‫اﻷوﻟﻰ‬ ‫اﻟﻌﻣﻠﯾﺔ‬ ‫اﻟﺟﻠﺳﺔ‬
‫أﻟﻔﯾن‬ ‫د.ﻓرﺣﺎن‬ :‫إﺷراف‬‫اﻟﻌﺎﺑدﯾن‬ ‫زﯾن‬ ‫ﻧور‬ .‫م‬
‫اﻟﺧطﯾﺔ‬ ‫اﻟﺑرﻣﺟﺔ‬ ‫ﺑﺎﺳﺗﺧدام‬ ‫اﻟﻌﻣﻠﯾﺔ‬ ‫وأﻣﺛﻠﯾﺔ‬ ‫اﻟﻣﻧﺗﺞ‬ ‫ﺗرﻛﯾب‬
Product Formulation and Process Optimization Using
Linear Programming
‫ﻋﻠ‬ ‫ﯾﺠﺐ‬ .‫ﺑﻌﻀﮭﺎ‬ ‫ﻣﻊ‬ ‫ﺗﻤﺰج‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻜﻮﻧﺎت‬ ‫ﻣﻦ‬ ‫ﻣﺰﯾﺞ‬ ‫ﻣﻦ‬ ‫اﻷﻏﺬﯾﺔ‬ ‫ﻣﻌﻈﻢ‬ ‫ﺗﺘﺮﻛﺐ‬‫ﺗﺤﻘﯿﻖ‬ ‫اﻟﻤﻨﺘﺞ‬ ‫ﻰ‬
‫ﻧﺴﺐ‬ ‫ﻣﻦ‬ ‫ﻛﺜﯿﺮ‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺘﺤﻘﻖ‬ ‫ﯾﻤﻜﻨﻚ‬ .‫اﻟﻤﻜﻮﻧﺎت‬ ‫وﺑﻘﯿﺔ‬ ‫واﻟﻤﺎء‬ ‫واﻟﺪﺳﻢ‬ ‫اﻟﺒﺮوﺗﯿﻦ‬ ‫ﻣﺤﺘﻮى‬ ‫ﻓﻲ‬ ‫ﺷﺮوط‬
.‫ﺗﻜﻠﻔﺔ‬ ‫أﻗﻞ‬ ‫ﺗﺤﻘﻖ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻨﺴﺐ‬ ‫ﻓﻲ‬ ‫ﺗﺮﻏﺐ‬ ‫ﻗﺪ‬ ‫وﻟﻜﻦ‬ .‫اﻟﺸﺮوط‬ ‫ھﺬه‬ ‫ﺗﺤﻘﻖ‬ ‫اﻟﺨﻠﻄﺎت‬
:‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻋﻦ‬ ‫ﻣﺜﺎل‬
.‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻓﻲ‬ ‫ﻣﻘﺪﻣﺔ‬ ‫ﯾﻌﺘﺒﺮ‬ ‫ﻣﺜﺎل‬ ‫ﯾﻠﻲ‬ ‫ﻓﯿﻤﺎ‬
:‫اﻟﻣﺳﺄﻟﺔ‬
‫اﻟ‬‫ﺗﺤ‬ ‫ﻤﻄﻠﻮب‬‫ﻏﺬاﺋﻲ‬ ‫ﺧﺒﺰ‬ ‫ﻣﻜﻮﻧﺎت‬ ‫ﻧﺴﺐ‬ ‫ﺪﯾﺪ‬‫اﻟﺼﻮﯾﺎ‬ ‫وﻓﻮل‬ ‫اﻟﻘﻤﺢ‬ ‫ﻣﻦ‬ ‫ﯾﺘﺄﻟﻒ‬‫اﻟﺪﺳﻢ‬ ‫ﻣﻨﺰوع‬‫ﺣﯿﺚ‬
‫ﻋﻠﻰ‬ ‫اﻟﺨﺒﺰ‬ ‫ﯾﺤﺘﻮي‬ ‫أن‬ ‫ﯾﺠﺐ‬‫اﻷﻗﻞ‬20‫اﻷﻛﺜﺮ‬ ‫وﻋﻠﻰ‬ ‫ﺑﺮوﺗﯿﻦ‬ %55‫دﻗﯿﻖ‬ ‫وﯾﺘﻜﻮن‬ .‫ﻛﺮﺑﻮھﯿﺪات‬ %
‫ﻣﻦ‬ ‫اﻟﻘﻤﺢ‬11‫و‬ ‫ﺑﺮوﺗﯿﻦ‬ %70%‫ﻣﻦ‬ ‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫ودﻗﯿﻖ‬ ‫ﻛﺮﺑﻮھﯿﺪات‬47‫و‬ ‫ﺑﺮوﺗﯿﻦ‬ %40%
. ‫ﻛﺮﺑﻮھﯿﺪات‬
‫أن‬ ‫ﻋﻠﻤﺎ‬‫ھﻮ‬ ‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﺳﻌﺮ‬20‫ھﻮ‬ ‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫ودﻗﯿﻖ‬ ‫/ﻛﻎ‬ ‫س‬ ‫ل‬30.‫س/ﻛﻎ‬ ‫ل‬
1-‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻣﺘﻐﯿﺮات‬Problem function‫ھﻤﺎ‬ ‫ﻣﺘﻐﯿﺮﯾﻦ‬ ‫ﺑﺘﺤﺪﯾﺪ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ھﺬه‬ ‫ﺗﺤﻞ‬ :
W‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﻧﺴﺒﺔ‬ :
S‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫دﻗﯿﻖ‬ ‫ﻧﺴﺒﺔ‬ :
2-‫اﻟﮭﺪف‬The objective function‫ﻟﻜﻞ‬ ‫اﻟﻜﯿﻠﻮﻏﺮام‬ ‫ﺳﻌﺮ‬ ‫أن‬ ‫ﺑﻤﺎ‬ :‫ودﻗﯿﻖ‬ ‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﻣﻦ‬
‫ھﻲ‬ ‫اﻟﺼﻮﯾﺎ‬20‫و‬ ‫س‬ ‫ل‬30:‫ھﻲ‬ ‫اﻟﺨﻠﻄﺔ‬ ‫ﺳﻌﺮ‬ ‫ﻓﺈن‬ ‫اﻟﺘﺮﺗﯿﺐ‬ ‫ﻋﻠﻰ‬ ‫س‬ ‫ل‬
C=20 W + 30 S)1-1(
‫ﻗﯿﻤﺔ‬ ‫إﯾﺠﺎد‬ ‫ھﻮ‬ ‫وھﺪﻓﻚ‬W‫و‬S‫اﻟﺘﻜﻠﻔﺔ‬ ‫ﺗﺠﻌﻞ‬ ‫اﻟﺘﻲ‬C‫وﺿﻌﮭﺎ‬ ‫ﯾﺠﺐ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ .‫اﻷدﻧﻰ‬ ‫ﺣﺪھﺎ‬ ‫ﻓﻲ‬
.‫اﻟﻤﺴﺄﻟﺔ‬ ‫ھﺪف‬ ‫ﺗﺎﺑﻊ‬ ‫ﺗﺪﻋﻰ‬ ‫اﻷدﻧﻰ‬ ‫أو‬ ‫اﻷﻋﻈﻤﻲ‬ ‫ﺣﺪھﺎ‬ ‫ﻓﻲ‬

3-‫اﻟﺴﻠﺒﯿﺔ‬ ‫ﻏﯿﺮ‬ ‫اﻟﻤﺤﺪدات‬Nonnegative Constraints‫ﻣﻦ‬ ‫ﻟﯿﺲ‬ ‫أﻧﮫ‬ ‫ﺑﻤﺎ‬ :‫ﺗﻜﻮن‬ ‫أن‬ ‫اﻟﻤﻨﻄﻖ‬
:‫ﺑﺎﻟﻘﯿﻢ‬ ‫ﻣﺤﺪد‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺤﻞ‬ ‫ﻓﺈن‬ ‫ﺳﺎﻟﺒﺔ‬ ‫ﻗﯿﻤﺔ‬ ‫اﻟﻤﻜﻮﻧﺎت‬ ‫ﻷﺣﺪ‬
W ≥ 0 S ≥ 0)1-2(
4-‫اﻹﺟﻤﺎﻟﻲ‬ ‫اﻟﻮزن‬ ‫ﻣﺤﺪد‬Combined weight constraint‫ﻣﻜﻮن‬ ‫ﻣﻦ‬ ‫ﺗﺘﻜﻮن‬ ‫اﻟﺘﺮﻛﯿﺒﺔ‬ ‫أن‬ ‫ﺑﻤﺎ‬ :
‫ﻟـ‬ ‫ﻣﺴﺎو‬ ‫اﻟﻨﺴﺐ‬ ‫ﻣﺠﻤﻮع‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫ﻓﺈﻧﮫ‬100:‫أي‬
W + S = 100)1-3(
5-‫ا‬ ‫اﻟﻤﺤﺪدات‬‫ﻷﺧﺮى‬Other Constraints‫اﻟﻤﺰﯾﺞ‬ ‫ﺑﺮوﺗﯿﻦ‬ ‫ﻧﺴﺒﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ :20‫وﺑﻤﺎ‬ %
‫ا‬ ‫دﻗﯿﻖ‬ ‫ﺑﺮوﺗﯿﻦ‬ ‫ﻧﺴﺒﺔ‬ ‫أن‬‫ﻟﻘﻤﺢ‬11‫اﻟﺼﻮﯾﺎ‬ ‫دﻗﯿﻖ‬ ‫وﺑﺮوﺗﯿﻦ‬ %47‫ﻓﺈﻧﮫ‬ %
= ‫اﻟﺒﺮوﺗﯿﻦ‬0.11 W + 0.47 S≤20)1-4(
‫ﻣﺸﺎﺑﮫ‬ ‫وﺑﺸﻜﻞ‬
= ‫ﻛﺮﺑﻮھﯿﺪات‬0.70 W + 0.40 S≥55)1-5(
6-‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺻﯿﺎﻏﺔ‬Problem Statement‫ﯾﻤﻜﻦ‬ :‫ﻟﻠﻤﺤﺪدات‬ ‫وﻓﻘﺎ‬ ً‫ﺎ‬‫رﯾﺎﺿﯿ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻋﻦ‬ ‫اﻟﺘﻌﺒﯿﺮ‬
‫ﻗﯿﻢ‬ ‫أوﺟﺪ‬ :‫ﯾﻠﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﮭﺪف‬ ‫وﺗﺎﺑﻊ‬W ≥ 0‫و‬S ≥ 0:‫ﺣﯿﺚ‬
= ‫اﻟﺒﺮوﺗﯿﻦ‬0.11 W + 0.47 S≤20
= ‫ﻛﺮﺑﻮھﯿﺪات‬0.70 W + 0.40 S≥55
W + S = 100
:‫اﻷدﻧﻰ‬ ‫ﺣﺪھﺎ‬ ‫ﻓﻲ‬ ‫اﻟﺘﻜﻠﻔﺔ‬ ‫ﺗﺠﻌﻞ‬ ‫ﺑﺤﯿﺚ‬
C=20 W + 30 S
‫اﻟﺑﯾﺎﻧﻲ‬ ‫اﻟﺣل‬Graphic Solution:
‫ﻣ‬ ‫ﺣﻞ‬ ‫ﯾﻤﻜﻦ‬‫ﯾﻠﻲ‬ ‫ﻓﯿﻤﺎ‬ .ً‫ﺎ‬‫ﺑﯿﺎﻧﯿ‬ ‫ﻓﻘﻂ‬ ‫ﻣﺘﻐﯿﺮﯾﻦ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺤﺘﻮي‬ ‫اﻟﺘﻲ‬ ‫اﻟﺒﺴﯿﻄﺔ‬ ‫ﻟﻠﻤﺴﺎﺋﻞ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﺴﺎﺋﻞ‬
:‫ذﻟﻚ‬ ‫ﯾﺘﻢ‬ ‫ﻛﯿﻒ‬ ‫ﻧﺸﺮح‬
1-‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺳﺎﺣﺔ‬The problem space‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻣﺘﻐﯿﺮي‬ ‫ﺗﻤﺜﻞ‬ ‫ﻣﺤﺎور‬ ‫ﺑﺮﺳﻢ‬ ‫ﻧﺒﺪأ‬ :
‫اﻟﺸﻜﻞ‬1.1.‫ﻣﻌﯿﻨﺔ‬ ‫ﻟﺘﺮﻛﯿﺒﺔ‬ ‫اﻟﻤﻜﻮﻧﺎت‬ ‫أوزان‬ ‫اﻹﺣﺪاﺛﯿﺎت‬ ‫ھﺬه‬ ‫ﻓﻲ‬ ‫ﻧﻘﻄﺔ‬ ‫أﯾﺔ‬ ‫ﺗﻤﺜﻞ‬ .
2-‫اﻟ‬ ‫اﻟﺤﻠﻮل‬‫ﻤﻤﻜﻨﺔ‬potential solution‫ھﻤﺎ‬ ‫ﻟﻤﺘﻐﯿﺮان‬ ‫ﻗﯿﻤﺘﯿﻦ‬ ‫ھﻲ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ھﺬه‬ ‫ﺣﻞ‬ :W‫و‬
S‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﻤﻜﻦ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺣﻞ‬ ‫ﻟﺬﻟﻚ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺳﺎﺣﺔ‬ ‫ﻓﻲ‬ ‫ﻧﻘﻄﺔ‬ ‫إﺣﺪاﺛﯿﺎت‬ ‫ﺗﻤﺜﻞ‬ ‫اﻟﻜﻤﯿﺎت‬ ‫وھﺬه‬
‫اﻟﺸﻜﻞ‬ ‫ﯾﺒﯿﻦ‬ .‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺳﺎﺣﺔ‬ ‫ﻓﻲ‬ ‫ﻧﻘﻄﺔ‬ ‫أي‬1.1‫وﻟﺬﻟﻚ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻟﮭﺬه‬ ‫اﻟﻤﻤﻜﻨﺔ‬ ‫اﻟﺤﻠﻮل‬ ‫ﺑﻌﺾ‬
.‫ﻟﻠﻤﺴﺄﻟﺔ‬ ‫ﻧﮭﺎﺋﯿﺔ‬ ‫ﻻ‬ ‫ﺣﻠﻮل‬ ‫ﻓﮭﻨﺎك‬

3-‫اﻟﺤﻞ‬ ‫ﺧﻄﺔ‬Strategy‫اﺳﺘﺨﺪام‬ ‫ھﻲ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﻓﻲ‬ ‫اﻟﺤﻞ‬ ‫أﯾﺠﺎد‬ ‫ﺧﻄﺔ‬ :
‫ﺟﻤﯿﻊ‬ ‫ﯾﺤﻘﻖ‬ ‫اﻟﺬي‬ ‫اﻟﺠﺰء‬ ‫ﯾﺪﻋﻰ‬ .‫اﻟﺤﻞ‬ ‫ﻓﯿﮭﺎ‬ ‫ﯾﺘﻮاﺟﺪ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺤﻞ‬ ‫ﺳﺎﺣﺔ‬ ‫ﻟﺘﻀﯿﯿﻖ‬ ‫اﻟﻤﺤﺪدات‬
‫اﻟﻤﻼﺋﻢ‬ ‫اﻟﻤﺠﺎل‬ ‫اﻟﻤﺤﺪدات‬feasible region‫اﻟﺤﻞ‬ ‫ﻹﯾﺠﺎد‬ ‫اﻟﮭﺪف‬ ‫اﻟﺘﺎﺑﻊ‬ ‫ﻧﺴﺘﺨﺪم‬ ‫ﺛﻢ‬ .
.‫اﻟﻤﻼﺋﻢ‬ ‫اﻟﻤﺠﺎل‬ ‫ﺿﻤﻦ‬ ‫اﻷﻣﺜﻞ‬
4-‫اﻟﻤﻨﺎطﻖ‬‫اﻟﻨﺼﻔﯿﺔ‬Half planes‫اﻟﺸﻜﻞ‬ ‫أن‬ ‫ﻧﻼﺣﻆ‬ :1.1‫ﻣﻦ‬ ‫ﻟﻜﻞ‬ ‫ﺳﺎﻟﺒﺔ‬ ‫ﻗﯿﻢ‬ ‫ﯾﻤﺜﻞ‬W‫و‬S.
‫اﺗﻲ‬ ‫اﻟﻤﻨﺎطﻖ‬ ‫ﻧﺄﺧﺬ‬ ‫أن‬ ‫وﯾﺠﺐ‬ ‫ﻣﻌﻨﺎ‬ ‫ﻏﯿﺮ‬ ‫ذات‬ ‫اﻟﺴﺎﻟﺒﺔ‬ ‫ﻓﺎﻟﻘﯿﻢ‬ ‫أوزان‬ ‫ﻋﻦ‬ ‫ﯾﻌﺒﺮ‬ ‫ﻣﻨﮭﺎ‬ ‫ﻛﻞ‬ ‫وﻛﻮن‬
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫ﻓﺎﻟﻤﺤﺪد‬ ‫اﻟﻤﺜﺎل‬ ‫ﺳﺒﯿﻞ‬ ‫ﻓﻌﻼ‬ .‫ﻣﻨﮭﺎ‬ ‫ﻟﻜﻞ‬ ‫اﻟﻤﻮﺟﺒﺔ‬ ‫اﻟﻘﯿﻢ‬ ‫ﺗﻤﺜﻞ‬1-2‫ﯾﺘﻀﻤﻦ‬ ‫واﻟﺬي‬
S≥0‫ا‬ ‫ﻓﻲ‬ ‫ﻣﻨﺤﺼﺮ‬ ‫اﻟﺤﻞ‬ ‫ﯾﺠﻌﻞ‬‫اﻟﻤﻈﻠﻠﺔ‬ ‫اﻟﻤﻨﻄﻘﺔ‬ ‫وھﻲ‬ ‫اﻟﺸﺎﻗﻮﻟﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻣﻦ‬ ‫اﻟﯿﻤﯿﻨﻲ‬ ‫ﻟﻘﺴﻢ‬
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬1-2A.‫اﻟﻨﺼﻔﯿﺔ‬ ‫اﻟﻤﻨﻄﻘﺔ‬ ‫ﯾﺪﻋﻰ‬ ‫واﻟﻤﺘﺒﻘﻲ‬ ‫اﻟﺴﺎﺣﺔ‬ ‫ﻧﺼﻒ‬ ‫اﻟﻤﺤﺪد‬ ‫ﯾﺤﺼﺮ‬ ‫وﺑﺬﻟﻚ‬ .
‫اﻟﺸﻜﻞ‬ ‫ﯾﺒﯿﻦ‬ ‫ﻣﺸﺎﺑﮫ‬ ‫وﺑﺸﻜﻞ‬1-2B‫اﻟﻤﺤﺪد‬ ‫ﯾﻤﺜﻞ‬W≥0.
5-‫اﻟﻤﻨﺎطﻖ‬ ‫ﺗﻘﺎطﻊ‬Intersection planes:‫ﯾﺠﺐ‬ ‫أﻧﮫ‬ ‫ﺑﻤﺎ‬‫ا‬ ‫ﻛﻼ‬ ‫ﺗﻄﺒﯿﻖ‬‫ﻟﻤﺤﺪدﯾﻦ‬W ≥ 0
‫و‬S ≥ 0‫ﺑﺘﻘﺎط‬ ‫ﺗﺨﺘﺼﺮ‬ ‫اﻟﺤﻞ‬ ‫ﺳﺎﺣﺔ‬ ‫ﻣﻨﻄﻘﺔ‬ ‫ﻓﺈن‬ ‫ﻟﺬﻟﻚ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﻤﺤﺪدﯾﻦ‬ ‫ﺳﺎﺣﺘﻲ‬ ‫ﻊ‬
1-2C.
W
S
*
*
*
*
*
‫اﻟﻤﻤﻜﻨﺔ‬ ‫اﻟﺤﻠﻮل‬ ‫ﺑﻌﺾ‬
‫اﻟﺸﻜﻞ‬1.1‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺳﺎﺣﺔ‬W‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬S‫اﻟﺼﻮﯾﺎ‬ ‫دﻗﯿﻖ‬
‫اﻟﺸﻜﻞ‬1-2.‫اﻟﻮزن‬ ‫ﻣﺤﺪدات‬

6-‫اﻹﺟﻤﺎﻟﻲ‬ ‫اﻟﻮزن‬ ‫ﻣﺤﺪد‬ ‫رﺳﻢ‬The combined weight constraint
‫ﻋﻨﮫ‬ ‫اﻟﻤﻌﺒﺮ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫ﯾﻘﻊ‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺤﻞ‬ ‫ﻓﺈن‬ ‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫ودﻗﯿﻖ‬ ‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﻛﻤﯿﺔ‬ ‫ﻣﺠﻤﻮع‬ ‫أن‬ ‫ﺑﻤﺎ‬
‫ﺑﺎﻟﻤﻌﺎدﻟﺔ‬1-3‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻣﻮﺿﺢ‬ ‫ھﻮ‬ ‫ﻣﺎ‬ ‫وھﺬا‬1-3
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S
W
‫اﻟﺸﻜﻞ‬1-3‫اﻟﺼﻮﯾﺎ‬ ‫ودﻗﯿﻖ‬ ‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﻛﻤﯿﺎت‬ ‫ﻣﺠﻤﻮع‬ ‫ﻣﺤﺪد‬
7-‫اﻟﺘﻐﺬﯾﺔ‬ ‫ﻣﺤﺪدات‬ ‫رﺳﻢ‬Graphing nutrient constraints‫ﻓﻲ‬ ‫اﻟﺒﺮوﺗﯿﻦ‬ ‫ﻣﺤﺪد‬ ‫ﻟﻨﺄﺧﺬ‬ :
‫اﻟﻤﻌﺎدﻟﺔ‬1-4
= ‫اﻟﺒﺮوﺗﯿﻦ‬0.11 W + 0.47 S≤20
‫ﻣﻌﺎدﻟﺔ‬ ‫ﺑﻤﺴﺘﻘﯿﻢ‬ ‫ﯾﺘﻤﺜﻞ‬ ‫اﻟﺒﺮوﺗﯿﻦ‬ ‫ﻣﺤﺪد‬
0.11 W + 0.47 S=20
‫ﻧﻌﯿﻦ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫ھﺬا‬ ‫ﻟﺮﺳﻢ‬‫وﺣﻞ‬ ‫اﻟﻤﺘﻐﯿﺮﯾﻦ‬ ‫ﻷﺣﺪ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫ﺑﺎﻓﺘﺮاض‬ ‫ﯾﺘﻢ‬ ‫وھﺬا‬ .‫اﻟﺨﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺘﯿﻦ‬
‫اﻟﺴﮭﻮﻟﺔ‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬ ‫وﻟﻜﻦ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫اﻓﺘﺮاض‬ ‫ﯾﻤﻜﻦ‬ ‫أﻧﮫ‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬ .‫اﻟﺜﺎﻧﻲ‬ ‫ﻟﻠﻤﺘﻐﯿﺮ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬
‫اﻟﻘﯿﻤﺔ‬ ‫ﻟﻨﻔﺘﺮض‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬S.
(0)0.11 W + 0.47=20
20
W= =181.8
0.11
‫ﯾﻤ‬ ‫اﻟﺨﻂ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﺮ‬181.8, 0(
‫اﻟﻘﯿﻤﺔ‬ ‫ﻧﻔﺮض‬ ‫ﺛﻢ‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬W‫ﻗﯿﻤﺔ‬ ‫وﻧﺤﺴﺐ‬S.
0.11 (0) + 0.40 S=20
20
S= =42.5
0.47

) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬0, 42.5(
‫ﻣﺴﺘﻘﯿ‬ ‫ﯾﻤﯿﻦ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻨﻄﻘﺔ‬ ‫ﺗﻌﺘﺒﺮ‬ ‫ﻟﺬﻟﻚ‬ ‫اﻟﻤﺤﺪدة‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ﻣﻦ‬ ‫أﻛﺒﺮ‬ ‫ھﻲ‬ ‫اﻟﻤﺤﺪد‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫أن‬ ‫وﺑﻤﺎ‬‫ﻢ‬
‫اﻟﺸﻜﻞ‬ .‫اﻟﺤﻞ‬ ‫ﺳﺎﺣﺔ‬ ‫ھﻲ‬ ‫اﻹﺟﻤﺎﻟﯿﺔ‬ ‫اﻟﻜﻤﯿﺔ‬ ‫ﻣﺤﺪد‬ ‫ﻣﺴﺘﻘﯿﻢ‬ ‫ﻋﻠﻰ‬ ‫واﻟﻮاﻗﻌﺔ‬ ‫اﻟﺒﺮوﺗﯿﻦ‬ ‫ﻣﺤﺪد‬1-4
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‫اﻟﺸﻜﻞ‬1-4
8-‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻜﺮﺑﻮھﯿﺪرات‬ ‫ﻣﺤﺪد‬1-5‫ﯾﻠﻲ‬ ‫ﻛﻤﺎ‬
= ‫ﻛﺮﺑﻮھﯿﺪات‬0.70 W + 0.40 S≥55
‫ﻣﺴﺘﻘﯿ‬ ‫ﻋﻠﻰ‬ ‫وإﻧﻤﺎ‬ ‫اﻟﻤﺤﺎور‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫ﻻ‬ ‫اﻟﻤﺤﺪد‬ ‫ھﺬا‬ ‫ﺣﺪود‬ ‫ﻓﺈن‬ ‫اﻟﺴﺎﺑﻘﯿﻦ‬ ‫ﻟﻠﻤﺤﺪدﯾﻦ‬ ‫ﻣﺸﺎﺑﮫ‬ ‫ﻏﯿﺮ‬ ‫ﺑﺸﻜﻞ‬‫ﻢ‬
:‫ﻣﻌﺎدﻟﺘﮫ‬
0.70 W + 0.40 S=55)1-6(
‫وﺣﻞ‬ ‫اﻟﻤﺘﻐﯿﺮﯾﻦ‬ ‫ﻷﺣﺪ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫ﺑﺎﻓﺘﺮاض‬ ‫ﯾﺘﻢ‬ ‫وھﺬا‬ .‫اﻟﺨﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺘﯿﻦ‬ ‫ﻧﻌﯿﻦ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫ھﺬا‬ ‫ﻟﺮﺳﻢ‬
‫اﻟﺴﮭﻮﻟﺔ‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬ ‫وﻟﻜﻦ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫اﻓﺘﺮاض‬ ‫ﯾﻤﻜﻦ‬ ‫أﻧﮫ‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬ .‫اﻟﺜﺎﻧﻲ‬ ‫ﻟﻠﻤﺘﻐﯿﺮ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬
‫اﻟﻘﯿﻤﺔ‬ ‫ﻟﻨﻔﺘﺮض‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬S.
(0)=0.70 W + 0.4055
55
W 64.3
0.70
= =
) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫اﻟﺨﻂ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬64.3,0(
‫اﻟﻘﯿﻤﺔ‬ ‫ﻧﻔﺮض‬ ‫ﺛﻢ‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬W‫ﻗﯿﻤﺔ‬ ‫وﻧﺤﺴﺐ‬S.
S0.70 (0) + 0.40=55
55
S 137.5
0.40
= =

) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬0,137.5(
‫ﻣ‬ ‫أﺻﻐﺮ‬ ‫ھﻲ‬ ‫اﻟﻤﺤﺪد‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫أن‬ ‫وﺑﻤﺎ‬‫ﯾﺴﺎر‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻨﻄﻘﺔ‬ ‫ﺗﻌﺘﺒﺮ‬ ‫ﻟﺬﻟﻚ‬ ‫اﻟﻤﺤﺪدة‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ﻦ‬
‫اﻟﺸﻜﻞ‬ .‫اﻟﺤﻞ‬ ‫ﺳﺎﺣﺔ‬ ‫ھﻲ‬ ‫اﻹﺟﻤﺎﻟﯿﺔ‬ ‫اﻟﻜﻤﯿﺔ‬ ‫ﻣﺤﺪد‬ ‫ﻣﺴﺘﻘﯿﻢ‬ ‫ﻋﻠﻰ‬ ‫واﻟﻮاﻗﻌﺔ‬ ‫اﻟﻜﺮﺑﻮھﯿﺪرات‬ ‫ﻣﺤﺪد‬ ‫ﻣﺴﺘﻘﯿﻢ‬
1-5
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0 20 40 60 80 100 120 140 160
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S
‫اﻟﺸﻜﻞ‬1-5
‫اﻟﺘﻜﻠﻔﺔ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫ﻧﺪﺧﻞ‬ ‫ﻟﺬﻟﻚ‬ ‫اﻟﺪﻧﯿﺎ‬ ‫ﺣﺪودھﺎ‬ ‫ﻓﻲ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺘﻜﻠﻔﺔ‬ ‫إن‬ ‫ﺑﻤﺎ‬
C=20 W + 30 S
‫ﺗﻜﻦ‬ ‫وﻟﻮ‬ ‫ﻟﻠﺘﻜﻠﻔﺔ‬ ‫ﻗﯿﻤﺔ‬ ‫ﻧﻔﺮض‬C=1000
‫وﺣﻞ‬ ‫اﻟﻤﺘﻐﯿﺮﯾﻦ‬ ‫ﻷﺣﺪ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫ﺑﺎﻓﺘﺮاض‬ ‫ﯾﺘﻢ‬ ‫وھﺬا‬ .‫اﻟﺨﻂ‬ ‫ﻋﻠﻰ‬ ‫ﻧﻘﻄﺘﯿﻦ‬ ‫ﻧﻌﯿﻦ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫ھﺬا‬ ‫ﻟﺮﺳﻢ‬
‫اﻟﺴﮭﻮﻟﺔ‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬ ‫وﻟﻜﻦ‬ ‫ﻗﯿﻤﺔ‬ ‫أﯾﺔ‬ ‫اﻓﺘﺮاض‬ ‫ﯾﻤﻜﻦ‬ ‫أﻧﮫ‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬ .‫اﻟﺜﺎﻧﻲ‬ ‫ﻟﻠﻤﺘﻐﯿﺮ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬
‫اﻟﻘﯿﻤﺔ‬ ‫ﻟﻨﻔﺘﺮض‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬S.
0 (0)20 W + 3=1000
1000
W 50
20
= =
) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫اﻟﺨﻂ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬,059(
‫اﻟﻘﯿﻤﺔ‬ ‫ﻧﻔﺮض‬ ‫ﺛﻢ‬0‫ﻟﻠﻤﺘﻐﯿﺮ‬W‫ﻗﯿﻤﺔ‬ ‫وﻧﺤﺴﺐ‬S.
17 (0) + 20 S=05
1000
S 33.3
30
= =
) ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬0,33.3(

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‫اﻟﺸﻜﻞ‬1-6
‫ﻣﻌﺎدﻟ‬ ‫ﻣﺴﺘﻘﯿﻢ‬ ‫ﺳﺤﺐ‬ ‫ﯾﺘﻢ‬.‫ﺑﻨﻘﻄﺔ‬ ‫اﻟﺤﻞ‬ ‫ﻣﻨﻄﻘﺔ‬ ‫ﯾﻘﻄﻊ‬ ‫ﺣﺘﻰ‬ ‫اﻟﻤﯿﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺤﻔﺎظ‬ ‫ﻣﻊ‬ ‫اﻟﻜﻠﻔﺔ‬ ‫ﺔ‬
0
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0 20 40 60 80 100 120 140 160
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S
‫اﻟﺸﻜﻞ‬1-7
: ‫ھﻮ‬ ‫اﻟﺤﻞ‬ ‫ﯾﻜﻮن‬ ‫وﺑﺎﻟﺘﺎﻟﻲ‬

0
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0 20 40 60 80 100 120 140 160
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‫اﻟﺸﻜﻞ‬1-8
:‫اﻟﺤﻞ‬ ‫ﻧﻘﻄﺔ‬ ‫أي‬)50, 50(

‫اﻟﺛﺎﻧﯾﺔ‬ ‫اﻟﻌﻣﻠﯾﺔ‬ ‫اﻟﺟﻠﺳﺔ‬
‫أﻟﻔﯾن‬ ‫د.ﻓرﺣﺎن‬ :‫إﺷراف‬‫اﻟﻌﺎﺑدﯾن‬ ‫زﯾن‬ ‫ﻧور‬ .‫م‬
‫اﻹﻛ‬ ‫ﺑﺮﻧﺎﻣﺞ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﺣﻞ‬‫ﺴﻞ‬
:‫اﻟﻣﺳﺄﻟﺔ‬
‫ﺣﯿﺚ‬ ‫اﻟﺪﺳﻢ‬ ‫ﻣﻨﺰوع‬ ‫اﻟﺼﻮﯾﺎ‬ ‫وﻓﻮل‬ ‫اﻟﻘﻤﺢ‬ ‫ﻣﻦ‬ ‫ﯾﺘﺄﻟﻒ‬ ‫ﻏﺬاﺋﻲ‬ ‫ﺧﺒﺰ‬ ‫ﻣﻜﻮﻧﺎت‬ ‫ﻧﺴﺐ‬ ‫ﺗﺤﺪﯾﺪ‬ ‫اﻟﻤﻄﻠﻮب‬
‫اﻷﻗﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺨﺒﺰ‬ ‫ﯾﺤﺘﻮي‬ ‫أن‬ ‫ﯾﺠﺐ‬20‫اﻷﻛﺜﺮ‬ ‫وﻋﻠﻰ‬ ‫ﺑﺮوﺗﯿﻦ‬ %55‫دﻗﯿﻖ‬ ‫وﯾﺘﻜﻮن‬ .‫ﻛﺮﺑﻮھﯿﺪات‬ %
‫ﻣﻦ‬ ‫اﻟﻘﻤﺢ‬11‫و‬ ‫ﺑﺮوﺗﯿﻦ‬ %70‫ﻣﻦ‬ ‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫ودﻗﯿﻖ‬ ‫ﻛﺮﺑﻮھﯿﺪات‬ %47‫و‬ ‫ﺑﺮوﺗﯿﻦ‬ %40%
. ‫ﻛﺮﺑﻮھﯿﺪات‬
‫ﻋﻠﻤﺎ‬‫ھﻮ‬ ‫اﻟﻘﻤﺢ‬ ‫دﻗﯿﻖ‬ ‫ﺳﻌﺮ‬ ‫أن‬20‫ھﻮ‬ ‫اﻟﺼﻮﯾﺎ‬ ‫ﻓﻮل‬ ‫ودﻗﯿﻖ‬ ‫/ﻛﻎ‬ ‫س‬ ‫ل‬30.‫س/ﻛﻎ‬ ‫ل‬
‫أو‬ ‫ﻣﺘﺮاﺟﺤﺎت‬ ‫ﺷﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻟﻤﺴﺄﻟﺔ‬ ‫اﻟﻤﺸﻜﻠﺔ‬ ‫اﻟﻘﯿﻮد‬ ‫ﻣﻦ‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﺑﻜﺘﺎﺑﺔ‬ ‫ﻧﻘﻮم‬
) ‫اﻟﮭﺪف‬ ‫ﺗﺎﺑﻊ‬ ‫ﻛﺘﺎﺑﺔ‬ ‫ﺗﺘﻢ‬ ‫ﻛﻤﺎ‬ ‫ﻣﺴﺎوﯾﺎت‬C‫اﻟﺒﺤﺚ‬ ‫ﻋﻠﯿﮫ‬ ‫ﺑﺎﻻﻋﺘﻤﺎد‬ ‫ﯾﺘﻢ‬ ‫اﻟﺬي‬ ( ‫اﻟﺪﻧﯿﺎ‬ ‫ﺑﻘﯿﻤﺘﮭﺎ‬ ‫اﻟﻜﻠﻔﺔ‬ ‫ﺗﺎﺑﻊ‬
‫ﻟ‬ ‫اﻟﻤﺜﺎﻟﻲ‬ ‫اﻟﺤﻞ‬ ‫ﻋﻦ‬. ‫اﻟﻤﻄﻠﻮﺑﺔ‬ ‫ﻠﻤﺴﺄﻟﺔ‬
: ‫اﻟﺴﺎﺑﻘﺔ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻟﻤﺴﺄﻟﺔ‬ ‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫إﯾﺠﺎد‬ ‫ﻧﺮﯾﺪ‬
Min C= 30 S + 20W
‫اﻟﺘﺎﻟﯿﺔ‬ ‫ﺑﺎﻟﺸﺮوط‬:
S ≥ 0 (1)
W ≥ 0 (2)
S + W = 100 (3)
0.47 S + 0.11W ≥ 20 (4)
0.40S + 0.7W ≤ 55 (5)
:‫اﻟﺤﻞ‬
‫اﻟﻤﺴﺄﻟﺔ‬ ‫ھﺬه‬ ‫ﻟﺤﻞ‬‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫ﻋﻤﻞ‬ ‫ﺑﻮرﻗﺔ‬ ‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﮭﺪف‬ ‫ﺗﺎﺑﻊ‬ ‫ﺑﻜﺘﺎﺑﺔ‬ ‫ﻧﻘﻮم‬1-2:

‫اﻟﺸﻜﻞ‬1-2‫اﻟﺨﻼﯾﺎ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺴﺘﺨﺪﻣﺔ‬ ‫اﻟﻌﻼﻗﺎت‬ ‫ﻓﯿﮭﺎ‬ ‫ﻣﺒﯿﻦ‬ ‫ﻋﻤﻞ‬ ‫ورﻗﺔ‬
‫اﻟﺸﻜﻞ‬2-2‫ﺗﺤﻀﯿﺮھﺎ‬ ‫ﺑﻌﺪ‬ ‫اﻟﻌﻤﻞ‬ ‫ورﻗﺔ‬
‫إﻋﻄﺎء‬ ‫ﯾﺠﺐ‬ ‫أﻧﮫ‬ ‫ﻻﺣﻆ‬‫اﺑﺘﺪاﺋﯿﺔ‬ ‫ﻗﯿﻢ‬) ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻟﻤﺘﺤﻮﻻت‬‫ﻣﺜﺎﻟﻨﺎ‬ ‫ﻓﻲ‬‫اﻻﺑﺘﺪاﺋﯿﺔ‬ ‫اﻟﻘﯿﻢ‬ ‫أﻋﻄﯿﻨﺎ‬ ،S=1,
W=1.‫ﺧﻄﯿﺔ‬ ‫ﺑﺮﻣﺠﺔ‬ ‫ﻟﻤﺴﺄﻟﺔ‬ ‫اﻻﺑﺘﺪاﺋﻲ‬ ‫اﻟﺤﻞ‬ ‫ﻟﺸﺮوط‬ ‫ﻣﺤﻘﻘﺔ‬ ‫ﻣﻨﻄﻘﯿﺔ‬ ‫اﺑﺘﺪاﺋﯿﺔ‬ ‫ﻗﯿﻢ‬ ‫ﻧﻌﻄﻲ‬ ‫أن‬ ‫ﯾﺠﺐ‬ (
‫اﻷﻣﺮ‬ ‫ﻧﻄﻠﺐ‬ ‫ذﻟﻚ‬ ‫ﺑﻌﺪ‬Solver‫ﻣﻦ‬‫اﻷدوات‬ ‫ﻗﺎﺋﻤﺔ‬Tools‫اﻟﻨﺎﻓﺬة‬ ‫ﻟﺪﯾﻨﺎ‬ ‫ﻓﺘﻈﮭﺮ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬3-2

‫اﻟﺸﻜﻞ‬3-2
‫ھﺬه‬ ‫ﻓﻲ‬‫اﻟﻨﺎﻓﺬ‬‫ﯾﺘﻢ‬ ‫ة‬‫اﻟﻤﺘﺤﻮﻻت‬ ‫ﻣﻮاﻗﻊ‬ ‫ﺗﺤﺪﯾﺪ‬S,W‫اﻟﻘﯿﻮد‬ ‫ﺗﻌﺮﯾﻒ‬ ‫وﯾﺘﻢ‬ ‫اﻟﻌﻤﻞ‬ ‫ورﻗﺔ‬ ‫ﻓﻲ‬
Constraints:‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﮭﺪف‬ ‫وﺗﺎﺑﻊ‬
•‫ﻓﻲ‬Set Target Cell.‫اﻟﮭﺪف‬ ‫ﺗﺎﺑﻊ‬ ‫ﻋﻼﻗﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺤﺘﻮي‬ ‫اﻟﺘﻲ‬ ‫اﻟﺨﻠﯿﺔ‬ ‫ﻣﻮﻗﻊ‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :
‫ﻣﺜﺎﻟﻨﺎ‬ ‫ﻓﻲ‬ ‫وھﻲ‬D9‫ﻣﻜﺎن‬ ‫ﻋﻠﻰ‬ ً‫ﻻ‬‫أو‬ ‫اﻟﻤﺆﺷﺮ‬ ‫ﺑﻮﺿﻊ‬ ‫اﻟﺨﻠﯿﺔ‬ ‫ھﺬه‬ ‫ﻋﻨﻮان‬ ‫ﻛﺘﺎﺑﺔ‬ ‫ﺗﺘﻢ‬ .‫ﻛﺘﺎﺑﺔ‬
Set Target Cell‫اﻟﻤﻘﺼﻮدة‬ ‫اﻟﺨﻠﯿﺔ‬ ‫ﻣﻮﻗﻊ‬ ‫ﻧﺨﺘﺎر‬ ‫ﺛﻢ‬D9‫اﻟﻤﻜﺎن‬ ‫ﻓﻲ‬ ‫ﻋﻨﻮاﻧﮭﺎ‬ ‫ﻛﺘﺎﺑﺔ‬ ‫ﻓﺘﺘﻢ‬
.‫اﻟﻤﻄﻠﻮب‬
•‫ﻓﻲ‬Equal To‫ﻛﺎن‬ ‫ﻓﺈذا‬ (‫أﺻﻐﺮي‬ ‫أم‬ ‫أﻋﻈﻤﻲ‬ ‫)ﺣﻞ‬ ‫اﻟﻤﻄﻠﻮب‬ ‫اﻟﺤﻞ‬ ‫ﻧﻮع‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :
‫اﻟﻤﺘﻐﯿﺮات‬ ‫إﯾﺠﺎد‬ ‫اﻟﻤﻄﻠﻮب‬S,W‫اﻟﮭﺪف‬ ‫ﻟﺘﺎﺑﻊ‬ ‫اﻟﺼﻐﺮى‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ﺗﻌﻄﻲ‬ ‫اﻟﺘﻲ‬C‫ﻧﺨﺘﺎر‬
Min‫ﻓﻲ‬ ‫اﻟﺤﺎل‬ ‫ھﻮ‬ ‫ﻛﻤﺎ‬‫اﻟﮭﺪف‬ ‫ﻟﺘﺎﺑﻊ‬ ‫اﻟﻌﻈﻤﻰ‬ ‫اﻟﻘﯿﻤﺔ‬ ‫إﯾﺠﺎد‬ ‫اﻟﻤﻄﻠﻮب‬ ‫ﻛﺎن‬ ‫وإذا‬ .‫ﻣﺜﺎﻟﻨﺎ‬
‫ﻧﺨﺘﺎر‬Max.
•‫ﻓﻲ‬By Changing Cell‫ﺑﮭﺪف‬ ‫ﺗﻐﯿﯿﺮھﺎ‬ ‫ﻧﺮﯾﺪ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﺘﻐﯿﺮات‬ ‫ﻣﻮاﻗﻊ‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :
‫ﻗﯿﻢ‬ ‫أي‬ ،‫اﻟﻤﻄﻠﻮب‬ ‫اﻟﺤﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺤﺼﻮل‬S,W‫ﻣﺜﺎﻟﻨﺎ‬ ‫ﻓﻲ‬ ‫اﻟﺨﻼﯾﺎ‬ ‫ﻣﺠﺎل‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺠﺐ‬ ‫وھﻨﺎ‬
C3:C4.
•‫اﻟﻤﺠﺎل‬ ‫ﻓﻲ‬Subject to the Constrains.‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﻔﺮوﺿﺔ‬ ‫اﻟﻘﯿﻮد‬ ‫ﺗﻌﺮﯾﻒ‬ ‫ﯾﺘﻢ‬ :
‫اﻟﺰر‬ ‫ﻋﻠﻰ‬ ‫ﻧﻀﻐﻂ‬ ‫اﻟﻌﻤﻞ‬ ‫ورﻗﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﻌﺮﻓﺔ‬ ‫اﻟﻘﯿﻮد‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫إﻟﻰ‬ ‫ﺟﺪﯾﺪ‬ ‫ﻗﯿﺪ‬ ‫ﻹﺿﺎﻓﺔ‬Add
:‫اﻟﻌﻤﻞ‬ ‫ورﻗﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻘﯿﻮد‬ ‫وﺿﻊ‬ ‫أﻣﺎﻛﻦ‬ ‫ﺑﺘﺤﺪﯾﺪ‬ ‫اﻟﺨﺎﺻﺔ‬ ‫اﻟﺘﺎﻟﯿﺔ‬ ‫اﻟﻨﺎﻓﺬة‬ ‫ﻟﺪﯾﻨﺎ‬ ‫ﻓﺘﻈﮭﺮ‬

‫اﻟﺸﻜﻞ‬4-2
•‫ﻣﻜﺎن‬ ‫ﻓﻲ‬Cell Reference‫ا‬ ‫ﻣﺠﺎل‬ ‫أو‬ ‫اﻟﺨﻠﯿﺔ‬ ‫ﻋﻨﻮان‬ ‫ﻧﻜﺘﺐ‬ :‫ﻟﺨﻼﯾﺎ‬‫ﻋﻠﻰ‬ ‫ﺗﺤﺘﻮي‬ ‫اﻟﺘﻲ‬
‫ﻣﺜﺎﻟﻨﺎ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺘﺮاﺟﺤﺔ‬ ‫ﻣﻦ‬ ‫اﻷﯾﺴﺮ‬ ‫اﻟﻄﺮف‬ ‫ﻧﺘﯿﺠﺔ‬E13.
•, = ‫اﻟﻤﺘﺮاﺟﺤﺔ‬ ‫اﺗﺠﺎه‬ ‫ﻧﺨﺘﺎر‬≤,≥‫ﯾﻤﻜﻦ‬ .‫اﻷﺳﻔﻞ‬ ‫ﻧﺤﻮ‬ ‫اﻟﻤﻮﺟﮫ‬ ‫اﻟﺴﮭﻢ‬ ‫ذا‬ ‫اﻟﺰر‬ ‫ﺑﺎﺳﺘﻌﻤﺎل‬
‫اﺳﺘﻌﻤﺎل‬ ً‫ﺎ‬‫أﯾﻀ‬int.‫ﺻﺤﯿﺤﺔ‬ ‫ﺑﻘﯿﻢ‬ ‫اﻟﺤﻞ‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﻧﺮﯾﺪ‬ ‫ﻋﻨﺪﻣﺎ‬
•‫ﻣﻜﺎن‬ ‫ﻓﻲ‬Constraint‫اﻟﺨﻠﯿﺔ‬ ‫ﻋﻨﻮان‬ ‫ﻧﻀﻊ‬ :‫ﻋﻠﻰ‬ ‫ﺗﺤﺘﻮي‬ ‫اﻟﺘﻲ‬‫ﻣﻦ‬ ‫اﻷﯾﻤﻦ‬ ‫اﻟﻄﺮف‬
.‫اﻟﻤﺘﺮاﺟﺤﺔ‬
•‫ﻧﺨﺘﺎر‬Add‫اﻻﻧﺘﮭﺎ‬ ‫وﻋﻨﺪ‬ ‫اﻟﺠﺪﯾﺪ‬ ‫اﻟﻘﯿﺪ‬ ‫ﻹﺿﺎﻓﺔ‬‫ء‬‫ﻧﻀﻐﻂ‬OK.
•‫اﻟﺰر‬Change.‫اﺧﺘﯿﺎرھﺎ‬ ‫ﺗﻢ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﯿﻮد‬ ‫أﺣﺪ‬ ‫ﻟﺘﻌﺪﯾﻞ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :
•‫اﻟﺰر‬Delete.‫اﻟﻤﻌﺮﻓﺔ‬ ‫اﻟﻘﯿﻮد‬ ‫أﺣﺪ‬ ‫ﻟﻤﺴﺢ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :
•‫اﻟﺰر‬Reset All‫ﻣﺘﻐﯿﺮا‬ ‫وﻋﻨﺎوﯾﻦ‬ ‫ﻗﯿﻮد‬ ‫ﻣﻦ‬ ‫اﻟﺴﺎﺑﻘﺔ‬ ‫اﻟﺘﻌﺮﯾﻔﺎت‬ ‫ﺟﻤﯿﻊ‬ ‫ﻟﻤﺴﺢ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :‫ت‬
.‫اﻟﻤﺴﺄﻟﺔ‬
•‫اﻟﺰر‬Guess‫ﻟﺠﻌﻞ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :Excel‫ﺧﻠﯿﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺴﺘﻌﻤﻠﺔ‬ ‫ﻟﻠﻤﺘﻐﯿﺮات‬ ‫اﻟﺨﻼﯾﺎ‬ ‫ﻣﻮاﻗﻊ‬ ‫ﯾﻘﺪر‬
‫ﻣﻜﺎن‬ ‫ﻓﻲ‬ ‫ﻟﻮﺿﻌﮭﺎ‬ ‫اﻟﮭﺪف‬ ‫ﺗﺎﺑﻊ‬By Changing Cell.
•‫اﻟﺰر‬Option‫ﻟﺘﻐﯿﯿﺮ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :.‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺄﻟﺔ‬ ‫ﺣﻞ‬ ‫طﺮﯾﻘﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺨﯿﺎرات‬ ‫ﺑﻌﺾ‬
‫ﻋ‬ ‫اﻟﻀﻐﻂ‬ ‫ﻋﻨﺪ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﻨﺎﻓﺬة‬ ‫ﺗﻈﮭﺮ‬ ‫اﻟﺰر‬ ‫ھﺬا‬ ‫ﻠﻰ‬5-2:

‫اﻟﺸﻜﻞ‬5-2
:‫ﯾﺘﻢ‬ ‫اﻟﻨﺎﻓﺬة‬ ‫ھﺬه‬ ‫ﻓﻲ‬
‫ﻓﻲ‬Max Time(‫اﻷﻋﻈﻢ‬ ‫)اﻟﺰﻣﻦ‬‫ﯾﺘﺠﺎوزه‬ ‫ﻻ‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺬي‬ ‫اﻷﻋﻈﻢ‬ ‫اﻟﺰﻣﻦ‬ ‫ﺗﺤﺪﯾﺪ‬ :
‫ﺣﻞ‬ ‫ﻋﻦ‬ ‫اﻟﺒﺤﺚ‬ ‫ﻋﻦ‬ ‫ﺳﯿﺘﻮﻗﻒ‬ ‫اﻟﺒﺮﻧﺎﻣﺞ‬ ‫أي‬ ،(‫ﺣﻞ‬ ‫وﺟﻮد‬ ‫ﻋﺪم‬ ‫ﺣﺎل‬ ‫)ﻓﻲ‬ ‫اﻟﺤﻞ‬ ‫ﻓﻲ‬ ‫اﻟﺒﺮﻧﺎﻣﺞ‬
.‫اﻟﺰﻣﻦ‬ ‫ھﺬا‬ ‫ﺗﺠﺎوز‬ ‫ﺣﺎل‬ ‫ﻓﻲ‬
‫ﻓﻲ‬Iterations(‫)اﻟﺘﻜﺮار‬‫ﻟ‬ ‫اﻷﻋﻈﻢ‬ ‫اﻟﻌﺪد‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :‫ﻠﻤﺤﺎ‬‫اﻟﺒﺮﻧﺎﻣﺞ‬ ‫ﯾﺠﺮﯾﮭﺎ‬ ‫اﻟﺘﻲ‬ ‫وﻻت‬
.‫اﻟﺤﻞ‬ ‫ﻹﯾﺠﺎد‬
‫ﻓﻲ‬Precision(‫)اﻟﺪﻗﺔ‬‫ﻟﻤﺘﺮاﺟﺤﺎت‬ ‫اﻷﯾﺴﺮ‬ ‫اﻟﻄﺮف‬ ‫اﻗﺘﺮاب‬ ‫ﻓﻲ‬ ‫اﻟﻤﻘﺒﻮﻟﺔ‬ ‫اﻟﺪﻗﺔ‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :
.‫اﻟﻤﻄﻠﻮب‬ ‫ﻟﻠﺤﻞ‬ ‫اﻟﻮﺻﻮل‬ ‫ﻋﻨﺪ‬ ‫وذﻟﻚ‬ ،‫ﻣﻨﮭﺎ‬ ‫اﻷﯾﻤﻦ‬ ‫اﻟﻄﺮف‬ ‫ﻗﯿﻢ‬ ‫ﻣﻦ‬ ‫اﻟﻘﯿﻮد‬
‫ﻓﻲ‬Tolerance(‫ﺑﮫ‬ ‫اﻟﻤﺴﻤﻮح‬ ‫)اﻟﺘﻔﺎوت‬‫ﻣﺌﻮ‬ ‫ﻛﻨﺴﺒﺔ‬ ‫اﻟﻤﻘﺒﻮﻟﺔ‬ ‫اﻟﺪﻗﺔ‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﯾﺘﻢ‬ :‫ﺣﺎل‬ ‫ﻓﻲ‬ ‫ﯾﺔ‬
.‫اﻷرﻗﺎم‬ ‫ﺻﺤﯿﺤﺔ‬ ‫ﻣﺘﻐﯿﺮات‬ ‫اﺳﺘﻌﻤﺎل‬
‫ﻓﻲ‬Convergence(‫)اﻟﺘﻘﺎرب‬‫اﻟﻤﺴﺘﮭﺪﻓﺔ‬ ‫اﻟﺨﻠﯿﺔ‬ ‫ﻗﯿﻤﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻨﺴﺒﻲ‬ ‫اﻟﺘﻐﯿﺮ‬ ‫ﯾﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ :
‫ﻓﻲ‬ ‫اﻟﻤﻌﺮف‬ ‫اﻟﺮﻗﻢ‬ ‫ﻣﻦ‬ ‫أﻗﻞ‬Convergence‫اﻟﺨﻤﺲ‬ ‫اﻟﺘﻜﺮار‬ ‫ﻋﻤﻠﯿﺎت‬ ‫إﻟﻰ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬
‫ﯾﺘﻮﻗﻒ‬ ‫اﻷﺧﯿﺮة‬solver.‫اﻟﻤﻄﻠﻮب‬ ‫اﻟﺤﻞ‬ ‫إﻟﻰ‬ ‫وﺻﻞ‬ ‫ﻗﺪ‬ ‫وﯾﻜﻮن‬‫ﻓﻘﻂ‬ ‫اﻟﺘﻘﺎرب‬ ‫وﯾﻨﻄﺒﻖ‬‫ﻋﻠﻰ‬
‫اﻟﺨﻄﯿﺔ‬ ‫ﻏﯿﺮ‬ ‫اﻟﻤﺴﺎﺋﻞ‬.
‫اﻟﺨﯿﺎر‬Assume Linear Model‫ﻣﻦ‬ ‫وذﻟﻚ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﺣﻞ‬ ‫ﻋﻨﺪ‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :
.‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫إﻟﻰ‬ ‫اﻟﻮﺻﻮل‬ ‫ﻋﻤﻠﯿﺔ‬ ‫ﺗﺴﺮﯾﻊ‬ ‫أﺟﻞ‬

‫اﻟﺨﯿﺎر‬Show Iteration Results‫ﺗﻘﺮﯾﺐ‬ ‫ﻛﻞ‬ ‫ﺑﻌﺪ‬ ‫اﻟﺤﺎﻟﯿﺔ‬ ‫اﻟﺤﻞ‬ ‫ﻗﯿﻢ‬ ‫ﻹظﮭﺎر‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :
.‫اﻟﺼﺤﯿﺢ‬ ‫اﻟﺤﻞ‬ ‫إﻟﻰ‬
‫اﻟﺨﯿﺎر‬Assume Non-Negative:‫ﻣﻔﺮوﺿﺔ‬ ‫اﻟﺴﻠﺒﯿﺔ‬ ‫ﻋﺪم‬ ‫ﺷﺮوط‬ ‫ﺗﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﯾﺴﺘﻌﻤﻞ‬
.‫ﻓﻘﻂ‬ ‫اﻟﻤﻮﺟﺒﺔ‬ ‫اﻟﻤﻨﻄﻘﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺤﻞ‬ ‫ﯾﻜﻮن‬ ‫أي‬ ،‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﻋﻠﻰ‬
‫اﻟﺨﯿﺎر‬Use Automatic Scaling‫ﻣﺘﻐﯿﺮات‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﻟﺠﻌﻞ‬ ‫اﻟﺨﯿﺎر‬ ‫ھﺬا‬ ‫ﯾﺴﺘﻌﻤﻞ‬ :
.‫اﻟﺤﻞ‬ ‫إﯾﺠﺎد‬ ‫ﻋﻨﺪ‬ ‫ﻣﺘﻨﺎﺳﺒﺔ‬ ‫أﺧﺮى‬ ‫ﺟﮭﺔ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺴﺄﻟﺔ‬
Estimates(‫)اﻟﺘﻘﺪﯾﺮات‬‫ﻋﻠ‬ ‫ﻟﻠﺤﺼﻮل‬ ‫اﻟﻤﺴﺘﺨﺪم‬ ‫اﻟﻤﻨﮭﺞ‬ ‫ﺗﺤﺪﯾﺪ‬‫اﺑﺘﺪاﺋﯿﺔ‬ ‫ﺗﻘﺪﯾﺮات‬ ‫ﻰ‬
.‫اﻷﺑﻌﺎد‬ ‫أﺣﺎدي‬ ‫ﺑﺤﺚ‬ ‫ﻛﻞ‬ ‫ﻓﻲ‬ ‫اﻟﺮﺋﯿﺴﯿﺔ‬ ‫ﻟﻠﻤﺘﻐﯿﺮات‬
o‫اﻟﻈﻞ‬.‫ﻣﻤﺎس‬ ‫ﻣﺘﺠﮫ‬ ‫ﻣﻦ‬ ‫اﻟﺨﻄﻲ‬ ‫اﻟﺨﺎرﺟﻲ‬ ‫اﻻﺳﺘﻜﻤﺎل‬ ‫ﯾﺴﺘﺨﺪم‬
o‫ﻣﺘﻄﺎﺑﻖ‬ ‫ﺗﻌﺎﻣﺪي‬‫ﯾﻤﻜﻨﮫ‬ ‫اﻟﺬي‬ ‫اﻟﻤﺘﻄﺎﺑﻖ‬ ‫اﻟﺘﻌﺎﻣﺪي‬ ‫اﻟﺨﺎرﺟﻲ‬ ‫اﻻﺳﺘﻜﻤﺎل‬ ‫ﯾﺴﺘﺨﺪم‬
.‫ﻋﺎﻟﯿﺔ‬ ‫ﺧﻄﯿﺔ‬ ‫ﻏﯿﺮ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻨﺘﺎﺋﺞ‬ ‫ﺗﺤﺴﯿﻦ‬
Derivatives(‫)اﻟﻤﺸﺘﻘﺎت‬‫اﻟﺘﺒ‬ ‫ﺗﺤﺪﯾﺪ‬‫ﻟﺘﻘﺪﯾﺮ‬ ‫اﻟﻤﺴﺘﺨﺪم‬ ‫ﺎﯾﻦ‬‫ﻟﺪاﻻت‬ ‫اﻟﺠﺰﺋﯿﺔ‬ ‫اﻟﻤﺸﺘﻘﺎت‬
.‫واﻷﻏﺮاض‬ ‫اﻟﻘﯿﻮد‬
o‫اﻷﻣﺎم‬.‫ﺑﺒﻄﻲء‬ ‫اﻟﻘﯿﺪ‬ ‫ﻗﯿﻢ‬ ً‫ﺎ‬‫ﻧﺴﺒﯿ‬ ‫ﻓﯿﮭﺎ‬ ‫ﺗﺘﻐﯿﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﺴﺎﺋﻞ‬ ‫ﻟﻤﻌﻈﻢ‬ ‫ﺗﺴﺘﺨﺪم‬
o‫ﻣﺮﻛﺰي‬‫ﻣﻦ‬ ‫ﺑﺎﻟﻘﺮب‬ ‫وﺧﺎﺻﺔ‬ ‫ﺳﺮﯾﻊ‬ ‫ﺑﺸﻜﻞ‬ ‫اﻟﻘﯿﻮد‬ ‫ﻓﯿﮭﺎ‬ ‫ﺗﺘﻐﯿﺮ‬ ‫اﻟﺘﻲ‬ ‫ﻟﻠﻤﺴﺎﺋﻞ‬ ‫ﯾﺴﺘﺨﺪم‬
،‫أﻛﺜﺮ‬ ‫ﺣﺴﺎﺑﺎت‬ ‫ﯾﺘﻄﻠﺐ‬ ‫اﻟﺨﯿﺎر‬ ‫ھﺬا‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫وﻋﻠﻰ‬ .‫اﻟﺤﺪود‬‫ﻋﻨﺪ‬ ً‫ا‬‫ﻣﻔﯿﺪ‬ ‫ﯾﻜﻮن‬ ‫ﻗﺪ‬
‫إرﺟﺎع‬Solver.‫اﻟﺤﻞ‬ ‫ﺗﺤﺴﯿﻦ‬ ‫ﯾﺴﺘﻄﯿﻊ‬ ‫ﻻ‬ ‫أﻧﮫ‬ ‫ﻟﺮﺳﺎﻟﺔ‬
Search(‫)ﺑﺤﺚ‬.‫ﻟﻠﺒﺤﺚ‬ ‫اﺗﺠﺎه‬ ‫ﻟﺘﺤﺪﯾﺪ‬ ‫ﺗﻜﺮار‬ ‫ﻛﻞ‬ ‫ﻋﻨﺪ‬ ‫اﻟﻤﺴﺘﺨﺪﻣﺔ‬ ‫اﻟﺨﻮارزﻣﯿﺔ‬ ‫ﺗﺤﺪد‬
o‫ﻧﯿﻮﺗﻦ‬‫اﻟﺬاﻛﺮة‬ ‫أﻛﺒﺮﻣﻦ‬ ‫ﻣﺴﺎﺣﺔ‬ ‫ﻋﺎدة‬ ‫ﯾﺘﻄﻠﺐ‬ ‫اﻟﺬي‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﻣﻦ‬ ‫ﯾﻘﺮب‬ ‫أﺳﻠﻮب‬ ‫ﯾﺴﺘﺨﺪم‬
.‫اﻟﻤﺮاﻓﻖ‬ ‫ﺗﺪرج‬ ‫أﺳﻠﻮب‬ ‫ﻣﻦ‬ ‫أﻗﻞ‬ ‫ﺗﻜﺮارات‬ ‫ﯾﺘﻄﻠﺐ‬ ‫ﻟﻜﻨﮫ‬
o‫اﻟ‬‫ﻤﺮاﻓﻖ‬‫أﻛﺜﺮ‬ ‫ﺗﻜﺮارات‬ ‫ﯾﺤﺘﺎج‬ ‫ﻋﺎدة‬ ‫ﻟﻜﻨﮫ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫أﺳﻠﻮب‬ ‫ﻣﻦ‬ ‫أﻗﻞ‬ ‫ذاﻛﺮة‬ ‫ﯾﺘﻄﻠﺐ‬
‫اﻟﺨﯿﺎر‬ ‫ھﺬا‬ ‫ﯾﺴﺘﺨﺪم‬ .‫اﻟﺪﻗﺔ‬ ‫ﻣﻦ‬ ‫ﻣﻌﯿﻦ‬ ‫ﻣﺴﺘﻮى‬ ‫إﻟﻰ‬ ‫ﻟﻠﻮﺻﻮل‬‫ﻣﺴﺄﻟﺔ‬ ‫ﻟﺪﯾﻚ‬ ‫ﻛﺎﻧﺖ‬ ‫إذا‬
‫أو‬ ‫اﻟﻤﺸﻜﻠﺔ‬ ‫ھﻮ‬ ‫اﻟﺬاﻛﺮة‬ ‫اﺳﺘﺨﺪام‬ ‫وﯾﻜﻮن‬ ‫ﻛﺒﯿﺮة‬‫اﻟﺘﻜﺮار‬ ‫ﻋﻤﻠﯿﺎت‬ ‫ﺗﻨﻔﯿﺬ‬ ‫ﯾﺘﺴﺒﺐ‬ ‫ﻋﻨﺪﻣﺎ‬
.‫اﻟﺘﻘﺪم‬ ‫إﺑﻄﺎء‬ ‫ﻓﻲ‬
‫اﻟﮭ‬ ‫وﺗﺎﺑﻊ‬ ‫اﻟﻘﯿﻮد‬ ‫ﺗﺤﺪﯾﺪ‬ ‫ﺑﻌﺪ‬‫ﻧﺎﻓﺬة‬ ‫ﺗﻈﮭﺮ‬ ‫واﻟﺨﯿﺎرات‬ ‫ﺪف‬Solver Parameters‫ﺑﺎﻟﺸﻜﻞ‬6-2:

‫اﻟﺸﻜﻞ‬6-2
‫اﻟﺰر‬ ‫ﻋﻠﻰ‬ ‫ﻧﻀﻐﻂ‬ ‫أن‬ ‫ﯾﻜﻔﻲ‬ ‫اﻟﻨﺎﻓﺬة‬ ‫ھﺬه‬ ‫ﻓﻲ‬ ‫اﻟﻤﻌﺮﻓﺔ‬ ‫اﻟﺸﺮوط‬ ‫ﺣﺴﺐ‬ ‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫ﻋﻠﻰ‬ ‫ﻟﻠﺤﺼﻮل‬
solve:‫اﻵﺗﯿﺔ‬ ‫اﻷرﺑﻊ‬ ‫اﻟﺮﺳﺎﺋﻞ‬ ‫ﻣﻦ‬ ‫واﺣﺪة‬ ‫ﻋﻠﻰ‬ ‫ﺳﻨﺤﺼﻞ‬ ‫وﺑﻌﺪھﺎ‬
1-"Solver found a solution. All constraints and optimality conditions are
satisfied‫اﻟـ‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫ﻣﻤﺎ‬ "Solver.‫ﻟﻠﻨﻤﻮذج‬ ‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫وﺟﺪ‬
2-“Cell values did not converge‫ﻣﺎ‬ ‫إﻟﻰ‬ ‫ﺗﺼﻞ‬ ‫اﻟﮭﺪف‬ ‫داﻟﺔ‬ ‫أن‬ ‫إﻟﻰ‬ ‫ھﺬا‬ ‫وﯾﺸﯿﺮ‬ ".‫ﻻﻧﮭﺎﺋﯿﺔ‬
.‫ﺧﺎطﺌﺔ‬ ‫داﻟﺔ‬ ‫إدﺧﺎل‬ ‫أو‬ ‫ﻗﯿﺪ‬ ‫ﻛﺘﺎﺑﺔ‬ ‫ﻧﺴﯿﺎن‬ ‫ﻋﻦ‬ ‫ﻧﺎﺗﺞ‬ ‫وھﺬا‬
3-"Solver could not find a feasible solution‫ﻋﻠﻰ‬ ‫اﻟﺤﺼﻮل‬ ‫ﻋﺪم‬ ‫إﻟﻰ‬ ‫ﯾﺸﯿﺮ‬ ‫وھﺬا‬ "
‫ﺣﻞ‬،‫ﻣﻤﻜﻦ‬. ‫اﻟﺪوال‬ ‫أو‬ ‫ﻟﻠﻘﯿﻮد‬ ‫ﺻﺤﯿﺢ‬ ‫ﻏﯿﺮ‬ ‫إدﺧﺎل‬ ‫ﻣﻦ‬ ‫وﯾﻨﺘﺞ‬
4-"Conditions for Assume Linear Model not Satisfied‫إﻟﻰ‬ ‫ﺗﺸﯿﺮ‬ ‫اﻟﺮﺳﺎﻟﺔ‬ ‫ھﺬه‬ "
.‫ﺧﻄﯿﺔ‬ ‫ﻏﯿﺮ‬ ‫ﺻﯿﻐﺔ‬ ‫أو‬ ‫داﻟﺔ‬ ‫إدﺧﺎل‬
‫اﻟﻨﻤﻮذﺟﻲ‬ ‫اﻟﺤﻞ‬ ‫إﯾﺠﺎد‬ ‫ﺣﺎل‬ ‫ﻓﻲ‬‫اﻟﻨﺎﻓﺬة‬ ‫ﺗﻈﮭﺮ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬7-2:
‫اﻟﺸﻜﻞ‬7-2

‫ﻟﻠﻤﺘﻐﯿﺮات‬ ‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫ﻗﯿﻢ‬ ‫ﺗﺜﺒﯿﺖ‬ ‫أردﻧﺎ‬ ‫إذا‬S,W‫اﺧﺘﯿﺎر‬ ‫ﻋﻠﯿﻨﺎ‬ ‫ﯾﺠﺐ‬Keep Solver Solution
‫ﻧﺨﺘﺎر‬ ‫اﻷﺳﺎﺳﯿﺔ‬ ‫ﻗﯿﻤﮭﺎ‬ ‫إﻟﻰ‬ ‫اﻟﻤﺘﺤﻮﻻت‬ ‫ﻗﯿﻢ‬ ‫إﻋﺎدة‬ ‫أردﻧﺎ‬ ‫إذا‬ ‫أﻣﺎ‬Restore Original Values.
‫اﻟﺰر‬ ‫ﻋﻠﻰ‬ ‫اﻟﻀﻐﻂ‬ ‫ﯾﻜﻔﻲ‬ ‫اﻟﺤﻞ‬ ‫إﻧﮭﺎء‬OK.
‫اﻟﺸﻜﻞ‬8-2
‫اﻟﻤ‬ ‫ﻟﮭﺬه‬ ‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫أن‬ ‫ﻧﻼﺣﻆ‬‫ھﻮ‬ ‫ﺴﺄﻟﺔ‬S=50,W=50
‫اﻟﺬي‬ ‫اﻟﺤﻞ‬ ‫ﻧﻔﺲ‬ ‫وھﻮ‬‫و‬.‫اﻟﺒﯿﺎﻧﯿﺔ‬ ‫اﻟﻄﺮﯾﻘﺔ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫ﻋﻠﯿﮫ‬ ‫ﺻﻠﻨﺎ‬‫ﺗﻜﻮن‬ ‫اﻟﺤﻞ‬ ‫ﻟﮭﺬا‬ ‫اﻟﻮﺻﻮل‬ ‫وﻋﻨﺪ‬‫ﻧﺴﺒﺔ‬
‫ھﻲ‬ ‫اﻟﻤﺰﯾﺞ‬ ‫ﻓﻲ‬ ‫اﻟﺒﺮوﺗﯿﻦ‬29‫ھﻲ‬ ‫اﻟﻜﺮﺑﻮھﯿﺪرات‬ ‫وﻧﺴﺒﺔ‬ %55.%
‫إذن‬‫ﺑﺮﻧﺎﻣﺞ‬ ‫ﻓﻲ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻧﻤﻮذج‬ ‫وﺣﻞ‬ ‫ﻟﺼﯿﺎﻏﺔ‬‫اﻹﻛﺴﻞ‬‫ﺑﺎﻟﺨﻄﻮات‬ ‫اﻟﻌﻤﻞ‬ ‫ﯾﻨﺒﻐﻲ‬‫اﻵﺗﯿﺔ‬:
1.‫اﻟﺒﯿﺎﻧﺎ‬ ‫إدﺧﺎل‬‫ﺻﺤﯿﺢ‬ ‫ﺑﺸﻜﻞ‬ ‫ت‬
2.‫اﻟﻤﻄﻠﻮﺑﺔ‬ ‫اﻟﺼﯿﻎ‬ ‫ﻛﺘﺎﺑﺔ‬
3.(‫اﻟﮭﺪف‬ ‫)ﺧﻠﯿﮫ‬ ‫اﻟﮭﺪف‬ ‫ﺧﻠﯿﺔ‬ ‫ﺗﻌﺮﯾﻒ‬
4.‫اﻟﻤﺘﻐﯿﺮة‬ ‫اﻟﺨﻼﯾﺎ‬ ‫ﺗﺤﺪﯾﺪ‬
5.‫إﺿﺎﻓﺔ‬‫اﻟﻘﯿﻮد‬
6.‫ﺧﯿﺎر‬‫ا‬‫اﻟﺤﻞ‬ ‫ت‬
7.‫اﻟﻨﻤﻮذج‬ ‫ﺣﻞ‬

‫اﻟﺤﻞ‬ ‫ﺑﻌﺪ‬ ‫)ﻣﺎ‬ ‫اﻟﺤﺴﺎﺳﯿﺔ‬ ‫ﺗﺤﻠﯿﻞ‬‫اﻷﻣﺜﻞ‬(Sensitivity analysis
‫إﺿﺎﻓﺔ‬‫إﻟﻰ‬‫اﻟﺤﻞ‬ ‫ﻣﻦ‬ ‫ﻋﻠﯿﮭﺎ‬ ‫ﺣﺼﻠﻨﺎ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻤﻌﻠﻮﻣﺎت‬‫اﻷﻣﺜﻞ‬‫ﺧﯿﺎرات‬ ‫ھﻨﺎك‬ ‫ﻓﺎن‬ ،‫أ‬‫ﺧﺮى‬‫ﻟﻨﺎ‬ ‫ﯾﻮﻓﺮھﺎ‬
Solver‫اﻟﺘﻘﺮﯾﺮ‬ ‫وھﻲ‬answer report. ‫اﻟﺤﺴﺎﺳﯿﺔ‬ ‫ﺗﺤﻠﯿﻞ‬ ‫و‬
‫ﻟﻠﺤﻞ‬ ‫ﺗﻔﺼﯿﻠﯿﺔ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫ﯾﻌﻄﯿﻨﺎ‬ ‫ﻓﺎﻟﺘﻘﺮﯾﺮ‬‫اﻷﻣﺜﻞ‬‫ﻟﻨﺎ‬ ‫ﯾﺒﯿﻦ‬ ‫وﻛﺬﻟﻚ‬ ‫ﺗﻘﺮﯾﺮ‬ ‫ﺷﻜﻞ‬ ‫ﻋﻠﻰ‬‫أي‬‫اﻟﻘﯿﻮد‬ ‫ﻣﻦ‬
. ‫ﻣﻠﺰﻣﺔ‬ ‫ﺗﻜﻮن‬‫وأﯾﻀﺎ‬‫ﻣﻠﺰﻣﺔ‬ ‫ﻏﯿﺮ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﯿﻮد‬ ‫ﻋﻦ‬ ‫ﺻﻮرة‬ ‫ﯾﻌﻄﯿﻨﺎ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬9-2
Microsoft Excel 12.0 Answer Report
Worksheet: [linear programming.xls]Data
Report Created: 27/07/2009 12:15:01 ‫ص‬
Target Cell (Min)
Cell Name Original Value Final Value
$D$9 W ‫اﻟﮭدف‬ ‫ﺗﺎﺑﻊ‬ ‫ﻗﯾﻣﺔ‬ 2500.000043 2500.000043
Adjustable Cells
Cell Name Original Value Final Value
$C$3 S= ‫اﺑﺗداﺋﯾﺔ‬ ‫ﻗﯾم‬ 50.00000233 50.00000233
$C$4 W= ‫اﺑﺗداﺋﯾﺔ‬ ‫ﻗﯾم‬ 49.99999867 49.99999867
Constraints
Cell Name Cell Value Formula Status Slack
$E$13 S ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 50.00000233 $E$13>=$F$13 Not Binding 50.00000233
$E$14 W ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 49.99999867 $E$14>=$F$14 Not Binding 49.99999867
$E$15 S+W ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 100.000001 $E$15=$F$15 Not Binding 0
$E$16 Protein ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 29.00000095 $E$16>=$F$16 Not Binding 9.00000095
$E$17 Carbohydrate ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 55 $E$17<=$F$17 Binding 0
) ‫اﻟﺸﻜﻞ‬9-2(
‫ﺗﺤ‬ ‫أﻣﺎ‬(‫ﻣﺜﻼ‬ ‫)اﻟﻘﯿﻮد‬ ‫اﻟﺒﯿﺎﻧﺎت‬ ‫ﺗﻐﯿﺮ‬ ‫ﯾﺆدي‬ ‫ﻛﻢ‬ ‫ﯾﺨﺒﺮﻧﺎ‬ ‫ﻓﮭﻮ‬ ‫اﻟﺤﺴﺎﺳﯿﺔ‬ ‫ﺗﺤﻠﯿﻞ‬ ‫ﺗﻘﺮﯾﺮ‬ ‫أو‬ ‫اﻟﺤﺴﺎﺳﯿﺔ‬ ‫ﻠﯿﻞ‬
‫اﻷﻣﺜﻞ‬ ‫اﻟﺤﻞ‬ ‫ﻧﺘﯿﺠﺔ‬ ‫ﻓﻲ‬) ‫اﻟﺸﻜﻞ‬10-2(

Microsoft Excel 12.0 Sensitivity Report
Worksheet: [linear programming.xls]Data
Report Created: 27/07/2009 12:15:01 ‫ص‬
Adjustable Cells
Final Reduced
Cell Name Value Gradient
$C$3 S= ‫اﺑﺗداﺋﯾﺔ‬ ‫ﻗﯾم‬ 50.00000233 0
$C$4 W= ‫اﺑﺗداﺋﯾﺔ‬ ‫ﻗﯾم‬ 49.99999867 0
Constraints
Final Lagrange
Cell Name Value Multiplier
$E$13 S ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 50.00000233 0
$E$14 W ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 49.99999867 0
$E$15 S+W ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 100.000001 43.33333433
$E$16 Protein ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 29.00000095 0
$E$17 Carbohydrate ‫ﻟﻠﻣﺗراﺟﺣﺔ‬ ‫اﻷﯾﺳر‬ ‫اﻟطرف‬ 55 -33.33333532
) ‫اﻟﺸﻜﻞ‬10-2(

‫ﺑﺮﻧﺎﻣﺞ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﺣﻞ‬Matlab
‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺎﺋﻞ‬ ‫ﺣﻞ‬ ‫ﻋﻤﻠﯿﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺘﺎﻟﯿﺔ‬ ‫اﻟﺼﯿﻐﺔ‬ ‫ﺑﺮﻧﺎﻣﺞ‬ ‫ﯾﺴﺘﺨﺪم‬
‫ﯾﺘﻢ‬‫اﻟﺘﺎﻟﯿﺔ‬ ‫اﻟﺘﻌﻠﯿﻤﺔ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫اﻟﺴﺎﺑﻘﺔ‬ ‫اﻟﻤﻌﺎدﻻت‬ ‫ﻓﻲ‬ ‫اﻟﻤﺒﯿﻨﺔ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺄﻟﺔ‬ ‫ﺣﻞ‬
; l; u):eq; beqAA; b;x = linprog(f;
:‫ﻣﺜﺎل‬
‫اﻟﺘﺎﻟﯿﺔ‬ ‫اﻟﻤﺴﺄﻟﺔ‬ ‫ﺣﻞ‬ ‫ﯾﺮاد‬
‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫ﻛﺘﺎﺑﺘﮭﺎ‬ ‫ﯾﻤﻜﻦ‬ ‫ﻣﺎﺗﻼب‬ ‫ﺻﯿﻎ‬ ‫إﻟﻰ‬ ‫اﻟﺨﻄﯿﺔ‬ ‫اﻟﺒﺮﻣﺠﺔ‬ ‫ﻣﺴﺄﻟﺔ‬ ‫ﺗﺤﻮﯾﻞ‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬

‫اﻟﻤﻌﻄﯿﺎت‬ ‫ادﺧﺎل‬
>> f = -[4;2;1];
>> A = [2 1 0;1 0 2];
>> b = [1;2];
>> Aeq = [1 1 1];
>> beq = [1];
>> l = [0;0;0];
>> u = [1;1;2];
>> x = linprog(f,A,b,Aeq,beq,l,u)
‫اﻟﻨﺘﯿﺠﺔ‬
Optimization terminated successfully.
x =
0.5000
0.0000
0.5000
‫ﺗﺘﺮ‬ ‫اﻟﺴﺎﺑﻘﺔ‬ ‫اﻟﻤﺤﺪدات‬ ‫أﺣﺪ‬ ‫وﺟﻮد‬ ‫ﻋﺪم‬ ‫ﺣﺎل‬ ‫ﻓﻲ‬‫وﺟﻮد‬ ‫ﻋﺪم‬ ‫ﺣﺎل‬ ‫ﻓﻲ‬ ‫اﻟﻤﺜﺎل‬ ‫ﺳﺒﯿﻞ‬ ‫ﻓﻌﻠﻰ‬ ، ‫ﻓﺎرﻏﺔ‬ ‫ك‬
:‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫ﻓﺎرﻏﺔ‬ ‫اﻷدﻧﻰ‬ ‫اﻟﺤﺪ‬ ‫ﻣﺼﻔﻮﻓﺔ‬ ‫ﺗﺘﺮك‬ ‫اﻷدﻧﻰ‬ ‫اﻟﺤﺪ‬ ‫ﻣﺤﺪدات‬
>> l = [];
‫اﻟﻨﺘﯿﺠﺔ‬
Optimization terminated successfully.
x =
0.6667

-0.3333
0.6667
‫اﻟﻤﺴﺎ‬ ‫ﻣﺼﻔﻮﻓﺎت‬ ‫ﺗﺘﺮك‬ ‫اﻟﻤﺴﺎواة‬ ‫ﻣﺤﺪدات‬ ‫وﺟﻮد‬ ‫ﻋﺪم‬ ‫ﺣﺎل‬ ‫ﻓﻲ‬ ‫ﻣﺸﺎﺑﮫ‬ ‫وﺑﺸﻜﻞ‬‫اﻟﺸﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫ﻓﺎرﻏﺔ‬ ‫واة‬
:‫اﻟﺘﺎﻟﻲ‬
>> Aeq = [];
>> beq = [];
‫واﻟﻨﺘﯿﺠﺔ‬
Optimization terminated successfully.
x =
0.0000
1.0000
1.0000
‫ﻣﺜﺎل‬‫اﻟﻤﺎﺗﻼب‬ ‫ﺑﺮﻧﺎﻣﺞ‬ ‫ﺑﺎﺳﺘﺨﺪام‬ ‫اﻟﻐﺬاﺋﻲ‬ ‫اﻟﺨﺒﺰ‬ ‫ﻣﺴﺄﻟﺔ‬ ‫ﺣﻞ‬ ‫ﯾﺮاد‬ :
1-‫اﻟﻤﻌﺎدﻻت‬ ‫ﺗﻨﺴﯿﻖ‬
Min 20 X1 + 30 X2
X1 + X2 = 100
X1 ≥ 0
X2 ≥ 0
X1 ≤ 100
X2 ≤ 100
-0.11 X1 - 0.47 X2 ≤ 20
0.70 X1 + 0.40 X2 ≤ 55
‫ﺗﻌﻠﻤﯿﺔ‬ ‫إن‬ :‫ﻣﻼﺣﻈﺔ‬linprog‫اﻟﺘﺎﻟﯿﺔ‬ ‫اﻟﺼﯿﻔﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻌﺘﻤﺪ‬
‫إﻟﻰ‬ ‫اﻟﻤﻘﯿﺪات‬ ‫ﺟﻤﯿﻊ‬ ‫ﺗﺤﻮﯾﻞ‬ ‫ﯾﺠﺐ‬ ‫ﻟﺬﻟﻚ‬ ‫ﻣﻌﯿﻨﺔ‬ ‫ﻗﯿﻤﺔ‬ ‫ﺗﺴﺎوي‬ ‫أو‬ ‫أﻛﺒﺮ‬ ‫ﺷﺮطﮭﺎ‬ ‫ﻣﻌﺎدﻻت‬ ‫ﯾﻮﺟﺪ‬ ‫ﻻ‬ ‫أﻧﮫ‬ ‫أي‬
‫وﺿﻊ‬ ‫ﻟﺬﻟﻚ‬ ‫اﻟﺼﯿﻐﺔ‬ ‫ھﺬه‬.‫اﻟﺒﺮوﺗﯿﻦ‬ ‫ﻣﺤﺘﻮى‬ ‫ﻣﻘﯿﺪ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻨﺎﻗﺺ‬ ‫إﺷﺎرة‬
2-‫اﻟﻤﺪﺧﻼت‬ ‫ﻣﺼﻔﻮﻓﺎت‬ ‫وﺿﻊ‬

[ ] [ ]
20
f=
30
-0.11 0.47 20
A= b=
0.70 0.40 55
Aeq= 1 1 beq= 100
100 0
u= l=
100 0
 
 
 
−   
   
   
   
   
   
3-‫اﻟﻤﻌﻄﯿﺎت‬ ‫إدﺧﺎل‬
>> f = [20;30];
>> A = [-0.11 -0.47; 0.70 0.40];
>> b = [20; 55];
>> Aeq = [1 1];
>> beq = [100];
>> l = [0; 0];
>> u = [100;100];
>> x = linprog(f, A, b, Aeq, beq, l, u)
‫اﻟﻨﺘﯿﺠﺔ‬
Optimization terminated successfully.
x =
50.000
50.000