Westmead International SchoolSchool of Economics Business and Accountancy CHAPTER 1 LINEAR PROGRAMMING Linear programming is a powerful quantitative technique (or operational research technique) designs to solve allocation problem. The term ‘linear programming’ consists of the two words ‘Linear’ and ‘Programming’. The word 'Linear' is used to describe the relationship between decision variables which are directly proportional. For example, if doubling (or tripling) the production of a product will exactly double (or triple) the profit and required resources, then it is linear relationship. The word 'programming' means planning of activities in a manner that achieves some 'optimal' result with available resources. A program is 'optimal' if it maximizes or minimizes some measure or criterion of effectiveness such as profit, contribution (i.e. sales-variable cost), sales, and cost. Thus, 'Linear Programming' indicates the planning of decision variables which are directly proportional, to achieve the 'optimal' result considering the limitations within which the problem is to be solved. The minimization model starts with an objective function with the purpose of minimizing a goal while maximization model starts with an objective function with the purpose of maximizing a goal. MAXIMIZATION You make three kinds of computers: Sony, Dell, and Apple. These sell for $1500, $2000, and $2400. The Sony model requires 3 hours for circuit board installation and 1 hour to fit the peripheral equipment. The Dell model requires 1 1 Qualitative Technique Westmead International School School of Economics Business and Accountancy hour for circuit boards and 5 hours for peripherals. The Apple model requires 3 hours for circuit boards and 2 hours for peripherals. You have 120 hours available for circuit board work and 60 hours for fitting peripherals. MAXIMIZE PROFIT. GRAPHICAL METHOD TABLE 1.1 DATA TABLE Kinds of Computers Constraints Maximize Sony Dell Apple x 120 3 1 3 2 Qualitative Technique Y 60 1 5 2 RHS <= <= <= 1500 2000 2400 Equation form Max 120x + 60y 3x + y <= 1500 x + 5y <= 2000 3x + 2y <= 2400 Westmead International School School of Economics Business and Accountancy TABLE 1.2 GRAPHING RESULT X 0 500 0 392.8571 Corner Points Y 0 0 400 321.4286 Z 0. 60,000. 24,000. 66,428.57 INTERPRETATION: The Company should use 393 circuit boards, 321 peripherals to meet the minimum cost of Php66, 429. See graph result above. SIMPLEX METHOD TABLE 1.3 ITERATIONS Kinds of Computer Solution Cj BasicVariables 120x 60 y 0slack1 0slack2 0slac3 Quantity Iteration cj-zj 120 60 0 0 0 1 0 slack 1 3 1 1 0 0 1,500 0 slack 2 1 5 0 1 0 2,000 0 slack 3 3 2 0 0 1 2,400 Iteration cj-zj 0 20.0 -40 0 0 2 120 X 1 0.3333 0.3333 0 0 500 0 slack 2 0 4.6667 1 0 1,500 0.3333 0 slack 3 0 1 -1 0 1 900 Iteration cj-zj 0 0 -38.57 -4.285 0 3 120 X 1 0 0.3571 -0.071 0 392.8571 60 Y 0 1 0.2143 0 321.4286 0.0714 3 Qualitative Technique 4286 0 60 Dual Slack/Surplus Original Value Val 38.Westmead International School School of Economics Business and Accountancy 0 slack 3 0 0 0.8571 0 120 321.4 DUALING Kinds of Computer Solution Maximize Original Problem x y Sony 3 1 <= 1500 Dell 1 5 <= 2000 Apple 3 2 <= 2400 Dual Problem Sony Dell Apple Minimize 1500 2000 2400 X 3 1 3 >= 120 Y 1 5 2 >= 60 TABLE 1.5714 4.5714 TABLE 1.2143 1 578.5 RANGING Variable X Y Constraint Sony Dell Apple Kinds of Computer Solution Value Reduced Original Cost Val 392.077 4700 Infinity .429 2123.2857 0 4 Qualitative Technique 0 0 578.5714 1500 2000 2400 Lower Bound 12 40 Lower Bound Upper Bound 180 600 Upper Bound 400 500.9286 -0.0001 1821. 4286 0 0 578. 5 Qualitative Technique .57 TABLE 1.2857 0 INTERPRETATION: The Company should use 393 circuit boards. 429.6 SOLUTION LIST Kinds of Computer Solution Variable Status Value X Y slack 1 slack 2 slack 3 Optimal Value (Z) Basic Basic NONBasic NONBasic Basic 392.4286 RHS Dual 1500 2000 2400 66428. See graph result above.Westmead International School School of Economics Business and Accountancy TABLE 1.8571 321.57 38. 321 peripherals to meet the maximum profit of Php66.5714 4.5714 66428.7 OVERALL LINEAR PROGRAMMING RESULTS Maximize Sony Dell Apple Solution-> Kinds of Computer Solution X Y 120 60 3 1 <= 1 5 <= 3 2 <= 392.8571 321. Westmead International School School of Economics Business and Accountancy MINIMIZATION You make three kinds of computers: Sony. The deluxe model requires 3 hours for circuit boards and 2 hours for peripherals. $2000. The total manufacturing cost for each computer is $1500. and $2400. and Apple. The Cheap model requires 3 hours for circuit board installation and 1 hour to fit the peripheral equipment. Dell. The Good model requires 1 hour for circuit boards and 5 hours for peripherals. TABLE 1.1 DATA TABLE Kinds of Computer x Minimize Sony Dell Apple 20 3 1 3 GRAPHICALMETHOD 6 Qualitative Technique Y 60 1 5 2 RHS >= >= >= 1500 2000 2400 Equation form Min 20x + 60y 3x + y >= 1500 x + 5y >= 2000 3x + 2y >= 2400 . You have 120 hours available for circuit board work and 60 hours for fitting peripherals Determine the best mix that will MINIMIZE your COST. 08 INTERPRETATION: The Company should use 615 of circuit boards 277 of peripherals to meet the minimum cost of Php29.9231 X 0 2000 200 615.6 0 0 1 0 0 1 0 0 -1 -1 0 0 -1.000. SIMPLEX METHOD TABLE 1.8 0.Westmead International School School of Economics Business and Accountancy TABLE 1.923.2 0. 40.2 -0.2 2.000.000 2.3846 Z 90. 28. 58.000.6 -0.4 2.6 0.2 -0.100 400 1.2 GRAPHING RESULT Corner Points Y 1500 0 900 276.4 0 artfcl 3 0 surplus 3 Quantity -1 0 -1 0 0 0 0 1 -1 0 0 -1 1.500 2.2 0.600 Iteration 1 0 0 0 Iteration 2 0 60 0 Iteration 7 Qualitative Technique .400 0.4 0 0 0 1 -1 0 0 -1 1.3 ITERATIONS Cj Kinds of Computer Solution 0 artfcl 0 0 artfcl 0 1 surplus 2 surplus 1 2 Basic Variables 20 x 60 y cj-zj artfcl 1 artfcl 2 artfcl 3 7 3 1 3 8 1 5 2 0 1 0 0 -1 -1 0 0 0 0 1 0 cj-zj artfcl 1 Y artfcl 3 5. 9231 surplus 1 0 0 -1 1 -0.9231 0 -1 1 -0.1538 0.0769 -1.2143 0.077 0 0 0 0 10.2308 0.9286 -1.3846 0.Westmead International School School of Economics Business and Accountancy 3 cj-zj 0 0 20 60 X Y 1 0 0 1 0 artfcl 3 0 0 20 60 cj-zj X Y 0 1 0 0 0 1 0 surplus 1 0 cj-zj 20 60 0 1.2143 0.1538 -0.0769 623.4286 0.3846 276.3571 0.2143 0 0 0 0 392.2143 0.2143 1 -1 578.0714 -0.5 RANGING Variable Value X 615.2308 0.0769 615.077 Iteration 4 Iteration 5 TABLE 1.3846 0.2308 -1.2308 0.5715 -1 0 0 0 0 0 -1.7692 0.1538 0.0769 1.2308 -0.0769 623.7692 3.0 0.4 DUALING Minimize Sony Dell Apple Maximize X Y Kinds of Computer Solution Original Problem X Y 3 1 >= 1 5 >= 3 2 >= Dual Problem Cheap Good Deluxe 1500 2000 2400 3 1 3 1 5 2 1500 2000 2400 <= <= 20 60 TABLE 1.9286 0.0769 1.0714 0.0769 X Y 1 0 0 1 0 0 0 0 -0.2308 10.3571 0.8571 321.2308 0 0.1538 -0.3846 Kinds of Computer Solution Reduced Original Lower Cost Val Bound 0 20 12 8 Qualitative Technique Upper Bound 90 .2308 0 -0.9286 0.0714 0.0769 -1.3846 0.3846 276.2143 0 -1 -0.3846 0.0769 615.0 -0.0769 -3.9286 -0.0714 -0. 3846 276.7692 -3.9231 623.077 0 0 28923.3846 276.08 Dual 0 -10.08 TABLE 1.9231 28923.0769 0 0 60 Original Val 1500 2000 2400 13.077 4700 6000 TABLE 1.9231 Dual Value 0 -10.0769 INTERPRETATION: The Company should use 615 of circuit boards 277 of peripherals to meet the minimum cost of Php29.3333 Lower Bound -Infinity 800 1821.0769 0 Slack/Surplus 623.Westmead International School School of Economics Business and Accountancy Y Constrain t Sony Dell Apple 276.7692 -3.6 SOLUTION SET Kinds of Computer Solution Variable Status X Basic Y Basic surplus 1 Basic surplus 2 NONBasic surplus 3 NONBasic Optimal Value (Z) Value 615. 9 Qualitative Technique .429 100 Upper Bound 2123.7 OVERALL LINEAR PROGRAMMING RESULT Minimize Sony Dell Apple Solution-> Kinds of Computer Solution X Y RHS 20 60 3 1 >= 1500 1 5 >= 2000 3 2 >= 2400 615. Westmead International School School of Economics Business and Accountancy CHAPTER 2 PROJECT EVALUATION REVIEW TECHNIQUE CRITICAL PATH METHOD PROJECT CRASHING PERT and CPM techniques both involve dividing a large project into a series of smaller tasks and activities. If project managers want to reduce total project time. according to Render and Stair. and using the network to monitor and manage the project. according to Render and Stair. any delay of an activity on that path will delay project completion. they must reduce the length of some activity on the critical path identified through PERT or CPM. drawing a network diagram that connects the activities. Meanwhile. developing a sequence of these activities. CPM approaches project costs through two estimates: normal and crash. These steps are defining the project and its significant activities. Project crashing is the time and cost required to complete a project on a deadline. A modification of this technique. Render and Stair cite six steps common to both methods. PERT is an excellent technique for monitoring project completion time but does not consider costs. sometimes through additional expenditures to reduce completion time. 10 Qualitative Technique . computing the longest time path through the network. assigning time or cost estimates. Although the two techniques differ in their original forms. Normal time and cost are estimates of project time and cost under normal conditions. PERT/Cost. known as the critical path. allows project managers to control costs and project time. 1 DATA TABLE CRITICAL PATH METHOD Activity time Prec 1 Prec 2 2 2 A 1 B 1 A 4 B D 2 C E 5 F 2 G 12 G 2 G 1 H I 1 K 2 L N 6 K 1 M 3 N 2 P 4 Q A B C D E F G H I J K L M N O P Q R Prec 3 J TABLE 1.1.Westmead International School School of Economics Business and Accountancy CRITICAL PATH METHOD (CPM) Sam is the Event Manager of MTV Music Summit and she is tasked to organize the event and finish the preparation at the earliest possible time. The figures are summarized in Table1. TABLE 1. She was able to determine the sequence of activities and the time in days needed to finish each activity.2 CPM RESULT Project A B C CRITICAL PATH METHOD SOLUTION Activity Early Early Late time Start Finish Start 43 2 0 2 0 2 2 4 2 1 4 5 7 11 Qualitative Technique Late Finish Slack 2 4 8 0 0 3 . 3 CHARTS OF CRITICAL PATH METHOD GANTT CHART-EARLY TIMES GANTT CHART-LATE TIMES 12 Qualitative Technique 4 8 10 15 27 27 27 28 40 42 34 43 37 39 43 1 0 0 0 10 0 10 0 11 6 0 6 0 0 0 .Westmead International School School of Economics Business and Accountancy D E F G H I J K L M N O P Q R 1 4 2 5 2 12 2 1 1 2 6 1 3 2 4 2 4 8 10 15 15 15 27 28 34 28 36 34 37 39 3 8 10 15 17 27 17 28 29 36 34 37 37 39 43 3 4 8 10 25 15 25 27 39 40 28 42 34 37 39 TABLE 1. PROGRAM EVALUATION AND REVIEW TECHNIQUE (PERT) 13 Qualitative Technique . the critical method is from A-B-E-F-G-I-K-O-P-Q-R.Westmead International School School of Economics Business and Accountancy GANTT CHART-EARLY AND LATE TIMES PRECEDENCE GRAPH INTERPRETATION: From above computations. See critical path method result and graphs above. The figures are summarized in Table2.2 PERT RESULT Project A B C D E F Activity time 248 36 29 21 40 54 49 Early Start 0 36 65 36 86 140 14 Qualitative Technique PERT SOLUTION Early Late Late Finish Start Finish 36 65 86 76 140 189 0 36 65 149 86 140 36 65 86 189 140 189 Slack 0 0 0 113 0 0 Standard Deviation 3.33 1 1. probable. and pessimistic) in days needed to finish each activity.33 .Westmead International School School of Economics Business and Accountancy Manny is the Project Manager of Barber Shop and he is tasked to open a new branch in Batangas as part of the company’s expansion program at the earliest possible time.33 1.67 1. He was able to determine the sequence of activities and the estimates (optimistic.1. When he is going to finish the project? What is the critical path? TABLE 2.56 1.1 DATA TABLE PROGRAM EVALUATION AND REVIEW TECHNIQUE Optimistic time Most Likely time Pessimistic time Prec 1 A 34 35 42 B 23 30 31 A C 19 20 27 B D 37 40 43 A E 47 55 57 C F 43 50 51 E G 38 45 46 D H 22 25 48 E I 12 15 18 G Prec 2 F H TABLE 2.33 1. 3 TASK TIME COMPUTATION Optimistic time A B C D E F G H I 34 23 19 37 47 43 38 22 12 PERT SOLUTION Most Pessimistic Activity Likely time time time 35 42 36 30 31 29 20 27 21 40 43 40 55 57 54 50 51 49 45 46 44 25 48 28.33 1.67 233 233 233 248 0 64.33 15 18 15 Project Results Total of critical Activities Square root of total Standard Deviation Variance 1.67 3.78 1.4 CHARTS OF PROGRAM EVALUATION AND REVIEW TCHNIQUE 15 Qualitative Technique .33 1 TABLE 2.33 4.33 1.33 15 189 140 233 233 168.78 1.78 18.Westmead International School School of Economics Business and Accountancy G H I 44 28.78 1 12.67 1.67 0 1.33 1 1.33 1.56 TABLE 2.78 1.33 1 1.78 1.33 4.78 1 2.33 248 189 204. Westmead International School School of Economics Business and Accountancy GANTT CHART-LATE TIMES GANNT CHART-EARLY AND LATE TIMES PRECEDENCE GRAPH 16 Qualitative Technique . The figures are summarized in Table3. See PERT result. PROJECT CRASHING Vj. What activities should be crashed while minimizing additional cost? TABLE 3. He was able to determine the normal and minimum crashed time possible with their corresponding costs. task time computation and its graph above.1 DATA TABLE CRASHING METHOD A B C D E F G H Normal time 5 4 10 5 7 5 4 6 17 Qualitative Technique Crash time 2 2 6 3 6 3 2 3 Normal Cost 25000 30000 45000 30000 30000 20000 35000 35000 Crash Cost Prec 1 34000 40000 81000 38000 37000 26000 44000 65000 A B C D E F G .Westmead International School School of Economics Business and Accountancy INTERPRETATION: Using Project evaluation and review technique critical path method is A-B-C-E-F-G-I. the Event Manager of MTV Music Summit was informed that the project must be finished in 86 days.1. Westmead International School School of Economics Business and Accountancy TABLE 3.2 CRASHING RESULT Project A B C D E F G H TOTALS Normal time 46 5 4 10 5 7 5 4 6 CRASHING SOLUTION Crash Normal Crash Crash time Cost Cost cost/pd 27 2 25000 34000 3000 2 30000 40000 5000 6 45000 81000 9000 3 30000 38000 4000 6 30000 37000 7000 3 20000 26000 3000 2 35000 44000 4500 3 35000 65000 10000 250000 Crash by Crashing cost 3 2 4 2 1 2 2 3 9000 10000 36000 8000 7000 6000 9000 30000 115000 TABLE 3.3 CRASH SCHEDULE CRASHING SOLUTION Project time Period cost Cumulative cost A 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 0 3000 3000 3000 3000 3000 4000 4000 4500 4500 5000 5000 7000 9000 9000 9000 9000 10000 10000 10000 0 3000 6000 9000 12000 15000 19000 23000 27500 32000 37000 42000 49000 58000 67000 76000 85000 95000 105000 115000 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 18 Qualitative Technique B 1 2 2 2 2 2 2 2 2 2 C 1 2 3 4 4 4 4 D 1 2 2 2 2 2 2 2 2 2 2 2 2 2 E F G H 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 2 3 . See crashing result and crash schedule above.Westmead International School School of Economics Business and Accountancy INTERPRETATION: From above computations the crash time would be 27 days. CHAPTER 3 DECISION THEORY 19 Qualitative Technique . formulating rules that may lead to aimed a at most advantageous course of action under the given circumstances. (Figures below are in thousands) What will be Boot’s decision? TABLE 1.Westmead International School School of Economics Business and Accountancy Framework of logical and helping managers in mathematical concepts. 4. 50% that the market is average. Decision theory divides decisions into three classes (1) Decisions under certainty: where a manager has far too much information to choose the best alternative. The figures are summarized in Table 1. 5. the Market Analyst was able to determine further the probability of occurrence of each type of market. 1. (3) Decisions under uncertainty: where a manager has to dig-up a lot of data to make sense of what is going on and what it is leading to. The probability of occurrence is 40% that the market is favourable.1 DATA TABLE PAYOFF SUMMARY Expansion for plan payoff Favorable 20 Qualitative Technique Average Unfavorable . 2. Five criteria that may be used: Maximax Maximin Hurwicz Criterion of Realism La Place (Equally Likely) Minimax Regret Bors. (2) Decisions under conflict: where a manager has to anticipate moves and countermoves of one or more competitors.1. and 10% that the market is unfavourable. 3. 6 21.5 4.1 53.3 EXPECTED VALUE MULTIPLICATIONS Probabilities Option 1 Option 2 Option 3 Decision Theory Solution Favorable Average Unfavorable .2 DECISION TABLE RESUTS Decision Theory Solution Favorable Average Unfavorable EMV Probabilities Option 1 Option 2 Option 3 .1 53.5 21.4 PERFECT INFORMATION Probabilities Decision Theory Solution Favorable Average Unfavorable .8 48.5 Row sum (Exp Val) 58.5 54 65 43 .7 16.4 61 42 54 .4 .8 32.1 24.1 21 Qualitative Technique Maximum .5 54 65 43 .5 .1 67 45 55 TABLE 1.6 TABLE 1.25 .5 5.4 .Westmead International School School of Economics Business and Accountancy Probabilities Construct a plant Open Distribution Center Do Nothing Alpha = .4 27 6.1 67 45 55 Maximum 58.6 58.5 .1 Best EV Row Min Row Max 54 67 42 65 43 55 54 67 maximin Maximax TABLE 1.4 61 42 54 .8 48. 43 44.08 45.61 55.45 56.21 45.13 42.36 43.22 55.16 .48 43.12 .53 55.39 42.76 55.2 44.04 .17 Decision Theory Solution Option 1 Option 2 54 42 54.82 45.23 54.03 .78 43.44 44.04 .91 22 Qualitative Technique Option 3 43 43.07 .72 43.3 55.04 43.96 44.3 44.68 56.10 .84 55.5 67 45 55 67 6.95 45.65 43.09 .Westmead International School School of Economics Business and Accountancy Option 1 Option 2 Option 3 Perfect Information Perfect*probability Best Expected Value Exp Value of Perfect Info 61 42 54 61 24.02 .92 45.06 .52 42.6 HURWICS TABLE Hurwicz Value .5 9.4 0 19 7 .92 54.32 44.14 .1 0 22 12 Minimaxregret Maximum Regret Expected Regret 11 22 22 11 5.11 .46 54.8 44.08 .5 TABLE 1.5 REGRET OR OPPORTUNITY LOSS Decision Theory Solution Average Unfavorable Favorable Probabilities Option 1 Option 2 Option 3 .08 44.12 43.56 44.8 15 TABLE 1.6 43.17 44.56 44.05 .24 43.91 43.13 .5 11 0 22 .01 .69 54.15 54.26 42.7 63.6 58.07 55.4 54 65 43 65 32.38 54.1 5.84 43.69 45 55.00 .15 .68 44. 49 .67 48.72 46.85 59.83 47.24 46.35 52.81 53.48 46.45 .28 48.65 54.40 .51 50.81 58.18 .32 47.55 58.14 46.28 46.4 45.02 61.04 48.52 47.12 52.37 .59 59.38 .33 .26 .42 58.29 58.46 .35 .21 48.48 .46 59.36 49.03 58.5 53.16 45.99 57.33 59.75 47.64 48.51 57.24 60.59 49.32 .6 49.04 53.41 .9 49.44 48.98 48.96 54.9 58.97 51.36 .27 53.Westmead International School School of Economics Business and Accountancy .47 .52 .2 51.07 59.28 45.88 45.30 .19 .12 46.68 47.2 59.38 57.76 60.24 .12 49.55 .28 50.56 47.73 56.13 49.21 .47 56.34 56.5 60.23 .52 48.39 .96 47.86 56.25 57.19 54.98 60.64 57.44 .66 51.54 .94 59.52 45.56 23 Qualitative Technique 56.6 46.06 47.76 48.29 47.16 48.58 52.11 60.16 58.42 54.63 60.74 50.72 59.88 49 49.89 52.44 47.31 .76 45.05 50.34 .2 47.72 .92 48.22 .27 .15 61.88 46 46.43 51.36 46.8 47.50 .89 61.51 .43 .42 .28 .37 46.25 .08 47.53 .20 .84 46.29 .6 56.6 46.37 60.24 49.77 57.82 50.68 58.4 48.12 57.64 45.48 49.73 53.36 49. 65 .92 65.16 54.19 62.41 57.90 .63 60.41 61.71 59.79 .24 52.35 55.83 65.84 62.25 59.4 51.87 58.63 .76 51.84 49.72 .8 61.05 65.04 51.08 53.68 53.11 55.62 63.32 53.72 52.12 52.75 63.02 59.61 .92 51.78 .2 53.91 .96 53.66 .56 50.8 50.78 62 62.09 61.94 60.74 .32 50.68 50.4 .73 .79 59.89 .7 62.66 64.96 50.76 .Westmead International School School of Economics Business and Accountancy .58 .62 63.81 .96 66.4 64.86 61.69 .49 56.44 53.56 53.64 .1 58.68 .53 64.6 52.57 65.84 .23 63.09 66.36 63.26 56.93 62.06 62.36 52.88 .60 .1 63.88 64.71 62.33 58.17 60.34 55.45 62.47 62.32 62.59 .52 51.16 63.48 52.83 .03 56.67 .2 50.56 58.95 24 Qualitative Technique 61.55 61.8 53.77 .44 65.79 64.93 63.14 64.93 .49 63.8 56.58 62.32 61.86 .04 54.97 63.64 51.18 65.18 57.31 65.72 56.85 .92 .44 50.16 51.64 57.70 .28 54.01 64.94 .39 63.95 57.4 60.71 .62 .28 51.84 52.85 49.27 64.24 62.67 61.7 65.75 .57 55.87 .54 61.88 52 52.22 66.80 .57 .82 .48 59.92 54.08 50. 61 66. Boot’s best decision is to construct a plant.64 54. and would maximize the company’s profit. Skilful inventory management could mean great savings for the company.Westmead International School School of Economics Business and Accountancy . Proper inventory management should be maintained in order to bring down the total annual cost of inventory.76 54. Annual cost of inventory is the sum of the annual ordering cost and the annual cost. A necessary function of all business operations.54 64. CHAPTER 4 INVENTORY MANAGEMENT The word “inventory” in business means a detailed list of things in stock for a period of time.96 .52 54.48 66.77 65 54. 25 Qualitative Technique .00 66.98 .74 66.31 64.08 64.97 .88 55 INTERPRETATION: Base on the result.99 1.87 67 64. There are two costs involved in annul inventory: the ordering cost and the carrying cost. The order size that minimizes the sum of ordering cost and carrying cost is known as Economic Order Quantity (EOQ). By the use of tables showing various lot sizes of quantity orders b. Carrying cost is usually expressed on an annual basis as percentage of the average annual inventory. Two ways of finding EOQ: a. The annual demand is approximately 1.200 batteries.Westmead International School School of Economics Business and Accountancy Carrying cost Carrying cost or holding cost increases as the size of the inventory increases. Economic Order Quantity (EOQ) To minimize the um of ordering cost and carrying cost. heat. lights. Ordering Cost Ordering costs include costs associated with getting an item into the firm’s inventory. Ordering cost increase as the number of order increases. refrigerator and wages of personnel needed to protect and keep records of the inventory. we must determine the proper quantity to order. Other sources include taxes and deterioration of goods. the interest on the money invested in the inventory is a major distribution to the carrying cost. The supplier pays $28 for each battery and estimates that the annual holding cost is 30 26 Qualitative Technique . Total ordering cost is composed of purchase order. Storage cost is another source of carrying cost which includes rent of space. receiving and setting up of equipment. By the use of the formula : EOQ = An auto parts supplier sells Hardy-brand batteries to car dealers and auto mechanics. shipping costs. If the firm borrows money to finance inventory. 1 DATA TABLE Hardy-Brand Batteries Parameter Demand rate(D) Setup/Ordering cost(S) Holding cost(H) Unit cost Value 1200 20 30% 28 TABLE 1.87 Annual Setup cost 317.8 Orders per period(year) 15.59 75.2 INVENTORY RESULTS Parameter Demand rate(D) Setup/Ordering cost(S) Holding cost(H)@30% Unit cost Hardy-Brand Batteries Solution Value Parameter 1200 Optimal order quantity (Q*) 20 Maximum Inventory Level (Imax) 8. The supplier currently orders 100 batteries per month.3 COST CURVE 27 Qualitative Technique Value 75.Westmead International School School of Economics Business and Accountancy percent of the battery's value.98 TABLE 1. It costs approximately $20 to place an order (managerial and clerical costs). TABLE 1.49 Annual Holding cost 317.4 28 Average inventory 37.49 Unit costs (PD) 33600 Total Cost 34234.59 . and trend projection. exponential smoothing. Delphi method. CHAPTER 5 FORECASTING A planning tool that helps management in its attempts to cope with the uncertainty of the future.Westmead International School School of Economics Business and Accountancy IINTERPRETATION: Inventory results and cost curve answers are given above. moving averages. Forecasting starts with certain assumptions based on the management’s experience and judgment. relying mainly on data from the past and present and analysis of trends. These estimates are projected into the coming months or years using one or more techniques such as Box-Jenkins models. Since any error in the assumptions will result in a 28 Qualitative Technique . regression analysis. A forecast should not be confused with a budget. prepare the forecast for the period 6 using naive approach. the technique of sensitivity analysis is used which assigns a range of values to the uncertain factors (variables). NAIVE APPROACH Given the following data below. NAÏVE FORECASTING APPROACH 1 2 3 4 5 Demand(y) 80 85 75 78 84 TABLE 1.Westmead International School School of Economics Business and Accountancy similar or magnified error in forecasting.1 NAIVE FORECASTING RESULT NAÏVE FORECASTING APPROACH Measure Error Measures Bias (Mean Error) MAD (Mean Absolute Deviation) 29 Qualitative Technique Value 1 6 . 5 .08 (MSE) (MAPE) Next period forecast 84 |Error| Error^2 (Bias) (MAD) Std err |Pct Error| 9.13 4 78 75 3 3 9 .08 84 TABLE 1.67 30 Qualitative Technique .3 NAIVE CONTROL (TRACKING SIGNAL) NAÏVE FORECASTING APPROACH 1 Demand(y Forecast Error RSFE |RSFE| ) 80 Cum Abs Cum MAD Track Signal 2 85 80 5 5 5 5 5 1 3 75 85 -10 -5 10 15 7.5 -.04 5 84 78 6 6 36 .2 NAIVE DETAILS AND ERROR ANALYSIS NAÏVE FORECASTING APPROACH Forecast Error 1 Demand(y ) 80 2 85 80 5 5 25 .Westmead International School School of Economics Business and Accountancy MSE (Mean Squared Error) Standard Error (denom=n-2=2) MAPE (Mean Absolute Percent Error) Forecast next period 42.06 3 75 85 -10 10 100 .07 TOTALS 402 4 24 170 .67 4 78 75 3 -2 3 18 6 -.22 TABLE 1.5 9.4 1 6 42.33 5 84 78 6 4 6 24 6 .22 .3 AVERAGE 80. Prepare the forecast for the period 6 using three period moving average approach. MOVING AVERAGE APPROACH 1 2 3 4 5 TABLE 2. error analysis.4 NAIVE GRAPH INTERPRETATION: Using naive approach the interpretation for period six would be 84. See naive details. naive control (tracking signal) and naive graph above.Westmead International School School of Economics Business and Accountancy TABLE 1.1 MOVING AVERAGE FORECASTING RESULT 31 Qualitative Technique Demand(y) 80 85 75 78 84 . MOVING AVERAGE APPROACH Given the following data below. 78 .22 .33 12.33 (MAD) 12.08 32 Qualitative Technique .04 (MAPE) Std err NA 79 TABLE 2.03 5 84 79.08 AVERAGE Next period forecast 80.5 9.Westmead International School School of Economics Business and Accountancy MOVING AVERAGE APPROACH Measure Error Measures Bias (Mean Error) MAD (Mean Absolute Deviation) MSE (Mean Squared Error) Standard Error (denom=n-2=0) MAPE (Mean Absolute Percent Error) Forecast next period Value 1.04 79 TABLE 2.3 ERRORS AS FUNCTION OF N MOVING AVERAGE APPROACH N Bias MAD MSE Standard error MAPE 1 1 6 42.89 (MSE) .67 6.67 21.4 1.33 (Bias) 3.67 4.06 TOTALS 402 2.67 25.78 .2 DETAILS AND ERROR ANALYSIS MOVING AVERAGE APPROACH 1 Demand(y ) 80 Forecast Error |Error| Error^2 |Pct Error| 2 85 3 4 75 78 80 -2 2 4 .33 3.89 NA .33 4. errors as function of n.83 10.8 TABLE 2. 33 Qualitative Technique .67 Cum MAD Track Signal 2 3.33 -1 . EXPONENTIAL SMOOTHING Given the following data below.67 5.5 MOVING AVERAGE GRAPH INTERPRETATION: Using moving average approach.33 4. the forecast for period six would be 79.Westmead International School School of Economics Business and Accountancy 2 -. prepare the forecast for the period 6 using exponential smoothing approach.33 12.67 2. See moving average forecasting results.5 20.07 3 1.67 6. control tracking signal and moving average graph above.4 CONTROL (TRACKING SIGNAL) 1 2 3 4 5 Moving Average approach Solution Demand(y Forecast Error RSFE |RSFE| Cum ) Abs 80 85 75 78 80 -2 -2 2 2 84 79.67 4.79 .05 TABLE 2.33 3.89 .67 38.5 4.25 .04 4 4. details and error analysis. 69 80.83 .48 701.38 .06 152 232.47 239 212.38 4523.69 -80.69 6510.13 .55 281.25 33.26 .75 1139.14 TABLE 3.11 .1 FORECASTING RESULT EXPONENTIAL SMOOTHING APPROACH Measure Error Measures Bias (Mean Error) MAD (Mean Absolute Deviation) MSE (Mean Squared Error) Standard Error (denom=n-2=3) MAPE (Mean Absolute Percent Error) Forecast next period Value 24.75 33.42 1174 120.14 (Bias) (MAD) (MSE) Std err 34 Qualitative Technique 86.48 26.2 DETAILS AND ERROR ANALYSIS 1 2 3 4 5 TOTALS AVERAGE Next period forecast EXPONENTIAL SOOTHING APPROACH Demand(y Forecast Error |Error| Error^2 ) 213 189 24 24 576 312 195 117 117 13689 258 224.11 56.92 22615.19 86.38 4523.25 (MAPE) .52 26.11 1.83 |Pct Error| .96 234.Westmead International School School of Economics Business and Accountancy EXPONENTIAL SMOOTHING APPRACH Demand(y) 213 312 258 152 239 1 2 3 4 5 Forecast 189 0 0 0 0 TABLE 3.25 219.19 219.53 .11 56.8 24. 82 86.18 .46 26.13 .82 58.47 56.84 4720.25 .25 .01 .01 4474.09 .3 4535.3 ERRORS AS FUNCTION OF ALPHA Alpha .02 4495.57 4501.02 86.55 4540.68 33.83 23.11 56.25 .71 86.4 31.57 88.49 56.26 .85 4486.25 .75 56.33 40.29 57.53 25.16 4613.16 86.82 34.Westmead International School School of Economics Business and Accountancy TABLE 3.27 .25 .2 57.64 86.78 4485.72 24.43 37.69 38.63 86.05 .25 .19 86.19 4507.95 22.21 87.76 87.7 41.63 55.6 4680.08 4764.64 87.42 59.25 .19 .14 4471.4 26.41 86.37 4522.25 .06 4579 87.12 57.28 .4 4471.25 .62 89.92 86.25 .17 .25 .15 86.17 56 4595.96 57.00 .25 .2 87.25 .11 42.03 86.46 86.25 .25 .51 20.07 22.29 .4 56.35 30.88 56.31 EXPONENTIAL SMOOTHING PPROACH Bias MAD MSE Standard error 45.45 88 39.86 86.25 .89 4479.12 .75 87 35.25 .73 56.08 87.10 .04 .47 58.19 36.24 .84 86.68 60.06 .35 89.51 59.22 .49 57.17 87.25 .25 .27 4470.36 21.25 .23 .25 .58 60.43 59.15 .8 60.33 28.94 57.25 .92 88.67 4492.55 4473.38 4645.6 58.82 86.02 .07 56.6 4867 90.7 58.25 .71 57.25 .32 28.43 58.08 .25 .25 .06 44.56 32.07 .36 27.25 .93 4612.3 56.48 4511.68 35 Qualitative Technique MAPE .32 29.14 .91 87.95 4586.21 .38 4523.20 .11 .25 .62 24.25 .25 .55 86.52 58.30 .34 4813.76 56.75 4561.25 .67 86.25 .03 .25 .14 4563.21 21.46 59.22 4549.7 4478.16 .47 32.57 43. 98 5516.01 90.7 93.26 .27 92.46 .18 94.38 88.43 55.8 90.26 .63 18.53 55.35 91.52 .26 .9 9.81 92.69 5057.3 .26 .69 13.26 .46 93.13 4837.43 .45 .71 5187.87 55.61 61.31 4667.71 60.69 11.26 .58 89.64 14.22 93.68 .14 4727.98 93.76 55.29 55.57 4956.25 .91 95.39 .12 91.58 .Westmead International School School of Economics Business and Accountancy .29 16.51 .77 88.6 19.3 .31 55.17 89.11 10 9.17 17.26 .26 .97 57.35 10.69 .49 5134.35 5294.87 16.60 .74 92.54 5031.07 5082.63 .63 10.63 5006.33 .43 5267.15 95.26 .93 62.83 4749 4770.02 88.16 60.8 59.33 14.70 20.68 4815.94 5487.34 .57 56.26 .4 13.64 95.29 .64 4908.2 88.88 96.46 55.61 55.02 5459.26 .26 .66 94.95 10.51 92.36 12.29 .47 .29 .23 90.40 .38 .68 5108.11 19.27 .11 5213.3 .42 5321.23 55.36 .78 10.27 .96 4861.07 15.79 63.96 4981.96 4707.11 18.65 5349.5 5160.92 4648.13 11.69 15.26 59.64 4629.68 5240.37 .49 61.25 55.23 10.27 .34 4686.89 91.28 .28 .56 5404.37 89.49 .89 58.31 14 13.13 4884.49 10.35 .35 58.4 55.49 55.42 .28 .67 .05 61.57 88.41 .67 90.50 .81 55.13 96.14 5544.45 90.57 .66 55.04 92.62 .5 56.03 56.37 .61 .43 57.97 89.59 .26 .12 10.39 95.53 .29 .36 62.57 55.26 .71 55.98 14.41 5572.58 91.56 .27 55.77 87.3 11.26 .27 .3 .46 4932.22 63.73 36 Qualitative Technique 55.26 .42 94.81 9.32 .86 12.26 .65 .34 55.22 5432.37 55.94 94.54 .48 .63 4792.03 5376.2 55.44 .28 .72 17.9 11.6 12.29 .66 .49 11.46 16.64 .28 .84 88.13 12.26 .26 .55 .26 . 33 .80 .68 5773.6 97.85 98.43 5831.46 9.37 68.21 73.56 6419.93 10 64.4 CONTROL (TRACKING SIGNAL) 1 2 Demand(y ) 213 312 Exponential Smoothing Solution Forecast Error RSFE |RSFE| Cum Abs 189 24 24 24 24 195 117 141 117 141 37 Qualitative Technique Cum MAD 24 70.31 .96 .3 9.6 6119.88 .5 Track Signal 1 2 .79 9.48 66.33 .94 .97 .55 6203.44 71.44 .27 9.89 .29 9.8 71.75 100.33 .93 .87 97.47 70.33 .73 9.47 64.87 67.34 98.90 .01 103.34 .84 .00 9.39 6259.91 102.11 5947 5975.05 72.65 9.26 9.95 .27 5889.85 6004.09 69.82 .69 66.31 9.79 70.88 65.34 .23 5629.33 .68 101.98 .8 100.71 .34 TABLE 3.22 101.34 .8 103.Westmead International School School of Economics Business and Accountancy .32 99.32 .81 .74 74 5601.56 99.56 6367.06 64.04 100.29 9.13 70.09 66.72 .75 .03 6286.6 96.74 .54 5802.46 9.34 .45 101.07 99.11 97.09 98.32 .86 9.33 .92 .31 .93 73.58 102.32 .35 72.52 100.63 68 68.56 9.31 .25 67.32 .42 9.78 .23 103.87 5744.83 99.32 .78 5658.3 .65 6033.73 .58 9.37 6175.52 9.31 .04 6090.75 72.29 65.79 .87 .33 .19 5918.36 102.28 100.86 .14 102.61 9.35 9.34 9.32 .39 9.83 .58 98.74 69.34 5860.45 6313.33 9.32 .31 .91 .11 5715.76 .12 71.48 73.99 1.62 96.85 .33 .39 6062.28 9.45 69.33 .37 9.64 72.34 .63 6340.41 5687.31 .99 101.36 97.42 9.27 9.51 9.3 .77 .67 9.05 6147.21 6393.27 9.56 6231. 38 2. with the assumptions of pure chance traffic.14 TABLE 3. and being served by the server at the front of the queue.25 232.69 5 239 212.86 3 1.69 174. It is applicable in transport and telecommunication and is occasionally linked to ride theory.75 80.48 120.47 26.5 EXPONENTIAL SMOOTHING GRAPH CHAPTER 6 QUEING / WAITING LINES Queuing theory is the mathematical study of waiting lines. Incoming traffic to queuing theory systems is modelled via Poisson distribution.44 58.25 63. Call arrivals and departures are random and independent events.52 33.55 33. All incoming traffic can be routed to any other customer within the network and congestion is cleared as soon as servers are free.92 56.Westmead International School School of Economics Business and Accountancy 3 4 258 152 224.75 174.75 255. waiting in the queue (essentially a storage process).06 80. arriving at the back of the queue.48 281. There are several related processes. Statistical equilibrium probabilities within the system do not change. 38 Qualitative Technique .75 94.69 26. Westmead International School School of Economics Business and Accountancy Queuing theory is finding the best level of service by trade-off between the cost of providing good service and the cost of customer waiting line. M/M/1 with finite source M/M/1 Bryan is the branch manager of East-West Bank and he wants to improve the service of the bank by reducing the average waiting time of the bank’s client. M/D/1 4.1 DATA TABLE EAST-WEST BANK Parameter M/M/1 (exponential service times) Arrival rate(lambda) Service rate(mu) Number of servers Server cost $/time Waiting cost $/time 39 Qualitative Technique Value 3 15 1 8 11 . M/M/1 2.1. He was able to determine the average arrival and the average number of clients serviced per hour. Bryan knows that there is also an opportunity cost for clients who are idle while waiting in line. We will also assume a first in first out (FIFO) disciple where customers are serviced according to order of their arrival. M/M/m 3. The figures are summarized in Table1. Four Models presented using Kendall notation: 1. He was able to determine the teller’s labour cost as well as the average opportunity cost of clients who are waiting. How many clients are in the bank at any given time? How much time does a client spent in the bank? How many clients are waiting to be served? How much a time does a client spent in waiting? What is the probability that the teller is busy? What is the probability that there are no clients? How much is the total cost per shift? TABLE 1. 2 WAITING LINES RESULTS Parameter M/M/1 (exponential service times) Arrival rate(lambda) Service rate(mu) EAST-WEST BANK Solution Value Parameter Value Minutes Seconds Average server .03 1 0 3 4 5 6 0 0 0 0 1 1 1 1 0 0 0 0 TABLE 1.2 1 .02 1 60 .3 TABLE OF PROBABILITIES EAST-WEST BANK Solution k 0 Prob (num in sys = k) .04 2 .08 5 300 8.Westmead International School School of Economics Business and Accountancy TABLE 1.96 .8 Prob (num in sys >k) .25 .75 TABLE 1.55 10.05 .4 GRAPHS OF PROBABILITIES 40 Qualitative Technique .2 utilization 3 Average number in the queue(Lq) Average number in the system(Ls) Average time in the queue(Wq) Average time in the system(Ws) Cost (Labor + # waiting*wait cost) Cost (Labor + # in system*wait cost) 15 Number of servers 1 Server cost $/time 8 Waiting cost $/time 11 .16 .8 Prob (num in sys <= k) . Westmead International School School of Economics Business and Accountancy 41 Qualitative Technique . She observed that the average arrival and the average number of clients serviced per teller per hour remains the same.Westmead International School School of Economics Business and Accountancy INTERPRETATION: Waiting lines and table of probabilities results using MM1 method are above. The figures are summarized in Table2.1 DATA TABLE EAST-WEST BANK Parameter 42 Qualitative Technique Value .1. Clair knows that with the additional teller. M/M/m Clair. M/M/s. the Bank Manager of EAST-WEST Bank wants to improve the service of the bank by hiring an additional teller. TABLE 2. the number of clients waiting in line will decrease. She observed that the teller’s labour costs as well as the average opportunity cost of clients who are waiting remain the same. 2 system(Ls) 2 Average time in the 0 .02 waiting*wait cost) Cost (Labor + # in 18.82 .07 4.Westmead International School School of Economics Business and Accountancy M/M/s Arrival rate(lambda) Service rate(mu) Number of servers Server cost $/time Waiting cost $/time 3 15 2 8 11 TABLE 2.2 WAITING LINES RESULT Parameter M/M/s Arrival rate(lambda) Service rate(mu) Number of servers Server cost $/time Waiting cost $/time EAST-WEST BANK Solution Value Parameter Value Minutes Seconds Average server .04 2.04 242.16 .42 system(Ws) 11 Cost (Labor + # 16.82 .98 .42 queue(Wq) 8 Average time in the .3 TABLE OF PROBABILITIES k 0 1 2 3 4 5 EAST-WEST BANK Solution Prob (num in sys = Prob (num in sys <= k) k) .22 system*wait cost) TABLE 2.18 .02 1 0 1 0 1 0 1 43 Qualitative Technique Prob (num in sys >k) .1 utilization 3 Average number in the 0 queue(Lq) 15 Average number in the .02 0 0 0 0 . 02 24 18. SERVERS EAST-WEST BANK Solution Number of servers 1 Total cost based on waiting 8.55 Total cost based on system 10.Westmead International School School of Economics Business and Accountancy TABLE 2.2 5 40 42.2 TABLE 2.4 COST VS.5 GRAPHS OF PROBABILITIES 44 Qualitative Technique .75 2 3 16.2 4 32 34.22 26. She observed that the average arrival and the average number of clients serviced per hour remain the same. she presumed that the service rate distribution is now constant.1. How many clients are waiting to be served? How much time does a client spend waiting? How many clients are at the ATM at any given time? How much time does a client spend at the ATM? What is the probability that the ATM is occupied? What is the probability that there are no clients? How much is the total cost per shift? TABLE 3.Westmead International School School of Economics Business and Accountancy INTERPRETATION: Waiting lines. Lea knows that with the ATM as the teller. table of probabilities and cost vs. M/D/1 Grace. Because an ATM processes according to a fixed cycle.1 DATA TABLE EAST WEST BANK Parameter M/D/1 (constant service times) Value Arrival rate(lambda) 3 Service rate(mu) Number of servers Server cost $/time 15 1 8 Waiting cost $/time 11 45 Qualitative Technique . server results using MMm/MMs method are above. She observed that the ATM’s service cost is equal to the labour cost of a teller and the average opportunity cost of clients who are waiting remain the same. the number of clients waiting in line will decrease. the Branch Manager of EAST-WEST Bank wants to improve the service of the bank by replacing the teller with an ATM. The figures are summarized in Table3. 5 270 system(Ws) 11 Cost (Labor + # 8.5 30 queue(Wq) 8 Average time in the . CHAPTER 7 46 Qualitative Technique .03 the queue(Lq) 15 Average number in .48 system*wait cost) INTERPRETATION: Waiting lines result using MM1 method is above.2 WAITING LINES RESULTS Parameter M/D/1 (constant service times) Arrival rate(lambda) Service rate(mu) Number of servers Server cost $/time Waiting cost $/time EAST WEST BANK Solution Value Parameter Value Minutes Seconds Average server .28 waiting*wait cost) Cost (Labor + # in 10.2 utilization 3 Average number in .23 the system(Ls) 1 Average time in the 0 .08 4.Westmead International School School of Economics Business and Accountancy TABLE 3. very efficient algorithm exists for the solution of the optimization problem. Each town must be connected from another town starting with Town A.Westmead International School School of Economics Business and Accountancy NETWORK The term network flow program describes a type of model that is a special case of the more general linear program. the maximum flow problem. The class of network flow programs includes such problems as the shortest path problem. using possible distance. and the generalized minimum cost flow problem. MINIMAL SPANNING TREE Blair is the Head Engineer of MY-DSL and he is tasked t provide a telephone lines to 15 towns in the province of Nasugbu. Maximal-Flow Technique Maximal Flow Technique seeks to determine the maximum quantity that can go from one node to another in a network at any given time. Maximal-Flow Technique 3. the pure minimum cost flow problem. Shortest-Route Technique Minimal-Spanning Tree Technique The objective of Minimal-Spanning Tree Technique is to connect each node to at least one node while minimizing the total distance. Minimal-Spanning Three Technique 2. He 47 Qualitative Technique . Shortest-Route Technique The shortest-route technique determines the minimum distance from one point of the network to another. When the situation can be entirely modelled as network. many times more efficient than linear programming in the utilization of computer time and space resources. Network models use nodes and arcs to make decision. Three techniques: 1. It is an important class because many aspects of actual situations are readily recognized as networks and the representation of model is much more compact than the general linear program. Westmead International School School of Economics Business and Accountancy was able to determine the distance in kilometres from one town to another. The distance figures are summarized in Table 1.1. How should the towns be cnnected? MY-DSL A B C D E F G H I J K L M N O TABLE 1.1 DATA TABLE MINIMAL SPANNING TREE Start node End node 1 2 1 3 1 4 2 3 2 5 3 4 3 5 3 6 3 7 3 8 4 7 5 8 6 7 7 9 8 9 Cost 15 10 18 17 12 13 8 21 19 22 16 14 9 20 11 TABLE 1.2 NETWORK RESULT MY-DSL A Start End node node 1 2 Cost Cost Y 10 15 B 1 3 10 C 1 4 18 D 2 3 17 48 Qualitative Technique Include Westmead International School School of Economics Business and Accountancy E 2 5 12 Y 12 F 3 4 13 Y 13 G 3 5 8 Y 8 H I 3 3 6 7 21 19 J 3 8 22 K 4 7 16 Y 16 L 5 8 14 Y 14 M 6 7 9 Y 9 N 7 9 20 O 8 9 11 Y 11 Total 93 TABLE 1.3 SOLUTION STEP MINIMAL SPANNING TREE SOLUTION MY-DSL Ending node 3 Cost B Starting node 1 10 Cumulative cost 10 G 3 5 8 18 E 2 5 12 30 F 3 4 13 43 L 5 8 14 57 O 8 9 11 68 K 4 7 16 84 49 Qualitative Technique Westmead International School School of Economics Business and Accountancy M 6 7 9 93 INTERPRETATION: Using minimal spanning tree method 15 towns in Nasugbu should be connected from B(1-3), G(3-5), E(2-5), F(3-4), L(5-8), O(8-9), K(4-7), M(6-7) for a total of 93 kilometres. See minimal spanning tree network result and solution step above. MAXIMAL FLOW TECHNIQUE Bagito is the Industrial Engineer of Sabang Water District and she is tasked to ensure that maximum amount of water flows from Water Pump J. She was able to determine the capacity of the outflow and inflow of water from one pump to another in litres per second. The capacity figures are summarized in Table 2.1. What is the total capacity of the water flow? TABLE 2.1 DATA TABLE MAXIMAL FLOW TECHNIQUE SABANG WATER DISTRICT A Start node 1 End node Capacity 2 15 Reverse capacity 9 B 1 4 11 13 C 2 3 10 16 D 2 5 8 12 E 3 4 16 7 F 3 6 14 8 G 4 6 12 14 H 5 6 7 11 I 5 7 9 15 J 6 7 13 10 50 Qualitative Technique 3 ITERATIONS Iteration 1 2 3 BATANGAS WATER DISTRICT Solution Path Flow Cumulative Flow 1-> 4-> 6-> 7 11 11 1-> 2-> 3-> 6-> 5-> 7 9 20 1-> 2-> 5-> 6-> 7 2 22 INTERPRETATION: Using Maximal flow technique. the total capacity of the water flow would be 22 liters per second.Westmead International School School of Economics Business and Accountancy TABLE 2. SHORTEST ROUTE TECHNIQUE Boy is the Electrical Engineer of Batelect and he is tasked to provide electricity to a new Electric post I from the generator located at Electric Post A at 51 Qualitative Technique . See maximal flow technique network result and iterations above.2 NETWORK RESULT SABANG WATER DISTRICT Maximal Network Flow A B C D E F G H I J MAXIMAL FLOW TECHNIQUE Start End Capacity node node 22 1 1 2 2 3 3 4 5 5 6 2 4 3 5 4 6 6 6 7 7 15 11 10 8 16 14 12 7 9 13 Reverse capacity Flow 9 13 16 12 7 8 14 11 15 10 11 11 9 2 0 9 11 -9 9 13 TABLE 2. He was to determine the distance in kilometres from on electric post to another.Westmead International School School of Economics Business and Accountancy the minimum possible distance. The distance figures are summarized in Table 1.1. What is the shortest route from Post A to Post I? TABLE 3.2 NETWORK RESULT Total distance = 36 B E H BATELECT Solution Start node End node Distance 1 3 7 3 5 13 5 7 16 Cumulative Distance 7 20 36 TABLE 3.3 MINIMUM DISTANCE MATRIX 1 2 1 0 12 2 12 0 52 Qualitative Technique BATELECT Solution 3 4 7 13 19 8 5 20 17 6 21 33 7 36 33 .1 DATA TABLE BATELECT Start node End node Distance A 1 2 12 B 1 3 7 C 1 4 13 D 2 4 8 E 3 5 13 F 3 6 14 G 4 5 9 H 5 7 16 I 6 7 15 TABLE 3. See shortest route network result and minimum distance matrix above. First it assumes that to disturb or change the idea being transported in any way will damage and reduce it somehow.Westmead International School School of Economics Business and Accountancy 3 4 5 6 7 7 13 20 21 36 19 8 17 33 33 0 20 13 14 29 20 0 9 34 25 13 9 0 27 16 14 34 27 0 15 29 25 16 15 0 INTERPRETATION: Using shortest route technique. the model is effective in determining resource allocation in existing business structures. The transportation model is a valuable tool in analyzing and modifying existing transportation systems or the implementation of new ones. like taking a hole from one place and inserting it in another without change. It also assumes that it is possible to take an idea from one person's mind into another person's so that the two people will then understand in exactly the same way. CHAPTER 8 TRANSPORTATION MODEL The transportation model uses the principle of 'transplanting' something. the shortest route from post A to Post I would be 36 kilometres. 53 Qualitative Technique . In addition. Items are homogeneous. 3. The use of this model for capacity planning is similar to the models used by engineers in the planning of waterways and highways. Nonetheless. demand.Westmead International School School of Economics Business and Accountancy The model requires a few keys pieces of information. In this case. Only one route is used from place of shipment to the destination. The transportation problem involves determining a minimum-cost plan for shipping from multiple sources to multiple destinations. A transportation model is used to determine how to distribute supplies to various destinations while minimizing total shipping cost. production planning. The main applications of the transportation model mention in the chapter are location decisions. a shipping plan is produced and is not changed unless factors such as supply. which include the following: Origin of the supply Destination of the supply Unit cost to ship The transportation model can also be used as a comparative tool providing business decision makers with the information they need to properly balance cost and supply. 54 Qualitative Technique . 2. or unit shipping costs change. Shipping cost per unit is the same no matter how many units are shipped. This will help for comparison when identifying alternatives in terms of their impact on the final cost for a system. the major assumptions of the transportation model are the following: 1. This model will help decide what the optimal shipping plan is by determining a minimum cost for shipping from numerous sources to numerous destinations. capacity planning and transhipment. Louis. Grain is shipped to the mills in railroad cars. The transportation model can be used to compare location alternatives in terms of their impact on the total distribution costs for a system. each car capable of holding one ton of wheat. and Cincinnati. Grain Elevator Kansas Omaha Des Moines Demand Chicago 6 7 4 200 St. it will show the unit cost of shipping goods from each origin to each destination. The table will have a list of origins and each one's capacity or supply quantity period. Wheat is harvested in the Midwest and stored in grain elevators in three different cities – Kansas City. Transportation costs play an important role in location decision. It is subject to demand satisfaction at markets supply constraints. Also. It also determines how to allocate the supplies available from the various factories to the warehouses that stock or demand those goods. It will also show a list of destinations and their respective demands per period. Louis 8 11 5 100 Cincinna ti 10 11 12 300 Supply 150 175 275 MAXIMIZATION NORTHWEST CORNER METHOD Transportation Shipments Optimal Profit A B 55 Qualitative Technique C . and Des Moines. in such a way that total shipping cost is minimized. The transportation problem involves finding the lowest-cost plan for distributing stocks of goods or supplies from multiple origins to multiple destinations that demand the goods. Each grain elevator is able to supply the following number of tons of wheat to the mills on a monthly basis. can be put into a transportation table.Westmead International School School of Economics Business and Accountancy The variables in this model have a linear relationship and therefore. St. located in Chicago. Omaha. These grain elevators supply three flour mills. Westmead International School School of Economics Business and Accountancy 5925 1 2 3 150 50 100 25 275 A B -2 C 0 -4 -7 Marginal Profit 1 2 3 Final Solution Table A B 150 (-2) 50 100 (-4) (-7) 1 2 3 C (0) 75 275 Iterations A 150 50 (-4) 1 2 3 B (-2) 100 (-7) C (0) 75 275 Shipment with Profit A B 150/900 50/350 100/1100 1 2 3 From To Shipping List Shipment 1 2 2 2 3 A A B B C 150 50 100 25 275 56 Qualitative Technique C 25/275 275/3300 Profit per Unit 6 7 11 11 12 Shipment Profit 900 350 1100 275 3300 . Westmead International School School of Economics Business and Accountancy MINIMUM COST METHOD Transportation Shipments Optimal Profit A B 5925 1 150 2 50 100 3 C 25 275 Marginal Profit 1 2 3 A B -2 -4 -7 C 0 Final Solution Table A B 150 (-2) 50 100 (-4) (-7) 1 2 3 C (0) 75 275 Iterations A 150 50 (-4) 1 2 3 B (-2) 100 (-7) C (0) 75 275 Shipment with Profit A B 150/900 50/350 100/1100 1 2 3 From To Shipping List Shipment 1 2 2 A A B 150 50 100 57 Qualitative Technique C 25/275 275/3300 Profit per Unit 6 7 11 Shipment Profit 900 350 1100 . Westmead International School School of Economics Business and Accountancy 2 3 B C 25 275 11 12 VOGEL’S APPROXIMATION METHOD Transportation Shipments Optimal Profit A B 5925 1 125 2 75 100 3 275 3300 C 25 275 Marginal Profit A 1 2 3 B -2 C 0 -4 -7 Final Solution Table A B 125 (-2) 75 100 (-4) (-7) 1 2 3 C 25 (0) 275 Iterations A 125 75 (-4) 1 2 3 Shipment with Profit A B 125/750 75/525 100/1100 1 2 3 From B (-2) 100 (-7) C 25 (0) 275 C 25/250 275/3300 To 58 Qualitative Technique Shipping List Shipment Profit per Unit Shipment Profit . Westmead International School School of Economics Business and Accountancy 1 1 2 2 3 A C A B C 125 25 75 100 275 6 10 7 11 12 750 250 525 1100 3300 MINIMIZATION NORTHWEST CORNER METHOD Transportation Shipments Optimal Cost A B 4525 1 25 2 3 175 100 C 125 175 Marginal Cost A 1 2 3 1 2 3 B 1 3 0 C 4 Final Solution Table A B 25 (1) (0) (3) 175 100 C 125 175 (4) Iterations Iteration 1 1 2 3 Iteration 2 1 2 3 Iteration 3 1 A 150 50 (-4) A 150 50 (-4) A 150 59 Qualitative Technique B (-2) 100 (-7) B (5) (7) 100 B (1) C (0) 25 275 C (0) 125 175 C (-4) . Westmead International School School of Economics Business and Accountancy 2 3 Iteration 4 1 2 3 (4) 50 A 25 (0) 175 (7) 100 B (1) (3) 100 175 125 C 125 175 (4) Shipment with Cost A B 25/150 1 2 3 175/700 From To 1 1 2 3 3 A C C A C 100/500 Shipping List Shipment Cost per Unit 25 125 175 175 100 MINIMUM COST METHOD Transportation Shipments Optimal Cost A B 4525 1 25 2 3 175 100 60 Qualitative Technique C 125/1250 175/1925 6 10 11 4 5 Shipment Cost 150 1250 1925 700 500 C 125 175 . Westmead International School School of Economics Business and Accountancy Marginal Cost A 1 2 3 1 2 3 B 1 3 0 C 4 Final Solution Table A B 25 (1) (0) (3) 175 100 C 125 175 (4) Iterations Iteration 1 1 2 3 Iteration 2 1 2 3 1 2 3 A (-1) (-1) 200 A 25 (0) 175 B 25 (2) 75 B (1) (3) 100 Shipment with Cost A B 25/150 175/700 100/500 Shipping List 61 Qualitative Technique C 125 175 (5) C 125 175 (4) C 125/1250 175/1925 . Westmead International School School of Economics Business and Accountancy From To Shipment Cost per Unit 1 1 2 3 3 A C C A B 25 125 175 175 100 6 10 11 4 5 Shipment Cost 150 1250 1925 700 500 VOGEL’S APPROXIMATION METHOD Optimal Cost4525 1 2 3 Transportation Shipments A B 25 175 62 Qualitative Technique C 150 150 100 . Westmead International School School of Economics Business and Accountancy Marginal Cost A 0 1 2 3 B 1 3 C 4 Final Solution Table A B (0) (1) 25 (3) 175 100 1 2 3 C 150 150 (4) Iterations Iteration 1 1 2 3 Iteration 2 1 2 3 A (4) 175 25 A (0) 25 175 B (5) (3) 100 B (1) (3) 100 C 150 (-4) 150 C 150 150 (4) Shipment with Cost A B 1 2 3 25/175 175/700 100/500 From To Shipping List Shipment 1 2 2 3 3 C A C A B 150 25 150 175 100 63 Qualitative Technique C 150/1500 150/1650 Cost per Unit 10 7 11 4 5 Shipment Cost 1500 175 1650 700 500 .