From e85c0306838330b4efe48a5a4aff559aa5a64114 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=F0=9F=9A=80=20=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Thu, 24 Apr 2025 21:21:50 -0300 Subject: [PATCH] Delete class_8- 1st HandMade EXAM -LP -MathModels - Simplex directory MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana 🚀 Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- .../1st Handmade Exam- Questions.pdf | Bin 254543 -> 0 bytes .../answer_1.md | 171 ------------------ .../answer_1.pdf | Bin 171460 -> 0 bytes .../question_1.png | Bin 305859 -> 0 bytes .../answer_2.md | 110 ----------- .../answer_2.pdf | Bin 133327 -> 0 bytes .../graphic.png | Bin 188009 -> 0 bytes .../question_2.png | Bin 70136 -> 0 bytes ...g Problem Solving by the Simplex Method.md | 139 -------------- .../answer_3.pages | Bin 735801 -> 0 bytes .../question_3.png | Bin 70833 -> 0 bytes 11 files 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Workflow/answer_1.md +++ /dev/null @@ -1,171 +0,0 @@ - - -## **Problema de Programação Linear** - -**Objetivo**: Maximizar o lucro semanal da empresa de brinquedos. -**Variáveis de decisão**: - -- \$ x_1 \$ = número de carrinhos produzidos por semana. -- \$ x_2 \$ = número de triciclos produzidos por semana. - -**Função objetivo**: - -$$ -\text{Max } Z = 12x_1 + 60x_2 -$$ - -**Restrições**: - -1. **Usinagem**: \$ 15x_1 + 30x_2 \leq 2160 \$ minutos (36 horas). -2. **Pintura**: \$ 6x_1 + 45x_2 \leq 1320 \$ minutos (22 horas). -3. **Montagem**: \$ 6x_1 + 24x_2 \leq 900 \$ minutos (15 horas). -4. **Não negatividade**: \$ x_1 \geq 0 \$, \$ x_2 \geq 0 \$. - ---- - -## **Passo 1: Converter para a Forma Padrão** - -Adicione variáveis de folga (\$ s_1, s_2, s_3 \$) para transformar as desigualdades em igualdades: - -1. \$ 15x_1 + 30x_2 + s_1 = 2160 \$ -2. \$ 6x_1 + 45x_2 + s_2 = 1320 \$ -3. \$ 6x_1 + 24x_2 + s_3 = 900 \$ -4. \$ x_1, x_2, s_1, s_2, s_3 \geq 0 \$ - -A função objetivo fica: - -$$ -Z - 12x_1 - 60x_2 = 0 -$$ - ---- - -## **Passo 2: Montar a Tabela Simplex Inicial** - -| Base | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | Solução | -| :-- | :-- | :-- | :-- | :-- | :-- | :-- | -| $s_1$ | 15 | 30 | 1 | 0 | 0 | 2160 | -| $s_2$ | 6 | 45 | 0 | 1 | 0 | 1320 | -| $s_3$ | 6 | 24 | 0 | 0 | 1 | 900 | -| **Z** | -12 | -60 | 0 | 0 | 0 | 0 | - ---- - -## **Passo 3: Identificar a Variável que Entra na Base** - -Na linha Z, o coeficiente mais negativo é **-60** (de $x_2$). Portanto, $x_2$ entra na base. - ---- - -## **Passo 4: Calcular a Razão Mínima (Variável que Sai)** - -Divida a coluna "Solução" pelos coeficientes de $x_2$ (apenas valores positivos): - -- $s_1$: \$ 2160 / 30 = 72 \$ -- $s_2$: \$ 1320 / 45 \approx 29,33 \$ -- $s_3$: \$ 900 / 24 = 37,5 \$ - -**Menor razão**: 29,33 (linha de $s_2$). Logo, $s_2$ sai da base. - ---- - -## **Passo 5: Pivotear na Linha de $s_2$** - -**Linha pivô (linha 2)**: -\$ 6x_1 + 45x_2 + s_2 = 1320 \$ -Divida por 45 para tornar o coeficiente de $x_2$ igual a 1: -\$ x_2 = \frac{1320}{45} - \frac{6}{45}x_1 - \frac{1}{45}s_2 \$ -Simplificando: -\$ x_2 = 29,33 - 0,133x_1 - 0,022s_2 \$ - -**Atualizar as outras linhas**: - -1. **Linha 1 ($s_1$)**: -\$ s_1 = 2160 - 15x_1 - 30x_2 \$ -Substitua $x_2$: -\$ s_1 = 2160 - 15x_1 - 30(29,33 - 0,133x_1 - 0,022s_2) \$ -\$ s_1 = 2160 - 15x_1 - 880 + 4x_1 + 0,66s_2 \$ -\$ s_1 = 1280 - 11x_1 + 0,66s_2 \$ -2. **Linha 3 ($s_3$)**: -\$ s_3 = 900 - 6x_1 - 24x_2 \$ -Substitua $x_2$: -\$ s_3 = 900 - 6x_1 - 24(29,33 - 0,133x_1 - 0,022s_2) \$ -\$ s_3 = 900 - 6x_1 - 704 + 3,2x_1 + 0,53s_2 \$ -\$ s_3 = 196 - 2,8x_1 + 0,53s_2 \$ -3. **Linha Z**: -\$ Z - 12x_1 - 60x_2 = 0 \$ -Substitua $x_2$: -\$ Z - 12x_1 - 60(29,33 - 0,133x_1 - 0,022s_2) = 0 \$ -\$ Z - 12x_1 - 1760 + 8x_1 + 1,32s_2 = 0 \$ -\$ Z = 1760 - 4x_1 + 1,32s_2 \$ - ---- - -## **Passo 6: Nova Tabela após Pivoteamento** - -| Base | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | Solução | -| :-- | :-- | :-- | :-- | :-- | :-- | :-- | -| $s_1$ | -11 | 0 | 1 | 0,66 | 0 | 1280 | -| $x_2$ | -0,133 | 1 | 0 | -0,022 | 0 | 29,33 | -| $s_3$ | -2,8 | 0 | 0 | 0,53 | 1 | 196 | -| **Z** | -4 | 0 | 0 | 1,32 | 0 | 1760 | - ---- - -## **Passo 7: Verificar Otimalidade** - -Na linha Z, ainda há coeficiente negativo: **-4** (de $x_1$). -**Variável que entra**: $x_1$. - ---- - -## **Passo 8: Calcular Nova Razão Mínima** - -Divida a coluna "Solução" pelos coeficientes de $x_1$ (apenas valores positivos): - -- $s_1$: \$ 1280 / (-11) \$ → **negativo** (ignorado). -- $x_2$: \$ 29,33 / (-0,133) \$ → **negativo** (ignorado). -- $s_3$: \$ 196 / (-2,8) \$ → **negativo** (ignorado). - -**Não há razões válidas**. Isso indica que **não há solução ótima finita** (problema ilimitado), mas como todas as restrições são limitantes, revisamos o cálculo. - ---- - -## **Passo 9: Correção e Conclusão** - -O erro ocorreu porque os coeficientes de $x_1$ nas linhas básicas são **negativos**, o que significa que aumentar $x_1$ não viola as restrições. Portanto, **a solução ótima já foi encontrada na iteração anterior**. - ---- - -## **Solução Ótima** - -- **Variáveis básicas**: -\$ x_2 = 29,33 \$ (triciclos), \$ s_1 = 1280 \$, \$ s_3 = 196 \$. -- **Variáveis não básicas**: -\$ x_1 = 0 \$ (carrinhos), \$ s_2 = 0 \$. - -**Lucro máximo**: - -$$ -Z = 1760 - 4(0) + 1,32(0) = 1760 \text{ reais} -$$ - ---- - -## **Validação das Restrições** - -1. **Usinagem**: -\$ 15(0) + 30(29,33) = 880 \leq 2160 \$ ✓ -2. **Pintura**: -\$ 6(0) + 45(29,33) = 1320 \leq 1320 \$ ✓ -3. **Montagem**: -\$ 6(0) + 24(29,33) = 704 \leq 900 \$ ✓ - ---- - -## **Resposta Final** - -A empresa deve produzir **29 triciclos** e **nenhum carrinho** para maximizar o lucro, atingindo **R\$ 1.760,00** por semana. - - - diff --git a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_1-Mathematical Modeling & Linear Optimization-Full Problem Solving Workflow/answer_1.pdf b/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_1-Mathematical Modeling & Linear Optimization-Full Problem Solving Workflow/answer_1.pdf deleted file mode 100644 index 8a49aa580d8f4c8b5f1a26b4a91dce99e600d669..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 171460 zcmd43Q&xUojxFcG=kZ$ z^>}76q<)2Y0@1~vEo(qE(EB2d2Uv-*&Lo&ZegyELefd6@c8^+}90z2_gBeb-sL=H+ z?6!{AEz7;}>=&e61~~j;5~f>R1^>%&S7a+e&6Iq#K-u2pAnKZFX&q^yt?-)`7na8>Y zAZ&GsyiG%@t*2z^R@+8loGx=;^+wY!R&N`!PH!>axvBlQ^t%(QH*in}RZ~;W^Way~ zHfy48PHbL$ItzNi+H#Cciv{7N7c*+xS6Y$h_r-Rb8lGApH@^H026K>#2&kYVU!$g5PLFSE~I8UesBc4#8qE96u??VE03 zR2QaLNUgZ>fRkey*}8rj6Hkc%3Q>J$p|L+{Go#Ke>57B}azchYZyX59tqfrgETwC zYg$?*!Pax1AVod41=@zx#a3Ue85U}m-POspbDU~dZXvNt?I>##0zf}<3SA$h=<5)g z5@HT6&36kwm^C;wQ`2eRj?c2Cp=>8IemgNlOlK*pNp#gjph;-MkY6BC&o-3nWPb)y ze%**QF4qB{glkz~u}eb@-SRn@WZ&`Ly2OR*rA^ekoljZ9>@I^xB}<);S1K;KET&_1 zEcqrNuWFQ&ft6>*ZIWGWDJ{6!>7nrdx5FU_H8?!Qr)HixSA)VFdac2?u9bkr$j_e+s_+y810?@tq zVdu6;lztT~w?4a?Sz?NMQt*Y*_u$n0;wjw~jE4zX!<($eMXN&O8IK2P2XiEy^0nOK z&Om;wm;jCq=B_bHRtU*GHt@WTczFYMyfl63MOZ=O__s_+My+8pS?O4OCg2Xx&t2ZSyAPJVH~V9NlQ2}ttq6*2NM zTqgu4oC697hsh+-@GMaeU`DxI*Ep|VC_|Rd7+f2CP1Uu<2r0E5UC=oq@xdQ>F@ume z-nHe~DaJ`Ns4X?ro5=0F-zkhYB_He%Ox;Lml(uQfXkD!MY6;={_*P`OO~Qi^6r?Ds z1#9;x}3ul(FCy+kR zMLP^047{mj7`6&a^0pV&S#DPbM7BqnWk1a47mT}L_m>os9WX>HaL_^gG??m+z_e#= zr+$D-M6T+8*PfALi*C5dH;sqx0sp2|;VMe@D3~`eUzB?p4}(ukD;dU_7(E<8=S_!f ze)5)ah`(fn5j9PME3huneu?JGCLbB=FK|^}WvDS7YHqes8wzu|Rzv)ynrz7?oz#6% zSmsH$GZ`e(U1ti(`hLL;Z(nDcnBGFeR~I~KQIzR6nWF#&!f4+AO+D3lQjeAlq%(1oQP5xn2LH$ zl?<4!GtDX0n+Qblsx!t3ZLRSuJ{0Ee34ocYvrOW=QvjuvlbJ9ww_NgR)cdK;JhDUs z5+%yTNO6nu*u&%(?U*WglxB!fpQ4J z!|fGwyAXjJA9KuSzEiD*jV%G*SFL?rVn%)I2vPvjb|H|3SpBNk4sFNl`ZMfTzmdx+ zLA4vylo=6v>4jkxk{^(K9{PQY(~MT7^kR@NQTbvkwuixoC2)g)R2)gbu6Tr&hR4A< zq{B90mt}?$;3H^t%bSpD1Fbuo2){TYQu@7Ph$;Kg!C=$SzZ&~pk;VYjHs_E#vGNq> z7*9H#7;L+WSXXSS5zp5$wU#m~m(VZ~3gPk0Rf_K!SKvA>t9iNV64_*Cf1ukS=Lg89 z%~!g>^lMxfR#+30vIEj`03pRT;1Je?_3aZ(3d?i4s!2b_mWXGot7y5gKk*0Stt!PQ5;z_>j%^?X~oWSL)2uoo}GJMPt*(X;FVh2=1dzANktvz}|vb zA7yGTv9@gRg<0vp&YUgbCUXyEU1ykzZ{BwA-Kc<*OzZCQtiW^Bx%pz>htYRAIc@rb z0q=Q%P=I%BwAV$ZSvQudLx`Ayup#>AlUiZ5Q5tG(k(k1ezd~i>WSfaifaDX$&;f|b z>=g$bbE2Or898@gny#5y&vI%b8u*X|HkM*~Zi<7vsO;zAMuq!s0xRcnJ|zSwE|>>Q zznf!fWfMgrq%1<%^0^Ic!f+Q@@ zlN6y>mgb;DyN)nRKzoz+~gV5y-%>MD~XkzP3!1SkEk$_&&#L3RZ(a6M!faRYLgzapd|MWW% z`~~>`43IG~wlEN~b0^SZ`162`k%@qvlSv2a55oV~d;XgDH!+ZRv@=pNaVF6E^D+@J z0(xZ=cV_|}0(xOPYdc3JdjlgAg1`8KuoEKz>%U&-#6-aO*UE~2NCN@AsJpYclJg%{ z@eiIM{%0H``+szb{R!3p8wp>{|NR^MuLA<{kOq? 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index 93bd29b..0000000 --- a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.md +++ /dev/null @@ -1,110 +0,0 @@ - - -## **Problema de Programação Linear** - -**Objetivo**: Maximizar a função objetivo. - -**Função objetivo**: - -$$ -\text{Max } Z = x_1 + 5x_2 -$$ - -**Restrições**: - -$$ -\begin{cases} -x_1 + x_2 \leq 5 \\ -x_1 + 2x_2 \leq 6 \\ -2x_1 + x_2 \leq 7 \\ -x_1 \geq 0, \quad x_2 \geq 0 -\end{cases} -$$ - ---- - -## **Passo 1: Converter as Desigualdades em Igualdades para Traçar as Retas** - -1. \$ x_1 + x_2 = 5 \$ - - Se \$ x_1 = 0 \$, então \$ x_2 = 5 \$ - - Se \$ x_2 = 0 \$, então \$ x_1 = 5 \$ -2. \$ x_1 + 2x_2 = 6 \$ - - Se \$ x_1 = 0 \$, então \$ 2x_2 = 6 \Rightarrow x_2 = 3 \$ - - Se \$ x_2 = 0 \$, então \$ x_1 = 6 \$ -3. \$ 2x_1 + x_2 = 7 \$ - - Se \$ x_1 = 0 \$, então \$ x_2 = 7 \$ - - Se \$ x_2 = 0 \$, então \$ 2x_1 = 7 \Rightarrow x_1 = 3,5 \$ - ---- - -## **Passo 2: Desenhar as Retas no Plano Cartesiano** - -1. Trace as retas no plano \$ (x_1, x_2) \$. -2. Identifique a região viável, que é a área delimitada pelas retas e pelos eixos \$ x_1 \geq 0 \$ e \$ x_2 \geq 0 \$. - ---- - -## **Passo 3: Determinar os Vértices da Região Viável** - -Os vértices são os pontos de interseção das retas. Vamos calcular: - -1. Interseção entre \$ x_1 + x_2 = 5 \$ e \$ x_1 + 2x_2 = 6 \$: - - Subtraindo a primeira equação da segunda: -\$ (x_1 + 2x_2) - (x_1 + x_2) = 6 - 5 \$ -\$ x_2 = 1 \$ - - Substituindo \$ x_2 = 1 \$ na primeira equação: -\$ x_1 + 1 = 5 \Rightarrow x_1 = 4 \$ - - Ponto: \$ (4, 1) \$ -2. Interseção entre \$ x_1 + x_2 = 5 \$ e \$ 2x_1 + x_2 = 7 \$: - - Subtraindo a primeira equação da segunda: -\$ (2x_1 + x_2) - (x_1 + x_2) = 7 - 5 \$ -\$ x_1 = 2 \$ - - Substituindo \$ x_1 = 2 \$ na primeira equação: -\$ 2 + x_2 = 5 \Rightarrow x_2 = 3 \$ - - Ponto: \$ (2, 3) \$ -3. Interseção entre \$ x_1 + 2x_2 = 6 \$ e \$ 2x_1 + x_2 = 7 \$: - - Multiplicando a primeira equação por 2: -\$ 2x_1 + 4x_2 = 12 \$ - - Subtraindo a segunda equação da resultante: -\$ (2x_1 + 4x_2) - (2x_1 + x_2) = 12 - 7 \$ -\$ 3x_2 = 5 \Rightarrow x_2 = \frac{5}{3} \$ - - Substituindo \$ x_2 = \frac{5}{3} \$ na primeira equação: -\$ x_1 + 2\left(\frac{5}{3}\right) = 6 \$ -\$ x_1 = 6 - \frac{10}{3} = \frac{18 - 10}{3} = \frac{8}{3} \$ - - Ponto: \$ \left(\frac{8}{3}, \frac{5}{3}\right) \approx (2.67, 1.67) \$ -4. Interseção com os eixos: - - \$ (0, 0) \$ - - \$ (3.5, 0) \$ - - \$ (0, 3) \$ - ---- - -## **Passo 4: Avaliar a Função Objetivo nos Vértices** - -1. \$ (0, 0) \Rightarrow Z = 0 + 5(0) = 0 \$ -2. \$ (3.5, 0) \Rightarrow Z = 3.5 + 5(0) = 3.5 \$ -3. \$ (0, 3) \Rightarrow Z = 0 + 5(3) = 15 \$ -4. \$ (4, 1) \Rightarrow Z = 4 + 5(1) = 9 \$ -5. \$ (2, 3) \Rightarrow Z = 2 + 5(3) = 17 \$ -6. \$ \left(\frac{8}{3}, \frac{5}{3}\right) \Rightarrow Z = \frac{8}{3} + 5\left(\frac{5}{3}\right) = \frac{8}{3} + \frac{25}{3} = \frac{33}{3} = 11 \$ - ---- - -## **Passo 5: Identificar a Solução Ótima** - -O maior valor de \$ Z \$ é 17, que ocorre no ponto \$ (2, 3) \$. - ---- - -## **Resposta Final** - -$$ -\boxed{ -x_1 = 2, \quad x_2 = 3, \quad Z_{\text{máximo}} = 17 -} -$$ - ---- - -Portanto, a solução ótima é \$ x_1 = 2 \$, \$ x_2 = 3 \$, e o valor máximo da função objetivo é 17. - diff --git a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear 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Z = 5x₁ + 4x₂ - -Sujeito a: - -- x₁ + 2x₂ ≤ 6 -- -2x₁ + x₂ ≤ 4 -- 5x₁ + 3x₂ ≤ 15 -- x₁, x₂ ≥ 0 - - -## Preparação do Problema para o Método Simplex - -### Introdução de Variáveis de Folga - -O primeiro passo é converter as restrições de desigualdade em equações, introduzindo variáveis de folga: - -- x₁ + 2x₂ + s₁ = 6 -- -2x₁ + x₂ + s₂ = 4 -- 5x₁ + 3x₂ + s₃ = 15 - -A função objetivo na forma padrão: -Z - 5x₁ - 4x₂ = 0 - -### Construção do Tableau Inicial - -O tableau inicial é formado pelos coeficientes das variáveis nas restrições, as variáveis de folga, e a função objetivo (com coeficientes negativos): - - -| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | -| :-- | :-- | :-- | :-- | :-- | :-- | :-- | -| s₁ | 1 | 2 | 1 | 0 | 0 | 6 | -| s₂ | -2 | 1 | 0 | 1 | 0 | 4 | -| s₃ | 5 | 3 | 0 | 0 | 1 | 15 | -| Z | -5 | -4 | 0 | 0 | 0 | 0 | - -## Resolução Passo a Passo pelo Método Simplex - -### Primeira Iteração - -**Passo 1: Identificar a variável que entra na base** - -Analisamos a linha Z e escolhemos a variável com coeficiente mais negativo. Neste caso, x₁ tem coeficiente -5, portanto x₁ entrará na base. - -**Passo 2: Identificar a variável que sai da base** - -Calculamos as razões para determinar qual variável sairá da base: - -- Linha 1 (s₁): 6/1 = 6 -- Linha 2 (s₂): não calculamos porque o coeficiente -2 não é positivo -- Linha 3 (s₃): 15/5 = 3 - -A menor razão positiva é 3, correspondente à linha 3, então s₃ sairá da base. - -**Passo 3: Operações de pivoteamento** - -O elemento pivô é 5 (linha 3, coluna x₁). - -1. Dividimos a linha do pivô (linha 3) pelo valor do pivô (5): --[^2][^1][^12] ÷ 5 = [1, 0.6, 0, 0, 0.2, 3] -2. Eliminamos os outros elementos da coluna pivô: - - Linha 1:[^1][^1][^3] - 1×[1, 0.6, 0, 0, 0.2, 3] = [0, 1.4, 1, 0, -0.2, 3] - - Linha 2: [-2, 1, 0, 1, 0, 4] - (-2)×[1, 0.6, 0, 0, 0.2, 3] = [0, 2.2, 0, 1, 0.4, 10] - - Linha Z: [-5, -4, 0, 0, 0, 0] - (-5)×[1, 0.6, 0, 0, 0.2, 3] = [0, -1, 0, 0, 1, 15] - -Obtemos o seguinte tableau: - - -| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | -| :-- | :-- | :-- | :-- | :-- | :-- | :-- | -| s₁ | 0 | 1.4 | 1 | 0 | -0.2 | 3 | -| s₂ | 0 | 2.2 | 0 | 1 | 0.4 | 10 | -| x₁ | 1 | 0.6 | 0 | 0 | 0.2 | 3 | -| Z | 0 | -1 | 0 | 0 | 1 | 15 | - -### Segunda Iteração - -**Passo 1: Identificar a variável que entra na base** - -Na linha Z, o coeficiente de x₂ é -1, então x₂ entrará na base. - -**Passo 2: Identificar a variável que sai da base** - -Calculamos as razões: - -- Linha 1 (s₁): 3/1.4 = 2.143 -- Linha 2 (s₂): 10/2.2 = 4.545 -- Linha 3 (x₁): 3/0.6 = 5 - -A menor razão positiva é 2.143, correspondente à linha 1, então s₁ sairá da base. - -**Passo 3: Operações de pivoteamento** - -O elemento pivô é 1.4 (linha 1, coluna x₂). - -1. Dividimos a linha do pivô (linha 1) pelo valor do pivô (1.4): - - [0, 1.4, 1, 0, -0.2, 3] ÷ 1.4 = [0, 1, 0.714, 0, -0.143, 2.143] -2. Eliminamos os outros elementos da coluna pivô: - - Linha 2: [0, 2.2, 0, 1, 0.4, 10] - 2.2×[0, 1, 0.714, 0, -0.143, 2.143] = [0, 0, -1.571, 1, 0.714, 5.286] - - Linha 3: [1, 0.6, 0, 0, 0.2, 3] - 0.6×[0, 1, 0.714, 0, -0.143, 2.143] = [1, 0, -0.429, 0, 0.286, 1.714] - - Linha Z: [0, -1, 0, 0, 1, 15] - (-1)×[0, 1, 0.714, 0, -0.143, 2.143] = [0, 0, 0.714, 0, 0.857, 17.143] - -Obtemos o seguinte tableau: - - -| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | -| :-- | :-- | :-- | :-- | :-- | :-- | :-- | -| x₂ | 0 | 1 | 0.714 | 0 | -0.143 | 2.143 | -| s₂ | 0 | 0 | -1.571 | 1 | 0.714 | 5.286 | -| x₁ | 1 | 0 | -0.429 | 0 | 0.286 | 1.714 | -| Z | 0 | 0 | 0.714 | 0 | 0.857 | 17.143 | - -### Verificação de Otimalidade - -Como todos os coeficientes na linha Z são não-negativos, atingimos a solução ótima. - -## Solução Ótima - -Expressando a solução em frações para maior precisão: - -- x₁ = 12/7 ≈ 1.714 -- x₂ = 15/7 ≈ 2.143 -- Valor máximo de Z = 120/7 ≈ 17.143 - - -## Conclusão - -Aplicando o método Simplex, encontramos que a solução ótima para o problema de programação linear é produzir aproximadamente 1.714 unidades do produto 1 e 2.143 unidades do produto 2, resultando em um valor máximo da função objetivo Z = 17.143. - -O método Simplex mostrou-se eficiente, convergindo para a solução ótima em apenas duas iterações. As variáveis de folga s₁ e s₃ são zero na solução ótima, indicando que as restrições correspondentes são ativas (estão no limite), enquanto s₂ = 5.286, indicando que a segunda restrição não é ativa. - diff --git a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3.pages b/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3.pages deleted file mode 100644 index 6e5ddbfe740564915e937d35ca43413b94a320cb..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 735801 zcmbTdcUTk4A1}NiflwkHOe8=cQ4kT6ppZyFiiBXHHx)f}!~>WFq=^_n5HW-#6k7mkfYF06r2M?zoPCIy*igI+o0t8Rr-H`v(wu1l0*Zw!E z|ID2EvK;7u%rLahe*DKX@3#Oj&;)%2eFcN80W|{<*Z}nVAz-<3Pj%3L@!!i<4j?r! zL|p@_sim#6Qs6IrKn(;2t3kl(>i@Y7=-5g=05MQEw8D947zIW{t!c*id``V4+Ou6~ z67+Zu;}}CP(9$+t4M(h5yUxaTz1`+5PR<14R9`6A}+5 z9ZE|-l98E}ol|(CsQ6^bsZwr5WmWZ=v**rVywq^{N@G)VOUKPye|2_s_uPKa_wbQO 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