Managerial Decision Making Chapter 3 Modelling and Solving LP Problems in a Spreadsheet

Introduction  Solving LP problems graphically is only possible when there are two decision variables.  Few real-world LPs have only two decision variables.  Fortunately, there are packages for solving LP problems  CPLEX (ILOG (IBM),  Xpress-MP (Dash Optimization,  NEOS Solver (  GAMS (General Algebraic Modeling System,  AMPL ( A Mathematical Programming Language,  COIN-OR (COmputational INfrastrut. for OR,  Many other  Relatively small LP problems can be solved using spreadsheets  Solver Add-in in Excel (Frontline Systems, 2

Implementing an LP Model in a Spreadsheet  Organize the data for the model on the spreadsheet.  Data-oriented model  General form of LP model  Reserve cells for all decision variables in the model.  Create a formula for the objective.  Create a formula for the left-hand-side (LHS) of each constraint.  Use Excel Solver to solve the problem. 3

Implementing Blue Ridge Model in Excel  X 1 = number of Aqua-Spa tubs to produce  X 2 = number of Hydro-Lux tubs to produce  Max350X X 2 (profit in dollars)  Subject toX 1 + X 2 <= 200(pumps)  9X 1 + 6X 2 <= 1566(hours of labor)  12X X 2 <= 2880(feet of tubing)  X 1, X 2 >= 0  Let’s implement the model in Excel (BlueRidge.xls, Fig3-1.xls)BlueRidge.xlsFig3-1.xls 4

Excel Solver  To use Excel Solver Add-in, go to:  In Excel 2003: Menu Tools  Solver  In Excel 2007: Data toolbar  Solver  If Solver is not available in Excel, it needs to be enabled:  In Excel 2003: Menu Tools  Add-Ins  Enable Solver Add-in  In Excel 2007: Main Menu  Excel Options  Add-Ins  Go  Enable Solver Add-in  If critical errors occur, disabling and enabling the Solver may help.  Solver requires the following parameters:  Target cell - the cell that represents the objective function  Changing cells - the cells representing the decision variables  Constraints – each constraint is defined by the LHS formula, the sign ( =), and the RHS value  For LP models, go to Options and select “Assume Linear Model” 5

Make vs. Buy: The Electro-Poly Problem  Electro-Poly is a leading maker of slip-rings.  A $750,000 order has just been received. Model 1Model 2Model 3 Number ordered3,0002, Hours of wiring per unit21.53 Hours of harnessing per unit121 Cost per unit to Make$50$83$130 Cost per unit to Buy$61$97$145  The company has to fulfill the order using 10,000 hours of wiring capacity and 5,000 hours of harnessing capacity.  Can we make all required slip-rings? If not, which slip-rings should we buy? 6

Make vs. Buy Model 7  Decision variables:  M 1 = number of model 1 slip rings to make in-house  M 2 = number of model 2 slip rings to make in-house  M 3 = number of model 3 slip rings to make in-house  B 1 = number of model 1 slip rings to buy from competitor  B 2 = number of model 2 slip rings to buy from competitor  B 3 = number of model 3 slip rings to buy from competitor  For short, we may write:  M i = number of model i slip rings to make in-house (i=1..3)  B i = number of model i slip rings to buy from competitor (i=1..3)

Make vs. Buy Model (continued) 8  Objective: Minimize the total cost of filling the order  Min 50M M M B B B 3 (cost in $)  Demand Constraints  M 1 + B 1 = 3,000(units of model 1)  M 2 + B 2 = 2,000(units of model 2)  M 3 + B 3 = 900(units of model 3)  Resource Constraints  2M M 2 + 3M 3 <= 10,000(hours of wiring)  1M 1 + 2M 2 + 1M 3 <= 5,000(hours of harnessing)  Nonnegativity Conditions  M 1, M 2, M 3, B 1, B 2, B 3 >= 0

Make vs. Buy: Excel Implementations  The data oriented model: Fig3-17.xlsFig3-17.xls  The same model in general LP form: ElectroPoly.xlsElectroPoly.xls  One row for all variables  One row for the objective function  One row for each constraint 9

An Investment Problem: Retirement Planning Services, Inc.  A client wishes to invest $750,000 in the following bonds. CompanyReturn Years to MaturityRating Acme Chemical8.65%111-Excellent DynaStar9.50%103-Good Eagle Vision10.00%64-Fair Micro Modeling8.75%101-Excellent OptiPro9.25%73-Good Sabre Systems9.00%132-Very Good  The following restrictions have to be satisfied:  No more than 25% can be invested in any single company.  At least 50% should be in long-term bonds (maturing in 10+ years).  At most 35% can be invested in DynaStar, Eagle Vision, and OptiPro. 10

Investment Model  Decision variables  X i = amount of money to invest in company i (i=1..6)  Objective: Maximize the total return from the investment  Max0.0865X X X X X X 6 (return in $)  Constraints  X 1 + X 2 + X 3 + X 4 + X 5 + X 6 = 750,000(amount invested in $)  X 1 + X 2 + X 4 + X 6 >= 375,000(min 50% long-term)  X 2 + X 3 + X 5 <= 262,500(max 35% in low-rating)  X i <= 0.25(750,000) for all i=1..6(max 25% in each company)  X i >= 0 for all i=1..6  Implementation: see file Fig3-20.xlsFig3-20.xls 11

Transportation Problem: Tropicsun  Tropicsun needs to transport citrus fruits from its groves to its processing plants at minimum cost.  Groves and available citrus fruits:  Mt. Dora – 275,000 bushels  Eustis – 400,000 bushels  Clermont – 300,000 bushels  Processing plants and their processing capacities:  Ocala – 200,000 bushels  Orlando – 600,000 bushels  Leesburg – 225,000 bushels  Trucking company charges a flat rate for every mile that each bushel of fruits is transported.

Transportation Problem: Tropicsun 13 Mt. Dora 1 Eustis 2 Clermont 3 Ocala 4 Orlando 5 Leesburg 6 Distances (in miles) Capacity Supply 275, , , , , ,000 Groves Processing Plants

Transportation Model 14  Decision variables  X ij = # of bushels shipped from node i to node j, i=1..3, j=4..6  Objective: Minimize the total number of bushel-miles  Min21X X X 16 +  35X X X 26 +  55X X X 36 (total bushel-miles)  Assuming the cost of transporting one bushel for one mile is constant, the above objective is equivalent to minimizing the total transportation cost.

Transportation Model (continued)  Supply constraints  X 14 + X 15 + X 16 = 275,000 (bushels taken from Mt. Dora)  X 24 + X 25 + X 26 = 400,000 (bushels taken from Eustis)  X 34 + X 35 + X 36 = 300,000 (bushels taken from Clermont)  Capacity constraints  X 14 + X 24 + X 34 <= 200,000 (bushels delivered to Ocala)  X 15 + X 25 + X 35 <= 600,000 (bushels delivered to Orlando)  X 16 + X 26 + X 36 <= 225,000 (bushels delivered to Leesburg)  Nonnegativity conditions  X ij >= 0 for all i=1..3 and j=4..6  Implementation: see file Fig3-24.xls, Tropicsun.xlsFig3-24.xlsTropicsun.xls 15

A Blending Problem: The Agri-Pro Company  Agri-Pro has received an order for 8,000 pounds of chicken feed to be mixed from the following four feeds. Percent of Nutrient in NutrientFeed 1Feed 2 Feed 3Feed 4 Corn30%5%20%10% Grain10%30%15%10% Minerals20%20%20%30% Cost per pound$0.25$0.30$0.32$0.15  The mix must contain at least 20% corn, 15% grain, and 15% minerals.  (Similar problems occur in the petrochemical industry.) 16

Blending Model  Decision variables  X i = thousands of pounds of feed i in the mix (i=1..4)  Objective: Minimize the total cost of the mix  Min 250X X X X 4 (cost in $)  Parameters of the mix must be satisfied  (0.30X X X X 4 )/8 >= 0.20(at least 20% of corn)  (0.10X X X X 4 )/8 >= 0.15(at least 15% of grain)  (0.20X X X X 4 )/8 >= 0.15(at least 15% of minerals)  Other constraints  X 1 + X 2 + X 3 + X 4 = 8(amount of the mix)  X i >= 0 for all i=1..4  Implementation: see file Fig3-28.xlsFig3-28.xls  Can you define the same problem using different variables? 17

Goals For Spreadsheet Design  Communication – A spreadsheet's primary business purpose is communicating information to managers.  Reliability – The output a spreadsheet generates should be correct and consistent.  Auditability – A manager should be able to retrace the steps followed to generate the different outputs from the model in order to understand and verify results.  Modifiability – A well-designed spreadsheet should be easy to change or enhance in order to meet dynamic user requirements. 18

Spreadsheet Design Guidelines - I  Organize the data, then build the model around the data (for data-oriented models).  Do not embed numeric constants in formulas.  Things which are logically related should be physically related.  Use formulas that can be copied.  Relative (without $) and absolute (with $) references  F4 key: B3  $B$3  B$3  $B3  back to B3  =SUMPRODUCT(coefficients, variables), =SUMPRODUCT(B5:C5,$B$3:$C$3)  Column/row total should be close to the column/row being totaled. 19

Spreadsheet Design Guidelines - II  The English-reading eye scans left to right, top to bottom.  Use color, shading, borders and protection to distinguish changeable parameters from other model elements.  Use text boxes and cell notes to document various elements of the model. 20

End of Chapter 3 21  Other problems in the chapter:  Production Planning Problem – scheduling production over 6 months to balance the production and inventory costs  Multi-Period Cash Flow Problem – establishing a sinking fund to pay for construction of a restaurant  Data Envelopment Analysis – comparing efficiency of divisions