3 - 2 OR (Operations Research)

Original source visual. Selectable source labels and wording: • Product Y is manufactured by using 3 kg of raw material A and 4 kg of raw material B per product. • A profit of $3,000 can be obtained from the sale of product X, and a profit of $4,000 from product Y. • Only 120 kg of raw material A and only 100 kg of raw material B can be used. Usage restriction Product X Product Y amount Raw material A (kg) 6 3 120 Raw material B (kg) 2 4 100 Profit ($) 3,000 4,000 1) Derive the objective function (i.e., the expression to calculate profit) and the constraint expressions, where the production volumes of product X and Y are x and y, respectively. [Constraint expressions] 6x + 3y ≤ 120 ··· Raw material A 2x + 4y ≤ 100 ··· Raw material B x ≥ 0, y ≥ 0 ··· The production volume must be an integer 0 or greater. [Objective function] Profit = 3x + 4y ··· Maximum (in units of $1,000) 2) Calculate x and y from the constraint expressions. x ≤ 10, y ≤ 20 3) Substitute the maximum values for x and y in the objective function, and calculate the maximum profit. Maximum profit = $3,000 per product × 10 products + $4,000 per product × 20 products = $110,000

• Product Y is manufactured by using 3 kg of raw material A and 4 kg of raw material B per product. • A profit of $3,000 can be obtained from the sale of product X, and a profit of $4,000 from product Y. • Only 120 kg of raw material A and only 100 kg of raw material B can be used. Usage restriction Product X Product Y amount Raw material A (kg) 6 3 120 Raw material B (kg) 2 4 100 Profit ($) 3,000 4,000 1) Derive the objective function (i.e., the expression to calculate profit) and the constraint expressions, where the production volumes of product X and Y are x and y, respectively. [Constraint expressions] 6x + 3y ≤ 120 ··· Raw material A 2x + 4y ≤ 100 ··· Raw material B x ≥ 0, y ≥ 0 ··· The production volume must be an integer 0 or greater. [Objective function] Profit = 3x + 4y ··· Maximum (in units of $1,000) 2) Calculate x and y from the constraint expressions. x ≤ 10, y ≤ 20 3) Substitute the maximum values for x and y in the objective function, and calculate the maximum profit. Maximum profit = $3,000 per product × 10 products + $4,000 per product × 20 products = $110,000

Original source visual. Selectable source labels and wording: Examples: Create an arrow diagram for a project that is composed of the following activities. Activity Number of days required Preceding activity A 3 None B 2 None C 1 A D 2 A E 4 A F 2 B, C G 1 D H 1 D, E, F

Examples: Create an arrow diagram for a project that is composed of the following activities. Activity Number of days required Preceding activity A 3 None B 2 None C 1 A D 2 A E 4 A F 2 B, C G 1 D H 1 D, E, F

Original arrow diagram illustrating earliest node time. It retains numbered nodes 1 through 6, activities A through H, dummy-activity arrows, activity durations and the original forward-pass node calculations and boxed results.

Original arrow diagram illustrating latest node time. It retains nodes 1 through 6, activities, durations, dummy activities, backward-pass expressions and paired original node-time boxes.

Original critical-path diagram. Thick arrows follow nodes 1, 2, 5 and 6 through activities A, E and H. The other activities, dummy activities, durations and each pair of node-time boxes remain visible.

Original Gantt chart: rows Activity 1 through Activity 3, columns Month 1 through 8 and continuation dots. The source legend distinguishes Plan from Result; the bars and original row lengths remain intact.

Original source visual. Selectable source labels and wording: • Total inventory cost This is the total cost that is required to perform inventory control. Total inventory cost = Storage cost + Ordering cost • Storage cost (Inventory cost, Inventory storage cost) This is the cost (e.g., warehouse cost, labor cost, utilities cost) that is required to store inventory. This is determined by the annual storage cost per stock, and the annual average inventory quantity. Storage cost = Annual average inventory quantity

• Total inventory cost This is the total cost that is required to perform inventory control. Total inventory cost = Storage cost + Ordering cost • Storage cost (Inventory cost, Inventory storage cost) This is the cost (e.g., warehouse cost, labor cost, utilities cost) that is required to store inventory. This is determined by the annual storage cost per stock, and the annual average inventory quantity. Storage cost = Annual average inventory quantity

Original source visual. Selectable source labels and wording: × Annual storage cost per stock • Ordering cost (Procurement cost) This is the cost (e.g., labor cost, communications cost, insurance premium) that is required for ordering a product. This is determined by the fixed cost for a single order and the number of orders. (This does not involve the order quantity.) Ordering cost = Number of orders × Ordering cost per order • EOQ (Economic Order Quantity) This is the order quantity that is required to minimize total inventory cost. • Safety stock This is a certain quantity of inventory that is set aside in advance in order to prevent out of stock. Safety stock is generally considered separately from total inventory cost or EOQ.

× Annual storage cost per stock • Ordering cost (Procurement cost) This is the cost (e.g., labor cost, communications cost, insurance premium) that is required for ordering a product. This is determined by the fixed cost for a single order and the number of orders. (This does not involve the order quantity.) Ordering cost = Number of orders × Ordering cost per order • EOQ (Economic Order Quantity) This is the order quantity that is required to minimize total inventory cost. • Safety stock This is a certain quantity of inventory that is set aside in advance in order to prevent out of stock. Safety stock is generally considered separately from total inventory cost or EOQ.

Original source visual. Selectable source labels and wording: [How to calculate total inventory cost] 1) Since the storage cost is determined by the average inventory quantity, first calculate the annual average inventory quantity. The following is a model of inventory changes. Therefore, the average inventory quantity is “Order quantity (Q) ÷ 2”.

[How to calculate total inventory cost] 1) Since the storage cost is determined by the average inventory quantity, first calculate the annual average inventory quantity. The following is a model of inventory changes. Therefore, the average inventory quantity is “Order quantity (Q) ÷ 2”.

Original source visual. Selectable source labels and wording: Storage cost = (Q ÷ 2) × P 2) Since the ordering cost is determined by the number of orders, first calculate the annual number of orders. It is necessary to order enough to meet the total annual demand, so the number of orders is calculated as “Total annual demand (D) ÷ Order quantity (Q)”. Ordering cost = (D ÷ Q) × H 3) The total inventory cost can then be calculated as “Storage cost + Ordering cost” as follows: Q D Total inventory cost = × P + × H 2 Q

Storage cost = (Q ÷ 2) × P 2) Since the ordering cost is determined by the number of orders, first calculate the annual number of orders. It is necessary to order enough to meet the total annual demand, so the number of orders is calculated as “Total annual demand (D) ÷ Order quantity (Q)”. Ordering cost = (D ÷ Q) × H 3) The total inventory cost can then be calculated as “Storage cost + Ordering cost” as follows: Q D Total inventory cost = × P + × H 2 Q

Original inventory-cost graph. The horizontal axis is Order quantity (Q), with Economic order quantity (Q) marked. Curves are Total inventory cost, Storage cost and Ordering cost; a dashed vertical line and horizontal guide identify the Minimum value of total inventory cost.

Original economic-order-quantity derivation box, retaining Storage cost = Ordering cost, the source equality using Q, D, P and H, and the final square-root expression with its original numerator and denominator.

Original source visual. Selectable source labels and wording: • Periodic ordering method With this ordering method, the ordering interval is set in advance and demand is forecast for each order to calculate the required order quantity. This ordering method is most suitable for high-ranking (Rank A) products that have a high unit price, make up a large portion of overall sales, and require careful management. Order quantity = Projected demand for ordering interval + Projected demand for procurement period − Inventory quantity at time of order + Safety stock quantity • Fixed quantity ordering method (Order point method) With this ordering method, the order quantity is fixed, but the ordering interval is not set. When the inventory falls below the order point, an order is placed for EOQ (i.e., the quantity that minimizes total inventory cost). This ordering method is most suitable for mid-ranking (Rank B) products that have a low unit price and make up a medium portion of overall sales. Order point = Projected demand for procurement period + Safety stock quantity

• Periodic ordering method With this ordering method, the ordering interval is set in advance and demand is forecast for each order to calculate the required order quantity. This ordering method is most suitable for high-ranking (Rank A) products that have a high unit price, make up a large portion of overall sales, and require careful management. Order quantity = Projected demand for ordering interval + Projected demand for procurement period − Inventory quantity at time of order + Safety stock quantity • Fixed quantity ordering method (Order point method) With this ordering method, the order quantity is fixed, but the ordering interval is not set. When the inventory falls below the order point, an order is placed for EOQ (i.e., the quantity that minimizes total inventory cost). This ordering method is most suitable for mid-ranking (Rank B) products that have a low unit price and make up a medium portion of overall sales. Order point = Projected demand for procurement period + Safety stock quantity

Original source visual. Selectable source labels and wording: • Two-bin method With this ordering method, inventory is kept on two shelves. When one shelf goes out of stock, an order is placed for the quantity on that shelf while the inventory from the other shelf is being used. This method is generally most suitable for inexpensive products that are consumed in large quantity.

• Two-bin method With this ordering method, inventory is kept on two shelves. When one shelf goes out of stock, an order is placed for the quantity on that shelf while the inventory from the other shelf is being used. This method is generally most suitable for inexpensive products that are consumed in large quantity.

Original source visual. Selectable source labels and wording: Forecasting by using past trends (1) This concept is to predict future demand by analyzing trends in past data. • Time series analysis This is an analytical method that identifies specific trends from past time series data. A variation that indicates a long-term increase Trend variations (or decline) trend A variation that indicates a periodic trend of Cyclical variations several years to a decade or so A variation because of natural conditions or Seasonal variations social customs An accidental variation because of the disaster Irregular variations or other unpredictable factors • Correlation analysis/Regression analysis

Forecasting by using past trends (1) This concept is to predict future demand by analyzing trends in past data. • Time series analysis This is an analytical method that identifies specific trends from past time series data. A variation that indicates a long-term increase Trend variations (or decline) trend A variation that indicates a periodic trend of Cyclical variations several years to a decade or so A variation because of natural conditions or Seasonal variations social customs An accidental variation because of the disaster Irregular variations or other unpredictable factors • Correlation analysis/Regression analysis

Original source visual. Selectable source labels and wording: • Least squares method This is a method that mathematically calculates trend lines such that the error between values on the graphed trend lines (i.e., predicted values) and actual values is minimized. It is used in cases such as finding regression lines. • Moving average method This is a method that performs forecasting by calculating the mean of partial time series with moving along the time axis. It levels variations in data in order to make trends easy to spot. • Exponential smoothing This is a method that calculates current period predicted values through the weighted average of the previous period actual value and previous period predicted value that are weighted by an exponential smoothing constant (α), as shown in the following expression. Current period predicted value = α × Previous period actual value + (1 − α) × Previous period predicted value

• Least squares method This is a method that mathematically calculates trend lines such that the error between values on the graphed trend lines (i.e., predicted values) and actual values is minimized. It is used in cases such as finding regression lines. • Moving average method This is a method that performs forecasting by calculating the mean of partial time series with moving along the time axis. It levels variations in data in order to make trends easy to spot. • Exponential smoothing This is a method that calculates current period predicted values through the weighted average of the previous period actual value and previous period predicted value that are weighted by an exponential smoothing constant (α), as shown in the following expression. Current period predicted value = α × Previous period actual value + (1 − α) × Previous period predicted value

Original Original source table, printed page 68, region 1. Original selectable labels and wording: Strategy of B Strategy of A B1 B2 A1 +10 −10 A2 −20 +30

Strategy of B B1 B2 Strategy of A A1 +10 −10 A2 −20 +30

Original source visual. Selectable source labels and wording: • Expectation principle This is a concept that calculates expected values for the case of taking each strategy from the probabilities that future conditions will occur, and sees the strategy for which the expected value is highest as the optimal strategy. Example: Calculate the expected value for each strategy. Expected value for strategy S1 = (+80) × 0.3 + (+20) × 0.5 + (−40) × 0.2 = +26 Expected value for strategy S2 = (+60) × 0.3 + (+10) × 0.5 + (−10) × 0.2 = +21 Expected value for strategy S3 = (+40) × 0.3 + (+30) × 0.5 + (−20) × 0.2 = +23 => The strategy S1 which offers the highest expected value is seen as the optimal strategy. • Most probable future principle This is a concept that sees the strategy, which obtains the maximum benefit under the conditions of the highest probability of occurrence from among the future conditions, as the optimal strategy. Example: The cloudy (50%) has the condition with the highest probability of occurrence in the future. Thus, strategy S3, which offers the maximum benefit (+30) when the weather is cloudy, is seen as the optimal strategy. • Aspiration level principle This is a concept that selects the optimal strategy (e.g., the strategy which yields the maximum benefit) from among strategies that meet the intention (i.e., criteria) of the decision-maker. Example: If the decision-maker has set criteria of “Select a strategy with the possibility of yielding the maximum profit, from among strategies with a maximum loss of not more than 20,” then between strategies S2 and S3 (i.e., both of which yield a maximum loss of not more than 20), strategy S2 with the possibility of a maximum profit of +60 is the optimal strategy.

• Expectation principle This is a concept that calculates expected values for the case of taking each strategy from the probabilities that future conditions will occur, and sees the strategy for which the expected value is highest as the optimal strategy. Example: Calculate the expected value for each strategy. Expected value for strategy S1 = (+80) × 0.3 + (+20) × 0.5 + (−40) × 0.2 = +26 Expected value for strategy S2 = (+60) × 0.3 + (+10) × 0.5 + (−10) × 0.2 = +21 Expected value for strategy S3 = (+40) × 0.3 + (+30) × 0.5 + (−20) × 0.2 = +23 => The strategy S1 which offers the highest expected value is seen as the optimal strategy. • Most probable future principle This is a concept that sees the strategy, which obtains the maximum benefit under the conditions of the highest probability of occurrence from among the future conditions, as the optimal strategy. Example: The cloudy (50%) has the condition with the highest probability of occurrence in the future. Thus, strategy S3, which offers the maximum benefit (+30) when the weather is cloudy, is seen as the optimal strategy. • Aspiration level principle This is a concept that selects the optimal strategy (e.g., the strategy which yields the maximum benefit) from among strategies that meet the intention (i.e., criteria) of the decision-maker. Example: If the decision-maker has set criteria of “Select a strategy with the possibility of yielding the maximum profit, from among strategies with a maximum loss of not more than 20,” then between strategies S2 and S3 (i.e., both of which yield a maximum loss of not more than 20), strategy S2 with the possibility of a maximum profit of +60 is the optimal strategy.

Original decision tree for the probable-future example. Branches retain strategies S1, S2 and S3, weather Sunny, Cloudy and Rainy, original probabilities, outcome amounts and the source decision/chance node symbols.

Original source visual. Selectable source labels and wording: • Laplace’s principle This is a concept that assumes that each condition has the same probability of occurrence, and calculates the optimal strategy on the basis of the expectation principle. Example: Assume that company B has equal probabilities of selecting strategy B1 and B2 to calculate the expected value. Strategy A1 = (+10) × 0.5 + (−10) × 0.5 = ±0 Strategy A2 = (−20) × 0.5 + (+30) × 0.5 = +5 => The strategy A2 with the maximum expected value is seen as the optimal strategy. • Maxi-max principle (or Mini-min principle) This is a concept that sees the strategy which, from among the best case (i.e., the maximum benefit) for each strategy, obtains the greatest benefit, as the optimal strategy. Example: Find the maximum benefit for each strategy. Strategy A1: When company B selects strategy B1, maximum benefit is +10 Strategy A2: When company B selects strategy B2, maximum benefit is +30 => The strategy A2 with the largest maximum value is seen as the optimal strategy. • Maxi-min principle (or Mini-max principle) This is a concept that determines the strategy which, from among the worst case (i.e., the minimum benefit) for each strategy, obtains the greatest benefit, as the optimal strategy. Example: Find the minimum benefit for each strategy. Strategy A1: When company B selects strategy B2, minimum benefit is −10 Strategy A2: When company B selects strategy B1, minimum benefit is −20

• Laplace’s principle This is a concept that assumes that each condition has the same probability of occurrence, and calculates the optimal strategy on the basis of the expectation principle. Example: Assume that company B has equal probabilities of selecting strategy B1 and B2 to calculate the expected value. Strategy A1 = (+10) × 0.5 + (−10) × 0.5 = ±0 Strategy A2 = (−20) × 0.5 + (+30) × 0.5 = +5 => The strategy A2 with the maximum expected value is seen as the optimal strategy. • Maxi-max principle (or Mini-min principle) This is a concept that sees the strategy which, from among the best case (i.e., the maximum benefit) for each strategy, obtains the greatest benefit, as the optimal strategy. Example: Find the maximum benefit for each strategy. Strategy A1: When company B selects strategy B1, maximum benefit is +10 Strategy A2: When company B selects strategy B2, maximum benefit is +30 => The strategy A2 with the largest maximum value is seen as the optimal strategy. • Maxi-min principle (or Mini-max principle) This is a concept that determines the strategy which, from among the worst case (i.e., the minimum benefit) for each strategy, obtains the greatest benefit, as the optimal strategy. Example: Find the minimum benefit for each strategy. Strategy A1: When company B selects strategy B2, minimum benefit is −10 Strategy A2: When company B selects strategy B1, minimum benefit is −20

Original source visual. Selectable source labels and wording: => The strategy A1 with the greatest minimum value is seen as the optimal strategy. • Mini-max regret principle This is a concept that sees, for each condition, the strategy with the smallest maximum value of the difference (i.e., regret) between the benefit that is predicted when the best-case strategy is selected, and the benefit of the strategy that is actually adopted. Example: Find the maximum value of the regret for each strategy. Strategy A1: When company B selects strategy B2, maximum regret is 40 Strategy A2: When company B selects strategy B1, maximum regret is 30 => The strategy A2 with the smallest maximum regret is seen as the optimal strategy.

=> The strategy A1 with the greatest minimum value is seen as the optimal strategy. • Mini-max regret principle This is a concept that sees, for each condition, the strategy with the smallest maximum value of the difference (i.e., regret) between the benefit that is predicted when the best-case strategy is selected, and the benefit of the strategy that is actually adopted. Example: Find the maximum value of the regret for each strategy. Strategy A1: When company B selects strategy B2, maximum regret is 40 Strategy A2: When company B selects strategy B1, maximum regret is 30 => The strategy A2 with the smallest maximum regret is seen as the optimal strategy.

Original Original source table, printed page 71, region 2. Original selectable labels and wording: Strategy of B Strategy of A B1 B2 A1 +20 +10 A2 +10 −10

Strategy of B B1 B2 Strategy of A A1 +20 +10 A2 +10 −10

Original shortest-path grid with A at the lower left, B at the upper right and intermediate X. All lattice nodes, paths and the marked A-X-B points are preserved.

Original paired shortest-path solutions. The divide-and-conquer solution retains the two combination calculations, 6 paths and 4 paths, and their product 24. The dynamic-programming solution retains the numbered incremental grids and every original node count.

[Solution using the divide-and-conquer approach] Divide into the shortest paths from A to X, and the shortest paths from X to B. 1) Calculate the number of shortest paths from A to X. Shortest paths are the number of combinations of vertical 2, horizontal 2: 4C2 = 6 paths 2) Calculate the number of shortest paths from X to B. Shortest paths are the number of combinations of vertical 1, horizontal 3: 4C1 = 4 paths. 3) Calculate the number of shortest paths from A to B via X. 4C2 × 4C1 = 6 × 4 = 24 paths [Solution using the dynamic programming] Fix the shortest paths from A, in order. (Each number indicates the number of paths.)