained.

BIS 517-Aslı Sencer27 Spreadsheet with Optimal Solution 2. Enter the data: the coefficients of the objective function in cells B4:F4, the right-hand sides in cells H5:H12, and the exchange coefficients in cells B5:F To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button. 4. For bigger problems insert additional rows or columns. Insert them in the middle of the table and not at the beginning or the end. Copy the formulas in column J to any new cells created by inserting rows. Check to make sure the ranges of the formulas and signs in the Solver Parameters dialog box are correct. 1. To solve other problems: