規劃求解 Linear Programming: https://youtu.be/eiv5AfVLfTk?list=PLQutejmQFO78NDjx56gJjBmGgPxQt4Gq7 Q0 demo 一個測驗分為甲、乙兩部，各有10題。 甲部：答對1題得7分，每題需時6分鐘； 乙部：答對1題得8分，每題需時7分鐘； 考生在甲乙兩部，必須各答4題或以上， 測驗時限為90分鐘。求最高分數多少？ You are about to take a test which contains two sections, A and B. There are 10 questions in each section. The marks and time required for each question in the two sections are given in the following table: You must attempt at least 4 questions from each section and the maximum time available for the test is 90 minutes. In order to maximize your score, how many questions in each section should you do? What is the maximum score? Q1 某團體利用飛機把衣服、藥物送到地震災區。 (1) 衣服放箱、藥物放盒 (2) 每個箱和盒皆佔0.9立方米 (3) 一箱衣服重27kg、一盒藥物重54kg 飛機可載重21,600 kg, 容量540立方米 一箱衣服可幫助12家庭、一盒藥物可幫助18家庭 求最多可幫助多少個家庭 Clothes and medicines are to be sent to victims of an earthquake in a certain country. (1) The clothes are put in boxes whereas the medicines are stored in containers. (2) Each box and each container occupy the same space of 0.9 cubic meters. (3) Each box of clothes weighs 27 kg whereas each container of medicines weighs 54 kg. An airplane is going to take these supplies to the country. The airplane can carry at most 21,600 kg of the supplies and has a storage space of 540 cubic meters. It is supposed that x boxes of clothes and y containers of medicines are taken. A box of clothes can help 12 families while a container of medicines can help 18 families. Assuming that each family can get either clothes or medicine, find the number of boxes of clothes and the number of containers of medicines that should be sent in order to maximize the number of families that can be helped. Q2 售價：單層巴士$20; 雙層巴士$45 用最多$400買不多於10輛巴士 求各買巴士多少? 最多支出多少? In a toy shop, there are 2 types of toy buses: 'single-deck' and 'double-deck'. The selling prices for the two types of buses are $20 each and $45 each respectively. A boy wants to buy x 'single-deck' and y 'double-deck' toy buses. If the total number of toy buses bought should not be greater than 10 and the boy has $400 to spend, how much money he will spend at most? Q3 游泳池可容納最多250成人及250小童，且不可多於450人。 入場費：成人$20、小童$15。每兩個小童必須有一個成人陪同。 求入場費的最大值。 A swimming pool in a certain estate can accommodate at most 250 adults and 250 children. There should be no more than 450 people in the pool. The admission fee is $20 for each adult and $15 for each child. It is required that for every two children admitted there is at least one adult. It is supposed that x adults and y children are admitted to the swimming pool. Find the total amount of admission fee. Q4 製餅師傅打算按以下條件，製造x kg蛋糕及y kg餅乾： (1) 造1kg蛋糕：成本$60; 造1kg餅乾：成本$30 製餅師傅最多花$1,200 (2) 製成的蛋糕、餅乾不可超過30kg (3) 利潤為：蛋糕$15/kg、餅乾$7/kg (4) 最少要造5kg蛋糕及5kg餅乾 求最大利潤。 A baker wants to bake x kg of cakes and y kg of biscuits subject to the following conditions: (1) The cost of baking 1 kg of cakes is $60 and the cost of baking 1 kg of biscuits is $30. The baker decides to spend at most $1,200. (2) The total weight of cakes and biscuits made should not exceed 30 kg. (3) The profits of selling cakes and biscuits are $15/kg and $7/kg respectively. (4) At least 5 kg of cakes and 5 kg or biscuits should be made. Find the maximum profit. Q5 把350個蘋果放進x個盒及y個籃內。 一個盒可放25個蘋果、而一個籃可放10個蘋果。 盒的數目不可多於籃的數目 成本：盒$12;籃$6 求x和y (最低成本多少?) 350 apples are packed in x boxes and y baskets. a box can hold 25 apples while a basket can hold 10 apples. It is known that the number of boxes is not greater than the number of baskets. The cost of a box is $12 and that of a basket is $6. How many boxes and how many baskets should be used to keep the cost to a minimum? Q6 醫院為每個病人提供的食物須符合以下要求: 碳水化合物: 至少120單位 蛋白質: 至少40單位 纖維: 至少30單位 食物 碳水化合物 蛋白質 纖維 成本 $/kg P 120 units 20 units 10 units $60 Q 40 units 20 units 30 units $30 求最低成本多少? A hospital has to provide a daily diet of at least 120 units of carbohydrate, 40 units of protein and 30 units of fibre for each patient. Two kinds of food, P and Q, are available and their contents per kg are shown in the table below. Food Carbo-hydrate Protein Fibre Unit cost $/kg P 120 units 20 units 10 units $60 Q 40 units 20 units 30 units $30 Food P costs $60/kg and food Q costs $30/kg. Now x kg of food P and y kg of food Q are provided for each patient daily. Find the values of x and y so as to minimize the cost of the diet. What is the minimum cost? Q7 生產商每天生產x個雪櫃及y座電視。 (1) 8<=x<=14 (2) 5<=y<=10 (3) x+y不超過20 利潤為：雪櫃$1600及電視$2400 求最大利潤 An electrical appliance manufacturer decides to produce x refrigerators and y TV sets each day subject to the following conditions: (1) x should be at least 8 and at most 14. (2) y should be at least 5 and at most 10. (3) The total number of electrical appliances produced each day should not exceed 20. The manufacturer has a profit of $1600 from each refrigerator sold and $2400 from each TV set sold. Find out the values of x and y so as to maximize the total profit and find the corresponding profit. Q8 產品A,B,C是由成份M和成份N製成 (1) 每噸M可造：4000件A, 20000件B, 6000件C (2) 每噸N可造：6000件A, 5000件B, 3000件C 工廠接獲訂單：24,000件A, 60,000件B, 24,000件C 成本：M(每噸$4,000); N(每噸$3,000) 要完成訂單，最低成本多少? A factory produces 3 products A, B and C from 2 material M and N. (1) Each tonne of M produces 4,000 pieces of A, 20,000 pieces of B and 6,000 pieces of C. (2) Each tonne of N produces 6,000 pieces of A, 5,000 pieces of B and 3,000 pieces of C. The factory has received an order for 24,000 pieces of A, 60,000 pieces of B and 24,000 pieces of C. The costs of M and N are respectively $4,000 and $3,000 per tonne. Determine the least cost of the materials used so as to meet the order. Q9 農夫想租田種生果及稻米。 租金：生果田(每平方米$300)及稻米田(每平方米$100) 地主要求：每租一平方米生果田，必須至少租三平方米稻米田 利潤：種生果$600/平方米; 種稻米$150/平方米 若租金不超過$9000，求最大利潤 A farmer wishes to rent fields to grow fruit and rice. The costs per sq meters for renting fruit-field and rice-field are $300 and $100 respectively. The farmer has $9,000 to spend on rent. The landlord requires that for every sq meter of fruit-field the farmer rents, he must rent at least 3 sq meters of rice-field. The farmer estimates that the profit per sq meter from the fruit-field and the rice-field will be $600 and $150 respectively. Find the areas of fruit-field and rice-field in order to maximize the profit. Q10 超級市場售賣橙汁及蘋果汁。 超級市場經理訂了橙汁的數量，至少是蘋果汁數量的兩倍。 超級市場只可容納450瓶果汁 利潤：橙汁$2.5; 蘋果汁$3.0 求最大利潤 A supermarket sells two kinds of juice: orange juice and apple juice. To meet the taste of the customers, the manager orders a number of bottles of orange juice which is at least twice the number of bottles of apple juice. The storage space available can keep at most 450 bottles of juice. The profits on selling a bottle of orange juice and on selling a bottle of apple juice are $2.5 and $3 respectively. How many bottles of orange juice and how many bottles of apple juice should be sold daily when the profit attains a maximum? What is the maximum daily profit on selling the juice? Q11 機器A: 每小時可生產120 nuts及100 bolts 運作成本：每小時$150 機器B: 每小時可生產 60 nuts及150 bolts 運作成本：每小時$180 若要完成至少1200 nuts及1500 bolts 機器A及B各需要運作多少小時? 最低成本多少? A factory uses machines A and B to produce nuts and bolts. In an hour's operation, machine A can produce 120 nuts and 100 bolts while machine B can produce 60 nuts and 150 bolts. Suppose that machine A and B are operated for x hours and y hours respectively so as to produce at least 1200 buts and 1500 bolts.(Note: x and y are real numbers) The hourly costs for operating machine A and machine B are $150 and $180 respectively. How long should each machine be operated in order to minimize the cost? What is the minimum cost? Q12 計劃中的新酒店有單人房及雙人房若干。 (1) 每4間雙人房，必須有1間單人房 (2) 房間總數最多200間 (3) 單人房至少20間；雙人房最多120間 每日租金：單人房$800；雙人房$1000 求單人房及雙人房各佔多少間，每日最多可收取租金多少? A company plans to build a hotel with x single rooms and y double rooms satisfying the following conditions: (1) For every four double rooms, there is at least one single room. (2) The total number of rooms is at most 200. (3) There are at least 20 single rooms and at most 120 double rooms. If the daily rent of a single room is $800 while that of a double room is $1000, find the number of single rooms and the number of double rooms that should be built in the hotel to maximize the total daily rent. What is the total daily rent? Answers: Q1. 400 200 Q2. 2 8 Q3. 250 200 Q4. 17 6 Q5. 10 10 Q6. 0.5 1.5 Q7. 10 10 Q8. 3 2 Q9. 15 45 Q10. 300 150 Q11. 7.5 5 Q12. 30 120