{"history":[],"article":{"id":"6717a0c9-ff88-4be8-bccc-7de11db48cf5","title":"Dynamic lot-size model","slug":"dynamic-lot-size-model","content":":::summary\n:::\n\n# \nThe **dynamic lot-size model** in [inventory theory](https://bliptext.com/articles/inventory-theory), is a generalization of the [economic order quantity](https://bliptext.com/articles/economic-order-quantity) model that takes into account that demand for the product varies over time.\nThe model was introduced by [Harvey M. Wagner](https://bliptext.com/articles/harvey-m-wagner) and [Thomson M. Whitin](https://bliptext.com/articles/thomson-m-whitin) in 1958.\n\n# Problem setup\nWe have available a [forecast of product demand](https://bliptext.com/articles/demand-forecasting)\nd_{t\nover a relevant time horizon t=1,2,...,N (for example we might know how many [widgets](https://bliptext.com/articles/widget-economics) will be needed each week for the next 52 weeks).\nThere is a [setup cost](https://bliptext.com/articles/setup-cost)\ns_{t\nincurred for each order and there is an inventory [holding cost](https://bliptext.com/articles/holding-cost)\ni_{t\nper item per period (\ns_{t\nand\ni_{t\ncan also vary with time if desired). The problem is how many units\nx_{t\nto order now to minimize the sum of setup cost and inventory cost.\nLet us denote [inventory](https://bliptext.com/articles/inventory):\nThe functional equation representing minimal cost policy is:\nWhere H is the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function).\nWagner and Whitin proved the following four theorems:\nx_{t\n=0; ∀t\nx_{t\n=0 or for some k (t≤k≤N)\nd_{t*\nis satisfied by some\nx_{t**\n, t**d_{t\n, t=t**+1,...,t*-1, is also satisfied by\nx_{t**\n\n# Planning Horizon Theorem\nThe precedent theorems are used in the proof of the Planning Horizon Theorem.\nLet\ndenote the minimal cost program for periods 1 to t. If at period t* the minimum in F(t) occurs for j = t** ≤ t*, then in periods t > t* it is sufficient to consider only t** ≤ j ≤ t. In particular, if t* = t**, then it is sufficient to consider programs such that\nx_{t*\n> 0.\n\n# The algorithm\nWagner and Whitin gave an [algorithm](https://bliptext.com/articles/algorithm) for finding the optimal solution by [dynamic programming](https://bliptext.com/articles/dynamic-programming).\nStart with t*=1:\nd_{t\n, t = t**, t** + 1, ...\n, t*, by this order\nx_{t**\n)\ns_{t**\n+\ni_{t**\nI_{t**\nto the costs of acting optimally for periods 1 to t**-1 determined in the previous iteration of the algorithm\nBecause this method was perceived by some as [too complex](https://bliptext.com/articles/computational-complexity-theory), a number of authors also developed approximate [heuristics](https://bliptext.com/articles/heuristics) (e.g., the [Silver-Meal heuristic](https://bliptext.com/articles/silver-meal-heuristic) ) for the problem.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:32:39.905Z","updated_at":"2025-01-26T11:32:39.905Z","search_vector":"'dynam':1 'lot':3 'lot-siz':2 'model':5 'size':4","search_content":" \n \n\n# \nThe **dynamic lot-size model** in [inventory theory](https://bliptext.com/articles/inventory-theory), is a generalization of the [economic order quantity](https://bliptext.com/articles/economic-order-quantity) model that takes into account that demand for the product varies over time.\nThe model was introduced by [Harvey M. Wagner](https://bliptext.com/articles/harvey-m-wagner) and [Thomson M. Whitin](https://bliptext.com/articles/thomson-m-whitin) in 1958.\n\n# Problem setup\nWe have available a [forecast of product demand](https://bliptext.com/articles/demand-forecasting)\n d t \nover a relevant time horizon t=1,2,...,N (for example we might know how many [widgets](https://bliptext.com/articles/widget-economics) will be needed each week for the next 52 weeks).\nThere is a [setup cost](https://bliptext.com/articles/setup-cost)\n s t \nincurred for each order and there is an inventory [holding cost](https://bliptext.com/articles/holding-cost)\n i t \nper item per period (\n s t \nand\n i t \ncan also vary with time if desired). The problem is how many units\n x t \nto order now to minimize the sum of setup cost and inventory cost.\nLet us denote [inventory](https://bliptext.com/articles/inventory):\nThe functional equation representing minimal cost policy is:\nWhere H is the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function).\nWagner and Whitin proved the following four theorems:\n x t \n=0; ∀t\n x t \n=0 or for some k (t≤k≤N)\n d t* \nis satisfied by some\n x t** \n, t** d t \n, t=t**+1,...,t*-1, is also satisfied by\n x t** \n\n# Planning Horizon Theorem\nThe precedent theorems are used in the proof of the Planning Horizon Theorem.\nLet\ndenote the minimal cost program for periods 1 to t. If at period t* the minimum in F(t) occurs for j = t** ≤ t*, then in periods t > t* it is sufficient to consider only t** ≤ j ≤ t. In particular, if t* = t**, then it is sufficient to consider programs such that\n x t* \n> 0.\n\n# The algorithm\nWagner and Whitin gave an [algorithm](https://bliptext.com/articles/algorithm) for finding the optimal solution by [dynamic programming](https://bliptext.com/articles/dynamic-programming).\nStart with t*=1:\n d t \n, t = t**, t** + 1, ...\n, t*, by this order\n x t** \n)\n s t** \n+\n i t** \n I t** \nto the costs of acting optimally for periods 1 to t**-1 determined in the previous iteration of the algorithm\nBecause this method was perceived by some as [too complex](https://bliptext.com/articles/computational-complexity-theory), a number of authors also developed approximate [heuristics](https://bliptext.com/articles/heuristics) (e.g., the [Silver-Meal heuristic](https://bliptext.com/articles/silver-meal-heuristic) ) for the problem.","revisionCount":0},"hasMore":false,"nextCursor":null}}}}}}}}}}}}}}}}}}}

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