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Session 6a Overview Multiple Objective Optimization • • • • • Two Dimensions More than Two Dimensions Finance and HR Examples Efficient Frontier Pre-emptive Goal Programming Intro to Decision Analysis Decision Models -- Prof. Juran 2 Scenario Approach Revisited Year 1983 1984 1985 1986 1987 1988 1989 1990 1991 Ford 14.13 15.21 19.33 28.13 37.69 50.50 43.63 26.63 28.13 Lilly Kellogg Merck HP 14.47 8.09 5.02 42.38 16.50 10.00 5.22 33.88 27.88 17.38 7.61 36.75 37.13 25.88 13.76 41.88 39.00 26.19 17.61 58.25 42.75 32.13 19.25 47.25 68.50 33.81 25.83 31.88 73.25 37.94 29.96 57.00 83.50 65.38 55.50 69.88 Use the scenario approach to determine the minimumrisk portfolio of these stocks that yields an expected return of at least 22%, without shorting. Decision Models -- Prof. Juran 3 Using the same notation as in the GMS case, the percent return on the portfolio is represented by the random variable R. 5 R ri xi i 1 In this model, xi is the proportion of the portfolio (i.e. a number between zero and one) allocated to investment i. (In the GMS case, we used thousands of dollars as the units.) Each investment i has a percent return under each scenario j, which we represent with the symbol rij. Decision Models -- Prof. Juran 4 We calculate the percent return on each of the stocks in each year: Year 1984 1985 1986 1987 1988 1989 1990 1991 Ford 0.076 0.271 0.455 0.340 0.340 -0.136 -0.390 0.056 Lilly Kellogg Merck HP 0.140 0.236 0.040 -0.201 0.690 0.738 0.458 0.085 0.332 0.489 0.808 0.140 0.050 0.012 0.280 0.391 0.096 0.227 0.093 -0.189 0.602 0.052 0.342 -0.325 0.069 0.122 0.160 0.788 0.140 0.723 0.852 0.226 For example, Ford went from $14.31 to $15.21 in 1984, so the return on Ford stock in 1984 was: r1 j S1 j S0 Decision Models -- Prof. Juran S0 15.21 14.13 0.076 14.13 5 The portfolio return under any scenario j is given by: 5 R j rij x i i 1 Decision Models -- Prof. Juran 6 Let Pj represent the probability of scenario j occurring. The expected value of R is given by: 8 R R j Pj j 1 The standard deviation of R is given by: R Decision Models -- Prof. Juran R 8 j 1 2 j R Pj 7 In this model, each scenario is considered to have an equal probability of occurring, so we can simplify the two expressions: 8 R R j 1 8 R 8 R Decision Models -- Prof. Juran j j 1 j R 2 8 8 Managerial Formulation Decision Variables We need to determine the proportion of our portfolio to invest in each of the five stocks. Objective Minimize risk. Constraints All of the money must be invested. The expected return must be at least 22%. No shorting. Decision Models -- Prof. Juran (1) (2) (3) 9 Mathematical Formulation Decision Variables x1, x2, x3, x4, and x5 (corresponding to Ford, Lilly, Kellogg, Merck, and HP). R 8 Objective Minimize Z = R Constraints j 1 i 1 R 2 8 5 x j i 1.0 (1) 8 R R j 1 8 j 0.22 For all i, xi ≥ 0 Decision Models -- Prof. Juran (2) (3) 10 A 1 2 Exp Return 3 StDev 4 5 req. return 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 B C = = 0.127 0.265 = 0.220 D E Total 1 F Ford 1.000 G Lilly 0.000 Ford 14.13 15.21 19.33 28.13 37.69 50.50 43.63 26.63 28.13 Lilly 14.47 16.50 27.88 37.13 39.00 42.75 68.50 73.25 83.50 Historical data on returns Year Ford 1984 0.076 1985 0.271 1986 0.455 1987 0.340 1988 0.340 1989 -0.136 1990 -0.390 1991 0.056 mean 0.127 stdevp 0.265 Lilly 0.140 0.690 0.332 0.050 0.096 0.602 0.069 0.140 0.265 0.235 H Kellogg 0.000 I Merck 0.000 J HP 0.000 Kellogg 8.09 10.00 17.38 25.88 26.19 32.13 33.81 37.94 65.38 Merck 5.02 5.22 7.61 13.76 17.61 19.25 25.83 29.96 55.50 HP 42.38 33.88 36.75 41.88 58.25 47.25 31.88 57.00 69.88 Kellogg 0.236 0.738 0.489 0.012 0.227 0.052 0.122 0.723 0.325 0.271 Merck 0.040 0.458 0.808 0.280 0.093 0.342 0.160 0.852 0.379 0.290 HP -0.201 0.085 0.140 0.391 -0.189 -0.325 0.788 0.226 0.114 0.341 =SUM(G2:K2) =AVERAGE(B19:B26) Historical data Year 1983 1984 1985 1986 1987 1988 1989 1990 1991 =SQRT(AVERAGE(C19:C26)) =SUMPRODUCT($F$2:$J$2,F19:J19) return 0.076 0.271 0.455 0.340 0.340 -0.136 -0.390 0.056 deviation^2 0.003 0.021 0.108 0.045 0.045 0.069 0.267 0.005 =(B19-$C$2)^2 Decision Models -- Prof. Juran 11 The decision variables are in F2:J2. The objective function is in C3. Cell E2 keeps track of constraint (1). Cells C2 and C5 keep track of constraint (2). Constraint (3) can be handled by checking the “assume non-negative” box in the Solver Options. Decision Models -- Prof. Juran 12 Decision Models -- Prof. Juran 13 A 1 2 Exp Return 3 StDev 4 5 req. return 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 B C = = 0.220 0.128 = 0.220 return 0.042 0.450 0.366 0.205 0.076 0.194 0.175 0.253 deviation^2 0.032 0.053 0.021 0.000 0.021 0.001 0.002 0.001 Decision Models -- Prof. Juran G F Ford 0.173 Lilly 0.426 Ford 14.13 15.21 19.33 28.13 37.69 50.50 43.63 26.63 28.13 Lilly 14.47 16.50 27.88 37.13 39.00 42.75 68.50 73.25 83.50 Historical data on returns Ford Year 0.076 1984 0.271 1985 0.455 1986 0.340 1987 0.340 1988 -0.136 1989 -0.390 1990 0.056 1991 0.127 mean 0.265 stdevp Lilly 0.140 0.690 0.332 0.050 0.096 0.602 0.069 0.140 0.265 0.235 D E Total 1 Historical data Year 1983 1984 1985 1986 1987 1988 1989 1990 1991 J I Merck 0.105 HP 0.241 Kellogg 8.09 10.00 17.38 25.88 26.19 32.13 33.81 37.94 65.38 Merck 5.02 5.22 7.61 13.76 17.61 19.25 25.83 29.96 55.50 HP 42.38 33.88 36.75 41.88 58.25 47.25 31.88 57.00 69.88 Kellogg 0.236 0.738 0.489 0.012 0.227 0.052 0.122 0.723 0.325 0.271 Merck 0.040 0.458 0.808 0.280 0.093 0.342 0.160 0.852 0.379 0.290 HP -0.201 0.085 0.140 0.391 -0.189 -0.325 0.788 0.226 0.114 0.341 H Kellogg 0.054 14 Conclusions Invest 17.3% in Ford, 42.6% in Lilly, 5.4% in Kellogg, 10.5% in Merck, and 24.1% in HP. The expected return will be 22%, and the standard deviation will be 12.8%. Decision Models -- Prof. Juran 15 2. Show how the optimal portfolio changes as the required return varies. Decision Models -- Prof. Juran 16 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 A Required Return 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.210 0.220 0.230 0.240 B Risk 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.115 0.116 0.119 0.123 0.128 0.133 0.139 Decision Models -- Prof. Juran C Return 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.179 0.180 0.190 0.200 0.210 0.220 0.230 0.240 D Ford 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.289 0.285 0.249 0.224 0.198 0.173 0.148 0.122 E Lilly 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.407 0.413 0.430 0.429 0.428 0.426 0.425 0.424 F Kellogg 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.029 0.038 0.046 0.054 0.063 0.071 G Merck 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.039 0.072 0.105 0.138 0.171 H HP 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.302 0.286 0.271 0.256 0.241 0.226 0.211 17 Optimal Portfolio 100% 90% Lilly Proportion of Portfolio 80% 70% 60% Kellogg 50% Ford 40% Merck 30% 20% HP 10% 0% 15% 20% 25% 30% 35% Required Return Decision Models -- Prof. Juran 18 3. Draw the efficient frontier for portfolios composed of these five stocks. Decision Models -- Prof. Juran 19 Efficient Frontier 50% 45% 40% Merck Expected Return 35% Kellogg 30% 25% Lilly 20% Ford 15% HP 10% 5% 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% Risk (Standard Deviation) Decision Models -- Prof. Juran 20 Repeat Part 2 with shorting allowed. Decision Models -- Prof. Juran 21 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 A Required Return 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200 0.210 0.220 0.230 0.240 B Risk 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.114 0.115 0.116 0.119 0.123 0.128 0.133 0.139 Decision Models -- Prof. Juran C Return 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.166 0.170 0.180 0.190 0.200 0.210 0.220 0.230 0.240 D Ford 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.311 0.300 0.274 0.249 0.224 0.198 0.173 0.148 0.122 E Lilly 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.432 0.431 0.430 0.429 0.428 0.426 0.425 0.424 F Kellogg 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.013 0.021 0.029 0.038 0.046 0.054 0.063 0.071 G Merck -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.074 -0.059 -0.026 0.007 0.039 0.072 0.105 0.138 0.171 H HP 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.322 0.315 0.300 0.286 0.271 0.256 0.241 0.226 0.211 22 Efficient Frontier 50% 45% 40% Merck 35% Expected Return Kellogg 30% 25% Lilly 20% Ford 15% HP 10% 5% 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% Risk (Standard Deviation) Decision Models -- Prof. Juran 23 GMS Case Revisited Assuming that Torelli's goal is to minimize the standard deviation of the portfolio return, what is the optimal portfolio that invests all $10 million? Decision Models -- Prof. Juran 24 Formulation Decision Variables The decision variables are four amounts: x1, x2, x3, and x4, representing GMS stock, Put Option A, Put Option B, and Put Option C, respectively. Objective Minimize Z = R R 7 j 1 j R P 2 j Constraints 4 x i 1 i 10 ,000 xi 0 for all investments i. Decision Models -- Prof. Juran 25 Optimal Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 A GMS price B C D E F G H I J K L 100 Scenarios for GMS stock in one month Scenario GMS price Probability 1 150 0.05 2 130 0.10 3 110 0.20 4 100 0.30 5 90 0.20 6 80 0.10 7 70 0.05 Returns from one unit of each investment GMS Put A Put B Put C 50% -100% -100% -100% 30% -100% -100% -100% 10% -100% -100% -100% 0% -100% -100% -20% -10% -100% 56% 60% -20% 355% 213% 140% -30% 809% 369% 220% Put options on GMS stock that expire in one month Option A B C Strike price 90 100 110 Option price $2.20 $6.40 $12.50 Investment decision (thousands of dollars spent on each investment) GMS Put A Put B Put C 8297 -8 -665 2376 Portfolio Return Sqdev 2446 5201674 786 385978 -873 1077810 198 1068 230 4237 225 3543 219 2912 Now the nonnegativity conditions for the changing cells is removed, and the investor sells short on the put A and B options. This lowers the standard deviation of the portfolio (and also increases its mean). Total 10000 = Budget 10000 Return from portfolio ($1000) Mean 165 Stdev 718 Units of investments purchased (shares for GMS, number of puts for options) GMS Put A Put B Put C 82972 -3798 -103843 190058 Decision Models -- Prof. Juran 26 Efficient Frontier for GMS GMS Risk vs. Return $600 Efficient Frontier with Shorting Expected Return (x 1000) $500 $400 Efficient Frontier - No Shorting $300 Minimum Risk with Shorting $200 Minimum Risk - No Shorting $100 GMS Stock Only "One-for-One" $$- $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 $2,000 Std Dev of Return (x 1000) Decision Models -- Prof. Juran 27 Parametric Approach Revisited From Session 5a: (a) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with no shorting of stocks allowed. (b) Draw the efficient frontier for portfolios composed of these three stocks. (c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with no shorting of stocks allowed. Decision Models -- Prof. Juran 28 Formulation Objective Minimize Z = 12 x 12 22 x 22 32 x 32 2 x 1 x 2 COV1 , 2 2 x 2 x 3COV2 , 3 2 x 1 x 3COV1 , 3 Constraints x 1 1 x 2 2 x 3 3 0.12 3 x i 1 i 1.0 For all i, x i 0 Decision Models -- Prof. Juran (1) (2) (3) 29 Optimal Solution A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Mean return Variance of return StDev of return B Stock 1 0.140 0.200 0.447 C Stock 2 0.110 0.080 0.283 D Stock 3 0.100 0.180 0.424 Stock 1 1.00 0.80 0.70 Stock 2 0.80 1.00 0.90 Stock 3 0.70 0.90 1.00 Stock 1 0.333 Stock 2 0.667 Stock 3 0.000 Actual 0.120 >= Required 0.120 E F Correlations Stock 1 Stock 2 Stock 3 G H I J Stock 1 0.2000 0.1012 0.1328 Stock 2 0.1012 0.0800 0.1080 Stock 3 0.1328 0.1080 0.1800 Covariances Stock 1 Stock 2 Stock 3 Investment decision Fractions to invest Total 1 = Required 1 Expected portfolio return Portfolio variance Portfolio stdev 0.103 0.321 Decision Models -- Prof. Juran 30 SolverTable Decision Models -- Prof. Juran 31 Decision Models -- Prof. Juran 32 SolverTable Output 0.100 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 Stock 1 Stock 2 Stock 3 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.244 1.861 -0.616 -0.240 1.859 -0.619 -0.212 1.849 -0.637 -0.185 1.840 -0.655 -0.158 1.830 -0.673 -0.130 1.821 -0.691 -0.103 1.811 -0.709 -0.075 1.802 -0.726 -0.048 1.792 -0.744 -0.021 1.783 -0.762 Decision Models -- Prof. Juran Risk 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Return A B C D E F 3 Input (cell $D$18) values along side, output cell(s) along top 0.109 0.109 0.109 0.109 0.109 0.109 0.109 0.109 0.109 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.240 0.240 0.240 0.240 0.240 0.240 0.240 0.240 0.240 0.240 0.240 0.241 0.241 0.242 0.244 0.245 0.247 0.249 33 Efficient Frontier Expected Return 16% 15% 14% Stock 1 13% 12% 11% Stock 2 10% Stock 3 9% 8% 20% 25% 30% 35% 40% 45% 50% Risk Decision Models -- Prof. Juran 34 Parametric Approach, cont. (c) Determine the minimum-variance portfolio that attains an expected annual return of at least 0.12, with shorting of stocks allowed. All we need to do here is remove the non-negativity constraint and re-run SolverTable. Decision Models -- Prof. Juran 35 Efficient Frontier Expected Return 16% 15% 14% 13% Stock 1 12% 11% Stock 2 10% 9% 8% 20% Stock 3 25% 30% 35% 40% 45% 50% Risk Decision Models -- Prof. Juran 36 Preemptive Goal Programming: Consulting Example The Touche Young accounting firm must complete three jobs during the next month. Job 1 will require 500 hours of work, job 2 will require 300 hours, and job 3 will require 100 hours. At present the firm consists of five partners, five senior employees, and five junior employees, each of whom can work up to 40 hours per month. Decision Models -- Prof. Juran 37 The dollar amount (per hour) that the company can bill depends on the type of accountant assigned to each job, as shown in the table below. (The "X" indicates that a junior employee does not have enough experience to work on job 1.) Partner Senior employee Junior employee Decision Models -- Prof. Juran Job 1 160 120 X Job 2 120 90 50 Job 3 110 70 40 38 All jobs must be completed. Touche Young has also set the following goals, listed in order of priority: •Goal 1: Monthly billings should exceed $74,000. •Goal 2: At most one partner should be hired. •Goal 3: At most three senior employees should be hired. •Goal 4: At most one junior employee should be hired. Decision Models -- Prof. Juran 39 Managerial Formulation Decision Variables There are three types of decisions here. First, we need to decide how many people to hire in each of the three employment categories. Second, we need to assign the available human resources (which depend on the first set of decisions) to the three jobs. Finally, since it is not apparent that we will be able to satisfy all of Touche Young’s goals, we need to decide which goals not to meet and by how much. Decision Models -- Prof. Juran 40 Objective In the long run we want to minimize any negative difference between actual results and each of the four goals. Of course, our optimization methods require that we only have one objective at a time, so we will use a variation of goal programming to solve the problem four times. The approach here will be to treat each of the goals as an objective until it is shown to be attainable, after which we will treat it as a constraint. For example, we will solve the model with the goal of minimizing any shortfall in the $74,000 revenue target. Once we find a solution that has no shortfall, we will solve the problem again, with an added constraint that the shortfall be zero. Decision Models -- Prof. Juran 41 Constraints The numbers of people hired must be integers. (1) Each project must receive its required number of man-hours. (2) Our model must take into account any difference between the actual performance of the plan and the four targets. (3) We can’t assign people to jobs unless we hire them. (4) Decision Models -- Prof. Juran 42 Mathematical Formulation Define tk to be the target amount for goal k. For example, the target for goal 1 (the billing goal) is $74,000; therefore t1 = 74,000. Define δk to be the “negative difference” between what we have achieved and the target for “current” goal k. (δ is the Greek letter delta.) In the case of the billing goal, the negative difference would be any amount less than 74,000. In the case of goal 2, the negative difference would be any amount of new partners hired greater than 1. If our current solution yields $60,000 in billings, then δ1 = 74,000 – 60,000 = 14,000. Define vk to be the best value for goal k that has previously been achieved in our model. For example, if the best solution we can find yields only $60,000 in billings, then v1 = 14,000. Define xi to be the number of new hires of type i. Define Aij to be the number of man-hours of type i assigned to job j. Define Rij to be the number of man-hours of type i required for job j. Decision Models -- Prof. Juran 43 Decision Variables xi (three decisions), Aij (nine decisions), δk (up to three decisions) Objective Minimize Z = k Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δ Goals ≠ k = vk for all goals < k (3) 3 A j 1 ij 40 x i for all i. Decision Models -- Prof. Juran (4) 44 A B C D E F G 1 Assigned Hours Job 1 Job 2 Job 3 Assigned Available 2 Partners 0 0 0 0 <= 200 3 Seniors 0 0 0 0 <= 200 4 Juniors 0 0 0 0 <= 200 5 Assigned 0 0 0 6 = = = =SUM(D2:D4) =SUM(B4:D4) 7 Needed 500 300 100 =B10+C10-D10 =SUMPRODUCT(J2:L4,B2:D4) 8 9 Goals Actual Under Over Net Goal 10 Billings 0 0 0 0 = 74000 11 Partners Hired 0 0 0 0 = 1 12 Seniors Hired 0 0 0 0 = 3 13 Juniors Hired 0 0 0 0 = 1 14 =C10 15 Deviation Already attained Priority 16 Billings 0 <= 0 1 17 Partners Hired 0 <= 0 2 18 Seniors Hired 0 <= 0 3 19 Juniors Hired 0 <= 0 4 =D11 20 Decision Models -- Prof. Juran H I J Billling Rate Job 1 Partners 160 =$J$11*L9 Seniors 120 Juniors NA Present staff Partners Seniors Juniors 5 5 5 Hours/month 40 K L Job 2 Job 3 120 110 90 70 50 40 Added Total 0 5 0 5 0 5 =J7+K7 45 The tk targets are in cells G10:G13. The δk “negative difference” variables will be in B16:B19. We use the range C10:D13 to track all deviations (both positive and negative), and then refer to the “undesirable” one in B16:B19. For example, it is undesirable to have billings under 74,000, so B16 refers to C10. It is undesirable for the number of new partners to be over 1, so B17 refers to D11. The vk “best achieved” variables will be in D16:D19. The xi are in K7:K9. The Aij assignments are in B2:D4. The Rij requirements are in B7:D7. Cells G2:G4 keep track of constraint (4). We constrain B4 to be zero. Decision Models -- Prof. Juran 46 First Iteration At first we’ll ignore all of the goals except the billing target of $74,000. Decision Variables xi (three decisions, cells K7:K9), Aij (nine decisions, cells B2:D4) Objective Minimize Z = 1 (the shortfall, if any, between planned billings and $74,000) Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) A11 = 0. (4) 3 A j 1 ij 40xi for all i. (5) Note that constraint (3) doesn’t matter in this iteration. Also note the balance equation constraint, forcing E10 = G10. Decision Models -- Prof. Juran 47 Decision Models -- Prof. Juran 48 A B C D E F G 1 Assigned Hours Job 1 Job 2 Job 3 Assigned Available 2 Partners 0 0 0 0 <= 320 3 Seniors 500 0 0 500 <= 520 4 Juniors 0 300 100 400 <= 480 5 Assigned 500 300 100 6 = = = 7 Needed 500 300 100 8 9 Goals Actual Under Over Net Goal 10 Billings 79000 0 5000 74000 = 74000 11 Partners Hired 3 0 0 3 = 1 12 Seniors Hired 8 0 0 8 = 3 13 Juniors Hired 7 0 0 7 = 1 14 15 Deviation Already attained Priority 16 Billings 0 <= 0 1 17 Partners Hired 0 <= 0 2 18 Seniors Hired 0 <= 0 3 19 Juniors Hired 0 <= 0 4 H I J Billling Rate Job 1 Partners 160 Seniors 120 Juniors NA K L Job 2 Job 3 120 110 90 70 50 40 Present staff Partners Seniors Juniors 5 5 5 Added Total 3 8 8 13 7 12 Hours/month 40 We have verified that it is feasible to have billings of $74,000. Decision Models -- Prof. Juran 49 Second Iteration Meeting the first goal will now be a constraint, and we’ll focus on the second goal. Decision Variables xi (as before), Aij (as before), δk (for the billings goal) Objective Minimize Z = 2 (the number of partners hired more than the goal of 1) Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δ 1 = v1 (forcing the 74,000 billing goal to be met) (3) A11 = 0. (4) 3 A j 1 ij 40 x i for all i. (5) Note that v1 = 0 from the previous iteration. Decision Models -- Prof. Juran 50 Decision Models -- Prof. Juran 51 A B C D E F G 1 Assigned Hours Job 1 Job 2 Job 3 Assigned Available 2 Partners 0 0 0 0 <= 240 3 Seniors 500 0 0 500 <= 720 4 Juniors 0 300 100 400 <= 400 5 Assigned 500 300 100 6 = = = 7 Needed 500 300 100 8 9 Goals Actual Under Over Net Goal 10 Billings 79000 0 5000.000171 74000 = 74000 11 Partners Hired 1 0 0 1 = 1 12 Seniors Hired 13 0 0 13 = 3 13 Juniors Hired 5 0 0 5 = 1 14 15 Deviation Already attained Priority 16 Billings 0 <= 0 1 17 Partners Hired 0 <= 0 2 18 Seniors Hired 0 <= 0 3 19 Juniors Hired 0 <= 0 4 H I J Billling Rate Job 1 Partners 160 Seniors 120 Juniors NA K L Job 2 Job 3 120 110 90 70 50 40 Present staff Partners Seniors Juniors 5 5 5 Added Total 1 6 13 18 5 10 Hours/month 40 Now we know that it is feasible to achieve both of the first two goals. Decision Models -- Prof. Juran 52 Third Iteration Meeting both the first and second goals will now be constraints, and we’ll focus on the third goal. Decision Variables xi (three decisions), Aij (nine decisions), δk (two decisions) Objective Minimize Z = 3 (the number of senior employees hired above the goal of 3) Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δ Goals k = vk (for goals 1 and 2) (3) A11 = 0. (4) 3 A j 1 ij 40xi for all i. (5) Note that v1 = v2 = 0 from the previous iteration. Decision Models -- Prof. Juran 53 Decision Models -- Prof. Juran 54 A B C D E F G 1 Assigned Hours Job 1 Job 2 Job 3 Assigned Available 2 Partners 195.25 6E-07 6.2152E-07 195.25 <= 200 3 Seniors 304.75 0 0 304.75 <= 320 4 Juniors 0 300 99.99999938 400 <= 400 5 Assigned 500 300 100 6 = = = 7 Needed 500 300 100 8 9 Goals Actual Under Over Net Goal 10 Billings 86810 0 12809.99833 74000 = 74000 11 Partners Hired 0 1 0 1 = 1 12 Seniors Hired 3 0 0 3 = 3 13 Juniors Hired 5 0 0 5 = 1 14 15 Deviation Already attained Priority 16 Billings 0 <= 0 1 17 Partners Hired 0 <= 0 2 18 Seniors Hired 0 <= 0 3 19 Juniors Hired 0 <= 0 4 H I J Billling Rate Job 1 Partners 160 Seniors 120 Juniors NA K L Job 2 Job 3 120 110 90 70 50 40 Present staff Partners Seniors Juniors 5 5 5 Added Total 0 5 3 8 5 10 Hours/month 40 All of the first three goals are feasible. Decision Models -- Prof. Juran 55 Fourth Iteration Meeting all of the first three goals will now be constraints, and we’ll focus on the fourth goal. Decision Variables xi (three decisions), Aij (nine decisions), δk (for the first three goals) Objective Minimize Z = 4 (the number of new juniors hired above the goal of 1) Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δ Goals k = vk for goals 1, 2, 3 (3) A11 = 0. (4) 3 A j 1 ij 40xi for all i. (5) Note that v1 = v2 = v3 = 0 from the previous iteration. Decision Models -- Prof. Juran 56 Decision Models -- Prof. Juran 57 A B C D E F G 1 Assigned Hours Job 1 Job 2 Job 3 Assigned Available 2 Partners 195 7 30 232 <= 240 3 Seniors 305 15 0 320 <= 320 4 Juniors 0 278 70 348 <= 360 5 Assigned 500 300 100 6 = = = 7 Needed 500 300 100 8 9 Goals Actual Under Over Net Goal 10 Billings 89990 0 15990 74000 = 74000 11 Partners Hired 1 0 0 1 = 1 12 Seniors Hired 3 0 0 3 = 3 13 Juniors Hired 4 0 3 1 = 1 14 15 Deviation Already attained Priority 16 Billings 0 <= 0 1 17 Partners Hired 0 <= 0 2 18 Seniors Hired 0 <= 0 3 19 Juniors Hired 3 <= 0 4 Decision Models -- Prof. Juran H I J K L Billling Rate Job 1 Job 2 Job 3 Partners 160 120 110 Seniors 120 90 70 Juniors NA 50 40 Present staff Partners Seniors Juniors 5 5 5 Hours/month 40 Added Total 1 6 3 8 4 9 58 Fifth Iteration We can do one more round here. Given that we are going to hire 1 partner, 3 seniors, and 4 juniors, why not maximize the revenue from that combination of workers? Decision Variables xi (three decisions), Aij (nine decisions), δk (for the 2nd, 3rd, and 4th goals) Objective Maximize Z = Revenue Constraints All xi are integers. (1) Aij = Rij for all i, j. (2) δk = vk for goals 2, 3, 4 (3) A11 = 0. (4) 3 A j 1 ij 40 x i for all i. (5) Note that v2 = v3 = 0, and v4 = 3, from the previous iteration. Decision Models -- Prof. Juran 59 Decision Models -- Prof. Juran 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 A Assigned Hours Partners Seniors Juniors Assigned C Job 2 0 60 240 300 = 300 D Job 3 0 0 100 100 = 100 E Assigned 240 320 340 Needed B Job 1 240 260 0 500 = 500 Goals Billings Partners Hired Seniors Hired Juniors Hired Actual 91000 1 3 4 Under 0 0 0 0 Over 0 0 0 3 Net 91000 1 3 1 <= <= <= <= Already attained 0 0 0 0 Deviation Billings 0 Partners Hired 0 Seniors Hired 0 Juniors Hired 3 F <= <= <= = = = = G Available 240 320 360 Goal 74000 1 3 1 H I J K L Billling Rate Job 1 Job 2 Job 3 Partners 160 120 110 Seniors 120 90 70 Juniors NA 50 40 Present staff Partners Seniors Juniors 5 5 5 Hours/month 40 Added Total 1 6 3 8 4 9 Priority 1 2 3 4 There are two constraints that don’t show in the window: $E$2:$E$4<=$G$2:$G$4, and $K$7:$K$9 = integer. Decision Models -- Prof. Juran 61 Conclusions It would appear that it is infeasible to reach all four of the goals. However, the first three can be met while delivering all of the jobs as required. The only goal that can’t be attained is the limit of hiring only one new junior-level employee. The best solution we found requires us to hire four new employees at this level. It is possible to have revenues of $91,000 under this hiring plan. Decision Models -- Prof. Juran 62 Decision Analysis is a tool for studying situations in which there are several decision alternatives and a set of uncertain future events. Decision Alternatives are elements of a set of possible choices, represented by d1, d2, d3, … di States of Nature are elements of a set of N random future events, represented by s1 , s 2 , s 3 , … sN A Payoff Table lists outcomes associated with some combination of decision alternative and state of nature. The payoff for decision alternative i under state of nature j is symbolized by vij. Decision analysis is a natural extension of our previous work with conditional probability and systems of probabilities and payoffs; the only new element here is the opportunity for a decision maker to make choices at certain discrete points in time. Decision Models -- Prof. Juran 63 A buyer for a large department store chain must place orders with an athletic shoe manufacturer 6 months prior to the time the shoes will be sold in the department stores. In particular, the buyer must decide on November 1 how many pairs of the manufacturer's newest model of tennis shoes to order for sale during the upcoming summer season. Assume that each pair of this new brand of tennis shoes costs the department store chain $45 per pair. Furthermore, assume that each pair of these shoes can then be sold to the chain's customers for $70 per pair. Any pairs of these shoes remaining unsold at the end of the summer season will be sold in a closeout sale next fall for $35 each. Finally, assume that the department store chain must purchase these tennis shoes from the manufacturer in lots of 100 pairs. Decision Models -- Prof. Juran 64 The probability distribution of consumer demand for these tennis shoes (in hundreds of pairs) during the upcoming summer season has been assessed by market research specialists and is provided in the table below. Consumer Demand 1 2 3 4 5 6 Decision Models -- Prof. Juran Probability 0.05 0.15 0.25 0.30 0.15 0.10 65 Basically, the contribution is given by the following formula, in which profit is symbolized by the Greek letter π, purchase quantities (in hundreds) are represented by p and demand states (in hundreds) are represented by d. ij = Revenue from Regular Price + Revenue from Closeout Price – Cost to Purchase $70 * 100 * min p i , d j $35 * 100 * max 0 , p i d j p i * 100 * 45 For example, if we purchase 400 shoes and demand is 200, then p = 4 and d = 2. ij $70 * 100 * min p i , d j $35 * 100 * max 0 , p i d j p i * 100 * 45 $70 * 100 * 2 $35 * 100 * 2 4 * 100 * 45 $14 ,000 $7 ,000 $18 ,000 $3 ,000 Decision Models -- Prof. Juran 66 1 2 3 4 A $45 $70 $35 B C D E Cost of each pair of new tennis shoes Selling price of each pair of shoes Closeout sale price of each leftover pair 5 F G H Payoff Table Consumer Demand 6 7 Purchased 1 2 3 4 5 6 8 1 $2,500 $2,500 $2,500 $2,500 $2,500 $2,500 9 2 $1,500 $5,000 $5,000 $5,000 $5,000 $5,000 10 3 $500 $4,000 $7,500 $7,500 $7,500 $7,500 11 4 ($500) $3,000 $6,500 $10,000 $10,000 $10,000 12 5 ($1,500) $2,000 $5,500 $9,000 $12,500 $12,500 13 6 ($2,500) $1,000 $4,500 $8,000 $11,500 $15,000 14 Probability 0.05 0.15 0.25 0.3 0.15 0.1 15 =($A$2*100*MIN(G$7,$A13))+($A$3*100*MAX(0,($A13-G$7)))-($A13*$A$1*100) 16 Decision Models -- Prof. Juran 67 For each possible strategy, we can calculate the expected revenue and the standard deviation of revenue. A 5 B C D E F G H I J K Payoff Table Consumer Demand 6 7 Purchased 1 2 3 4 5 6 Expected Revenue Std Dev of Revenue Variance 8 1 $2,500 $2,500 $2,500 $2,500 $2,500 $2,500 $2,500 $0 0 9 2 $1,500 $5,000 $5,000 $5,000 $5,000 $5,000 $4,825 $184 33993.75 10 3 $500 $4,000 $7,500 $7,500 $7,500 $7,500 $6,625 $625 390468.8 11 4 ($500) $3,000 $6,500 $10,000 $10,000 $10,000 $7,550 $1,197 1432025 12 5 ($1,500) $2,000 $5,500 $9,000 $12,500 $12,500 $7,425 $1,467 2153244 13 6 ($2,500) $1,000 $4,500 $8,000 $11,500 $15,000 $6,775 $1,613 2602819 14 Probability 0.05 0.15 0.25 0.3 0.15 0.1 15 =SUMPRODUCT(B13:G13,$B$14:$G$14) =SQRT(J13) 16 17 =(($B$14*(B13-H13))^2)+(($C$14*(C13-H13))^2)+(($D$14*(D13-H13))^2)+(($E$14*(E13-H13))^2)+(($F$14*(F13-H13))^2)+(($G$14*(G13-H13))^2) 18 The variance formula looks ugly, but it works. Decision Models -- Prof. Juran 68 Risk Profile $8,000 Expected Revenue $7,000 Buy 400 Pairs Buy 500 Pairs Buy 600 Pairs Buy 300 Pairs $6,000 $5,000 Buy 200 Pairs $4,000 $3,000 Buy 100 Pairs $2,000 $1,000 $0 $0 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 Std. Deviation of Revenue Decision Models -- Prof. Juran 69 Summary Multiple Objective Optimization • • • • • Two Dimensions More than Two Dimensions Finance and HR Examples Efficient Frontier Pre-emptive Goal Programming Intro to Decision Analysis Decision Models -- Prof. Juran 70