OR확정모델2002-중간고사해답.doc - OR 확정모델 2002(중간고사 학부 학번 이름 2002 5.1(수 < 만점 140 점 전체 6 문제 총 6

OR확정모델2002-중간고사해답.doc - OR 확정모델...

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OR 확정모델 2002 ( 중간고사 ) 2002. 5.1.( ) ( ) 학부 , 학번 ( ) 이름 ( ) < 만점 140 , 전체 6 문제 , 6 페이지 , 제한시간 10 0 , 지면이 부족하면 뒷면을 사용 > 1. (25 ) Formulate the following “Cyclic Scheduling” problem as a linear program. A bakery has a known demand for its cakes and pies. This demand varies from day to day during the week but does not vary from week to week. Also the number of workers employed varies by the day of the week but is constant from week to week. Suppose that there are w j workers employed and c j orders for cakes and p j orders for pies on Monday ( j =1) through Saturday ( j =6). The bakery is closed on Sunday. Cakes that are not used to satisfy orders on the day that they are baked can be used to satisfy the next day’s orders, i.e. , they can be stored for one day. Pies can be used to satisfy demands on the day that they are baked, or for satisfying orders during the next two days, i.e ., they can be stored for two days. In particular, pies, but not cakes, baked on Saturday can be used on Monday. Suppose that each worker can prepare k pies or cakes per day and that the total workforce is just sufficient to prepare all cakes and pies ordered during the week., i.e., 6 1 6 1 ) ( j j j j j w k p c . How should the workers be assigned to tasks so as to minimize total inventory, i.e., the number of pies and cakes stored, summed over all seven days of the week? <Solution> x 0j = # cakes produced in day j to satisfy today’s orders x 1j = # cakes produced in day j to satisfy next day’s orders y 0j = # pies produced in day j to satisfy today’s orders y 1j = # pies produced in day j to satisfy next day’s orders y 2j = # pies produced in day j to satisfy the day after next day’s orders w j ’ = # workers in day j assigned to cakes w j ’’ = # workers in day j assigned to pies
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Min 6 1 j ( x 1j + y 1j + 2 y 2j ) s.t workers x 0j + x 1j = kw j y 0j + y 1j + y 2j = k w j ’’ w j ’ + w j ’’ = w j cakes x 01 = c 1 x 0j + x 1j-1 = c j j = 2,…,6 pies y 01 + y 26 = p 1 y 02 + y 12 = p 2 y 0j + y 1j-1 + y 2j-2 = p j j = 3,…,6 all variables 0
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2. (25 ) 다음 LP 문제에 대하여 Simplex 법을 이용하여 해를 구하시오 . Max 3 2 1 2 x x x Z s.t . 4 3 2 3 2 1 x x x 10 2 2 1 x x 7 3 2 1 x x x 0 , 0 , 0 3 2 1 x x x <Solution> Max Z = 2x1 – x2 + x3 s.t x 1 – 2 x 2 + 3 x 3 4 2 x 1 + x 2 = 10 x 1 x 2 x 3 7 x j 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 RHS –2 1
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